PPacking in Protein Cores
J. C. Gaines , , A. H. Clark , L. Regan , , and C. S. O’Hern , , , , Program in Computational Biology and Bioinformatics, Yale University, New Haven,Connecticut, 06520 Integrated Graduate Program in Physical and Engineering Biology (IGPPEB), YaleUniversity, New Haven, Connecticut, 06520 Department of Mechanical Engineering & Materials Science, Yale University, NewHaven, Connecticut, 06520 Department of Molecular Biophysics & Biochemistry, Yale University, New Haven,Connecticut, 06520 Department of Chemistry, Yale University, New Haven, Connecticut, 06520 Department of Physics, Yale University, New Haven, Connecticut, 06520 Department of Applied Physics, Yale University, New Haven, Connecticut, 06520
Abstract.
Proteins are biological polymers that underlie all cellular functions. The firsthigh-resolution protein structures were determined by x-ray crystallography in the 1960s.Since then, there has been continued interest in understanding and predicting proteinstructure and stability. It is well-established that a large contribution to protein stabilityoriginates from the sequestration from solvent of hydrophobic residues in the protein core.How are such hydrophobic residues arranged in the core? And how can one best model thepacking of these residues? Here we show that to properly model the packing of residues inprotein cores it is essential that amino acids are represented by appropriately calibrated atomsizes, and that hydrogen atoms are explicitly included. We show that protein cores possessa packing fraction of φ ≈ .
56, which is significantly less than the typically quoted valueof 0 .
74 obtained using the extended atom representation. We also compare the results forthe packing of amino acids in protein cores to results obtained for jammed packings fromdisrete element simulations composed of spheres, elongated particles, and particles withbumpy surfaces. We show that amino acids in protein cores pack as densely as disorderedjammed packings of particles with similar values for the aspect ratio and bumpiness as foundfor amino acids. Knowing the structural properties of protein cores is of both fundamentaland practical importance. Practically, it enables the assessment of changes in the structureand stability of proteins arising from amino acid mutations (such as those identified as aresult of the massive human genome sequencing efforts) and the design of new folded, stableproteins and protein-protein interactions with tunable specificity and affinity.
Keywords : proteins, random close packing, jamming a r X i v : . [ c ond - m a t . s o f t ] J a n acking in Protein Cores
1. Introduction
Proteins are biological polymers that play important roles in cellular processes rangingfrom the purely structural to the actively catalytic. Proteins are linear chains of differentcombinations of the 20 naturally occurring amino acid residues with variable chain lengthsfrom tens to tens of thousands. A key feature that distinguishes proteins from otherpolymers is that each folds into a unique three-dimensional structure. Proteins typicallyfold spontaneously in aqueous solution at room temperature. The amino acid sequence isthe only information required to specify a protein’s unique structure [1, 2].The amino acids can be grouped into two main categories: hydrophobic and hydrophilic.Hydrophobic residues form the solvent-inaccessible core of a protein and hydrophilic residues,both polar and charged, are on the solvent-accessible surface. As of 2017, the structures ofmore than 125,000 proteins have been determined, primarily by x-ray crystallography, witha median resolution of ≈ . α -C-O, and different combinations of side chain atoms that branch from the C α atom (Fig. 1). The repeating units are joined by a peptide bond between the carboxyl carbon(C) of a given amino acid and the nitrogen (N) of the next. All bond lengths and bond anglesare specified by the same basic stereochemistry that defines the structures of small molecules[15, 16]. The three-dimensional structure that a protein adopts is specified by the aminoacid dihedral angles. For each amino acid in the protein chain, there are two backbonedihedral angle degrees of freedom, φ and ψ , and N s side chain dihedral angle degrees offreedom, χ , . . . , χ N s . (See Fig. 1.) N s ranges from zero (for alanine and glycine) to five(for arginine). The third backbone dihedral angle is typically constrained to be ω = 180 ◦ or0 ◦ . Repetition of certain backbone φ and ψ values in a stretch of amino acids gives rise tospecific secondary structures, such as α -helices and β -sheets [17, 18]. All proteins are formedfrom different combinations of α -helix, β -sheet, and ‘random coil’ structures. InteractionsFigure 1: Stick representation of a valine(Val) residue with each atom shown in adifferent color: C (green), N (blue), O (red),and H (white). The heavy (non-hydrogen)atoms are also labeled. The two backbonedihedral angles φ and ψ and one side chaindihedral angle χ (defined by the atoms N-C α -C β -C γ ) are indicated. acking in Protein Cores Figure (2) (left) Illustration of a Val residuewith each atom represented as a sphere: C(green), O (red), N (blue), and H (grey).(right) Val and Ile residues with connectedbackbones taken from PDB 1K5C. In bothpanels, heavy atoms are labeled. between different elements of secondary structure are stabilized by interactions between theside chains [19, 20, 21]. In addition, side chain interactions on the surfaces of proteins alsospecify how different proteins bind to each other and to other molecules [6].A minimal physical model for an amino acid is a composite particle formed fromconnected spheres with stereochemical constraints (Fig. 2). As is clear from Fig. 2, aminoacids are non-spherical objects with complex shapes. Thus, we can imagine proteins as stringsof interconnected non-spherical objects that fold into compact three-dimensional structures.Many prior studies have argued that the cores of folded proteins are tightly packed. Forexample, several studies have measured the ratio between the volume of a core amino acidand its Voronoi volume to be greater than 70%, which suggests dense crystalline packing[22, 23]. In addition, experimental studies find that mutations in protein cores from smallto large residues typically destabilize the protein, suggesting that there is very little emptyspace present to accommodate additional atoms [24, 25].In this review, we summarize prior work on the structural properties of protein coresand provide strong evidence that although protein cores are densely packed, they are notas densely packed as crystalline solids. Instead, protein cores possess packing fractions of ∼ .
56 [14]. Even though this value is lower than that for crystalline solids ( e.g. .
74 forface-centered-cubic crystals), protein cores are solid-like with very little free volume thatwould allow side chain motion. We also show that static packings of particles with complex,non-spherical shapes possess packing fractions below 0 .
6, yet still display solid-like propertiesand that the amino acids in protein cores can be modeled as random, densely packed non-spherical objects. We then relate our computational studies of dense packing in proteincores to experimental studies of mutations that are able to alter the structure and stabilityof proteins.
2. Packing efficiency in protein cores
By determining the packing fraction of protein cores one can begin to understand theirstructural and mechanical properties. For example, the shear modulus ( i.e. the material acking in Protein Cores @ P ( @ )
8% 74% 18%13% 69% 18% @ @
0% 0%
13% 6% 60%1% 3% 1% @ @
0% 0%
11% 12% 54%0% 5% 3%
Figure 3: (left) The observed side chain dihedral angle probability distribution (black dottedline), P ( χ ), for Val residues in a database of high resolution protein crystal structures(described in [14, 32, 33]) compared to P ( χ ) predicted by the hard-sphere dipeptide mimeticmodel for Val using the explicit hydrogen atom representation (blue solid line). (center) Theobserved side chain dihedral angle probability distribution P ( χ , χ ) for Ile. (right) Thepredicted side chain dihedral angle distribution for Ile using the hard-sphere model. Theprobabilities increase from light to dark. The percentages give the fractional probabilitiesthat occur in each of the three and nine rotamer bins in the left panel and center/right panels,respectively. The center and right panels are reprinted with permission from [J. C. Gaines,W. W. Smith, L. Regan, and C. S. O’Hern, Phys. Rev. E, 93, 032415, 2016.] Copyright(2016) by the American Physical Society.response to applied shear stress) typically increases monotonically with the packing fractionsince the number of stress-bearing interatomic contacts increases with the packing fraction[26]. Thus, the rigidity of proteins is strongly correlated with the packing density [27, 28].In addition, knowing the packing density is vital for predicting changes in stability frommutations to protein cores, many of which are disease-associated [29]. Accurate calculationsof the packing density are also necessary to predict structure from sequence and to designnew stable proteins [10, 30, 31].One of the first studies of the packing density of protein cores was performed by Richardsin 1974. At this time, only a few protein crystal structures were available. Richards focusedon two proteins: lysozyme and ribonuclease S [22]. When a protein structure is obtainedfrom x-ray crystallography, the resolution of the structure typically does not allow for theplacement of the hydrogen atoms in the protein. In the past, researchers circumvented thisproblem by implementing an “extended atom” model, where the atomic radii of each heavyatom are increased by a factor that depends on the number of hydrogen atoms that arebound to it [22, 23, 34]. New computational techniques allow for the accurate placementof hydrogen atoms in a protein crystal structure [35, 36], which provides a more detailed“explicit hydrogen” model of proteins. Since hydrogen atoms comprise ∼
50% of the atomsin a protein, the extended atom approximation can have major effects on the accuracy of acking in Protein Cores sp aromatic O C sp , C aromatic : 1 . C O : 1 . O : 1 . N : 1 . H : 1 .
10 ˚A; and S : 1 .
75 ˚A. In Fig. 3, we show that theside chain dihedral angle distributions predicted using the hard-sphere model for a Val andIle dipeptide agree with the observed side chain dihedral angle distributions. We have shownsimilar agreement between the observed and predicted side chain dihedral angle distributionsfor Cys, Leu, Met, Phe, Thr, Trp, Tyr, and Ser [38]. The atomic radii are similar to valuesof van der Waals radii reported in other studies, and typically smaller than those used inextended atom models (Table 1) [18, 22, 34, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47].The packing fraction of each residue in a protein core can be calculated using φ r = (cid:80) i V i (cid:80) i V vi , (1)where V i is the ‘non-overlapping’ volume of atom i , V vi is the Voronoi volume of atom i , andthe summations are over all atoms of a particular residue. We also calculate the packingfraction of a protein core, φ c , where both summations are over all atoms of all residuesin a particular protein core. Voronoi cells were obtained for each atom using Laguerretessellation, where the placement of the Voronoi cell wall is based on the relative radii ofneighboring atoms (which is the same as the location of the plane that separates overlappingatoms) [14, 48]. V i was calculated by splitting overlapping atoms by the plane of intersectionbetween the two atoms.Our analysis focuses on residues in protein cores. We have identified all core residuesin a database of high resolution crystal structures (described in [14, 32, 33]) using a method acking in Protein Cores φ c of protein cores as a function of thenumber of core residues ( N R ) using the explicit hydrogen (blue circles) and extended atom(red squares) representations. More residues are designated as core using the extend atommodel (25 on average) than using the explicit hydrogen model (15 on average). The dashedand solid horizontal lines indicate the average packing fraction of each system, φ c = 0 . φ c = 0 .
56 for explicit hydrogen. (b) The probability distribution(red dotted line) of packing fractions at jamming onset P ( φ ) from simulations of mixturesof individual residues found in protein cores. The results were obtained by simulating 100jammed packings of N R = 24 residues with amino acid frequencies that match protein cores.The probability distribution of packing fractions of protein cores is shown by the solid blackline. Panels (a) and (b) are reprinted with permission from [J. C. Gaines, W. W. Smith,L. Regan, and C. S. O’Hern, Phys. Rev. E, 93, 032415, 2016.] Copyright (2016) by theAmerican Physical Society.described previously [14, 49]. In brief, non-core atoms are identified as those that are on thesurface of the protein or near an interior void with a radius ≥ . φ c ≈ .
56 [14].This value is much closer to packing fractions obtained for jammed packings of frictional orelongated particles rather than φ c = 0 . .
74 for packings with significant FCC crystallineorder as proposed in earlier studies [22, 23, 34]. (See Section 4.) The most significantdifference between the recent and prior studies is the use of a well-calibrated explicit hydrogenmodel instead of an extended atom model. acking in Protein Cores i.e. force and torque-balanced) jammed packings. (See the Appendix for a more detaileddescription of the packing-generation protocol.) We initialized the system by randomlyinserting N R residues into a cubic box (with periodic boundary conditions). We assumedthat the residues, which are composed of rigidly connected spherical atoms of differentsizes, interact via purely repulsive linear spring forces. We then compress the system bysmall packing fraction increments ∆ φ , followed by energy minimization. For sufficientlysmall ∆ φ , the form of the purely repulsive potential does not influence the structure ofthe final packings. For jammed packings, the total potential energy per residue U/N R > U/N R = 0after energy minimization. In this case, atomic motions can occur in the system without aconcomitant increase in the total potential energy. Thus, we can identify the packing fractionat jamming onset φ J as the one at which the minimized total potential increases above asmall threshold [50].We studied mixtures of N R residues with the fractions of Ala, Ile, Leu, Met, Phe, andVal residues matching the percentages found in protein cores. (We focused on non-polarresidues, but because Gly has no side chain and Cys can form disulfide bonds, these werenot included in the simulations.) These simulations generate disordered jammed packingswith φ = 0 .
56 similar to that found in protein cores (Fig. 4 (b)). These results indicate thatthe connectivity of the protein backbone does not impose significant constraints on the freevolume in protein cores. φ P ( φ ) Figure 5: The distribution of packingfractions P ( φ ) for core (solid line) andinterface (dotted line) residues from high-resolution protein crystal structures.To further analyze the packing efficiency inprotein cores, we also calculated the distributionof the local packing fractions ( i.e. φ foreach residue type) in protein cores for bothprotein crystal structures and simulations. Wefind that the distributions of the local packingfractions for each residue type have similaraverage values, differing by < acking in Protein Cores ≤ . i.e. primarily hydrophobic residues with few charged and polar residues). We find that 73%and 68% of the residues in protein cores and interfaces, respectively, are hydrophobic withsimilar frequencies for each amino acid. In addition, both the distribution of core packingfractions and interface packing fractions are peaked near 0 .
56 as shown in Fig. 5. This resultdemonstrates that protein-protein interfaces are packed similarly to protein cores.
3. Protein core repacking
Computational protein core repacking allows investigation of the uniqueness of the side chainconformation of residues in protein cores. Unique side chain conformations for core residueswould imply that protein cores are jammed with very little free volume for rearrangementsof side chains. There are two categories of protein core repacking investigations: one startswith all possible sequences and seeks to recover the wild type sequence [52, 53] and the otherstarts with the wild type sequence and seeks to recover the observed combination of sidechain dihedral angles and determine if alternative combinations are possible. Here we focuson the second, where the side chains of core residues are removed simultaneously and all sidechain dihedral angle combinations of the starting sequence are sampled. The energy of eachconformation is evaluated, the optimal conformation is predicted, and then compared to theobserved structure.To study repacking of protein cores, we again use a hard-sphere plus stereochemistrymodel. The cores of 221 proteins in the Dunbrack Database [32, 33] were studied. As away to model the system at non-zero temperature and to improve the statistics, variationsin bond lengths and angles are implemented by replacing each side chain with differentinstances of the side chain taken from high-resolution protein crystal structures [4]. Coreresidues were identified as described in Section 2. As described in previous work [38, 14],the hard-sphere model treats each atom i as a sphere that interacts pairwise with all othernon-bonded atoms j via the purely repulsive Lennard-Jones potential: U RLJ ( r ij ) = (cid:15) (cid:34) − (cid:18) σ ij r ij (cid:19) (cid:35) Θ( σ ij − r ij ) , (2) acking in Protein Cores ILE LEU MET PHE SER THR TRP TYR VAL00.10.20.30.40.50.60.70.80.91 F ( ∆ χ ) ILE LEU MET PHE SER THR TRP TYR VAL00.10.20.30.40.50.60.70.80.91 F ( ∆ χ ) Figure 6: (left) Single and (right) combined residue rotations in the context of the proteincore: The fraction ( F (∆ χ )) of each residue type for which the hard-sphere model predictionof the side chain conformation deviates by ∆ χ < o (yellow), 20 o (red), or 30 o (blue) fromthe crystal structure.where r ij is the center-to-center separation between atoms i and j , σ ij = ( σ i + σ j ) / σ i / i , Θ( σ ij − r ij ) is the Heaviside step function, and (cid:15) is the strength of therepulsive interactions. Values for the atomic radii are listed in Section 2.Predictions of the side chain conformations of single amino acids are obtained by rotatingeach of the side chain dihedral angles χ , χ , ..., χ n (with a fixed backbone conformation[54]), and finding the lowest energy conformation of the residue, where the total energy U ( χ , ..., χ n ) includes both intra- and inter-residue steric repulsive interactions. We thencalculate the Boltzmann weight of the lowest energy side chain conformation of the residue, P i ( χ , ...., χ n ) ∝ e − U ( χ ,...,χ n ) /k B T , where the small temperature, T /(cid:15) =10 − , approximateshard-sphere-like interactions. We select 50 bond length and angle variants, and for each wefind the lowest energy dihedral angle conformation and corresponding P i ( χ , ...., χ n ) values.We average P i over the variants to obtain P m ( χ , ...., χ n ). We then compare the particulardihedral angle combination { χ HS , ..., χ HSn } associated with the highest value of P m to theside chain of the crystal structure { χ xtal , ..., χ xtaln } . To assess the accuracy of the hard-spheremodel in predicting the side chain dihedral angles of residues in protein cores, we calculate∆ χ = (cid:113) ( χ xtal − χ HS ) + . . . + ( χ xtaln − χ HSn ) . (3)We determine the fraction F (∆ χ ) of residues of each type with ∆ χ less than 10 o , 20 o , and30 o . (See Fig. 6.)In Fig. 6 (left), we investigate the accuracy of the hard-sphere model in predicting theside chain dihedral angles of single residues in protein cores. For each amino acid (Ile, Leu,Met, Phe, Ser, Thr, Trp, Tyr, and Val), we calculate the fraction of residues, F (∆ χ ), forwhich the predicted side chain dihedral angle conformation is within 10 o , 20 o and 30 o of the acking in Protein Cores ILE LEU MET PHE SER THR TRP TYR VAL00.10.20.30.40.50.60.70.80.91 F ( ∆ χ ) Figure 7: Comparison of the accuracy ofsingle and combined rotations for coreresidues in 221 proteins [32, 33]. Eachbar shows the fraction of residues, F (∆ χ ),for which the hard-sphere model predictionof the side chain conformation has ∆ χ < o for single (blue) or combined (red)rotations.crystal structure value. Consistent with our prior results, the hard-sphere model accuratelypredicts the side chain dihedral angle combinations of single residues in the context of theprotein for Ile, Leu, Phe, Thr, Trp, Tyr, and Val ( ≥
90% within 30 o ) [49]. This resultemphasizes that the purely repulsive hard-sphere model can accurately predict the side chaindihedral angle combinations for nonpolar and uncharged amino acids.We find that the hard-sphere model is unable to predict with high accuracy the observedside chain conformations for two residues that we studied: Ser and Met. Our results for Metare consistent with those found in Virrueta et al. [55]. In this prior work, we found thatlocal steric interactions were insufficient to predict the shape of the P ( χ ) distribution forMet. It was necessary to add attractive atomic interactions to the hard-sphere model toreproduce the observed P ( χ ). Here, using only repulsive interactions, we predict ≈
80% ofMet residues are within 30 o of the crystal structure. Our results for Ser (only 38% within 30 o )are also consistent with our prior work in Caballero et al. [49]. We speculate that becausethe side chain of Ser is small, hydrogen-bonding interactions must be included to correctlyplace its side chain. In contrast, we suggest that the more bulky Thr and Tyr side chainscause steric interactions to determine the positioning of their side chains, even though theyare able to form hydrogen bonds [37].In addition to single residue rotations, we performed core repacking using combinedrotations of interacting core residues in each protein [56]. For the combined rotationmethod, all residues in an interacting cluster are rotated simultaneously (with fixed backboneconformations), and the global minimum energy conformation is identified. A cluster ofinteracting residues is defined such that side chain atoms of each residue in the clusterinteract with one or more other residues in the cluster, but do not interact with the sidechains of other core residues in the protein.Single and combined rotations have the same prediction accuracy (Figs. 6 and 7), whichshows that there are very few arrangements of the residues in a protein core that are stericallyallowed and that the side chain conformations of most core residues are dominated by packingconstraints. This result implies that there are no alternative sterically allowed conformationsof core residues other than those in the crystal structure. If alternative sterically allowed acking in Protein Cores et al. [35], where they found that “in a well-packedcore region, it is rare that a bond angle can be rotated much in either direction withoutproducing clashes.”
4. Jammed packings of spherical and nonspherical particles Q φ Figure 8: Global bond orientational orderparameter Q versus packing fraction φ for 100 jammed packings of monodispersespheres.A strict definition of jamming means that adisordered system is solid-like and possesses astatic shear modulus [26]. However, jammingalso implies that a system is confined to a smallregion of configuration space, such that little orno motion of the constituent particles can occur.The results presented in Secs. 2 and 3 provideseveral indications that residues in protein coresare jammed in this latter sense. First, fornearly all protein cores, single and collectiverepacking give the same side chain dihedralangle combinations found in the protein crystalstructures. This result emphasizes that thereare no alternative low energy conformations forcore residues. Second, the packing fraction ofprotein cores is ≈ .
56, which is similar to thosereported for disordered jammed packings of frictional [57] and elongated particles [58, 59, 60].In this section, we present the results of simulations of jammed packings in threespatial dimensions (3D) for a wide variety of particle shapes including monodisperse spheres,polydisperse spheres, spheres with varying sizes of asperities (or “bumps”), ellipsoids,ellipsoids with varying sizes of asperities, and non-axisymmetric, elongated particles. Thisrange of shapes allows us to study the influence of the particle aspect ratio and surfacebumpiness on the packing fraction and determine which particle shapes produce packingfractions that match the packing fraction of residues in protein cores.We start the discussion with jammed packings of monodisperse spheres. In monodispersesystems, the packing fraction depends on the degree of order that is present in the system.For example, in Fig. 8, we show that the packing fraction varies with the global bondorientational order parameter Q [61, 62], which measures the degree to which the separationvectors connecting a given particle and its nearest neighbors are consistent with icosohedralsymmetry. Q ≈ .
57 for perfect FCC crystalline sphere packings with φ ≈ .
74. Thepacking fraction for jammed packings of monodisperse spheres decreases as Q decreases, acking in Protein Cores α φ Figure 9: Jammed packing fraction φ versus aspect ratio α for frictional spheres (blueasterisks) from Ref. [57], bumpy (green triangles) spheres, smooth, prolate ellipsoids ofrevolution from Refs. [59] (dotted line) and [60] (solid line) and spherocylinders (dashedline) from Ref. [58]. The static friction coefficient for the frictional spheres varies from µ = 10 − to 10 from top to bottom. For the bumpy spheres (Fig. 10 (a) and (b)), twelvebumps are placed on the vertices of an icosohedron, and the relative sizes of the bumps aredecreased to increase the bumpiness B from ≈ − to 0 .
15 from top to bottom. We also showthe packing fraction and aspect ratio for Ala (open diamond), Ile (open leftward triangle),Leu (open circle), Met (open square), Phe (x), and Val (open upward triangle) residues inprotein cores. The error bars indicate the root-mean-square fluctuations from averaging overinstances of each residue with different backbone and side chain conformations. Results forbumpy ellipsoids are indicated by the filled rightward and upward triangles and results forthe non-axisymmetric shapes in Fig. 10 (g) and (h) are indicated by the filled diamond andcircle, respectively.reaching random close packing φ ≈ .
64 in the limit Q → Q can be obtained by varying the rate at which kinetic energy is drainedfrom the system [64]. For the present studies, we consider the limit of fast quenching rates,which gives rise to disordered packings.Particle size differences can strongly decrease a system’s tendency to order. In previousstudies, we focused on jammed packings of bidisperse spheres with half large spheres, halfsmall spheres, and a modest diameter ratio of d = 1 . d (cid:38) .
4) can also increase the packing fraction of jammedpackings of polydisperse spheres. In this case, small spheres can fill in the gaps betweencontacting larger spheres. For example, Apollonian sphere packings [66] characterized by a acking in Protein Cores d = 1 . d = 1 .
5) is between sp carbon and hydrogen atoms.Thus, we expect that jammed sphere packings composed of mixtures of atoms with the samesizes and number fractions as in protein cores will have packing fraction φ ≈ .
64. This resultwas shown previously in Ref. [14]. Thus, jammed packings composed of individual sphereswith polydispersity that matches atom size differences in protein cores possess packingfractions that are larger than the values we observe in protein cores (Sec. 2).We now consider jammed packings of symmetric elongated particles, i.e. spherocylindersand ellipsoids, as a function of the aspect ratio α . In Fig. 9, we show that the packingfraction φ ( α ) is qualitatively the same for jammed packings of spherocylinders and ellipsoids. φ ≈ .
64 for spherical particles with α = 1, increases for α >
1, reaches a peak near α ≈ . φ > .
7, and then decreases to a plateau value of φ ≈ .
68 at large α .To compare the results for jammed packings of symmetric, elongated particles topackings of amino acids presented in Sec. 2, we define a generalized aspect ratio and surfacebumpiness to characterize the shape of composite particles made from collections of spheres.We define bumpiness by B = (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) (cid:90) d ˆ u (cid:16) (cid:126)R (ˆ u ) − (cid:126) R (ˆ u ) (cid:17) R (ˆ u ) , (4)where ˆ u is a unit vector with an origin at the geometric center of the composite particle, theintegral is over all orientations of ˆ u , (cid:126)R (ˆ u ) gives the location on the surface of the compositeparticle along ˆ u , and (cid:126) R (ˆ u ) gives the location on the surface of a reference prolate ellipsoidof revolution along ˆ u . The bumpiness B for a given composite particle will depend on theorientation of the reference prolate ellipsoid axis ˆ e and the values of the major a and minor b axes.To define the aspect ratio α for composite particles, we find the reference prolate ellipsoidof revolution that yields the smallest bumpiness. We first fix the reference ellipsoid axis ˆ e to be in the direction that gives the largest distance between the geometric center and thesurface of the composite particle. We then minimize B (ˆ e, a, b ) over a and b at fixed ˆ e , anddefine α = a/b for the optimal values of the major and minor axes of the reference ellipsoid.Fig. 9 shows the packing fraction versus aspect ratio for Ala, Val, Ile, Leu, Met, and Pheresidues in protein cores. As discussed in Sec. 2, most core residues have packing fractionsnear 0 . .
56. The aspect ratios of amino acids depend on the amino acid type and theirbackbone and side chain conformations. The average aspect ratios vary from α ≈ . ≈ . φ and α are obtained from the root-mean-squarefluctuations over different instances of each residue in protein cores.The packing fraction φ ≈ . .
56 observed for amino acids in protein cores withnominal aspect ratios in the range 1 . (cid:46) α (cid:46) . acking in Protein Cores B = 0 . α = 1 .
00 and (b) B = 0 . α = 1 .
00; bumpy ellipsoidswith (c) B = 0 . α = 1 .
40 and (e) B = 0 . α = 1 .
40; (e) Ala and (f) Phe residues; and(g,h) two examples of non-axisymmetric composite particles. φ ≈ . φ ≈ .
64 to ≈ .
55 as the static friction coefficient µ increases from 10 − to 10.We find similar results for bumpy spheres (green squares) in Fig. 9. Here, the bumpyspheres are composite particles made from twelve spheres arranged on the vertices of anicosohedron. We decrease the ratio r of the size of each sphere to the size of the icosohedronto increase the bumpiness B . We show in Fig. 11 that for bumpy spheres formed from anicosohedron, we can generate 0 (cid:46) B (cid:46) .
15 (corresponding to 5 (cid:38) r (cid:38) . α ≈ α > acking in Protein Cores B (cid:46) .
17 over a wide range of aspect ratios using thismethod for constructing bumpy axisymmetric elongated particles.In Fig. 9, we show the packing fraction for jammed packings of bumpy ellipsoids over arange of bumpiness values for two aspect ratios, α ≈ . .
25, which spans the range ofaspect ratios calculated for amino acids in protein cores. For both aspect ratios, the packingfraction decreases from the values obtained from packings of smooth elongated particles to φ ≈ .
55 as the bumpiness is increased from B = 0 .
01 to 0 . e.g. Ala and Phe in Fig. 10 (e) and (f)) possess bumpiness values between B = 0 . .
3, whereas bumpy axisymmetric shapes have B (cid:46) .
17. Thus, we also studied jammed α B Figure 11: Surface bumpiness B versus aspect ratio α for several particle shapes consideredin the packing simulations. For bumpy spheres (green squares) with α = 1 created byplacing spheres on the vertices of an icosohedron, bumpiness can be varied over the range0 (cid:46) B (cid:46) .
15. For prolate ellipsoids (black dots) with 8 or 14 spherical bumps (black dots),we can achieve maximum bumpiness values B ≈ .
17 over a wide range of α indicated by thegrey rectangle. We also show bumpiness versus aspect ratio for Ala (diamond), Ile (leftwardtriangle), Leu (circle), Met (square), Phe (x), and Val (upward triangle) residues in proteincores. B and α for the non-axisymmetric particles in Fig. 10 (g) and (h) are given by thered diamond and magenta circle, respectively. acking in Protein Cores ◦ . The composite particle in panel(h) contains 7 spheres with two spheres each placed at the top and bottom of the particlein planes perpendicular to the long axis and in staggered orientations. The bumpiness andaspect ratio of these non-axisymmetric composite particles is varied by changing the sizeof the bumps compared to the size of the sphere that circumscribes the composite particle.For these two types of non-axisymmetric particles, we were able to increase the maximumbumpiness to B ≈ .
4, which is even larger than that of any of the core amino acids (Fig. 11).As shown in Fig. 9, the packing fractions for jammed packings of the non-axisymmetricparticles in Fig. 10 (g) and (h) (with B = 0 .
33 and 0 .
39) are φ ≈ .
56. These results showthat jammed packings of particles with the same B and α as those found for amino acidsyield the same packing fraction as amino acids in protein cores.
5. Mutations in protein cores
Additional insight into the packing efficiency in protein cores can be obtained by examiningthe results from experimental studies of protein core mutations. Several groups haveexperimentally investigated the potential plasticity of protein cores by performing mutations, i.e. by changing the identities core amino acids. Lim and Sauer simultaneously mutatedseveral hydrophobic residues in the core of a small protein, and used a genetic screen toidentify those that were functional and stable. They found that very few combinationsof amino acids other than the wildtype set resulted in a stable, folded protein [24]. Thefunctional new cores were dominated by hydrophobic amino acids and the total side chainvolumes were within 10% of the original core volume. Combinations of residues outsideof these requirements were nonfunctional. Moreover, stereochemical constraints furtherrestricted the allowed sequence space. For example, although many permutations of coreresidues can maintain the same total volume and hydrophobicity in the core, they do notresult in a protein with the same structure and stability [24]. As a result of hydrophobic,volume, and steric constraints, only 0.3% of 60,000 sequences sampled are fully functional[24, 25]. These observations provide experimental support for the dominance of stericinteractions in protein cores. Similar experimental results have been found in other proteins[68, 69, 70, 71].Liu, et al. investigated how mutations from small to large residues in the core affectprotein stability [72]. This work illustrates the difficulty in generalizing the effects of aparticular type of mutation at different locations and in different proteins. In this work,three Ala residues in the core of a small protein were mutated, individually, to either Cys,Ile, Leu, Met, Phe, Trp, or Val, and the resulting effect on protein stability was determined.They also solved the crystal structures of several of the mutated proteins. They foundthat in all cases, to varying degrees, to accommodate the larger amino acid side chain, the acking in Protein Cores et al. hypothesized that this behavior was due to a cavityin the protein near the mutation site, which allowed for more flexibility in this region of theprotein [72]. (See also Sec. 6.)This work shows that the protein core is not able to accommodate mutations to largerresidues without significant rearrangement and subsequent destabilization of the originalstructure. If substantial empty space existed in the protein core, then mutations of thistype would likely have small effects because they would fill the existing empty space andnot require backbone rearrangements. Instead, backbone rearrangements are necessary toaccommodate larger amino acids, supporting the idea that protein cores are tightly packed[72]. This example also illustrates that much is still unknown about protein core packingand how it controls protein stability. The current state of knowledge is such that one canpredict neither the backbone movements in response to the incorporation of a larger sidechain, nor the changes in stability that result from these structural changes.
6. Conclusions and Future Directions
Our computational studies have established that protein cores are comprised of irregularlyshaped objects that are packed into disordered jammed arrangements with φ ≈ .
56 [14].For a given core, there are no alternative arrangements of the same amino acids that areconsistent with a well-packed core with no atomic overlaps [49, 56]. It has also been shown,both experimentally and computationally, that there are a small number of combinationsof different core residues that can properly fit in and fill a given core, and thus give rise toa stable folded protein [24, 25, 72, 73, 74]. There are also experimental examples in whichamino acids in the core are substituted with ones that are either smaller or larger. Oftensuch substitutions result in changes in the backbone positions. With the current state ofunderstanding in the field, it is not possible to reliably predict such movements. For somemutations, the rearranged protein is as stable as the starting protein, for others it is lessstable. Again, the state of the art in computational modeling is such that it is not possibleto predict either the structure or the stability of the repacked, rearranged protein.Even dense packing of amino acids in protein cores results in some void space notoccupied by amino acids. There has been some analysis of voids in proteins using a range ofprobe sizes [75, 23]. Various probe sizes are used to identify void connectivity in the proteinand to remove small physically irrelevant voids. Obviously, an exceedingly small probe ( e.g. radius (cid:46) .
05 ˚A) will identify a large amount of void space, because even the very smallestvoids will be counted. Conversely, a large probe ( e.g. radius (cid:38) . . et al. examined void statistics in a dataset of high-resolution protein structures [75]. acking in Protein Cores ≈ per residue. To put this intoperspective, a CH group and a water molecule have a volume of ≈ , which indicatesthat the voids in protein cores are small. In future studies, we will consider the location andsize of buried voids to predict the consequences of changes of amino acid size and sequencein protein cores.
7. Acknowledgments
We gratefully acknowledge the support of the Raymond and Beverly Sackler Institute forBiological, Physical, and Engineering Sciences (to L.R., C.S.O., and J.C.G.), NationalLibrary of Medicine Training Grant No. T15LM00705628 (to J.C.G.), and National ScienceFoundation (NSF-PHY-1522467 to L.R., C.S.O. and J.C.G.). This work also benefited fromthe facilities and staff of the Yale University Faculty of Arts and Sciences High PerformanceComputing Center and the National Science Foundation (Grant No. CNS-0821132), whichin part funded acquisition of the computational facilities. We also thank Mark D. Shattuckfor his input on measurements of void volumes in protein cores.
8. Appendix
In this Appendix, we provide additional details that support the results presented in themain text. In Table 2, we provide the volume of the 11 residues that occur most frequentlyin protein cores using the explicit hydrogen representation. Gly and Ala have the smallestvolumes and Tyr and Trp have the largest. These values differ quantitatively from thoseobtained using the extended atom model.Residue Volume (˚A )Ala 48.8Cys 64.3Gly 35.6Ile 88.1Leu 88.1Met 92.7Phe 100.7Thr 69.0Trp 121.9Tyr 107.5Val 75.0Table 2: Volumes for the 11 residues that occur most frequently in protein cores using theexplicit hydrogen representation. acking in Protein Cores F (∆ χ ) of residues forwhich the prediction of the hard-sphere model is less than ∆ χ from the observed side chainconformation that are shown in Figs. 6 and 7. To assess the accuracy of the hard-spheremodel in predicting the side chain dihedral angle conformations of residues in protein cores,repacking calculations were performed using N v = 300 bond length and angle variants foreach core residue. For each residue, we randomly select M bond length and angle variantsout of the N v variants. For each set of variants, we identified the optimal side chain dihedralangle combination and calculated ∆ χ . We then repeat this process N times, which yields aset of N ∆ χ values. We then calculated the mean fraction of residues F (∆ χ ), which satisfy∆ χ < ◦ , 20 ◦ , or 30 ◦ , and the standard deviation. We used N = 50 and M = 50 for singleresidue rotations and N = 50 and M = 30 for combined rotations.To understand how particle elongation and surface bumpiness affect packing properties,we generated jammed packings of composite particles formed from spheres. Each compositeparticle is composed of n spherical asperities placed on the vertices of an icosohedron orlocations on the surface of a prolate ellipsoid of revolution. Spherical asperities i and j oncomposite particles C and C ’ interact via the pairwise potential U CC (cid:48) ij = (cid:15) (1 − r ij /σ ij ) Θ( σ ij − r ij ), where (cid:15) is the energy scale of the interaction, r ij is the distance between the centers ofasperities i and j , σ ij = ( σ i + σ j ) / i and j , and Θ is theHeaviside step function. Thus, composite particles C and C ’ interact via U CC (cid:48) = (cid:80) i,j U CC (cid:48) ij .The total potential energy of the system is U = (cid:80) C>C (cid:48) U CC (cid:48) .To find jammed packings, we employ a packing-generation protocol similar to thatin Ref. [60]. We first place N composite particles randomly in a cubic periodic cell ofunit size. At each step we increase the asperity sizes σ i and bond lengths δ ij betweenasperities (fixing the ratios between σ i and δ ij ) corresponding to ∆ φ ≈ − , then we relaxthe system to the nearest potential energy minimum using dissipative dynamics, where thedissipative forces are proportional to the composite particle velocities. If the potential energyis zero after energy minimization ( i.e. below a small threshold U/N < − ), we continuecompressing; otherwise, we decompress the system, where ∆ φ is halved each time we switchfrom compression to decompression. We stop the packing-generation protocol when thepotential energy is nonzero and the average particle overlaps are between 0 .
01% and 0 . φ and the overlap threshold, provided they are sufficiently small.
9. References [1] K.A. Dill. Dominant forces in protein folding.
Biochemistry , 29:7133, 1990.[2] G.D. Rose, P.J. Fleming, J.R. Banavar, and A. Maritan. A backbone-based theory of protein folding.
Proc. Natl. Acad. Sci. USA , 103:16623, 2006.[3] H.M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T.N. Bhat, H. Weissig, I.N. Shindyalov, and P.E.Bourne. The Protein Data Bank.
Nucleic Acids Res. , 28:235, 2000.[4] R.L. Dunbrack and F.E. Cohen. Bayesian statistical analysis of protein side-chain rotamer preferences.
Prot. Sci. , 6:1661, 1997. acking in Protein Cores [5] L. LoConte, C. Chothia, and J. Janin. The atomic structure of protein-protein recognition sites. J.Mol. Biol. , 285:2117, 1999.[6] F. Glaser, D.M. Steinberg, I.A. Vakser, and N. Ben-Tal. Residue frequencies and pairing preferencesat protein-protein interfaces.
Proteins: Struct., Funct., Bioinf. , 43:89, 2001.[7] O. Keskin, C-J Tsa, H. Wolfson, and R. Nussinov. A new, structurally nonredundant, diverse data setof protein-protein interfaces and its implications.
Protein Sci. , 13:1043, 2004.[8] A.J. Bordner and R. Abagyan. Statistical analysis and prediction of protein-protein interfaces.
Proteins ,60:353, 2005.[9] D. Reichmann, O. Rahat, M. Cohen, H. Neuvirth, and G. Schreiber. The molecular architecture ofprotein-protein binding sites.
Cur. Opin. Struc. Biol. , 17(1):67, 2007.[10] W. Sheffler and D. Baker. RosettaHoles: Rapid assessment of protein core packing for structureprediction, refinement, design and validation.
Protein Sci. , 18:229, 2009.[11] N. London, D. Movshovitz-Attias, and O. Schueler-Furman. The structural basis of peptide-proteinbinding strategies.
Structure , 18(2):188, 2010.[12] A.Q. Zhou, C.S. O’Hern, and L. Regan. Revisiting the Ramachandran plot from a new angle.
ProteinSci. , 20:1166, 2011.[13] A.Q. Zhou, D. Caballero, C.S. O’Hern, and L. Regan. New insights into the interdependence betweenamino acid stereochemistry and protein structure.
Biophys. J. , 105:2403, 2013.[14] J.C. Gaines, W.W. Smith, L. Regan, and C.S. O’Hern. Random close packing in protein cores.
PhysicalReview E , 93, 2016.[15] R.A. Engh and R. Huber. Accurate bond and angle parameters for X-ray protein structure refinement.
Acta Crystallogr. A , 47:392, 1991.[16] F.H. Allen. The Cambridge Structural Database: A quarter of a million crystal structures and rising.
Acta. Crystallogr. B , 58:380, 2002.[17] G.N. Ramachandran, C. Ramakrishnan, and V. Sasisekharan. Stereochemistry of polypeptide chainconfigurations.
J. Mol. Biol. , page 95, 1963.[18] C. Ramakrishnan and G. N. Ramachandran. Stereochemical criteria for polypeptide and protein chainconformations.
Biophys. J. , 5:909, 1965.[19] J.W. Bryson, S.F. Betz, H.S. Lu, D.J. Suich, H.X. Zhou, K.T. O’Neil, and W.F. DeGrado. Proteindesign: A hierarchic approach.
Science , 270:935, 1995.[20] M. Munson, S. Balasubramanian, K.G. Fleming, A.D. Nagi, R. O’Brien, J.M. Sturtevant, and L. Regan.What makes a protein a protein? Hydrophobic core designs that specify stability and structuralproperties.
Protein Sci. , 5:1584, 1996.[21] C.K. Smith and L. Regan. Guidelines for protein design: The energetics of beta sheet side chaininteractions.
Science , 270:980, 1995.[22] F.M. Richards. The interpretation of protein structures: Total volume, group volume distributions andpacking density.
J. Mol. Biol. , 82:1, 1974.[23] J. Liang and K. Dill. Are proteins well-packed?
Biophys. J. , 81:751, 2001.[24] W.A. Lim and R.T. Sauer. Alternative packing arrangements in the hydrophobic core of lambdarepressor.
Nature , 339:31, 1989.[25] W.A. Lim and R.T. Sauer. The role of internal packing interactions in determining the structure andstability of a protein.
J. Mol. Biol. , 219(2):359, 1991.[26] C.S. O’Hern, L.E. Silbert, A.J. Liu, and S.R. Nagel. Jamming at zero temperature and zero appliedstress: The epitome of disorder.
Phys. Rev. E , 68:011306, 2003.[27] K. Gekko.
Volume and Compressibility of Proteins , page 75. Springer Netherlands, Dordrecht, 2015.[28] T.V. Chalikian, V.S. Gindikin, and K.J. Breslauer. Volumetric characterizations of the native, moltenglobule and unfolded states of cytochromecat acidic pH.
J. Mol. Biol. , 250:291, 1995.[29] M. Gao, H. Zhou, and J. Skolnick. Insights into disease-associated mutations in the human proteomethrough protein structural analysis.
Structure , 23:1362, 2015.[30] L. Regan, D. Caballero, M. R. Hinrichsen, A. Virrueta, D. M. Williams, and C. S. O’Hern. Protein acking in Protein Cores design: Past, present, and future. Biopolymers Peptide Science , 104:334, 2015.[31] W. Sheffler and D. Baker. RosettaHoles2: A volumetric packing measure for protein structure refinementand validation.
Protein Sci. , 19:1991, 2010.[32] G. Wang and R.L. Dunbrack Jr. PISCES: A protein sequence culling server.
Bioinformatics , 19:1589,2003.[33] G. Wang and R.L. Dunbrack Jr. PISCES: Recent improvements to a PDB sequence culling server.
Nucleic Acids Res. , 33:W94, 2005.[34] J. Tsai, R. Taylor, C. Chothia, and M. Gerstein. The packing density in proteins: Standard radii andvolumes.
J. Mol. Biol. , 290:253, 1999.[35] J.M. Word, S.C. Lovell, J.S. Richardson, and D.C. Richardson. Asparagine and glutamine: Usinghydrogen atom contacts in the choice of side-chain amide orientation.
J. Mol. Biol. , 285:1735, 1999.[36] J.M. Word, S.C. Lovell, J.S. Richardson, and D.C. Richardson. Visualizing and quantifying moleculargoodness-of-fit: Small-probe contact dots with explicit hydrogen atoms.
J. Mol. Biol. , 285:1735, 1999.[37] A.Q. Zhou, C.S. O’Hern, and L. Regan. The power of hard-sphere models: Explaining side-chaindihedral angle distributions of Thr and Val.
Biophys. J. , 102:2345, 2012.[38] A.Q. Zhou, C.S. O’Hern, and L. Regan. Predicting the side-chain dihedral angle distributions of non-polar, aromatic, and polar amino acids using hard sphere models.
Proteins , 82:2574, 2014.[39] A. Bondi. Vdw volumes and radii.
J. Phys. Chem.
Proteins Struct. Funct. Bioinf. , 68:595, 2007.[42] L. Pauling.
The Nature of the Chemical Bond . Cornell University Press, Ithaca, NY, 1948.[43] L.L. Porter and G.D. Rose. Redrawing the Ramachandran plot after inclusion of hydrogen-bondingconstraints.
Proc. Natl. Acad. Sci. USA. , 108:109, 2011.[44] C. Chothia. Structural invariants in protein folding.
Nature , 254:304, 1975.[45] A.J. Li and R. Nussinov. A set of van der Waals and coulombic radii of protein atoms for molecularand solvent-accessible surface calculation, packing evaluation, and docking.
Proteins Struct. Funct.Bioinf. , 32:111, 1998.[46] F.A. Mamony, L.M. Carruthers, and H.A. Scheraga. Intermolecular potentials from crystal data.III. Determination of empirical potentials and application to the packing configurations and latticeenergies in crystals of hydrocarbons, carboxylic acids, amines, and amides.
J. Phys. Chem. , 78:1595,1974.[47] N.L. Allinger and Y.H. Yuh.
Quantum Chemistry Program Exchange , 12:395, 1980.[48] C.H. Rycroft. Voro++: A three-dimensional Voronoi cell library in C++.
Chaos , 19:041111, 2009.[49] D. Caballero, A. Virrueta, C.S. O’Hern, and L. Regan. Steric interactions determine side-chainconformation in protein cores.
Protein Eng., Des. Sel. , 29:367, 2016.[50] G.-J. Gao, J. Blawzdziewicz, , and C.S. O’Hern. Studies of the frequency distribution of mechanicallystable disk packings.
Phys. Rev. E , 74:061304, 2006.[51] C.J. Tsai, S.L. Lin, H.J. Wolfson, and R. Nussinov. Studies of protein-protein interfaces: A statisticalanalysis of the hydrophobic effect.
Protein Sci. , 6:53, 1997.[52] G. Dantas, C. Corrent, S.L. Reichow, J.J. Havranek, Z.M. Eletr, N.G. Isern, B. Kuhlman, G. Varani,E.A. Merritt, and D. Baker. High-resolution structural and thermodynamic analysis of extremestabilization of human procarboxypeptidase by computational protein design.
J Mol. Biol. , 366:1209,2007.[53] N. Dobson, G. Dantas, D. Baker, and G. Varani. High-resolution structural validation of thecomputational redesign of human U1A protein.
Structure , 14:847, 2006.[54] H. Liu and Q. Chen. Computational protein design for given backbone: recent progresses in generalmethod-related aspects.
Curr. Opin. Struct. Biol. , 39:89, 2016.[55] A. Virrueta, C. S. O’Hern, and L. Regan. Understanding the physical basis for the side chain acking in Protein Cores conformational preferences of met. Proteins: Struct., Funct., Bioinf. , 84:900, 2016.[56] J. C. Gaines, A. Virrueta, S.J. Fleishman, C. S. O’Hern, and L. Regan. Collective repacking revealsthat the structure of protein cores are uniquely specified by steric repulsive interactions.
Protein Eng.Des. Sel. , 2017.[57] L.E. Silbert. Jamming of frictional spheres and random loose packing.
Soft Matter , 6:2918, 2010.[58] J. Zhao, S. Li, R. Zou, and A. Yu. Dense random packings of spherocylinders.
Soft Matter , 8:1003,2012.[59] A. Donev, R. Connelly, F.H. Stillinger, and S. Torquato. Underconstrained jammed packings ofnonspherical hard particles: Ellipses and ellipsoids.
Phys. Rev. E , 75:051304, 2007.[60] C. F. Schreck, M. Mailman, B. Chakraborty, and C. S. O’Hern. Constraints and vibrations in staticpackings of ellipsoidal particles.
Phys. Rev. E , 85:061305, 2012.[61] Y. Jin and H. A. Makse. A first-order phase transition defines the random close packing of hard spheres.
Physica A , 389:5362, 2010.[62] T. M. Truskett, S. Torquato, and P. G. Debenedetti. Quantifying disorder in equilibrium and glassysphere packings.
Phys. Rev. E , 62:993, 2000.[63] K. Zhang, W.W. Smith, M. Wang, Y. Liu, J. Schroers, M.D. Shattuck, and C.S. O’Hern. Connectionbetween the packing efficiency of binary hard spheres and the glass-forming ability of bulk metallicglasses.
Phys. Rev. E , 90:032311, 2014.[64] S. S. Ashwin, J. Blwzdziewicz, C. S. O’Hern, and M. D. Shattuck. Calculations of the basin volumesfor mechanically stable packings.
Phys. Rev. E , 85:061307, 2012.[65] N. Xu, J. Blawzdziewicz, and C.S. O’Hern. Reexamination of random close packing: Ways to packfrictionless disks.
Phys. Rev. E , 71:061306, 2005.[66] R.S. Farr and E. Griffiths. Estimate for the fractal dimension of the Apollonian gasket in d dimensions.
Phys Rev E Stat Nonlin Soft Matter Phys , 81, 2010.[67] S. Papanikolaou, C.S. O’Hern, and M.D. Shattuck. Isostaticity at frictional jamming.
Phys. Rev. Lett. ,110:198002, 2013.[68] A.E. Eriksson, W.A. Baase, X.J. Zhang, D.W. Heinz, M. Blaber, E.P. Baldwin, and B.W. Matthews.Response of a protein structure to cavity-creating mutations and its relation to the hydrophobic effect.
Science , 255:178, 1992.[69] A.E. Eriksson, W.A. Baase, and B.W. Matthews. Similar hydrophobic replacements of Leu99 andPhe153 within the core of T4 lysozyme have different structural and thermodynamic consequences.
J. Mol. Biol. , 229:747, 1993.[70] K. Ishikawa, H. Nakamura, K. Morikawa, and S. Kanaya. Stabilization of Escherichia coli ribonucleaseHI by cavity-filling mutations within a hydrophobic core.
Biochemistry , 32:6171, 1993.[71] J. Xu, W.A. Baase, E. Baldwin, and B.W. Matthews. The response of T4 lysozyme to large-to-smallsubstitutions within the core and its relation to the hydrophobic effect.
Protein Sci. , 7:158, 1998.[72] R. Liu, W.A. Baase, and B.W. Matthews. The introduction of strain and its effects on the structureand stability of T4 lysozyme.
J. Mol. Biol. , 295:127, 2000.[73] B.I. Dahiyat and S.L. Mayo. Probing the role of packing specificity in protein design.
Proc. Natl. Acad.Sci. U.S.A. , 94(19):10172, 1997.[74] B. Kuhlman and D. Baker. Native protein sequences are close to optimal for their structures.
Proc.Natl. Acad. Sci. USA. , 97:10383, 2000.[75] A.L. Cuff and A.C.R. Martin. Analysis of void volumes in proteins and application to stability of thep53 tumour suppressor protein.