Volumetric Physics of Polypeptide Coil−Helix TransitionsHeinrich Krobath, Tao Chen, and Hue Sun Chan*
Departments of Biochemistry and Molecular Genetics, Faculty of Medicine, University of Toronto, Toronto, Ontario M5S 1A8,Canada
*S Supporting Information
ABSTRACT: Volumetric properties of proteins bear directly on their biological functionsin hyperbaric environments and are useful in general as a biophysical probe. To gain insightinto conformation-dependent protein volume, we developed an implicit-solvent atomicchain model that transparently embodies two physical origins of volume: (1) a fundamentalgeometric term capturing the van der Waals volume of the protein and the particulate,finite-size nature of the water molecules, modeled together by the volume encased by theprotein’s molecular surface, and (2) a physicochemical term for other solvation effects, accounted for by empirical proportionalityrelationships between experimental partial molar volumes and solvent-accessible surface areas of model compounds. We testedthis construct by Langevin dynamics simulations of a 16-residue polyalanine. The simulated trajectories indicate an averagevolume decrease of 1.73 ± 0.1 Å3/residue for coil−helix transition, ∼80% of which is caused by a decrease in geometric void/cavity volume, and a robust positive activation volume for helical hydrogen bond formation originating from the transient voidcreated by an approaching donor−acceptor pair and nearby atoms. These findings are consistent with prior experiments withalanine-rich peptides and offer an atomistic analysis of the observed overall volume changes. The results suggest, in general, thathydrostatic pressure likely stabilizes helical conformations of short peptides but slows the process of helix formation. In contrast,hydrostatic pressure is more likely to destabilize natural globular proteins because of the void volume entrapped in their foldedstructures. The conceptual framework of our model thus affords a coherent physical rationalization for experiments.
Volumetric properties of proteins, i.e., how they respond tohydrostatic pressure in Nature and in the laboratory a la
Le Chatelier’s principle, are an important window into thephysics of these biomolecules and how they functionphysiologically or malfunction in diseases.1−9 Pressure haslong been known to affect the stability of globular proteins,leading to denaturation and/or unfolding of certain orderedstructures.10,11 The effects of pressure on proteins and peptideshave a direct impact on biological adaptation in the deepocean12 and this environment’s possible role in the origin oflife.13 While pressure at the deepest reaches of the Earth’soceans, ≈1100 atm at the Mariana Trench, is low compared tothe pressures of ≥3000 atm that are routinely used to probeproteins in the laboratory, the pressure sustained by subseafloorsedimentary microbes14 could be higher. Closer to home, in ourbodies, neuronal excitability is very sensitive to pressures ofeven a few atmospheres.15 Although the biomolecular basis ofthis medical observation remains to be elucidated, it illustratesthe broad relevance of volumetric effects to biomedicalsciences.8
Pressure, by itself or in conjunction with temperature16 anddenaturant,17,18 is a useful but perhaps underutilized probe inprotein biophysics. In principle, pressure can facilitate eitherprotein unfolding or folding, depending on which conforma-tional state has a smaller partial molar volume.1−9,19−21 Highpressure has also been applied to study folding intermedi-ates22,23 and to dissociate multimeric proteins and proteinaggregates to study protein−protein interactions. These includenative multiprotein complexes as well as those implicated inamyloidosis and neurodegenerative diseases.8,9 Pressure can
induce other structural transitions,24−26 affect conformationaldynamics24 and enzymatic activity,9 and thus can be used toinvestigate a wide variety of protein properties and theirimplications, including structural aspects16 of protein evolu-tion.27
Overall volume changes accompanying unfolding of globularproteins are mostly negative. Although the partial molar volumecan increase upon unfolding of some proteins at low pressure, italways decreases at high pressure.2,28 This trend likelyoriginates from the void volume trapped inside foldedstructures.19,21,29−31 In contrast, spectroscopic experimentsindicate that the ordered helical conformations of nonglobularlong polylysine (∼2000 residues) and short alanine-basedpeptides of 16−21 residues32−34 are stabilized, not destabilized,by pressure;20,35−38 i.e., these helices occupy less volume thanrandom-coil-like disordered conformations. This observationneed not contradict the void-volume idea because isolatedaqueous helices presumably do not form a sequestered corethat can harbor appreciable empty space.21 In support of thisperspective is a recent experiment by Kiefhaber and co-workerson helical hydrogen bond formation,20 long been seen as anelementary step in the coil−helix transition.39−42 They showedthat adding a helical segment to an existing helical structure inan alanine-rich peptide entails a negative reaction volume and apositive activation volume that they recognized can involve asteric component,20 which is to say that while pressure
Received: August 5, 2016Revised: September 16, 2016Published: October 24, 2016
kinetically slows helix formation, it thermodynamically favorsthe helical state.Besides geometric voids, other physicochemical effects of
aqueous solvation3,5,43−46 also factor into the volumetricproperties of proteins. Seeking a better delineation of theirrespective roles and to provide an atomistic description and/orinterpretation that is currently not accessible by experiment, wedeveloped an implicit-water atomic chain model embeddedwith these two volume contributions to account for theconformation-dependent partial molar volume of polypeptides.Briefly, our formulation uses molecular surface volume47−50 forexcluded volume and water-inaccessible voids; it also uses anempirical volume contribution that is proportional to solvent-accessible surface area47,51 and based on model-compoundexperiments44 to account for the remaining, more “chemical”aspects of solvation volumes. This model is complementary toexplicit-water molecular dynamics, which can tackle pressuredependence directly by using NPT simulations52−56 or bycombining NVT simulation in different fixed volumes.57−60
Several explicit-water simulations posited that the helical stateof alanine-rich peptides is destabilized by pressure,54,55,57 whichis not in accord with experiment.20 In contrast, a recentmolecular dynamics study reproduced the experimentalpressure dependence by using a different water model.56
These results indicate that volumetric properties predicted byexplicit-water simulations can be highly sensitive to watermodel parametrization.56 Such sensitivity has indeed beensuggested by several investigations of simplified models.61−64 Inthis context, our approach, as a synthesis of intuitive geometricconsiderations47−51 and empirical volumetric data on modelcompounds,3,5,43−45 is hereby offered as a tool for conceptualadvancement.Hydrostatic pressure affects every conformation, not only
those belonging to well-populated ensembles such as the foldedand unfolded states. Activation volumes of sparsely populatedtransition states dictate how kinetics is affected by pressure.Because our model computes a volume for every conformation,it can address the nature of activation volumes. For globularproteins, experimental activation volume is almost alwayspositive for folding but is often negative for unfolding, thoughthe latter can be positive for some proteins.65,66 Experimentalactivation volumes depend on denaturant concentration, as hasbeen shown for Tendamistat,67 suggesting a contribution toactivation volumes from nongeometric chemical effects. At thesame time, explicit-water simulations of small nonpolarsolutes21,68 and secondary structure elements52,53 suggestedthat a part of the volume barrier to folding likely originatesgeometrically from the transient voids21 caused by imperfectpacking and steric dewetting52 in the folding transition state.Transient voids are associated with the desolvation free energybarrier52,53,68 as well as the transition-state heat capacity52,69
and enthalpy.52,70 This connection allows pressure-dependentparametrizations of implicit-water potentials with desolvationbarriers70−72 to be used to model pressure effects infolding.73,74 Our implicit-water approach, which goes beyondthis technique by treating volume independently, was inspiredby recent direct computation of volume in explicit-watersimulations21,52,53,75−77 indicating that molecular surfacevolume provides a good description of the volume barrier tothe association of two methanes.21 The present application ofour model to (Ala)16 (16-residue polyalanine) producesquantitatively reasonable agreement with the experimentalnegative reaction volume and reproduces the experimental
finding of a volume barrier for hydrogen bond formation inshort Ala-based peptides.20 The approach allows dissection ofthe observed volume barrier into contributions from variousgeometric and chemical effects. Within the same modelingframework, an increase in volume upon folding is deduced tobe likely for naturally occurring globular proteins, as would beexpected from the void-volume perspective.19 These findingsand their ramifications are detailed below.
■ MODEL AND METHODSForce Field and Langevin Dynamics. Langevin dynamics
simulations78 of the 16-residue polyalanine peptide wereconducted using a new model modified from that of Knottand Chan79 by replacing its Lennard-Jones side chain−sidechain hydrophobic potential by one with a desolvation barrier, aphysics-based feature21,68,69 demonstrably important forcapturing real protein behaviors.70−74,80−82 Other terms ofthe interaction potential follow those of the original simpleatomic force field introduced by Irback and co-workers forMonte Carlo sampling.83 Both the original83 and the presentmodified force field can fold selected peptide sequences, basedon a three-letter alphabet {G, H, P} of glycine (G),hydrophobic (H), and polar (P) residues, into α-helices andhelix bundles. The polyalanine peptide here is modeled as(H)16, i.e., a homopolymer of 16 hydrophobic residues. Detailsof the interaction potential and the simulation methodology areprovided in the Methods Text of the Supporting Informationwith additional references.84−89 Although the study presentedhere used only three-letter sequences, it is worth noting that theforce field of Irback et al.83 has been extended to cover all 20types of amino acid residues. Notable recent applications ofthese simple atomic force fields include studies of mutationaleffects and conformational switches in protein evolution.90−92
Because the primary focus of this effort is volume, notenergetics, the main service provided by the model force fieldto our investigation is as a tool for sampling conformations withdifferent volumes. With this in mind, the conclusions in thiswork regarding volumetric properties of different polypeptide/protein conformations are expected to be robust againstphysically reasonable variations of the energy parameters aslong as the van der Waals radii in our model are realistic(Methods Text of the Supporting Information).
Geometric Component of Conformational Volume.We considered three volumes that are uniquely defined by thegeometry of a given protein conformation and the size of awater probe. (i) The van der Waals volume (vdW-V) does notinvolve a water probe. It is the volume of the union ofoverlapping atomic spheres, defined by the center positions {ri}and the vdW radii {ai} of all the atoms in a given conformation,as illustrated by the space-filling diagram in panels A and G ofFigure 1.93−96 (ii) The molecular surface volume (MSV), asdefined by Connolly,48 was computed by first analyticallyconstructing49 the molecular surface of the given conformationusing a spherical water probe with a radius (rp) of 1.4 Å. MSV isthe volume enveloped by this surface and thus is inaccessible toany part of the water probe (Figure 1A). MSV is larger thanvdW-V because MSV also accounts for cavity (void) volumesand surface effects arising from the finite size of the probe,including the geometric effect of a volume barrier to solvation/desolvation.21 (iii) The solvent-excluded volume (SV) is thevolume inaccessible to the center of the water probe due to thepresence of the given conformation,47,97 defined as the volumeof the union of spheres centered at all atomic positions {ri} of
the conformation with radii {ai+rp}. (Thus, vdW-V ismathematically equivalent to SV with a hypothetical rp = 0.)The surface enveloping SV is the solvent-accessible surface area[SASA (Figure 1A)].51 The SV − MSV difference is envelopevolume Venv.
50
In addition to these volume measures, the Voronoi−Delaunay formalism98 has provided important volumetricinsights as well by tracking nonoverlapping Voronoi cells ofindividual atoms in a protein.99−101 Depending on the positionsof solvent molecules, Voronoi volumes tend to be larger atsolvent-exposed than at buried positions,102,103 a packing effectthat is also captured in part by vdW-V and MSVconsiderations.96,104,105 Because the Voronoi approach ac-counts explicitly for solvent molecules, it can be used toidentify hydration shells. For the same reason, however, it is lessuseful than MSV for our implicit-water model. Voronoivolumes were not used in the analysis presented here.The Chemical Component and an Overall Implicit-
Solvent Model of Conformational Volume. To apply ourimplicit-water model of conformational volume to experimentalhelix−coil systems, we first calibrated it against experimentalpartial molar volumes (PMVs) of model compounds.3,5,43−46
The rationale is to partition experimental PMV, in a physicallywell-motivated manner, into a geometric MSV and a chemicalcomponent corresponding to PMV −MSV. PMV and MSV areexpected to be different, because in addition to the geometriceffects captured by MSV, PMV embodies additional structuraleffects such as void volumes in the solvent caused by thermalfluctuations as well as other physicochemical effects of solvent−
protein interactions such as water ordering near theprotein.3,106−108
We determined the chemical components of peptidebackbone (BB) and alanine side chain (SC) PMVs by firstsimulating (Gly)n and (Ala)n oligopeptides using our proteinchain representation, yielding MSV(Gly) = MSVBB = 53.32 ±0.04 Å3 and MSVSC = 31.47 ± 0.1 Å3 (SFigure 1). Thesecomputed MSVs were then compared with the PMVs obtainedusing vibrational densimetry by Makhatadze et al.,44 viz.,PMV(Gly) = 37.6 cm3/mol (62.7 Å3/molecule) and PMV(Ala-SC) = 27.2 cm3/mol (45.3 Å3/molecule) at 25 °C, as reportedin Tables II and V of ref 44. Thus, the PMV − MSV volumedifference for a backbone unit is PMVBB −MSVBB ≡ δVBB = 9.4Å3, and that for the alanine side chain is PMVSC − MSVSC ≡δVSC = 13.9 Å3. For the sake of simplicity, the experimentalPMV of the glycine peptide unit (H2CαC′ONH) was used todetermine δVBB, although Cα hydrogens are not represented inour chain model.79,83 The resulting overestimation of δVBB,however, is expected to be small. On the basis of this analysis,the conformation-dependent PMV of our protein model isdefined as
δ= + VPMV MSV (1)
∑δ δ
δ
=Θ =
+Θ =
⎡⎣⎢
⎤⎦⎥
V V
V
SASASASA ( 0)
SASASASA ( 0)
i
i
i
i
i
BB
BB
BB
SC
SC
SC(2)
where PMV, MSV, δV, SASAiBB, and SASAi
SC are computed forthe same protein conformation and the summation is over theresidues (labeled by i) along the protein sequence. SASAi
BB andSASAi
SC are the SASA values of the BB and SC, respectively, ofresidue i. MSV is seen here as the volume intrinsic to a proteinconformation plus other associated volume geometricallyinaccessible to water, whereas δV is the volumetric effect ofprotein−solvent contactlike interactions other than simpleexcluded volume. As such, the δV from a chemical group isintuitively expected to be roughly proportional to the numberof water molecules in contact with (i.e., its hydrationnumber109) and, thus, the SASA of the group.110 BecauseδVBB and δVSC are δV contributions from a BB and a SC group,respectively, when they are maximally exposed while positionedin the middle of a peptide sequence,44 the conformation-dependent δV contributions from the BB and SC in eq 2 areweighted by SASAi
BB/⟨SASAiBB(Θ = 0)⟩ and SASAi
SC/⟨SASAi
SC(Θ = 0)⟩, respectively, where ⟨SASAiBB(Θ = 0)⟩ and
⟨SASAiSC(Θ = 0)⟩ are the averaged residue-specific SASA
values of the groups in disordered chains with no helicalcontent. A precise definition of Θ will be given in the Results.To assess the merit of the formulation given above, as a
control we have compared results computed using the PMVdefined by eqs 1 and 2 against those computed using an ad-hocdefinition of PMV without the MSV baseline, viz.
∑=Θ =
+Θ =
‐
⎡⎣⎢
⎤⎦⎥
PMV PMVSASA
SASA ( 0)
PMVSASA
SASA ( 0)
i
i
i
i
i
ad hoc BB
BB
BB
SC
SC
SC(3)
Figure 1. Geometric volumes in the coil−helix transition. (A) Model(Ala)16 peptide conformation in a space-filling depiction in which thevdW surfaces of N, Cα and C′, and O atoms, and SC united atoms arecolored differently. A spherical water probe (drawn to scale) rolls overthe vdW surfaces, defining direct-contact and re-entrant surfaces, thesmoothly connected set of which is the molecular surface (translucent)that envelops MSV. The solvent-accessible surface (black curve)envelops SV. (B−F) Ribbon diagrams of snapshots along an examplecoil−helix transition trajectory at t/107τS values of 7.4, 8.0, 8.2, 8.3, and8.32, respectively. The preceding conformation in panel A wasextracted from the same trajectory at a t/107τS of 6.8. (G) vdW surfacerepresentation of the fully formed helix in panel F.
where PMVBB = 62.7 Å3, PMVSC = 45.3 Å3 as reported byMakhatadze et al.,44 and the SASA-dependent weighting factorshere are the same as those in eq 2.Reaction Volumes and Volume Maxima during
Hydrogen Bond Formation and Rupture. We investigatedthe volumetric kinetics of hydrogen bond formation andrupture in our model by first identifying all such events in theequilibrated parts of all of our simulated trajectories, in thefollowing manner. Let the beginning time of the equilibratedpart of a trajectory be denoted t00. We examined time segmentsof the trajectory defined by time intervals [t00, t00 + L], [t00 +Δt, t00 + L + Δt], [t00 + 2Δt, t00 + L + 2Δt], ..., where L is achosen segment length, Δt = 250τS is the time interval betweentwo successive volume computations in most of oursimulations, and τS is the Langevin dynamics time step, untilwe encountered a time segment with s and s + 1 hydrogenbonds at the start and end points of the time segment,respectively, and s = 4, 5, 6, 7, or 8. We saved such a timesegment, which contains a hydrogen bond formation event, forfurther analysis. We then continued scanning the trajectory asbefore, but with a new starting time point equal to the endpoint of the saved time segment. The process was repeateduntil all trajectories were scanned. We applied this procedurefor L = 1000τS, 1250τS, and 1500τS and repeated the entireprocedure to study hydrogen bond rupture by identifying andsaving time segments with s and s − 1 hydrogen bonds at thestart and end points of the time segment, respectively. For eachof the saved time segments, we recorded the starting (S), final(F), and maximal (M) volumes. Volume barriers are defined asthe M volume minus the S volume. The sharpness of thevolume maximum (at t*) may be characterized by ΔV+ ≡ V(t*)− V(t* + Δt) and ΔV− ≡ V(t*) − V(t* − Δt), as t* was neverat the end points of the time segments. We report thesequantities using the notation ξ1 = min(ΔV+, ΔV−) and ξ2 =max(ΔV+, ΔV−). Statistics resulting from this analysis areprovided in STables 1−6.
■ RESULTSAs outlined above and detailed in the SI Methods Text, weadopted a computationally efficient Langevin dynamics modelof polypeptides79 in which all backbone atoms except the Cα
hydrogens are represented explicitly83 to allow for structuralanalyses of backbone hydrogen bonds. On this basis, desolva-tion barriers were added to the side chain hydrophobicinteractions for more physical, real protein-like behaviors.68−72
More significantly, a new direct geometric/chemical model ofconformation-dependent partial molar volume was incorpo-rated. We now apply this construct primarily to (Ala)16, amodel system chosen to provide theoretical insights into recentexperimental volumetric data on similar short peptides such asthe 16-residue AK16,37 20-residue AK20,38 and 21-residueXan/Nal-containing20 Ala-based peptides.Coil−Helix Transition of a Short Model Peptide.
Starting from random-coil-like conformations, our model canreliably access the fully helical state at the simulationtemperature of 0.29T0 (in model units) that correspondsroughly to room temperature (SI Methods Text). Throughoutthis work, the coil−helix transition is monitored by the progressvariable Θ, also termed the reaction coordinate, which is thefractional number of helical hydrogen bonds in a givenconformation normalized by the maximal number in the fullyhelical state. For the sake of terminological simplicity, we oftenrefer to the process of achieving Θ = 1 kinetically as “folding”.
Equilibrium sampling of our model at different temperaturesranging from strongly helix-favoring (low T) to strongly helix-disfavoring (high T) indicates no free energy barrier along Θbetween the coil-like low-Θ state and the fully helical Θ = 1state. At T = 0.29T0, the (Ala)16 peptide contains an average of79 ± 20% helical content in equilibrium (SFigure 1). Thisbehavior is consistent with the Zimm−Bragg39 and relatedbiophysical pictures40−42 that posit the coil−helix transition asnoncooperative, with fraying from ends for example, rather than“all or none”. It follows that folding of our model (Ala)16peptide does not involve a major free energy barrier111 along Θunder strongly folding conditions, such as the T = 0.29T0condition for our kinetic simulations. Thus, the process is“downhill” as understood in the protein folding literature,111
though smaller free energy barriers such as those arising fromdesolvation and hydrogen bond formation always exist. Thefolding kinetics of (Ala)16 is approximately single-exponentialnonetheless (SFigure 2), consistent with the fact that essentiallysingle-exponential relaxation and apparent barrierless foldingare not mutually exclusive.111 However, because folding kineticsis downhill, the relationship between kinetic and thermody-namic profiles of (Ala)16 is different from that for cooperativeprotein folding. Unlike the preequilibrium with substantiallyoverlapping profiles in the unfolded basin for cooperativefolding,112 the kinetic profile for the conformational populationsampled during (Ala)16 folding is entirely distinct from thethermodynamic profile at the same T = 0.29T0. Instead, itoverlaps significantly with the thermodynamic profile for amuch higher T = 0.4T0 (SFigure 1).When T is sufficiently low, (Ala)16 is driven to the lowest-
energy fully helical state (SFigure 3A). Because the Ala sidechains are not in close contact in a helix, there are morehydrophobic contacts in the coil-like disordered state around Θ= 0.25 (SFigure 3B) than in the more helical conformations.The decrease in total energy E with increasing Θ is thus apartial trade-off between a decreasing hydrogen bonding energyEHB that is not fully compensated by an increasing hydrophobicenergy EHP (SFigure 3A). Significant nonhelical hydrogenbonding is present in the coil state (SFigure 3C). Thesehydrogen bonds hinder helix folding because they have to befirst broken before the full helix can be formed (SFigure 3D).
A Polypeptide Volumetric Dynamics Based on anInterplay of Connolly Volumes and Experimental Small-Molecule Partial Molar Volumes. Our volumetric model isbased on the molecular surface volume (MSV) enveloped bythe Connolly molecular surface and the solvent-accessiblesurface (Figure 1A). As rationalized in Model and Methods andthe SI Text, we express the partial molar volume of a givenpolypeptide conformation as PMV = MSV + δV, where MSVand δV are identified as the geometric and chemicalcomponents of PMV, respectively; δV is taken to be a sumof backbone (BB) and side chain (SC) contributions, eachproportional to its respective solvent-accessible surface area(SASA). Parameters for computing δV were deduced frompublished experimental data44 by subtracting MSV baselines(SFigure 4). As the coil−helix transition progresses (Figure1A−F), the SCs are placed increasingly on the outside,shielding most of the BB from the solvent (Figure 1G).Specifically, as the Θ of (Ala)16 increases from 0 to 1, theaverage SASA increases by ≈39% for SC but decreases ≈50%for BB, netting a total average SASA increase of ≈7% (Figure2). This trend implies that as the helix folds, δV has anincreasing contribution from the SC relative to the BB.
In view of the decreasing overall BB SASA with helixformation, a closer examination of the individual SASAs of thecarbonyl (C′O) and amino (N−H) groups is instructive.Helix formation is effectuated by desolvation of hydrogen bonddonor (N−H) and acceptor (C′O) groups. In the coil state,acceptors are more solvent-exposed than donors by a factor of≈10 in terms of absolute SASA because carbonyl C′ and Oatoms are larger than amino N and H atoms, and the C′Obond is longer than the N−H bond.79 For most of the residuesexcept at the chain ends, the coil−helix transition drasticallyreduces the level of solvent exposure for both donor andacceptor by up to 90%, whereas the exposure of Cα remainslargely unchanged (SFigure 5).The theoretical construct presented here allows for
simultaneous tracking of energies and volumes of a foldingtrajectory (Figure 3). Consistent with a substantial presence ofnonhelical hydrogen bonds in the coil state (SFigure 3C,D), theexample trajectory shows that successful formation of acomplete helix often requires rupturing of existing hydrogenbonds and breaking of favorable hydrophobic side chaincontacts, as is indicated in panels A and B of Figure 3 by thehigh EHB and EHP peaks around t/107τS = 7.6 before completehelix formation at t/107τS ≈ 8.3. Apparently, at least for thistrajectory, the kinetic bottleneck in the coil−helix transition isfirst to access a conformation without nonhelical hydrogenbonds to allow for helix propagation, but such conformationsare relatively rare in the initial conformations with no or lowhelical content. For instance, they make up only ≈10% of theconformations at Θ = 0.25 (SFigure 3D).Along the volume trajectories in Figure 3C, vdW-V fluctuates
the least. This is expected because conformation-dependent
variations in atomic overlap, which are quite minimal, are theonly source of vdW-V variations. In contrast, MSV alsoaccounts for void-volume and finite-size effects of water. Theseeffects are more sensitive to conformational change. As a result,MSV is approximately 10% larger than vdW-V and has a largerfluctuation. With the incorporation of other aspects of solvationbeyond simple excluded volume, PMV of (Ala)16 isapproximately 28% larger than MSV, meaning that the chemicalcomponent δV adds ≈28% volume to PMV on top of thegeometric MSV. Because δV entails an extra element ofvariability by virtue of the proportionalities of its contributionsto conformation-dependent SASAs (Figure 3D), PMV exhibitsmost fluctuation among the three volumes we tracked in Figure3C. A comparison of panels B and D of Figure 3 indicatesfurther that transient hydrophobic-contact formation anddisruption is a major source of δV variability and hence PMVfluctuation.
The Geometric−Chemical Formulation Predicts aDecrease in Volume upon Formation of an Isolated α-Helix in Water. The trajectory depicted in Figure 3C indicatesthat MSV is likely smaller in the fully helical state than in thecoil state, but the PMV trend is less clear because of significantfluctuation. To ascertain the general trend, we computedvolumetric properties, averaged over all 122 kinetic trajectorieswe simulated, to determine their dependence on helicity Θ(Figure 4). Average reaction volumes can be estimated fromthis data set because it provides average volumes V(Θ) forindividual conformational ensembles with a given Θ. Becausethe dominant coil-like population is centered around Θ = 0.25(SFigure 3B), coil−helix reaction volumes per residue providedbelow are defined by the quantity V(Θ = 1) − V(Θ = 0.25)divided by the total number of residues (n = 16) unlessspecified otherwise. Alternatively defined reaction volumes suchas [V(Θ = 1) − V(Θ = 0)]/n can also be obtained from thedata plotted in Figure 4.
Figure 2. Solvent exposure in the coil−helix transition. Results shownare averages over all simulated (Ala)16 trajectories. (A) The total SASAincreases with helicity for Θ > 0.5 (top). The residue-specificnormalized SASAi averaged over all 16 residues increases monotoni-cally with Θ for the SC but decreases monotonically for the BB(bottom). (B) Residue-specific SASAi values for the BB (top) and SC(bottom) for Θ = 0 (blue), Θ = 0.5 (green), and Θ = 1 (red) exhibitstrong position dependence, especially near the two termini of thepeptide when BB shielding is minimal (top). As helicity Θ increases(indicated by small solid arrows), SASAi decreases for the BB andincreases for the SC monotonically (long solid arrows) for most butnot all residue positions i. Note that the Θ = 0 blue curves in panel Bcorrespond to the ⟨SASAi
BB(Θ = 0)⟩ and ⟨SASAiSC(Θ = 0)⟩
normalization factors in this formulation.
Figure 3. Energetic and volumetric trajectories of a coil−helixtransition. Time-dependent properties for the final part of thetrajectory in Figure 1 show that helix formation correlates withdecreased (more favorable) hydrogen bond energy (A) and increased(less favorable) hydrophobic energy (B) and is associated with variousdegrees of fluctuation in volume measures (C). Variation of chemicalcomponent δV of the conformational volume (D) is seen to correlatestrongly with that of hydrophobic side chain interactions (B). Shownproperties are sliding averages over time windows of 100τS.
For Θ > 0.25, the average vdW-V decreases monotonicallywith increasing Θ. The average vdW-V for fully helicalconformations is ≈0.4% smaller than those of coil con-formations (Figure 4A). This corresponds to a helix-formationvdW-V reaction volume of ΔvdW-V0 = −0.32 ± 0.01 Å3/residue (−0.19 ± 0.01 cm3 mol−1 residue−1) or −0.57 ± 0.02Å3 (−0.34 ± 0.01 cm3/mol) per helical hydrogen bond,suggesting strongly that a local increase in atomic overlap isassociated with the formation of a helical hydrogen bond. Overthe same Θ range, MSV also decreases monotonically (Figure4B), registering a decrease of 3.2% for the entire coil−helixtransition, or a MSV reaction volume of ΔMSV0 = −2.68 ±
0.03 Å3/residue (−1.61 ± 0.02 cm3 mol−1 residue−1) or −4.76± 0.05 Å3 (−2.86 ± 0.04 cm3/mol) per hydrogen bond.Indeed, both the mean and standard deviation of the MSVdistribution decrease with increasing Θ, but there are largeoverlaps in the distributions for different Θ values (SFigure 6),as is expected from the large fluctuations seen in the examplefolding trajectory (Figure 3C). Because the decrease in MSV ismuch larger than that in vdW-V, void-volume reduction is thedominant contribution to the negative ΔMSV0; only 12% ofΔMSV0 is due to vdW-V reduction. In contrast, the averagesolvent-excluded volume SV of the helical conformation is 1.5%larger than that for coil structures (Figure 4C) because of theincreasing envelope volume Venv with increasing Θ (Figure 4D),resulting in a positive reaction volume of ΔSV0 = 2.62 ± 0.3Å3/residue (1.57 ± 0.2 cm3 mol−1 residue−1).In light of the opposite Θ-dependent trends of MSV and SV,
it is worth emphasizing that MSV is a more appropriatebaseline for a PMV model than SV because MSV represents apure, bare-bones geometric effect of excluded volume (Modeland Methods). In contrast, the additional Venv part of SVaccounts for a volume with significant nonsteric solute−solventinteractions, the volumetric effects of which, however, are moredesirable to be captured by a separate δV term alone. In otherwords, we view the PMV = MSV + δV partition of volumes asphysically more sensible than some other partitions such asPMV = SV + δV′ because MSV can be equal to PMV (i.e., δV =0) for the pure state of a hypothetical solute that allows for100% packing density at T = 0 (such as a cube), but SV doesnot share this property.For our model (Ala)16, PMV adds 27−29% to MSV. The δV
addition is larger for more helical conformations because oftheir higher levels of hydrophobic exposure (see above). ThisΘ-dependent effect reduces the magnitude of the negativereaction volume relative to that of MSV, but only slightly. Theaverage model PMV decreases monotonically with increasing Θ(Figure 4E). This trend is consistent with experiment (seebelow). Relative to the two dominant coil populations at Θ =0.167 and 0.25 (SFigure 3B), the reaction volumes for helixformation are ΔPMV0 = [PMV(Θ = 1) − PMV(Θ = 0.167)]/16 = −1.81 ± 0.07 Å3/residue (−1.09 ± 0.04 cm3 mol−1
residue−1), respectively. These two values average to a ΔPMV0of −1.73 ± 0.1 Å3/residue (−1.04 ± 0.06 cm3 mol−1 residue−1).Although a volume barrier to the formation of a helical segmentwas observed experimentally,20 the average PMV profile inFigure 4E does not exhibit a barrier along Θ. We will addressthis apparent mismatch below by examining single-moleculeevents. The average MSV profile in Figure 4B also lacks abarrier, as has also been observed in a recent explicit-watermolecular dynamics study.56
Because |ΔvdW-V0| < |ΔMSV0| or |ΔPMV0|, the presentcoil−helix transition increases the level of packing according toa geometric measure of atomic packing density50 (ρ = vdW-V/MSV) as well as an apparent packing density (ρ* = vdW-V/PMV) (Figure 4F), where ρ = 89.3 and 91.8% and ρ* = 70.0and 71.1% for Θ = 0.25 and 1.0, respectively. These measures,related but distinct from other definitions of packing density inmathematical investigations113,114 and studies of globularproteins,47,115 are useful for characterizing packing densitiesof isolated small peptides with relatively large boundary/surfaceeffects. In this context, it is noteworthy that the helix−coilvolume differential is grossly overestimated by SV. As a fraction
Figure 4. Volume changes induced by the coil−helix transition. Meanvolumetric quantities averaged over all simulated (Ala)16 trajectoriesare shown as functions of Θ. Reaction volumes of coil−helix transitionwere computed as the value of a given volume measure at Θ = 1 minusthat at Θ = 0.25; the latter corresponds to the most helical part of thedisordered conformational population (SFigure 3B). The reactionvolume is very slightly negative for vdW-V (0.4% decrease) due toincreased atomic overlaps upon helix formation (A), more negative forMSV (B), but positive for SV (C) and Venv (D). Following the MSVtrend, PMV decreases with increasing Θ (E). Geometry-based andapparent packing densities (ρ and ρ*, respectively) both increase withincreasing Θ (F). The positive chemical contribution of δV to PMV(=MSV + δV) as a fraction of MSV is larger for Θ > 0.5 than for Θ ≤0.5 (G). SV exhibits a similar trend, but the increase in SV/MSV uponhelix formation can be as large as 5 times that of the correspondingincrease in PMV/MSV (H). Statistical uncertainties of the plottedmean values are smaller than the size of the plotting symbol exceptwhere indicated by error bars.
of MSV, experiment-based PMV is 2% larger (Figure 4G), butSV is ∼10% larger (Figure 4H) for helices than for coils.The Volume Barrier to Hydrogen Bond Formation
Largely Originates from Transient Voids Created by theApproaching Donor−Acceptor Pair and Nearby Atoms.To gain deeper insight into volume barriers and reactionvolumes in coil−helix transitions, we now shift our attentionfrom averaged volumes of conformational ensembles (Figure 4)to volumetric properties of single-molecule helical hydrogenbond formation and rupture events. As described in Model andMethods, we assembled a collection of these kinetic eventsfrom short time segments of the simulated (Ala)16 foldingtrajectories. General properties were gleaned from the statisticsof a total of 148782 formation and 439214 rupture events(STables 1−6), and in-depth analysis of one helical hydrogenbond formation event is used as illustration (Figures 5 and 6).
The helical hydrogen bond formation and rupture statisticsindicate that PMV and MSV barriers are a very robust feature ofthese kinetic processes. A PMV and a MSV maximum (M)larger than both the start (S) and final (F) volumes wereobserved without exception in all formation and rupture eventswe examined, as illustrated by panels B and C of Figure 5. AnMSV peak is expected21 from geometrical consideration(SFigure 7A). vdW-V also exhibits a peak in this case (Figure5A), though it was not expected in general (SFigure 7A); onlythe PMVad-hoc we introduced in Model and Methods as acontrol does not (Figure 5D). Echoing the general trend forhelical hydrogen bond formation (STables 13), the reactionvolumes in Figure 5 are all negative. (Note that the reaction
Figure 5. Activation volume of single-helical hydrogen bondformation. The changes in four volume measures are shown for theformation of an additional helical hydrogen bond in a conformationthat already has five such bonds (Θ increases from 0.417 to 0.50). Thekinetic event spanning 1000τS was taken from the trajectory analyzedin Figures 1 and 2. A negative reaction volume is seen for all fourvolume measures considered here (ΔvdW-V0 = −6.08 Å3, ΔMSV0 =−1.39 Å3, ΔPMV0 = −4.9 Å3, and ΔPMVad-hoc,0 = −47.6 Å3). Avolume maximum (indicated with an asterisk) with a sharpnesscharacterized by ξ1 and ξ2 is observed for vdW-V, MSV, and PMV (A−C), but not for the PMVad-hoc (D) that we introduced as a control (eq3 in Model and Methods). vdW-V*, MSV*, and PMV* are volumebarriers defined as the peak value of the given volume measure minusthat at the start (S) conformation.
Figure 6. Details of conformational and volumetric changes during theformation of a single helical hydrogen bond. Structural aspects of theevent in Figure 5 are analyzed here, with the hydrogen bond donor(D) and acceptor (A) labeled. (A) The S, M, and F conformations arecolored green, blue, and yellow, respectively. Previously establishedhydrogen bonds in the vicinity of D and A are shown by thin yellowlines. Local conformational changes during the S−M (left) and M−F(right) transitions are indicated by white arrows. (B) Changes in thevdW spherical overlaps between D and A. The vdW surfaces of thecarbonyl oxygen (red), amino nitrogen (blue), and donor hydrogen(white) involved in the hydrogen bond being formed are shown withthe vdW surface of another carbonyl oxygen (of residue 3, O3) in thevicinity. In this example, the process started (S) with no overlapamong these atoms. No overlap developed at the volume maximum(M) either. The final hydrogen bond formation (F) buried thehydrogen entirely, leading to a substantial vdW overlap between D andA. As a side effect in this particular case, an additional vdW-V-reducingoverlap is seen between the two carbonyl oxygens. (C) Space-fillingrepresentation of the peptide fragment comprising residues 3−9,shown in approximately the same orientation as in panel B with thesame color code for N, O, and H atoms, but with C atoms coloredlight brown and SCs gray. The re-entrant part of the molecular surfaceis translucent. The D−A pair yet to be hydrogen bonded is connectedby a yellow line (S and M). An increased void volume develops from Sto M around the D−A pair because a more curved backbone (A, S−M) deepens the groove and the movement of the side chain of residue8 (SC) further enlarges the cavity, as may be visualized by the smallerwater probe-contact surfaces for A and O3 (bright red patches) and aless curved re-entrant surface (arrows) in M than in S. Close-upimages of these structural drawings viewed at different orientations areprovided in SFigure 9.
volumes obtained from Figure 5 and STables 1−3 are definedas averages over individual hydrogen bond formation eventsinstead of as differences in ensemble-averaged volumes as in theanalysis of Figure 4 described above.) The PMV/MSV ratio atM is equal to 1.278, which is in line with the aforementionedobservation that PMV typically adds 27−29% to MSV,implying that this PMV peak is not caused by an extraordinaryhigh degree of exposure of hydrophobic residues. In fact, thechemical contribution to the PMV peak is negligible in this casebecause the volume barrier ratio PMV*/MSV* = 1.01. In otherwords, 99% of the PMV peak in Figure 5 originated from thegeometric component of PMV. Because vdW-V*/MSV* = 0.4,roughly 40 and 60% of this PMV peak were caused by reducedvdW atomic overlaps and increased desolvated void/cavityvolume, respectively.We now look into structural details of the S, M, and F
conformations (Figure 6). In this particular case, the donor(residue 8) and acceptor (residue 4) of the hydrogen bondbeing formed are in a highly helical environment, withhydrogen bonds between residues 3 and 7, 5 and 9, 6 and10, 10 and 14, and 11 and 15 already established. As thehydrogen bond between residues 4 and 8 materializes, thedonor−acceptor distance, rDA (Figure 6A), first decreases from4.7 Å (S) to 4.1 Å (M) and then to 1.9 Å (F) (see also SFigure7B). The accompanying helix-enhancing conformationalchange from S to M entails an increased backbone curvaturebetween residues 4 and 8, contributing more void volume tothe peak MSV in addition to the void volume created by theapproaching donor−acceptor pair. The existence of an MSVand a PMV peak as observed in panels B and C of Figure 6 forthe full peptide is a robust feature. When one focuses locally ononly residues 3−9 of the (Ala)16 peptide and examines the 4−8hydrogen bond formation process at a higher time resolution(SFigure 7B−F), MSV increases by 6.3 Å3 from 599.1 Å3 at S to605.4 Å3 at M (SFigure 7D). The corresponding ratio PMV/MSV of 1.265 at M is very similar to the value of 1.278 reportedabove for the entire (Ala)16 peptide, indicating an essentiallyneutral effect from the chemical component of PMV at M. Incontrast, vdW-V of residues 3−9 fluctuates widely from S to M(SFigure 7C). Unlike MSV, there is no intrinsic vdW-V barrierto hydrogen bond formation (SFigure 7A). Thus, vdW-V is notexpected to contribute much, if at all, to the volume barrier tohydrogen bonding in general. The vdW-V peak for the full-length peptide in this particular case (Figure 5A) was caused bythe rupture of a non-native hydrogen bond between residues 6and 11 (SFigure 8), a chance occurrence not intrinsic to thehelical hydrogen bond formation process. The two primaryorigins of the PMV barrier to hydrogen bond formation arethus (i) the increase in void volume created by local backboneconformational changes between the donor and acceptorpositions and (ii) the void volume created between theapproaching donor−acceptor pair itself (Figure 6C). The firsteffect is seen to be dominant because while the hydrogenbonding partners have to overcome a maximal volumetricdesolvation barrier of 2.9 Å3 at d = 3.2 Å (SFigure 7A), thePMV barrier for full-length (Ala)16, at more than 9 Å3 onaverage (STables 1−3), is almost always much higher. Theevent in Figure 6 is a good illustration, as the increased voidvolume from decreasing H−O distance rDA from 4.7 Å at S to4.1 Å at M here accounts for only 1.25 Å3 (SFigure 7A) of the6.3 Å3 MSV barrier for the peptide fragment of residues 3−9.The extensive hydrogen bond formation and rupture
statistics in STables 46 for full-length (Ala)16 indicate that
these processes always entail a sharp PMV barrier that is at least7 Å3 higher than its preceding or succeeding volumemeasurement. The PMV barrier always coincides with a MSVbarrier of at least 5.5 Å3. Because no donor−acceptor contactexists at the time point of the PMV/MSV barrier, the vdW-Vvalues at M relative to those at S or F, which amounts to at least2.7 Å3 by the same ξ1/ξ2 sharpness measure, may be used toquantify volumetric noise due to conformational changes notessential to hydrogen bond formation or rupture. Notably, theobserved sharpness of both the PMV and MSV barriers ishigher than this level of noise. The picture remains the same ifone focuses on the volumetric barrier heights (S to M) ofhydrogen bond formation or rupture themselves (STables 13): ⟨PMV*⟩ ranges from 11.7 to 16.9 Å3, whereas ⟨MSV*⟩ranges from 9.3 to 13.3 Å3; both are significantly larger than therange of 3.76−4.98 Å3 for ⟨vdW-V*⟩. The significantvolumetric barrier of hydrogen bond formation and rupture isa major reason for the wide MSV distribution for any given Θ(SFigure 6). In all cases, the barrier PMV/MSV ratios suggestno special role of the chemical component of PMV in the PMVbarriers.As the chain descends the volume barrier (M) to form the
hydrogen bond (F), the MSV of the peptide fragment ofresidues 3−9 decreases from 605.4 Å3 at M to 587.3 Å3 at F(SFigure 7D), netting a reaction volume of −11.8 Å3 relative tothe MSV of 599.1 Å3 at S. Approximately one-half of thisdecrease is contributed by the vdW-V difference (542.7 Å3 −548.2 Å3 = 5.5 Å3) between F and S (SFigure 7C). NegativeMSV and vdW-V reaction volumes for helical hydrogen bondformation are expected because of the tight packing of thehelical backbone with a negligible void volume along the helicalaxis (Figure 6B,C) as well as the fact that the vdW overlap ofthe hydrogen bond donor and acceptor itself at rDA ≈ 2 Åcontributes approximately −0.78 Å3 (−0.47 cm3/mol) to thenegative reaction volume (SFigure 7A).Statistics of hydrogen bond formation and rupture reaction
volumes indicate that they depend on the number of helicalhydrogen bonds already present (STables 13). With differentinitial helicities, the F−S partial molar volume difference of anindividual hydrogen bond formation event varies from ΔPMV0= −0.8 Å3 for Θ = 0.33 to ΔPMV0 = −2.9 Å3 for Θ = 0.67. Incomparison, hydrogen bond rupture can entail a ΔPMV0 ofboth signs ranging from −0.49 to 0.27 Å3. These subtletiesaside, it is clear that helical hydrogen bond formation is themain cause of the overall negative reaction volume of the coil−helix transition, averaging to ΔPMV0 = −2.31 ± 0.1 Å3 (−1.39± 0.08 cm3/mol) per hydrogen bond formed (Figure 4E).While both the average MSV and PMV reaction volumes forhydrogen bond formation are negative, ⟨ΔMSV0⟩ is alwaysmore negative than ⟨ΔPMV0⟩ by 5−25%, indicating that thelevel of nonpolar exposure is increased by hydrogen bondformation. The average ⟨vdW-V0⟩ for (Ala)16 is alsoconsistently negative but amounts to only ≈20% of the totalMSV reaction volume, implying that ≈80% of the negativeMSV reaction volume for the (Ala)16 coil−helix transition iscaused by the reduced void volume.
■ DISCUSSIONOur Predictions of MSV and PMV for (Ala)16 Are
Consistent with Experiment. The theory developed in thiswork affords quantitatively reasonable agreements with recentexperiments on short alanine-rich peptides. Using Fouriertransform infrared spectroscopy (FTIR), Imamura and Kato38
reported a positive per-residue reaction volume (ΔV0) of 0.45± 0.02 Å3 (0.27 ± 0.01 cm3/mol) or 1.53 ± 0.05 Å3 (0.92 ±0.03 cm3/mol) for the helix to coil unfolding transition of thealanine-rich peptide AK20 at 25.4 °C by assuming a strictlytwo-state or Zimm−Bragg noncooperative transition, respec-tively.38 Because a strictly two-state helix−coil transition isphysically unrealistic for polypeptides as noted (SFigure 1), weinterpret the data in ref 38 as a partial molar volume change of−1.53 ± 0.05 Å3/residue (−0.92 ± 0.03 cm3 mol−1 residue−1)for helix formation (hence the change in sign). Considering thesimplicity of our model, the present computed average ΔPMV0of −1.73 ± 0.1 Å3/residue (−1.04 ± 0.06 cm3 mol−1 residue−1)is in good agreement with this experimental measurement. Thisagreement lends support to our theoretical approach ofextracting δV values from experimental model-compoundPMVs using MSV baselines. In contrast, as a control, if MSVbaselines were discarded and PMVs thought to be entirelyproportional to SASAs as in the quantity PMVad-hoc (Model andMethods), the predicted ΔPMVad-hoc of −10.8 ± 0.3 Å3/residue(SFigure 10) would grossly overestimate the coil−helix reactionvolume.Consistent with the finding of Imamura and Kato,38 a more
recent triplet−triplet energy transfer (TTET) study of a 21-residue Xan- and Nal-containing Ala-based peptide byNeumaier et al. reported a negative reaction volume for helixformation, as well. Their measured reaction volume at 5 °C is−0.38 Å3 (−0.23 cm3/mol) per helical segment.20 As suggestedby the authors, a possible reason for the smaller magnitude oftheir negative reaction volume vis-a-vis that of Imamura andKato is that different temperatures were used in the TTET (5°C) and FTIR (25.4 °C) experiments.20 This considerationmay apply to the larger computed magnitudes of negativereaction volumes of −0.8 Å3 (−0.48 cm3/mol) per helicalhydrogen bond from our trajectory analysis (STable 1). Ourmodel result of a positive PMV barrier to helical hydrogenbond formation is consistent with Neumaier et al.’s reportedpositive activation volume of 3.7 Å3 (2.2 cm3/mol) for theformation of a single helical hydrogen bond,20 although thePMV barrier of at least 11.7 Å3 (7.4 cm3/mol) deduced fromour trajectory analysis (Figure 5C and STable 1) is higher.Aside from the difference between experimental and effectivesimulation temperatures, another possible contributing reasonfor the higher volume barriers in our simulation is that no termfor hydrostatic pressure was applied in the currentimplementation of our model, whereas activation volumes ofatomic associations likely decrease with an increase inpressure.21 Also of note is the fact that the volume barrier tohydrogen bonding we identified is kinetic in nature and doesnot manifest as a global PMV/MSV barrier along the progressvariable Θ. The volume barrier for the establishment of a helicalsegment, now confirmed by both experiment and theory, islikely connected to a free energy barrier as is evident by ratherslow experimental time constants of ≈50−60 ns for adding orremoving a helical segment.116 Helical hydrogen bondformation likely entails an early stage with energeticallyunfavorable bond angles, which can become a kineticbottleneck in the formation of a helical segment.The MSV/PMV Perspective Should Also Account for
Increases in Volume upon Folding of NaturallyOccurring Globular Proteins. On the basis of our presentfindings and previous considerations,21 we concur with the viewthat the observed positive reaction volumes for the folding ofglobular proteins are not caused by the formation of α-helices
but rather by the void volumes or packing defects in theproteins’ folded cores.19,20 Whether other secondary structureelements such as π-helices that are less tightly packed cancontribute positively to the folding reaction volume is an openquestion that can also be addressed by our formulation in thefuture. Although a detailed investigation of volumetric proper-ties of globular proteins is beyond the scope of this article, it isinstructive to take a first step to extend our formalism to a 54-residue model three-helix bundle (Figure 7 and SFigure 11)
that undergoes reversible folding and unfolding.79,83 Despitethe structural artificiality of this model, which contains onlyglycines and one type each of Ala-sized nonpolar and polarresidues, its MSV increases upon folding (Figure 7A). Thisresult is in line with our expectation that the volumetric trendof a folding globular protein is opposite to that of the coil−helixtransition of an isolated α-helix. This MSV increasing trend isquite robust, as it is observed in a modified version of themodel with a larger side chain vdW radius, as well (SFigure11B), even though a larger residue size might be expected toreduce the void volume in the sequestered core. However,because of the extraordinarily high packing density (ρ > 0.86)of the folded state of this model three-helix bundle, its foldingreaction volume is negative when the chemical component ofPMV is taken into consideration (black dots in Figure 7B). Inother words, the decreasing trend of the δV component uponfolding (SFigure 11A) overcompensates for the increasing MSV
Figure 7. Larger void and/or cavity volumes in the folded states ofnaturally occurring globular proteins lead to larger PMVs relative totheir unfolded-state PMVs. (A) MSV of our model three-helix bundleincreases with the fractional number of native contacts Q as packingdefects build up toward the hydrophobic core of Q = 1 foldedconformations (a snapshot is shown by the ribbon diagram). (B) Thechange in PMV upon folding, ΔPMV, of this model was computedusing conformations sampled in the shaded unfolded (uf) and folded(f) regions in panel A. Plotted as a function of atomic packing densityρ, each small black dot for ΔPMV in the scatter plot (B) is the PMVfof a folded conformation minus the ensemble average ⟨PMV⟩uf = 5478Å3 of the unfolded region (SFigure 11A). Extrapolation of the model-predicted ρ dependence of ΔPMV for ΔPMV < 0 to smaller ρ valuesthat are typical of natural globular proteins is indicated by the twostraight lines. Included for comparison are the published ΔPMV > 0(ref 2) and ρ values of a selection of natural proteins [larger circles andstars (see STable 7)], the volumetric properties of which are seen to beconsistent with the present extrapolation. We computed the shown ρvalues by MSROLL117 (circles) for CI2 [Protein Data Bank (PDB)entry 2CI2, green], BSCspB (PDB entry 1CSQ, blue), lysozyme (PDBentry 1DPX, cyan), azurin (PDB entry 1AZU, magenta), chymo-trypsinogen (PDB entry 1CHG, yellow), thermolysin (PDB entry1KEI, olive), metmyoglobin (PDB entry 1YMB, dark blue),cytochrome c (PDB entry 1HRC, deep purple), and SNase WT(PDB entry 1EY0, brown), and also by ProteinVolume50 for CI2 andBSCspB (stars).
trend (Figure 7A). Nonetheless, when the predicted depend-ence of the folding reaction volume on atomic packing densityρ is extrapolated to ρ < 0.82 values that are typical of naturallyoccurring globular proteins,50,117 it is seen to be consistent withpositive folding reaction volumes as exemplified by severalnatural proteins (Figure 7B) for which pertinent data areavailable.2 Two related lessons were learned here. (i) Globularpolypeptide structures do not necessarily have a positivereaction volume upon folding, and the reaction volume can benegative if tight packing comparable to that in our model three-helix bundle can be achieved. (ii) Natural globular proteinshave a positive folding reaction volume because of the choice ofresidues in their sequences, as required by structure andfunction, that tends to leave relatively large cavities in theirfolded cores and also encode rough surfaces with significant re-entrant void volumes.Cavities and/or voids generally destabilize the folded state
even under constant atmospheric pressure because of the lossof favorable van der Waals contacts.118 Cavities can be empty(void) or partly filled with water. Consistent with theperspective described above, while hydrostatic pressure onthe L99A mutant of T4 lysozyme with a large internalcavity119,120 leads to hydration of the cavity and increasedconformational fluctuation,121 the pressure-induced conforma-tional fluctuation is reduced by filling the empty cavity with abenzene molecule.122 Thus, one may expect a decreasing trendin the pressure-induced destabilizing effect for folded proteinswith smaller and smaller void volumes. However, because manyother effects also contribute to protein stability, the overalleffects of void-volume changes can be subtle.122
In summary, we have presented a coherent theoreticalframework for understanding conformation-dependent volu-metric effects in peptides and proteins. Aside from theconceptual clarity it provides, the present formulation alsosidesteps uncertainties about the applicability of current explicitwater models to pressure-related simulations.56,57 Because ourapproach is based in part on experimental PMV measurements,it offers a complementary and practical approach to studyingthe effects of pressure on polypeptide properties. To achievethis goal more comprehensively, an obvious extension would beto include pressure (P) coupling either as a PV term in MonteCarlo conformational sampling or as a derived P-dependentforce in Langevin dynamics simulation. Effects of pressure oncoil−helix transition paths can be subtle. For instance, thevolume barriers observed in our simulation are consistent witha pressure-induced slowdown of the coil−helix transition.20
However, the helix formation time under the presentsimulation conditions is not correlated with any appreciablebiases in PMV-increasing versus PMV-decreasing kinetictransitions (SFigure 12), although such biases can be inducedwhen pressure is applied. Another natural extension of ourmodel is the incorporation of all amino acid residue types.45
Temperature effects such as expansivity of proteins123 can alsobe addressed by combining multiple-temperature simulationswith experimental small-compound PMV data measured atdifferent temperatures44 to address combined pressure−temperature effects.100,124 Denaturant effects67 can also beencompassed by additional SASA-dependent terms.82 In theforeseeable future, these developments should allow for moresystematic theoretical investigations to gain insights into a widerange of effects of pressure on protein behaviors.7−9
■ ASSOCIATED CONTENT
*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.bio-chem.6b00802.
Details of the model and the simulation procedure (SIMethods Text), thermodynamic and kinetic profiles of(Ala)16 (SFigure 1), approximate single-exponentialrelaxation of helix formation kinetics (SFigure 2),properties of the model (Ala)16 coil−helix transition(SFigure 3), chain-length dependence of MSV of (Gly)nand (Ala)n (SFigure 4), residue-specific SASAs of peptidebackbone constituents (SFigure 5), helicity-dependentMSV distribution (SFigure 6), detailed analysis of thevolumetric barrier to hydrogen bond formation (SFigure7), an example of nonhelical hydrogen bond rupture andre-establishment (SFigure 8), S to M change in themolecular surface around a hydrogen bond donor andacceptor (SFigure 9), MSV baseline is essential for aviable physical picture in our model formulation (SFigure10), a model three-helix bundle with an atomic packingdensity significantly higher than those of naturallyoccurring proteins (SFigure 11), statistics of kineticPMV changes are nearly invariant with respect to foldingtime (SFigure 12), statistics of volumetric three-stateanalysis of the formation and rupture of a single helicalhydrogen bond occurring in different time segments(STables 1−3), statistics of the sharpness of the volumemaximum encountered during hydrogen bond formationand rupture (STables 4−6), and experimental foldingreaction volumes of natural proteins (STable 7) (PDF)
FundingThis work was funded by Canadian Institutes of HealthResearch Grant MOP-84281 to H.S.C. We are grateful for thisfinancial support and also for the generous allotment ofcomputational resources by SciNet of Compute Canada.
NotesThe authors declare no competing financial interest.
■ ACKNOWLEDGMENTS
We thank Cathy Royer for helpful discussion. Part of this workwas presented earlier this year by H.K. at the NSF ProteinFolding Consortium 2016 Annual Meeting at WashingtonUniversity in St. Louis and the 9th International Conference onHigh Pressure Bioscience and Biotechnology (2016) inToronto. We thank the interested participants of theseconferences for helpful input.
■ ABBREVIATIONS
FTIR, Fourier transform infrared spectroscopy; MSV, molec-ular surface volume; PMV, partial molar volume; SV, solvent-excluded volume; SASA, solvent-accessible surface area; TTET,triplet−triplet energy transfer; vdW-V, van der Waals volume.
■ REFERENCES(1) Silva, J. L., and Weber, G. (1993) Pressure Stability of Proteins.Annu. Rev. Phys. Chem. 44, 89−113.(2) Royer, C. A. (2002) Revisiting volume changes in pressure-induced protein unfolding. Biochim. Biophys. Acta, Protein Struct. Mol.Enzymol. 1595, 201−209.(3) Chalikian, T. V. (2003) Volumetric properties of proteins. Annu.Rev. Biophys. Biomol. Struct. 32, 207−235.(4) Meersman, F., Smeller, L., and Heremans, K. (2006) Proteinstability and dynamics in the pressure-temperature plane. Biochim.Biophys. Acta, Proteins Proteomics 1764, 346−354.(5) Chalikian, T. V., and Macgregor, R. B. (2009) Origins ofpressure-induced protein transitions. J. Mol. Biol. 394, 834−842.(6) Larios, E., and Gruebele, M. (2010) Protein stability at negativepressure. Methods 52, 51−56.(7) Royer, C. A., and Winter, R. (2011) Protein hydration andvolumetric properties. Curr. Opin. Colloid Interface Sci. 16, 568−571.(8) Silva, J. L., Oliveira, A. C., Vieira, T. C. R. G., de Oliveira, G. A. P.,Suarez, M. C., and Foguel, D. (2014) High-pressure chemical biologyand biotechnology. Chem. Rev. 114, 7239−7267.(9) Luong, T. Q., Kapoor, S., and Winter, R. (2015) Pressure-AGateway to Fundamental Insights into Protein Solvation, Dynamics,and Function. ChemPhysChem 16, 3555−3571.(10) Bridgman, P. W. (1914) The Coagulation of Albumen byPressure. J. Biol. Chem. 19, 511−512.(11) Wu, H. (1931) Studies on denaturation of proteins. 13. A theoryof denaturation. Chin. J. Physiol., Rep. Ser. 5, 321−344; (1995) reprintin Adv. Protein Chem. 46, 6−26.(12) Somero, G. N. (1992) Adaptations to high hydrostatic pressure.Annu. Rev. Physiol. 54, 557−577.(13) Daniel, I., Oger, P., and Winter, R. (2006) Origins of life andbiochemistry under high-pressure conditions. Chem. Soc. Rev. 35, 858−875.(14) Kallmeyer, J., Pockalny, R., Adhikari, R. R., Smith, D. C., andD’Hondt, S. (2012) Global distribution of microbial abundance andbiomass in subseafloor sediment. Proc. Natl. Acad. Sci. U. S. A. 109,16213−16216.(15) Garcia, A. J., III, Putnam, R. W., and Dean, J. B. (2010)Hyperbaric hyperoxia and normobaric reoxygenation increaseexcitability and activate oxygen-induced potentiation in CA1 hippo-campal neurons. J. Appl. Physiol. 109, 804−819.(16) Dellarole, M., Caro, J. A., Roche, J., Fossat, M., Barthe, P.,García-Moreno E, B., Royer, C. A., and Roumestand, C. (2015)Evolutionarily Conserved Pattern of Interactions in a Protein Revealedby Local Thermal Expansion Properties. J. Am. Chem. Soc. 137, 9354−9362.(17) Son, I., Shek, Y. L., Tikhomirova, A., Baltasar, E. H., andChalikian, T. V. (2014) Interactions of urea with native and unfoldedproteins: a volumetric study. J. Phys. Chem. B 118, 13554−13563.(18) de Oliveira, G. A. P., and Silva, J. L. (2015) A hypothesis toreconcile the physical and chemical unfolding of proteins. Proc. Natl.Acad. Sci. U. S. A. 112, E2775−E2784.(19) Roche, J., Caro, J. A., Norberto, D. R., Barthe, P., Roumestand,C., Schlessman, J. L., Garcia, A. E., García-Moreno E, B., and Royer, C.A. (2012) Cavities determine the pressure unfolding of proteins. Proc.Natl. Acad. Sci. U. S. A. 109, 6945−6950.(20) Neumaier, S., Buttner, M., Bachmann, A., and Kiefhaber, T.(2013) Transition state and ground state properties of the helix-coiltransition in peptides deduced from high-pressure studies. Proc. Natl.Acad. Sci. U. S. A. 110, 20988−20993.(21) Dias, C. L., and Chan, H. S. (2014) Pressure-DependentProperties of Elementary Hydrophobic Interactions: Ramifications forActivation Properties of Protein Folding. J. Phys. Chem. B 118, 7488−7509.(22) Prigozhin, M. B., Liu, Y., Wirth, A. J., Kapoor, S., Winter, R.,Schulten, K., and Gruebele, M. (2013) Misplaced helix slows downultrafast pressure-jump protein folding. Proc. Natl. Acad. Sci. U. S. A.110, 8087−8092.
(23) Tugarinov, V., Libich, D. S., Meyer, V., Roche, J., and Clore, G.M. (2015) The Energetics of a Three-State Protein Folding SystemProbed by High-Pressure Relaxation Dispersion NMR Spectroscopy.Angew. Chem., Int. Ed. 54, 11157−11161.(24) Roche, J., Louis, J. M., Bax, A., and Best, R. B. (2015) Pressure-induced structural transition of mature HIV-1 protease from acombined NMR/MD simulation approach. Proteins: Struct., Funct.,Genet. 83, 2117−2123.(25) Johnson, Q. R., Lindsay, R. J., Nellas, R. B., and Shen, T. (2016)Pressure-induced conformational switch of an interfacial protein.Proteins: Struct., Funct., Genet. 84, 820−827.(26) Akasaka, K. (2014) Pressure and protein dynamism. HighPressure Res. 34, 222−235.(27) Sikosek, T., and Chan, H. S. (2014) Biophysics of proteinevolution and evolutionary protein biophysics. J. R. Soc., Interface 11,20140419.(28) Hinz, H. J., Vogl, T., and Meyer, R. (1994) An alternativeinterpretation of the heat capacity changes associated with proteinunfolding. Biophys. Chem. 52, 275−285 [Erratum (1995) 53, 291]..(29) Rouget, J.-B., Aksel, T., Roche, J., Saldana, J.-L., Garcia, A. E.,Barrick, D., and Royer, C. A. (2011) Size and sequence and thevolume change of protein folding. J. Am. Chem. Soc. 133, 6020−6027.(30) Roche, J., Dellarole, M., Caro, J. A., Norberto, D. R., Garcia, A.E., Garcia-Moreno, B., Roumestand, C., and Royer, C. A. (2013) Effectof internal cavities on folding rates and routes revealed by real-timepressure-jump NMR spectroscopy. J. Am. Chem. Soc. 135, 14610−14618.(31) Graziano, G. (2015) On the effect of hydrostatic pressure on theconformational stability of globular proteins. Biopolymers 103, 711−718.(32) Marqusee, S., Robbins, V. H., and Baldwin, R. L. (1989)Unusually stable helix formation in short alanine-based peptides. Proc.Natl. Acad. Sci. U. S. A. 86, 5286−5290.(33) Chakrabartty, A., Kortemme, T., and Baldwin, R. L. (1994)Helix propensities of the amino acids measured in alanine-basedpeptides without helix-stabilizing side-chain interactions. Protein Sci. 3,843−852.(34) Vila, J. A., Ripoll, D. R., and Scheraga, H. A. (2000) Physicalreasons for the unusual alpha-helix stabilization afforded by charged orneutral polar residues in alanine-rich peptides. Proc. Natl. Acad. Sci. U.S. A. 97, 13075−13079.(35) Yu, T. J., Lippert, J. L., and Peticolas, W. L. (1973) Laser Ramanstudies of conformational variations of poly-L-lysine. Biopolymers 12,2161−2175.(36) Carrier, D., Mantsch, H. H., and Wong, P. T. T. (1990)Pressure-induced reversible changes in secondary structure of poly(L-lysine): An ir spectroscopic study. Biopolymers 29, 837−844.(37) Takekiyo, T., Shimizu, A., Kato, M., and Taniguchi, Y. (2005)Pressure-tuning FT-IR spectroscopic study on the helix-coil transitionof Ala-rich oligopeptide in aqueous solution. Biochim. Biophys. Acta,Proteins Proteomics 1750, 1−4.(38) Imamura, H., and Kato, M. (2009) Effect of pressure on helix-coil transition of an alanine-based peptide: an FTIR study. Proteins:Struct., Funct., Genet. 75, 911−918.(39) Zimm, B. H., and Bragg, J. K. (1958) Theory of the One-Dimensional Phase Transition in Polypeptide Chains. J. Chem. Phys.28, 1246−1247.(40) Lifson, S., and Roig, A. (1961) On the Theory of HelixCoilTransition in Polypeptides. J. Chem. Phys. 34, 1963−1974.(41) Qian, H., and Schellman, J. A. (1992) Helix-coil theories: acomparative study for finite length polypeptides. J. Phys. Chem. 96,3987−3994.(42) Chakrabartty, A., and Baldwin, R. L. (1995) Stability of alpha-helices. Adv. Protein Chem. 46, 141−176.(43) Bøje, L., and Hvidt, A. (1972) Volume effects in aqueoussolutions of macromolecules containing non-polar groups. Biopolymers11, 2357−2364.(44) Makhatadze, G. I., Medvedkin, V. N., and Privalov, P. L. (1990)Partial molar volumes of polypeptides and their constituent groups in
aqueous solution over a broad temperature range. Biopolymers 30,1001−1010.(45) Noudeh, G. D., Taulier, N., and Chalikian, T. V. (2003)Volumetric characterization of homopolymeric amino acids. Biopol-ymers 70, 563−574.(46) Strazdaite, S., Versluis, J., Backus, E. H. G., and Bakker, H. J.(2014) Enhanced ordering of water at hydrophobic surfaces. J. Chem.Phys. 140, 054711.(47) Richards, F. M. (1977) Areas, volumes, packing and proteinstructure. Annu. Rev. Biophys. Bioeng. 6, 151−176.(48) Connolly, M. L. (1983) Analytical molecular surface calculation.J. Appl. Crystallogr. 16, 548−558.(49) Connolly, M. L. (1985) Computation of molecular volume. J.Am. Chem. Soc. 107, 1118−1124.(50) Chen, C. R., and Makhatadze, G. I. (2015) ProteinVolume:calculating molecular van der Waals and void volumes in proteins.BMC Bioinf. 16, 101.(51) Lee, B., and Richards, F. M. (1971) The interpretation ofprotein structures: Estimation of static accessibility. J. Mol. Biol. 55,379−400.(52) MacCallum, J. L., Moghaddam, M. S., Chan, H. S., andTieleman, D. P. (2007) Hydrophobic association of alpha-helices,steric dewetting, and enthalpic barriers to protein folding. Proc. Natl.Acad. Sci. U. S. A. 104, 6206−6210.(53) Narayanan, C., and Dias, C. L. (2013) Hydrophobic interactionsand hydrogen bonds in β-sheet formation. J. Chem. Phys. 139, 115103.(54) Hatch, H. W., Stillinger, F. H., and Debenedetti, P. G. (2014)Computational study of the stability of the miniprotein trp-cage, theGB1 β-hairpin, and the AK16 peptide, under negative pressure. J. Phys.Chem. B 118, 7761−7769.(55) Mori, Y., and Okumura, H. (2014) Molecular dynamics of thestructural changes of helical peptides induced by pressure. Proteins:Struct., Funct., Genet. 82, 2970−2981.(56) Best, R. B., Miller, C., and Mittal, J. (2014) Role of solvation inpressure-induced helix stabilization. J. Chem. Phys. 141, 22D522.(57) Paschek, D., Gnanakaran, S., and Garcia, A. E. (2005)Simulations of the pressure and temperature unfolding of an alpha-helical peptide. Proc. Natl. Acad. Sci. U. S. A. 102, 6765−6770.(58) Paschek, D., Hempel, S., and García, A. E. (2008) Computingthe stability diagram of the Trp-cage miniprotein. Proc. Natl. Acad. Sci.U. S. A. 105, 17754−17759.(59) Day, R., Paschek, D., and Garcia, A. E. (2010) Microsecondsimulations of the folding/unfolding thermodynamics of the Trp-cageminiprotein. Proteins: Struct., Funct., Genet. 78, 1889−1899.(60) English, C. A., and García, A. E. (2014) Folding and unfoldingthermodynamics of the TC10b Trp-cage miniprotein. Phys. Chem.Chem. Phys. 16, 2748−2757.(61) Das, P., and Matysiak, S. (2012) Direct characterization ofhydrophobic hydration during cold and pressure denaturation. J. Phys.Chem. B 116, 5342−5348.(62) Dias, C. L. (2012) Unifying microscopic mechanism forpressure and cold denaturations of proteins. Phys. Rev. Lett. 109,048104.(63) Badasyan, A. V., Tonoyan, S. A., Mamasakhlisov, Y. S.,Giacometti, A., Benight, A. S., and Morozov, V. F. (2011) Competitionfor hydrogen-bond formation in the helix-coil transition and proteinfolding. Phys. Rev. E 83, 051903.(64) Bianco, V., and Franzese, G. (2015) Contribution of Water toPressure and Cold Denaturation of Proteins. Phys. Rev. Lett. 115,108101.(65) Vidugiris, G. J. A., Markley, J. L., and Royer, C. A. (1995)Evidence for a molten globule-like transition state in protein foldingfrom determination of activation volumes. Biochemistry 34, 4909−4912.(66) Jacob, M. H., Saudan, C., Holtermann, G., Martin, A., Perl, D.,Merbach, A. E., and Schmid, F. X. (2002) Water contributes actively tothe rapid crossing of a protein unfolding barrier. J. Mol. Biol. 318, 837−845.
(67) Pappenberger, G., Saudan, C., Becker, M., Merbach, A. E., andKiefhaber, T. (2000) Denaturant-induced movement of the transitionstate of protein folding revealed by high-pressure stopped-flowmeasurements. Proc. Natl. Acad. Sci. U. S. A. 97, 17−22.(68) Hummer, G., Garde, S., García, A. E., Paulaitis, M. E., and Pratt,L. R. (1998) The pressure dependence of hydrophobic interactions isconsistent with the observed pressure denaturation of proteins. Proc.Natl. Acad. Sci. U. S. A. 95, 1552−1555.(69) Shimizu, S., and Chan, H. S. (2001) Configuration-DependentHeat Capacity of Pairwise Hydrophobic Interactions. J. Am. Chem. Soc.123, 2083−2084.(70) Chan, H. S., Zhang, Z., Wallin, S., and Liu, Z. (2011)Cooperativity, local-nonlocal coupling, and nonnative interactions:principles of protein folding from coarse-grained models. Annu. Rev.Phys. Chem. 62, 301−326.(71) Cheung, M. S., García, A. E., and Onuchic, J. N. (2002) Proteinfolding mediated by solvation: water expulsion and formation of thehydrophobic core occur after the structural collapse. Proc. Natl. Acad.Sci. U. S. A. 99, 685−690.(72) Levy, Y., and Onuchic, J. N. (2006) Water mediation in proteinfolding and molecular recognition. Annu. Rev. Biophys. Biomol. Struct.35, 389−415.(73) Hillson, N., Onuchic, J. N., and Garcia, A. E. (1999) Pressure-induced protein-folding/unfolding kinetics. Proc. Natl. Acad. Sci. U. S.A. 96, 14848−14853.(74) Sirovetz, B. J., Schafer, N. P., and Wolynes, P. G. (2015) WaterMediated Interactions and the Protein Folding Phase Diagram in theTemperature-Pressure Plane. J. Phys. Chem. B 119, 11416−11427.(75) Floris, F. M. (2004) Nonideal Effects on the Excess Volumefrom Small to Large Cavities in TIP4P Water. J. Phys. Chem. B 108,16244−16249.(76) Moghaddam, M. S., and Chan, H. S. (2007) Pressure andtemperature dependence of hydrophobic hydration: volumetric,compressibility, and thermodynamic signatures. J. Chem. Phys. 126,114507.(77) Vilseck, J. Z., Tirado-Rives, J., and Jorgensen, W. L. (2015)Determination of partial molar volumes from free energy perturbationtheory. Phys. Chem. Chem. Phys. 17, 8407−8415.(78) Veitshans, T., Klimov, D., and Thirumalai, D. (1997) Proteinfolding kinetics: timescales, pathways and energy landscapes in termsof sequence-dependent properties. Folding Des. 2, 1−22.(79) Knott, M., and Chan, H. S. (2004) Exploring the effects ofhydrogen bonding and hydrophobic interactions on the foldability andcooperativity of helical proteins using a simplified atomic model.Chem. Phys. 307, 187−199.(80) Liu, Z., and Chan, H. S. (2005) Solvation and desolvationeffects in protein folding: native flexibility, kinetic cooperativity andenthalpic barriers under isostability conditions. Phys. Biol. 2, S75−S85.(81) Badasyan, A., Liu, Z., and Chan, H. S. (2008) Probing possibledownhill folding: native contact topology likely places a significantconstraint on the folding cooperativity of proteins with approximately40 residues. J. Mol. Biol. 384, 512−530.(82) Chen, T., and Chan, H. S. (2014) Effects of desolvation barriersand sidechains on local-nonlocal coupling and chevron behaviors incoarse-grained models of protein folding. Phys. Chem. Chem. Phys. 16,6460−6479.(83) Irback, A., Sjunnesson, F., and Wallin, S. (2000) Three-helix-bundle protein in a Ramachandran model. Proc. Natl. Acad. Sci. U. S. A.97, 13614−13618.(84) Brunger, A., Brooks, C. L., III, and Karplus, M. (1984)Stochastic boundary conditions for molecular dynamics simulations ofST2 water. Chem. Phys. Lett. 105, 495−500.(85) Kundrot, C. E., Ponder, J. W., and Richards, F. M. (1991)Algorithms for calculating excluded volume and its derivatives as afunction of molecular conformation and their use in energyminimization. J. Comput. Chem. 12, 402−409.(86) L’Ecuyer, P. (1988) Efficient and portable combined randomnumber generators. Commun. ACM 31, 742−751.
(87) Pettersen, E. F., Goddard, T. D., Huang, C. C., Couch, G. S.,Greenblatt, D. M., Meng, E. C., and Ferrin, T. E. (2004) UCSFChimera–a visualization system for exploratory research and analysis. J.Comput. Chem. 25, 1605−1612.(88) Daura, X., Mark, A. E., and Van Gunsteren, W. F. (1998)Parametrization of aliphatic CHn united atoms of GROMOS96 forcefield. J. Comput. Chem. 19, 535−547.(89) Kaya, H., Uzunog lu, Z., and Chan, H. S. (2013) Spatial ranges ofdriving forces are a key determinant of protein folding cooperativityand rate diversity. Phys. Rev. E 88, 044701.(90) Holzgrafe, C., and Wallin, S. (2015) Local versus global foldswitching in protein evolution: insight from a three-letter continuousmodel. Phys. Biol. 12, 026002.(91) Holzgrafe, C., and Wallin, S. (2014) Smooth functionaltransition along a mutational pathway with an abrupt protein foldswitch. Biophys. J. 107, 1217−1225.(92) Sikosek, T., Krobath, H., and Chan, H. S. (2016) TheoreticalInsights into the Biophysics of Protein Bi-Stability and EvolutionarySwitches. PLoS Comput. Biol. 12, e1004960.(93) Edelsbrunner, H., and Koehl, P. (2003) The weighted-volumederivative of a space-filling diagram. Proc. Natl. Acad. Sci. U. S. A. 100,2203−2208.(94) Edelsbrunner, H., Kirkpatrick, D., and Seidel, R. (1983) On theshape of a set of points in the plane. IEEE Trans. Inf. Theory 29, 551−559.(95) Edelsbrunner, H. (1995) The union of balls and its dual shape.Discrete Comput. Geom. 13, 415−440.(96) Petitjean, M. (1994) On the analytical calculation of van derWaals surfaces and volumes: Some numerical aspects. J. Comput.Chem. 15, 507−523.(97) Richmond, T. J. (1984) Solvent accessible surface area andexcluded volume in proteins. Analytical equations for overlappingspheres and implications for the hydrophobic effect. J. Mol. Biol. 178,63−89.(98) Voronoi, G. (1908) Nouvelles applications des parametrescontinus a la theorie des formes quadratiques. Deuxieme memoire.Recherches sur les parallelloedres primitifs. J. fu r die reine und Angew.Math. 134, 198−287.(99) Voloshin, V. P., Medvedev, N. N., Andrews, M. N., Burri, R. R.,Winter, R., and Geiger, A. (2011) Volumetric properties of hydratedpeptides: Voronoi-Delaunay analysis of molecular simulation runs. J.Phys. Chem. B 115, 14217−14228.(100) Voloshin, V. P., Medvedev, N. N., Smolin, N., Geiger, A., andWinter, R. (2015) Disentangling volumetric and hydrational propertiesof proteins. J. Phys. Chem. B 119, 1881−1890.(101) Paci, E., and Marchi, M. (1996) Intrinsic compressibility andvolume compression in solvated proteins by molecular dynamicssimulation at high pressure. Proc. Natl. Acad. Sci. U. S. A. 93, 11609−11614.(102) Gerstein, M., Tsai, J., and Levitt, M. (1995) The volume ofatoms on the protein surface: calculated from simulation, usingVoronoi polyhedra. J. Mol. Biol. 249, 955−966.(103) Gerstein, M., and Chothia, C. (1996) Packing at the protein-water interface. Proc. Natl. Acad. Sci. U. S. A. 93, 10167−10172.(104) Kratky, K. (1978) Area of intersection of n equal circular disks.J. Phys. A: Math. Gen. 11, 1017−1024.(105) Gibson, K., and Scheraga, H. (1987) Exact calculation of thevolume and the surface area of fused hard sphere molecules withunequal atomic radii. Mol. Phys. 62, 1247−1265.(106) Chopra, G., and Levitt, M. (2011) Remarkable patterns ofsurface water ordering around polarized buckminsterfullerene. Proc.Natl. Acad. Sci. U. S. A. 108, 14455−14460.(107) Galamba, N. (2013) Water’s structure around hydrophobicsolutes and the iceberg model. J. Phys. Chem. B 117, 2153−2159.(108) Mateus, M. P. S., Galamba, N., and Cabral, B. J. C. (2012)Structure and electronic properties of a benzene-water solution. J.Chem. Phys. 136, 014507.(109) Chalikian, T. V. (2008) On the molecular origins of volumetricdata. J. Phys. Chem. B 112, 911−917.
(110) Prehoda, K., and Markley, J. (1996) Use of partial molarvolumes of model compounds in the interpretation of high-pressureeffects on proteins. In High Pressure Effects in Biophysics andEnzymology (Markley, J. L., Northrop, D. B., and Royer, C. A., Eds.)pp 33−43. Oxford University Press, New York.(111) Knott, M., and Chan, H. S. (2006) Criteria for downhillprotein folding: calorimetry, chevron plot, kinetic relaxation, andsingle-molecule radius of gyration in chain models with subdueddegrees of cooperativity. Proteins: Struct., Funct., Genet. 65, 373−391.(112) Zhang, Z., and Chan, H. S. (2012) Transition paths, diffusiveprocesses, and preequilibria of protein folding. Proc. Natl. Acad. Sci. U.S. A. 109, 20919−20924.(113) Hales, T. (2005) A proof of the Kepler conjecture. Ann. Math.162, 1065−1185.(114) De Laat, D., De Oliveira Filho, F. M., and Vallentin, F. (2014)Upper bounds for packings of spheres of several radii. Forum ofMathematics, Sigma 2, 1−42.(115) Richards, F. M. (1974) The interpretation of proteinstructures: total volume, group volume distributions and packingdensity. J. Mol. Biol. 82, 1−14.(116) Fierz, B., Reiner, A., and Kiefhaber, T. (2009) Localconformational dynamics in α-helices measured by fast triplet transfer.Proc. Natl. Acad. Sci. U. S. A. 106, 1057−1062.(117) Connolly, M. L. (1993) The molecular surface package. J. Mol.Graphics 11, 139−141.(118) Eriksson, A. E., Baase, W. A., Zhang, X. J., Heinz, D. W., Blaber,M., Baldwin, E. P., and Matthews, B. W. (1992) Response of a proteinstructure to cavity-creating mutations and its relation to thehydrophobic effect. Science 255, 178−183.(119) Collins, M. D., Quillin, M. L., Hummer, G., Matthews, B. W.,and Gruner, S. M. (2007) Structural rigidity of a large cavity-containing protein revealed by high-pressure crystallography. J. Mol.Biol. 367, 752−763.(120) Collins, M. D., Hummer, G., Quillin, M. L., Matthews, B. W.,and Gruner, S. M. (2005) Cooperative water filling of a nonpolarprotein cavity observed by high-pressure crystallography andsimulation. Proc. Natl. Acad. Sci. U. S. A. 102, 16668−16671.(121) Maeno, A., Sindhikara, D., Hirata, F., Otten, R., Dahlquist, F.W., Yokoyama, S., Akasaka, K., Mulder, F. A. A., and Kitahara, R.(2015) Cavity as a Source of Conformational Fluctuation and High-Energy State: High-Pressure NMR Study of a Cavity-Enlarged Mutantof T4 Lysozyme. Biophys. J. 108, 133−145.(122) Nucci, N. V., Fuglestad, B., Athanasoula, E. A., and Wand, A. J.(2014) Role of cavities and hydration in the pressure unfolding of T4lysozyme. Proc. Natl. Acad. Sci. U. S. A. 111, 13846−13851.(123) Dellarole, M., Kobayashi, K., Rouget, J.-B., Caro, J. A., Roche,J., Islam, M. M., Garcia-Moreno E, B., Kuroda, Y., and Royer, C. A.(2013) Probing the Physical Determinants of Thermal Expansion ofFolded Proteins. J. Phys. Chem. B 117, 12742−12749.(124) Pandharipande, P. P., and Makhatadze, G. I. (2016)Applications of pressure perturbation calorimetry to study factorscontributing to the volume changes upon protein unfolding. Biochim.Biophys. Acta, Gen. Subj. 1860, 1036−1042.
All heavy (non-hydrogen) backbone atoms are represented in our model polyalanine
peptide chain except at the two termini. Backbone constituents are the amino group
(N-H), Cα atom, and the carbonyl group (C’=O). The positions of all of these atoms
are explicitly accounted for. This formulation thus allows for the analysis of backbone
hydrogen bonding between amino hydrogens and carbonyl oxygens, although Cα
hydrogens are not represented explicitly1. For simplicity, the amino group (NH2) at
the N-terminus is reduced to a single nitrogen atom, whereas the carboxyl group
C’OOH at the C-terminus is reduced to a single C’ atom in the model. The alanine
sidechain, which is a methyl group (CβH3), is represented by a single sidechain bead
at the position of the Cβ atom with increased radius rSC = 2.2 Å. This van der Waals
radius is nearly identical to the 21/6(3.910 Å)/2 = 2.19 Å value implemented in the
OPLS forcefield for a CH3 group bound to a CH2. Other geometric parameters of our
model follow the list of atomic vdW radii, bond lengths and bond angles provided in
2
Table 1 of ref.1. By this construction, the present (Ala)16 peptide model consists of 94
explicitly represented atoms.
Simple atomic forcefield
The potential energy function of our model is identical to that in ref.1 except for the
sidechain-sidechain hydrophobic interaction term. Briefly, the total potential function
V in ref.1, given in the original form V = Vl + VΘ + Vχ + Vω + Vdh + Vhc + Vhp + Vhb, is a
sum of harmonic bond length and bond angle terms (Vl + VΘ + Vχ), a term enforcing
correct chirality (Vχ), a single-minimum angular potential around the NC’O double
bond and a three-minimum potential for the dihedral angles φ, ψ (Vω + Vdh), a
repulsive excluded-volume term for all pairs of backbone and sidechain atoms that
are separated by three or more covalent bonds (Vhc), a term for sidechain-sidechain
hydrophobic interaction (Vhp), and a term for backbone hydrogen bonding (Vhb).
Here we replace the Lennard-Jones form of the sidechain-sidechain hydrophobic
interaction potential Vhp in ref.1 by an interaction potential EHP(r; rSC, rw, εcm, εdb, εssm)
that involves a desolvation barrier3–6, where r is the distance between a pair of Cβ
positions. This desolvation-barrier potential is characterized by a contact minimum at
a pair distance of rcm = 2rSC, a desolvation barrier at rdb = rcm + rw, and a solvent-
separated minimum at rssm = rcm + 2rw, where rw = 1.5 Å is approximately the radius of
a water molecule modeled as a sphere. The functional form of EHP is given by U(r;
rcm, ε, εdb, εssm) in Eqs. (2) and (3) of ref.4 with rcm = 2rSC. As in our recent
simulations6, we set the energy at the contact minimum to ‒εcm = ‒ε (ε > 0), the
desolvation barrier height εdb at rdb to 0.1ε, and the solvent-separated energy
minimum at rssm to ‒ssm = ‒0.2ε. Sidechain-sidechain excluded volume is accounted
for by the r < rcm part of EHP. Here, as in ref.1, we buttressed hardcore repulsion by
scaling the EHP > 0 values for r/rcm < 2‒1/6 = 0.891 by a factor of 2.2 (EHP = 0 at r =
2‒1/6 rcm and EHP < 0 for r > 2‒1/6 rcm). A long distance cutoff at r = 8 Å was also
applied to EHP.
Our potential function for backbone hydrogen bonding between a carbonyl oxygen
atom (acceptor; A) and an amino hydrogen (donor; D) takes exactly the same form
as the Vhb term in ref.1. Because of the centrality of this term in our analysis, we re-
display it here with some notational alterations:
3
𝐸HB({𝑟DA}) = HB ∑ [5 (DA
𝑟DA)
12
− 6 (DA
𝑟DA)
10
] cos2()cos2()
D−A pairs
. (1)
Here the total hydrogen bond energy EHB({rDA}) is a function of all donor-acceptor
distances {rDA} in a given polypeptide conformation, εHB and σDA correspond,
respectively, to εhb and σhb in Eq. (9) of ref.1. We used the same interaction
parameters for EHB as those for Vhb in ref.1, viz., εHB = 2.8ε (ε = 1 in the present study)
and σDA = 2.0 Å. EHB is strongly directional. The angles α and β are defined by α ≡
180° ‒ ζ1 where ζ1 is the N-H-O angle, and β ≡ 180° ‒ ζ2 where ζ2 is the H-O-C’
angle. A hydrogen bond is considered formed if 1.5 Å < rDA < 3.0 Å and both α, β <
45°. The hydrogen bond is regarded as helical if the sequence separation between A
and D is four (i, i + 4), and non-helical otherwise. Interaction parameters for all
energy terms other than EHP are also the same as those in ref.1.
The above-described scheme is capable of folding our model (Ala)16 peptide into a
right-handed α-helix with 12 helical hydrogen bonds. We define a “folded” helix as
one containing 12 helical hydrogen bonds and in which every distance between (i, i +
4) pairs of Cα atoms (i = 1, 2, …, 12) is smaller than 6.5 Å. A useful reaction
coordinate for coil-helix transition is the number of helical hydrogen bonds normalized
by the maximum number (12 for (Ala)16). We refer to this quantity as helicity or helical
content and denote it by Θ.
Energy and time scales
A quantitative connection between our Langevin time and real time may be estimated
by recognizing that the model hydrophobic interaction strength should correspond
to ε ≈ 1 - 2 kcal/mol for real polypeptides7. Taking ε = 2 kcal/mol, the temperature
scale T0 = ε/kB = 1015 K, where kB is Boltzmann’s constant. Dynamics of our model
(Ala)16 were obtained by numeric integration of the Langevin equation in the ballistic
regime using the Brünger-Brooks-Karplus (BBK) integrator8 applied to protein
folding1,7. In the ballistic regime, the characteristic timescale of the simulation is given
by 𝜏𝑙 = √𝑚0𝑎02 ⁄ ≈ 0.12 ps, where m0 = 12 atomic mass units (amu) and a0 = 1 Å.
Here we adopted an integration time step τS = 0.02τl = 2.4 fs and a friction coefficient
γ/m0 = 0.0125/τl so that γ = 2 x 10-12 g s-1, and γτS/m0 = 2.5 x 10-4 << 1 holds, thus
4
satisfying the condition for the ballistic regime. In view of the near invariance of shape
of the model chevron plots simulated using Langevin dynamics with widely different
friction coefficients5, the fact that γ is three orders of magnitude smaller than the
Stokes friction γw of a spherical object with radius a0 in water characterized by a
dynamic viscosity of µw = 1 cP at 20°C is expected to be of negligible impact on our
conclusions regarding equilibrium properties and ordering (but not speed) of kinetic
events.
Other simulation details
Model peptide simulation data presented in this work were obtained from trajectory
statistics of 122 successful “folding” simulations of coil-helix transition for (Ala)16 that
ended with a complete helix. Simulations were carried out at constant temperature T
= 0.29T0. According to the above consideration, this temperature corresponds to
approximately 25°C, the temperature at which the experimental PMV data used for
our model δV were originally determined. Simulations started from a randomly
generated extended conformation with time-dependent initialization of the random
number generator provided in the TINKER molecular modeling package9.
Equilibration of the Langevin thermostat was observed after approximately 1.5 x 105
Langevin time steps. Our data collection and statistical averaging commenced after 3
x 105 time steps. After that point, hydrophobic energy EHP, hydrogen bond energy
EHB, hydrogen bond count, helical hydrogen bond count, geometric volumetric
observables, and backbone as well as sidechain SASAs were recorded every Δt =
250 time steps. Geometry-based volumes were computed by routines provided in the
TINKER package9. In 36 of these simulations, atomic SASAs were computed in
addition to residue-specific backbone and sidechain SASAs (SFig.5). Statistical
analyses, PMV modeling, and volumetric analysis of single hydrogen bond
formations were carried out using in-house MATLAB (MATLAB R2008b, The
MathWorks Inc., Natick, MA, USA) scripts. ORIGIN (OriginLab, Northampton, MA,
USA) was used for data visualization. Conformational data was converted into
standard PDB format using an in-house MATLAB routine and visualized using the
CHIMERA 1.10.1 program package10. For comparison, temperature replica-
exchange simulation was also used to obtain thermodynamic distributions (SFig.1).
5
References for SI Methods Text:
(1) Knott, M., and Chan, H. S. (2004) Exploring the effects of hydrogen bonding and hydrophobic
interactions on the foldability and cooperativity of helical proteins using a simplified atomic model.
Chem. Phys. 307, 187–199.
(2) Daura, X., Mark, A. E., and Van Gunsteren, W. F. (1998) Parametrization of aliphatic CHn united
atoms of GROMOS96 force field. J. Comput. Chem. 19, 535–547.
(3) Liu, Z., and Chan, H. S. (2005) Desolvation is a likely origin of robust enthalpic barriers to protein
folding. J. Mol. Biol. 349, 872–89.
(4) Liu, Z., and Chan, H. S. (2005) Solvation and desolvation effects in protein folding: native flexibility,
kinetic cooperativity and enthalpic barriers under isostability conditions. Phys. Biol. 2, 75–85.
(5) Badasyan, A., Liu, Z., and Chan, H. S. (2008) Probing possible downhill folding: native contact
topology likely places a significant constraint on the folding cooperativity of proteins with approximately
40 residues. J. Mol. Biol. 384, 512–530.
(6) Chen, T., and Chan, H. S. (2014) Effects of desolvation barriers and sidechains on local-nonlocal
coupling and chevron behaviors in coarse-grained models of protein folding. Phys. Chem. Chem.
Phys. 16, 6460–6479.
(7) Veitshans, T., Klimov, D., and Thirumalai, D. (1997) Protein folding kinetics: timescales, pathways
and energy landscapes in terms of sequence-dependent properties. Fold. Des. 2, 1–22.
(8) Brünger, A., Brooks III, C. L., and Karplus, M. (1984) Stochastic boundary conditions for molecular
dynamics simulations of ST2 water. Chem. Phys. Lett. 105, 495–500.
(9) Kundrot, C. E., Ponder, J. W., and Richards, F. M. (1991) Algorithms for calculating excluded
volume and its derivatives as a function of molecular conformation and their use in energy
minimization. J. Comput. Chem. 12, 402–409.
(10) Pettersen, E. F., Goddard, T. D., Huang, C. C., Couch, G. S., Greenblatt, D. M., Meng, E. C., and
Ferrin, T. E. (2004) UCSF Chimera--a visualization system for exploratory research and analysis. J.
Comput. Chem. 25, 1605–1612.
6
SFigure 1: Thermodynamic and kinetic profiles of model (Ala)16. Negative logarithm of
normalized population density P(Θ), −lnP(Θ), as a function of helicity Θ of the conformations
visited by the 122 kinetic trajectories (red data points) at model temperature T = 0.29 T0; and
−lnP(Θ) of the conformations sampled under equilibrium conditions at various model
temperatures as indicated (black data points). Thermodynamic data were collected from a
temperature replica-exchange simulation lasted for 109 Langevin time steps. Lines
connecting data points are merely guides for the eye. The definition for the red kinetic profile
here follows that for the first passage profile –lnPFP in Zhang and Chan [Zhang Z, Chan HS
(2012) Transition paths, diffusive processes, and preequilibria of protein folding. Proc Natl
Acad Sci USA 109:20919-20924].
7
SFigure 2: Approximate single-exponential relaxation of helix formation kinetics. The
plot of natural logarithm of the fraction of model (Ala)16 peptides that have not arrived at the
fully helical state (vertical axis) as a function of time (horizontal axis) is approximately linear.
Here N(uf) and N(f) denote “unfolded” and “folded” populations, respectively, that we
obtained from the first passage times of complete helix formation (“folding”) among 122
kinetic trajectories initiated from randomly generated conformations. The mean first passage
time for complete helix formation is equal to 1.07 x 108 ± 8.54 x 106 τS. An approximate first-
order full-helix formation (folding) rate of kf = (8.6 ± 0.1) x 10-9 τS-1 was estimated by a linear
fitting of data for small folding times (t < 108 τS) with intercept set at t = 0.
8
SFigure 3: Properties of the model (Ala)16 coil-helix transition. Energetic, structural, and dynamic characteristics were determined as functions of helicity Θ from all 122 trajectories we simulated. A. Average backbone hydrogen bond (red), sidechain hydrophobic (blue) interaction energy, and their sum total (black). The coil-helix transition entails minimization of hydrogen bond energy (increasing favorability) and maximization of sidechain interaction energy (decreasing favorability). Minimum of the sum of these two energies is achieved by the fully helical structure in our model. B. Histogram of sampled helical contents. Among the relatively more disordered conformations that sample helicities between 0 and 0.5, the most probable conformations in this ensemble are those with Θ = 0.167 and Θ = 0.25. Thus, the typical coil-helix transition amounts to increasing Θ from Θ = 0.25 to Θ = 1. C. Average number of nonnative (non-helical) backbone hydrogen bonds. D. Here P(on-pathway) is defined to be the fraction of sampled conformations with no non-helical hydrogen bonds. In our model, the rate-limiting step in the coil-helix transition is the kinetic search for “on-pathway” conformations devoid of non-helical hydrogen bonds. Results shown here indicate that on-pathway conformations are improbable for Θ ≤ 0.4.
9
SFigure 4: Chain length dependence of MSV of polyglycine (Gly)n and polyalanine (Ala)n. To determine the geometric parameters for fully exposed BB and SC groups in our model, average MSV was computed for (A) polyglycine and (B) polyalanine as a function of chain length (n = numbers of amino acid residues, 6 ≤ n ≤ 15) using a modified version of our model with disabled hydrogen bonding and disabled hydrophobic sidechain interactions (εHB = εHP = 0). The simulations were conducted at T = 0.29T0; 107 Langevin time steps were used for each data point plotted. As stated in Methods of main text, both the peptide backbone (BB) unit (CαHC’ONH) and glycine (CαH2C’ONH) are represented as CαC’ONH in the present model. Accordingly, MSV of an alanine residue is identified with the MSV sum of BB and SC, hence MSV(SC) = MSV(Ala) − MSV(Gly), where MSV(Gly) and MSV(Ala) were determined by the slope of the linear fits, respectively, in A and B. Least-squares fitting
r2 = 0.999 in both fits). It follows that MSV(SC) = 31.47 0.1 Å3.
.
10
SFigure 5: Residue-specific solvent exposure of peptide backbone constituents. A. Amino (N-H); B. carbonyl (C=O); C. Cα. Average SASAs of these groups in non-helical ((Θ = 0, blue) and fully helical (Θ = 1, red) conformations were obtained from 36 of the simulated (Ala)16 trajectories for which atomic SASAs were recorded in addition to BB and SC SASAs. Apart from boundary effects, both the amino and carbonyl groups are shielded by up to 90% in the fully helical (red) relatively to the non-helical (blue) conformations, while carbonyl groups are generally more solvent-exposed than amino groups (A, B). In contrast, solvent exposure of α-carbons is largely unaffected by coil-helix transition.
11
SFigure 6: Helicity-dependent MSV distribution. Normalized distributions of MSV for
and 0.66 (brown). Standard deviation of the distribution decreases with increasing Θ, ranging
from 17 Å3 for Θ = 0.33 to 12 Å3 for Θ = 0.66.
12
SFigure 7: Detailed analysis of volumetric barrier to hydrogen bond formation. A. Total vdW-V and MSV of an isolated amino (N-H) group and a carbonyl oxygen atom (O) as a function of H-O distance rDA. The plotted vdW-V and MSV values are relative to the same large-rDA vdW-V baseline. A barrier to H-O approach exists for MSV but not vdW-V. B-F. Geometric variables of the hydrogen bond formation process in Figure 5 of main text were computed every 10 Langevin time steps, shown here as functions of t ‒ t0, where t0 = 8.195 x 106τS is the starting time of the trajectory segment. B. Distance rDA between the amino hydrogen of residue 4 and the carbonyl oxygen of residue 8, indicating a gradual decrease in the donor-acceptor distance rDA as the hydrogen bond forms. C,D. vdW-V and MSV of the polypeptide environment of the hydrogen bond being formed, comprising residues 3 – 9. The highest vDW-V and MSV barriers for this fragment are, respectively, 6.95 Å3 and 17.4 Å3 (D). E,F. MSV and PMV for the entire (Ala)16 peptide. Note that the temporal positions of volume peaks and minima for the 3 – 9 peptide fragment (C,D) remain largely unchanged for the full-length peptide (E,F).
13
SFigure 8: An example of non-helical hydrogen bond rupture and re-establishment. Shown conformations (S, M, F) were selected from the three-state analysis of the trajectory segment in main-text Fig. 5 and SFig.7. Solid red lines are established hydrogen bonds, marked by their corresponding donor-acceptor distances in Å, whereas dashed red lines are for indicating donor (D, white)-acceptor (A, red) distances (between residues 8 and 4 in S, M and between residues 11 and 6 in M). The carbonyl oxygen atom of residue 6 (r6) is involved in a helical hydrogen bond with residue 10 (r10) and in a non-helical hydrogen bond with residue 11 (r11) in both S and F. In this case, the vdW-V maximum observed in M (Fig. 5A and SFig.7C) is due in large measure to the transient rupture of the non-helical hydrogen bond (6-11) as reflected by a donor-acceptor distance increase from 2 to 2.2 Å in M compared to that in S and F.
14
SFigure 9: S to M change in molecular surface around a hydrogen bond donor and accepter. Same conformations as those depicted in the S and M parts of Fig.6C in the main text, drawn in the same style, but now in close-up display in varied orientations to further illustrate the increase in local MSV from S to M. In this particular hydrogen bond formation event, the mutually approaching donor (D) and acceptor (A) atoms from S (left) to M (right) is concomitant with an approach of the sidechain of residue 8 (SC) to the acceptor A atom (A). The water-excluding effect of this SC-A approach causes a reduction in water-probe contact surfaces of A and O3 and an increase in re-entrant surface around the about-to-be-formed hydrogen bond. This change can be visualized here, as in Fig.6C of the main text, by the reduced areas of bright red for A and O3 in (A) because more of their vdW surfaces are behind the translucent molecular surface in M than that in S when viewed in the orientation shown in (A). This leads to an increase of local MSV because of an enlarged re-entrant volume as indicated by the arrows in (B), with the D-A pair further behind the re-entrant surface in M than in S (B). As noted in the main text, the cavity behind the D-A pair also deepens because of a more curved backbone between D and A (Fig.6A of main text).
15
SFigure 10: MSV baseline is essential for a viable physical picture in our model formulation. As discussed in the main text, PMVad-hoc was considered as a control. PMVad-hoc as a function of Θ of our (Ala)16 model captures the negative reaction volume of the coil-helix transition but grossly overestimates the effect, yielding, e.g., PMVad-hoc(Θ = 1) – PMVad-hoc(Θ
Å3/residue. These values are several times larger than the order ‒1 Å3/residue reaction volume determined by experiment.
16
SFigure 11: A model three-helix bundle with a significantly higher atomic packing
density than naturally occurring proteins. Our three-helix bundle model is based on the
54-residue, 3-letter (hydrophobic, polar, glycine) sequence studied by Knott and Chan [Knott
M, Chan HS (2004) Exploring the effects of hydrogen bonding and hydrophobic interactions
on the foldability and cooperativity of helical proteins using a simplified atomic model. Chem
Phys 307:187–199]. A. PMV as a function of Q for the three-helix bundle model studied in
the main text with rSC = 2.2 Å for both the Ala-like hydrophobic (nonpolar) and polar
sidechains (SCs), and a desolvation barrier potential for nonpolar SC interactions as for our
model (Ala)16. Native contacts are defined between residue pairs with Cα-Cα distance less
than or equal to 6.5 Å in the folded structure. The total number of native contacts in this
three-helix bundle model as determined from a low-energy conformer is 54, 36 of which are
between pairs of residues belonging to the same helix. The SASA-dependent δVs
contributed by the nonpolar SCs were computed in exactly the same manner as the SC δVs
for (Ala)16, whereas δVs contributed by the polar SCs were computed in a similar manner by
using experimental data for serine from Makhatadze et al. [Makhatadze GI, Medvedkin VN,
Privalov PL (1990) Partial molar volumes of polypeptides and their constituent groups in
aqueous solution over a broad temperature range. Biopolymers 30:1001–1010]. As
discussed in the main text, because of the high atomic packing density of the folded state of
this model three-helix bundle, its folding reaction volume PMV(0.75 ≤ Q ≤ 1) − <PMV(0 ≤ Q ≤
0.25)> = (5,125 – 5,478) Å3 = −353 Å3 is negative. B. Included for comparison here is the
MSV profile, as a function of Q, for a modified three-helix bundle model using rSC = 2.5 Å as
in Knott and Chan (2004), εHP = 1, and a short-range interaction potential (SRIM) introduced
by Kaya et al. for the nonpolar SC interactions [Kaya H, Uzunoğlu Z, Chan HS (2013) Spatial
ranges of driving forces are a key determinant of protein folding cooperativity and rate
diversity. Phys Rev E 88:044701]. This modified three-helix bundle model has a total of 59
native contacts. The ribbon diagram is a snapshot of its folded state. It is noteworthy that this
model’s ΔMSV of folding, about −150 Å3, is very similar to that of the model in (A) and Fig.7A
of the main text. Results here were computed by replica-exchange simulations using 32
temperatures ranging from 0.3T0 to 0.61T0 for (A) and 0.24T0 to 0.48T0 for (B). Each data
point was obtained by averaging over about 500 conformations with the same given Q.
17
SFigure 12: Statistics of kinetic PMV changes is nearly invariant with respect to folding time. ΔtPMV here is the difference in PMV between two consecutive volume measurements Δt = 250τS apart along a given (Ala)16 trajectory we simulated, i.e., ΔtPMV = PMV(t + Δt) ‒ PMV(t), which can be positive, negative, or zero. While the average <ΔPMV> over all of our 122 simulated trajectories is essentially zero at ‒7 x 10-4 ± 1.6 x 10-3 Å3 because of cancellations between positive and negative ΔtPMVs, standard deviations σ(ΔtPMV)s of ΔtPMV distributions, each for a trajectory (data points in A), is large at about 18.7 Å3. Notably, σ(ΔtPMV) exhibits no systematic increase or decrease with folding time tf of the trajectory (A), i.e., first passage time to the fully helical conformation is not correlated with σ(ΔtPMV). The quantity ΔtPMV2sgn(ΔtPMV), where sgn(ΔtPMV) is the sign (±1) of ΔtPMV, was used to detect potential biases in ΔtPMV magnitude with respect whether ΔtPMV is positive or negative. Thus, <ΔtPMV2sgn(ΔtPMV)> > 0 indicates that the positive ΔtPMV steps are on average larger than negative ΔtPMV steps for the given trajectory, whereas <ΔtPMV2sgn(ΔtPMV)> < 0 indicates the opposite. The lack of correlation between <ΔtPMV2sgn(ΔtPMV)> and tf (B) means that the speed of complete helix formation, at least under a low pressure regime consistent with our modeling setup, is unlikely to be associated with either larger positive volume jumps or larger negative volume jumps.
18
STable 1: Statistics of volumetric three-state analysis of the formation and rupture of a single hydrogen bond in a model (Ala)16 conformation. N trajectory segments with individual length L= 1,000τS each covering either a hydrogen bond formation (Θ to Θ + 1/12) or rupture (Θ to Θ ‒ 1/12) event were analyzed. Average pairwise differences in PMV, MSV, and vdW-V values among start (S), maximum (M), and final (F for formation, R for rupture) states are tabulated. The X-Y notation for S-M, F-M etc. is shorthand for “volume of Y minus volume of X”. Standard deviations of the mean are reported in parentheses under the corresponding mean values. <…> denotes overall average, weighted by N, over Θ = 4/12, 5/12, 6/12, 7/12, and 8/12. Height of volume barrier (S-M) is indicated by an * superscript, reaction volume (F-S or R-S) is denoted as Δ…0 (see Figure 5 of main text).
Change in PMV (Å3)
Formation
<PMV*> = 11.66 0.08 Å3
<ΔPMV0> = -1.08 0.05 Å3
Rupture
<PMV*> = 11.95 0.05 Å3
<ΔPMV0> = -0.38 0.06 Å3
S-M F-M F-S N S-M R-M R-S N
0.33 11.81 (0.12)
12.63 (0.12)
-0.82 (0.12)
35,765 0.33 12.10 (0.08)
12.59 (0.08)
-0.49 (0.08)
81,708
0.42 11.74 (0.15)
13.01 (0.15)
-1.26 (0.15)
19,955 0.42 12.02 (0.09)
12.33 (0.09)
-0.32 (0.09)
54,469
0.50 11.32 (0.23)
12.73 (0.23)
-1.4 (0.23)
8,370 0.50 11.81 (0.13)
12.11 (0.13)
-0.3 (0.13)
29,890
0.58 11.16 (0.33)
12.76 (0.33)
-1.59 (0.33)
3,710 0.58 11.47 (0.18)
11.71 (0.18)
-0.24 (0.18)
13,732
0.67 10.43 (0.53)
12.49 (0.53)
-2.05 (0.53)
1,309 0.67 11.17 (0.26)
11.26 (0.26)
-0.09 (0.26)
6,020
Change in MSV (Å3)
Formation
<MSV*> = 9.34 0.07 Å3
<ΔMSV0> = -1.35 0.07 Å3
Rupture
<MSV*> = 9.51 0.04 Å3
<ΔMSV0> = 0.14 0.04 Å3
S-M F-M F-S N S-M R-M R-S N
0.33 9.51 (0.10)
10.62 (0.09)
-1.11 (0.10)
35,765 0.33 9.68 (0.07)
9.66 (0.07)
0.02 (0.07)
81,708
0.42 9.39 (0.13)
10.93 (0.13)
-1.54 (0.13)
19,955 0.42 9.56 (0.08)
9.36 (0.08)
0.2 (0.08)
54,469
0.50 8.92 (0.18)
10.49 (0.18)
-1.57 (0.18)
8,370 0.50 9.37 (0.10)
9.13 (0.10)
0.23 (0.10)
29,890
0.58 8.71 (0.26)
10.62 (0.26)
-1.91 (0.26)
3,,710 0.58 8.97 (0.14)
8.68 (0.14)
0.3 (0.14)
13732
0.67 7.97 (0.42)
10.06 (0.42)
-2.09 (0.42)
1,309 0.67 8.68 (0.20)
8.18 (0.20)
0.49 (0.20)
6,020
Change in vdW-V (Å3)
Formation
<vdW-V*> = 3.76 0.04 Å3
<ΔvdW-V0> = -0.35 0.04 Å3
Rupture
<vdW-V*> = 3.69 0.03 Å3
<ΔvdW-V0> = 0.02 0.03 Å3
S-M F-M F-S N S-M R-M R-S N
0.33 3.68 (0.06)
4.00 (0.06)
-0.32 (0.06)
35,765 0.33 3.62 (0.04)
3.62 (0.04)
0.01 (0.04)
81,708
0.42 3.82 (0.08)
4.20 (0.08)
-0.38 (0.08)
19,955 0.42 3.68 (0.05)
3.65 (0.05)
0.03 (0.05)
54,469
0.50 3.80 (0.13)
4.15 (0.13)
-0.35 (0.13)
8,370 0.50 3.81 (0.07)
3.78 (0.07)
0.02 (0.07)
29,890
0.58 4.04 (0.20)
4.33 (0.20)
-0.29 (0.29)
3,710 0.58 3.81 (0.10)
3.80 (0.10)
0.01 (0.10)
13,732
0.67 3.80 (0.26)
4.51 (0.26)
-0.71 (0.26)
1,309 0.67 3.89 (0.15)
3.85 (0.15)
0.04 (0.15)
6,020
19
STable 2: Statistics of volumetric three-state analysis of the formation and rupture of a
single hydrogen bond in a model (Ala)16 conformation. Same as STable1 except for L=
1,250τS.
Change in PMV (Å3)
Formation
<PMV*> = 14.33 0.04 Å3
<ΔPMV0> = -1.86 0.04 Å3
Rupture
<PMV*> = 14.87 0.02 Å3
<ΔPMV0> = -0.28 0.02 Å3
S-M F-M F-S N S-M R-M R-S N
0.33 14.60 (0.12)
16.20 (0.12)
-1.68 (0.12)
23,861 0.33 15.10 (0.07)
15.49 (0.07)
-0.39 (0.07)
61,587
0.42 14.20 (0.16)
16.11 (0.16)
-1.86 (0.16)
13,202 0.42 14.90 (0.10)
15.13 (0.10)
-0.24 (0.10)
41,232
0.50 13.99 (0.23)
16.28 (0.23)
-2.28 (0.23)
5,697 0.50 14.64 (0.12)
14.83 (0.12)
-0.19 (0.12)
23,025
0.58 13.53 (0.35)
16.04 (0.35)
-2.5 (0.35)
2,389 0.58 14.32 (0.17)
14.58 (0.17)
-0.26 (0.17)
10,329
0.67 13.26 (1.14)
15.64 (1.14)
-2.38 (1.14)
958 0.67 13.86 (0.25)
13.59 (0.25)
0.27 (0.25)
4,729
Change in MSV (Å3)
Formation
<MSV*> = 11.45 0.08 Å3
<ΔMSV0> = -1.95 0.08 Å3
Rupture
<MSV*> = 11.79 0.05 Å3
<ΔMSV0> = 0.17 0.05 Å3
S-M F-M F-S N S-M R-M R-S N
0.33 11.74 (0.12)
13.50 (0.12)
-1.76 (0.12)
23,861 0.33 12.03 (0.08)
11.96 (0.08)
0.08 (0.08)
61,587
0.42 11.39 (0.16)
13.34 (0.16)
-1.95 (0.16)
13202 0.42 11.83 (0.09)
11.65 (0.09)
0.18 (0.09)
41,232
0.50 10.97 (0.23)
13.41 (0.23)
-2.44 (0.23)
5,697 0.50 11.55 (0.12)
11.35 (0.12)
0.20 (0.12)
23,025
0.58 10.44 (0.33)
12.99 (0.33)
-2.56 (0.33)
2,389 0.58 11.19 (0.17)
10.92 (0.17)
0.27 (0.17)
10,329
0.67 10.22 (0.49)
12.63 (0.49)
-2.41 (0.49)
958 0.67 10.67 (0.24)
9.84 (0.24)
0.83 (0.24)
4,729
Change in vdW-V (Å3)
Formation
<vdW-V*> = 4.51 0.05 Å3
<ΔvdW-V0> = -0.39 0.05 Å3
Rupture
<vdW-V*> = 4.44 0.03 Å3
<ΔvdW-V0> = 0.06 0.03 Å3
S-M F-M F-S N S-M R-M R-S N
0.33 4.52 (0.07)
4.84 (0.07)
-0.32 (0.07)
23,861 0.33 4.41 (0.04)
4.36 (0.04)
0.05 (0.04)
61,587
0.42 4.41 (0.10)
4.85 (0.10)
-0.43 (0.10)
13,202 0.42 4.43 (0.05)
4.37 (0.05)
0.06 (0.05)
41,232
0.50 4.65 (0.16)
5.08 (0.16)
-0.43 (0.16)
5,697 0.50 4.48 (0.08)
4.50 (0.08)
-0.02 (0.08)
23,025
0.58 4.70 (0.24)
5.21 (0.24)
-0.52 (0.24)
2,389 0.58 4.60 (0.12)
4.48 (0.12)
0.12 (0.12)
10,329
0.67 4.57 (0.33)
5.45 (0.33)
-0.88 (0.33)
958 0.67 4.44 (0.18)
4.13 (0.18)
0.30 (0.18)
4,729
20
STable 3: Statistics of volumetric three-state analysis of the formation and rupture of a
single hydrogen bond in a model (Ala)16 conformation. Same as STable1 except for L=
1,500τS.
Change in PMV (Å3)
Formation
<PMV*> = 16.55 0.13 Å3
<ΔPMV0> = -1.96 0.13 Å3
Rupture
<PMV*> = 16.89 0.07 Å3
<ΔPMV0> = -0.30 0.07 Å3
S-M F-M F-S N S-M R-M R-S N
0.33 16.93 (0.18)
18.6 (0.18)
-1.66 (0.18)
17,287 0.33 17.22 (0.10)
17.53 (0.10)
-0.30 (0.10)
49,135
0.42 16.55 (0.23)
18.59 (0.23)
-2.04 (0.23)
9,731 0.42 16.92 (0.13)
17.29 (0.13)
-0.37 (0.10)
32,975
0.50 15.49 (0.34)
18.03 (0.34)
-2.54 (0.34)
4,063 0.50 16.73 (0.16)
16.84 (0.16)
-0.11 (0.16)
18,426
0.58 15.85 (0.49)
18.54 (0.49)
-2.69 (0.49)
1,792 0.58 15.78 (0.24)
16.33 (0.24)
-0.55 (0.24)
8,192
0.67 15.12 (0.79)
18.01 (0.79)
-2.89 (0.79)
694 0.67 15.62 (0.33)
15.73 (0.33)
-0.11 (0.33)
3,765
Change in MSV (Å3)
Formation
<MSV*> = 13.13 0.10 Å3
<ΔMSV0> = -2.08 0.10 Å3
Rupture
<MSV*> = 13.33 0.06 Å3
<ΔMSV0> = 0.17 0.06 Å3
S-M F-M F-S N S-M R-M R-S N
0.33 13.47 (0.14)
15.36 (0.14)
-1.89 (0.14)
17,287 0.33 13.66 (0.09)
13.52 (0.09)
0.14 (0.09)
49,135
0.42 13.18 (0.18)
15.28 (0.18)
-2.10 (0.18)
9,731 0.42 13.38 (0.10)
13.26 (0.10)
0.12 (0.10)
32,975
0.50 12.24 (0.28)
14.63 (0.28)
-2.40 (0.28)
4,063 0.50 13.12 (0.13)
12.78 (0.13)
0.34 (0.13)
18,426
0.58 12.31 (0.38)
14.99 (0.38)
-2.67 (0.38)
1,792 0.58 12.24 (0.19)
12.14 (0.19)
0.10 (0.19)
8,192
0.67 11.30 (0.63)
14.41 (0.63)
-3.11 (0.63)
694 0.67 11.94 (0.26)
11.52 (0.26)
0.42 (0.26)
3,765
Change in vdW-V (Å3)
Formation
<vdW-V*> = 4.98 0.07 Å3
<ΔvdW-V0> = -0.33 0.07 Å3
Rupture
<vdW-V*> = 4.90 0.03 Å3
<ΔvdW-V0> = 0.01 0.03 Å3
S-M F-M F-S N S-M R-M R-S N
0.33 5.04 (0.08)
5.34 (0.08)
-0.29 (0.08)
17,287 0.33 4.86 (0.05)
4.87 (0.05)
-0.01 (0.05)
49,135
0.42 4.92 (0.12)
5.17 (0.12)
-0.26 (0.12)
9,731 0.42 4.91 (0.06)
4.96 (0.06)
-0.04 (0.06)
32,975
0.50 5.02 (0.19)
5.21 (0.19)
-0.19 (0.19)
4,063 0.50 5.00 (0.09)
4.85 (0.09)
0.15 (0.09)
18,426
0.58 4.80 (0.27)
5.94 (0.27)
-1.14 (0.27)
1,792 0.58 4.84 (0.13)
4.87 (0.13)
-0.04 (0.13)
8,192
0.67 4.79 (0.35)
5.78 (0.35)
-0.99 (0.35)
694 0.67 5.04 (0.19)
4.83 (0.19)
0.21 (0.19)
3,765
21
STable 4: Statistics on the sharpness of PMV, MSV and vdW-V maxima. Analyzed trajectory segments and their classification are the same as those in STable 1 (L= 1,000τS). The sharpness measures ξ1 and ξ2 are defined in main text (see Figure 5). Averaged ξ1 and ξ2 over conformations with a given Θ are tabulated. Standard deviations of the mean are reported in parentheses under the corresponding average values. As in STable 1, <…> denotes overall average over different Θs.
PMV (Å3)
Formation
<1> = 6.99 0.05 Å3
<2> = 20.84 0.28 Å3
Rupture
<1> = 6.9 0.03 Å3
<2> =21.01 0.26 Å3
1 2 1 2 0.33 6.88
(0.07) 21.53 (0.07)
0.33 6.9 (0.04)
21.62 (0.04)
0.42 7.13 (0.08)
21.24 (0.08)
0.42 6.96 (0.04)
21.42 (0.04)
0.50 6.89 (0.13)
20.95 (0.13)
0.50 6.95 (0.06)
21.17 (0.06)
0.58 7.05 (0.19)
20.57 (0.18)
0.58 6.91 (0.08)
20.69 (0.08)
0.67 7.03 (0.30)
19.91 (0.30)
0.67 6.78 (0.10)
20.16 (0.14)
MSV (Å3)
Formation
<1> = 5.49 0.06 ų
<2> = 16.77 0.37 Å3
Rupture
<1> = 5.22 0.07 Å3
<2> = 16.93 0.32 Å3
1 2 1 2 0.33 5.45
(0.06) 17.66 (0.06)
0.33 5.31 (0.03)
17.71 (0.03)
0.42 5.67 (0.07)
17.35 (0.07)
0.42 5.33 (0.03)
17.43 (0.03)
0.50 5.38 (0.11)
16.88 (0.11)
0.50 5.29 (0.05)
17.09 (0.05)
0.58 5.57 (0.16)
16.45 (0.16)
0.58 5.18 (0.07)
16.53 (0.07)
0.67 5.38 (0.25)
15.53 (0.25)
0.67 4.98 (0.11)
15.91 (0.11)
vdW-V (Å3)
Formation
<1> = 1.58 0.09 Å3
<2> = 8.34 0.11 Å3
Rupture
<1> = 1.45 0.04 Å3
<2> = 8.26 0.13 Å3
1 2 1 2 0.33 1.46
(0.06) 8.06
(0.04) 0.33 1.48
(0.03) 7.87
(0.02) 0.42 1.56
(0.08) 8.22
(0.06) 0.42 1.43
(0.04) 8.06
(0.03) 0.50 1.37
(0.13) 8.49
(0.10) 0.50 1.41
(0.06) 8.29
(0.04) 0.58 1.61
(0.21) 8.67
(0.15) 0.58 1.42
(0.10) 8.54
(0.07) 0.67 1.90
(0.27) 8.26
(0.20) 0.67 1.53
(0.14) 8.55
(0.10)
22
STable 5: Statistics on the sharpness of PMV, MSV and vdW-V maxima. Analyzed
trajectory segments and their classification are the same as those in STable 2 (L= 1,250τS).
PMV (Å3)
Formation
<1> = 9.15 0.04 Å3
<2> = 22.68 0.24 Å3
Rupture
<1> = 9.15 0.02 Å3
<2> = 22.68 0.24 Å3
1 2 1 2 0.33 9.13
(0.08) 23.37 (0.08)
0.33 9.17 (0.04)
23.39 (0.04)
0.42 9.04 (0.10)
22.91 (0.10)
0.42 9.19 (0.05)
23.13 (0.05)
0.50 9.19 (0.16)
22.79 (0.16)
0.50 9.17 (0.07)
22.88 (0.07)
0.58 9.29 (0.23)
22.27 (0.23)
0.58 9.13 (0.11)
22.53 (0.11)
0.67 9.12 (0.30)
22.05 (0.37)
0.67 9.07 (0.16)
22.02 (0.16)
MSV (Å3)
Formation
<1> = 7.15 0.04 ų
<2> = 18.28 0.30 Å3
Rupture
<1> = 7.02 0.07 Å3
<2> = 18.39 0.32 Å3
1 2 1 2 0.33 7.26
(0.07) 19.11 (0.07)
0.33 7.15 (0.04)
19.15 (0.04)
0.42 7.12 (0.08)
18.67 (0.09)
0.42 7.13 (0.04)
18.85 (0.04)
0.50 7.20 (0.13)
18.39 (0.13)
0.50 7.07 (0.06)
18.54 (0.06)
0.58 7.12 (0.20)
17.78 (0.20)
0.58 6.98 (0.09)
18.03 (0.09)
0.67 7.06 (0.29)
17.44 (0.31)
0.67 6.75 (0.12)
17.36 (0.13)
vdW-V (Å3)
Formation
<1> = 2.25 0.12 Å3
<2> = 8.95 0.13 Å3
Rupture
<1> = 2.06 0.04 Å3
<2> = 8.79 0.12 Å3
1 2 1 2 0.33 2.12
(0.07) 8.60
(0.05) 0.33 2.13
(0.03) 8.43
(0.03) 0.42 2.10
(0.10) 8.69
(0.07) 0.42 2.08
(0.05) 8.58
(0.05) 0.50 2.34
(0.15) 9.11
(0.12) 0.50 2.11
(0.07) 8.85
(0.05) 0.58 2.03
(0.26) 9.14
(0.18) 0.58 1.93
(0.11) 9.05
(0.08) 0.67 2.68
(0.34) 9.20
(0.24) 0.67 2.03
(0.16) 9.02
(0.11)
23
STable 6: Statistics on the sharpness of PMV, MSV and vdW-V maxima. Analyzed
trajectory segments and their classification are the same as those in STable 3 (L= 1,500τS).
PMV (Å3)
Formation
<1> = 10.56 0.07 Å3
<2> = 23.70 0.31 Å3
Rupture
<1> = 10.40 0.07 Å3
<2> = 23.82 0.28 Å3
1 2 1 2 0.33 10.47
(0.08) 24.48 (0.09)
0.33 10.53 (0.04)
24.48 (0.04)
0.42 10.44 (0.11)
24.07 (0.12)
0.42 10.53 (0.05)
24.31 (0.06)
0.50 10.57 (0.17)
23.74 (0.19)
0.50 10.32 (0.07)
23.81 (0.08)
0.58 10.83 (0.25)
23.56 (0.27)
0.58 10.49 (0.11)
23.61 (0.12)
0.67 10.49 (0.41)
22.64 (0.43)
0.67 10.15 (0.16)
22.91 (0.17)
MSV (Å3)
Formation
<1> = 8.24 0.07 ų
<2> = 19.08 0.36 Å3
Rupture
<1> = 8.01 0.09 Å3
<2> = 19.21 0.35 Å3
1 2 1 2 0.33 8.27
(0.07) 20.04 (0.08)
0.33 8.20 (0.04)
20.07 (0.04)
0.42 8.15 (0.10)
19.58 (0.10)
0.42 8.17 (0.05)
19.81 (0.05)
0.50 8.29 (0.14)
19.12 (0.16)
0.50 7.97 (0.06)
19.25 (0.07)
0.58 8.45 (0.21)
18.76 (0.23)
0.58 8.05 (0.09)
18.83 (0.10)
0.67 8.06 (0.34)
17.92 (0.35)
0.67 7.68 (0.14)
18.09 (0.15)
vdW-V (Å3)
Formation
<1> = 2.71 0.13 Å3
<2> = 9.13 0.13 Å3
Rupture
<1> = 2.45 0.04 Å3
<2> = 9.06 0.12 Å3
1 2 1 2 0.33 2.55
(0.08) 8.94
(0.06) 0.33 2.51
(0.04) 8.74
(0.03) 0.42 2.34
(0.11) 8.82
(0.08) 0.42 2.49
(0.05) 8.88
(0.04) 0.50 2.64
(0.18) 9.36
(0.13) 0.50 2.38
(0.08) 9.04
(0.06) 0.58 3.00
(0.26) 9.51
(0.19) 0.58 2.55
(0.12) 9.38
(0.09) 0.67 3.03
(0.34) 9.00
(0.26) 0.67 2.31
(0.18) 9.28
(0.12)
24
STable 7: Summary of measured folding reaction volumes, ΔPMV, of nine natural proteins
plotted in Fig.7 of main text as examples and the experimental conditions under which the
ΔPMVs were determined. Information tabulated here was taken from Table 1 of Royer
[Royer CA (2002) Revisiting volume changes in pressure-induced protein unfolding. Biochim
Biophys Acta - Protein Struct Mol Enzymol 1595:201–209].
Overview of experimental reaction volumes for specific proteins
Name PDB ID ΔPMV [cm3 / mol] ΔPMV [Å3 / molecule] Conditions
