Numerical calculations of the pH of maximal protein stability: The effect of the sequence composition and three-dimensional structure

Numerical calculations of the pH of maximal protein stabilityThe effect of the sequence composition and three-dimensional structure

Emil Alexov

Howard Hughes Medical Institute and Columbia University, Biochemistry Department, New York, USA

A large number of proteins, found experimentally to havedifferent optimum pH of maximal stability, were studied toreveal the basic principles of their preferenence for a par-ticular pH. The pH-dependent free energy of folding wasmodeled numerically as a function of pH as well as the netcharge of the protein. The optimum pH was determined inthe numerical calculations as the pH of the minimum freeenergy of folding. The experimental data for the pH ofmaximal stability (experimental optimum pH) was repro-ducible (rmsd ¼ 0.73). It was shown that the optimum pHresults from two factors – amino acid composition and theorganization of the titratable groups with the 3D structure.It was demonstrated that the optimum pH and isoelectric

point could be quite different. In many cases, the optimumpHwas found at a pH corresponding to a large net charge ofthe protein. At the same time, there was a tendency forproteins having acidic optimum pHs to have a base/acidratio smaller than one and vice versa. The correlationbetween the optimum pH and base/acid ratio is significant ifonly buried groups are taken into account. Itwas shown thata protein that provides a favorable electrostatic environmentfor acids anddisfavors the bases tends to have high optimumpH and vice versa.

Keywords: electrostatics; pH stability; pKa; optimum pH.

The concentration of hydrogen ions (pH) is an importantfactor that affects protein function and stability in differentlocations in the cell and in the body [1]. Physiological pHvaries in different organs in human body: the pH in thedigestive tract ranges from 1.5 to 7.0, in the kidney it rangesfrom 4.5 to 8.0, and body liquids have a pH of 7.2–7.4 [2]. Itwas shown that the interstitial fluid of solid tumors havepH ¼ 6.5–6.8, which differs from the physiological pH ofnormal tissue and thus can be used for the design of pHselective drugs [3].

The structure and function of most macromolecules areinfluenced by pH, and most proteins operate optimally at aparticular pH (optimum pH) [4]. On the basis of indirectmeasurements, it has been found that the intracellular pHusually ranges between 4.5 and 7.4 in different cells [5]. Theorganelles’ pH affects protein function and variation of pHaway from normal could be responsible for drug resistance[6]. Lysosomal enzymes function best at the low pH of 5found in lysosomes, whereas cytosolic enzymes functionbest at the close to neutral pH of 7.2 [1].

Experimental studies of pH-dependent properties [7–11]such as stability, solubility andactivity, provide the benchmarksfor numerical simulation. Experiments revealed that altho-

ugh the net charge of ribonuclease Sa does affect thesolubility, it does not affect the pH of maximal stability oractivity [12]. Another experimental technique as acidic orbasic denaturation [13–15] demonstrates the importance ofelectrostatic interactions on protein stability.

pH-dependent phenomena have been extensively mode-led using numerical approaches [16–19]. A typical task is tocompute the pKas of ionizable groups [20–26], the isoelectricpoint [27,28] or the electrostatic potential distributionaround the active site [29]. It was shown that activity ofnine lipases correlates with the pH dependence of theelectrostatic potential mapped on the molecular surface ofthe molecules [29]. pH dependence of unfolding energy wasmodeled extensively and the models reproduced reasonablethe experimental denaturation free energy as a function ofpH [19,30–36].

The success of the numerical protocol to compute thepH dependence of the free energy depends on the modelof the unfolded state, the model of folded state and thuson the calculated pKas. It is well recognized that theunfolded state is compact and native-like, but the magni-tude of the residual pairwise interactions and the desol-vation energies has been debated. Some of the studiesfound that any residual structure of the unfolded state hasnegligible effect on the calculated pH dependence ofunfolding free energy [31], while others found the opposite[33–36]. It was estimated that the pKas of the acidicgroups in unfolded state are shifted by – 0.3 pK units inrespect to the pKas of model compounds. Althoughincluding the measured and simulated pK shifts into themodel of unfolded state changes the pH dependence ofthe unfolding free energy, it most of the cases it does notchange the pH of maximal stability [33–36]. Much more

Correspondence to E. Alexov, Howard Hughes Medical Institute and

Columbia University, Biochemistry Department, 630 W 168 Street,

New York, NY 10032, USA.

Fax: + 1 212 305 6926, Tel.: + 1 212 305 0265,

E-mail: [email protected]

Abbreviations: MCCE, multi-conformation continuum electrostatic;

SAS, solvent accessible surface.

(Received 15 September 2003, accepted 11 November 2003)

Eur. J. Biochem. 271, 173–185 (2004) � FEBS 2003 doi:10.1046/j.1432-1033.2003.03917.x

important is the modeling of the folded state, where theerrors of computing pKas could be significantly largerthan 0.3 units. Over the years it has been a continuouseffort to develop methods for accurate pKa predictions[20,21]. These include empirical methods [37], macroscopicmethods [38–41], finite difference Poisson–Boltzmann(FDPB)-based methods [20–22,42], FDPB and moleculardynamics [43–45], FDPB and molecular mechanics[25,46,47] and Warshel’s microscopic methods (e.g.,[16,17]). The predicted pKas were benchmarked againstthe experimental data and the average rmsd were found tovary from the best value of 0.5pK [38], to 0.7pK [48], to0.83pK [25] and to 0.89 [22]. Multi-Conformation Con-tinuum Electrostatics (MCCE) [25] method was shown tobe among the best pKas predictors and it will beemployed in this work.

In the present work we compute the pH dependence ofthe free energy of folding and the net charge. The optimumpH was identified as the pH at which the free energy offolding has minimum. A large number of proteins havingdifferent optimum pH [49] were studied to find the effect ofthe amino acid composition and 3D structure on theoptimum pH.

Experimental procedures

Methods

Calculations were carried out using available 3D structuresof selected proteins. A text search was performed onBRENDA database [49] in the field of �pH of stability�. Fol-lowing search strings were used: �maximal stability�, �maxi-mum stability�, �optimal stability�, �optimum stability�, �beststability�, �highest stability� and �greatest stability�. Thisrevealed 168 proteins with experimentally determined pHsofmaximal stability. Then a search of the ProteinData Base(PDB) was performed to find available structures for theseproteins. An attempt was made to select PDB structures ofproteins from the same species as those used in theexperiment (43 structures). Structures with missing residueswere omitted as well as the structures of proteins participa-ting in large complexes resulting in the final set of 28 proteinstructures. The protein names, the PDB file names and theexperimental pH of maximal stability are provided inTable 1. The source of the data is BRENDA database andthus the present study is limited to the proteins listed there.Therewillalwaysbeproteinswithexperimentallydetermined

Table 1. Proteins and corresponding PDB [57] files used in the paper. The experimental optimum pH (pH of optimal stability) is taken from

BRENDAwebsite [49]. The calculated optimumpH (the pHof theminimumof free energy of folding) is given in the forth column. The difference is

the calculated optimum pH minus the experimental number (fifth column). Bases/acid ratio for all ionizable groups is in sixth column, while the

seventh shows the bases/acids ratio for 66% buried groups. The last three columns show the averaged intrinsic pK shift, the averaged pKa shift and

the net charge of the folded protein at pH optimum, respectively.

Protein pdb code

Experimental

optimum

pH

Calculated

optimum

pH Difference

Base/acid

ratio

Buried

base/acid

ratio

Averaged

intrinsic

pK shift

Averaged

pKa shift

Net charge at

optimum pH

Dioxygenase 1b4u 8.0 8.0 0.0 0.94 1.33 0.08 ) 0.51 ) 3.0

Transferase 1f8x 6.5 5.0 ) 1.5 0.72 0.28 0.40 0.34 ) 5.5

Glutathione synthetase 1sga 8.0 7.5 ) 0.5 0.87 0.88 0.41 ) 0.58 ) 10.0

Isomerase 1b0z 6.0 6.0 0.0 1.02 0.90 0.05 ) 0.48 2.1

Coenzyme A 1bdo 6.5 7.0 0.5 0.67 1.50 0.22 0.03 ) 4.1

Dienelactone hydrolase 1din 7.0 6.5 ) 0.5 1.04 1.17 0.26 ) 0.36 ) 2.7

Dehydrogenase 1dpg 6.2 6.0 ) 0.2 0.79 1.05 0.38 ) 0.41 ) 13.0

Endothiapepsin 1gvx 4.15 4.0 ) 0.15 0.52 0.07 1.45 2.06 6.5

Dehydratase 1aw5 9.0 9.0 0.0 1.07 0.85 0.17 ) 0.48 ) 6.8

Cathepsin B 1huc 5.15 5.0 ) 0.15 0.90 0.73 1.28 0.11 5.8

Alginate lyase 1hv6 7.0 7.0 0.0 1.17 0.93 0.63 ) 0.72 2.7

Xylanase 1igo 5.5 6.5 1.0 1.41 1.00 0.60 ) 0.74 7.3

Hydrolase 1iun 7.5 7.0 ) 0.0 0.86 1.50 0.11 ) 1.15 ) 1.1

Aspartic protease 1j71 4.15 3.0 ) 1.15 0.54 0.33 0.98 1.32 9.4

Aldolase 1jcj 8.5 8.5 0.0 0.97 0.54 0.55 ) 0.19 ) 5.1

L-Asparaginase 1jsl 8.5 7.0 ) 1.5 1.17 1.85 ) 0.12 ) 0.83 ) 0.1

Amylase 1lop 5.9 6.0 0.1 0.81 1.00 0.33 ) 0.42 ) 8.2

c-Glutamil hydrolase 1l9x 7.0 7.5 0.5 1.19 0.77 0.45 ) 0.02 2.8

Mutase 1m1b 7.0 6.0 ) 1.0 0.95 0.86 0.25 0.13 ) 3.2

Methapyrogatechase 1mpy 7.7 7.0 ) 0.7 1.0 1.33 0.11 ) 1.35 ) 12.0

Pyrovate oxidase 1pow 5.7 6.0 0.3 0.91 0.78 0.60 ) 0.51 ) 2.0

Chitosanase 1qgi 6.0 6.5 0.5 1.09 0.54 0.29 ) 0.31 5.0

Xylose isomerase 1qt1 8.0 8.0 0.0 0.84 1.50 0.24 ) 0.30 ) 16.0

Pyruvate decarboxylase 1zpd 6.0 7.0 1.0 1.02 0.83 0.47 ) 0.24 3.8

Acid a-amylase 2aaa 4.9 4.0 ) 0.90 0.51 0.64 1.53 1.48 ) 1.7

Formate dehydrogenase 2nac 5.6 7.0 1.40 1.11 1.42 0.06 ) 1.1 2.4

Phosphorylase 2tpt 6.0 5.0 ) 1.0 0.91 0.93 0.38 ) 0.34 ) 3.8

b-Amylase 5bca 5.5 5.0 ) 0.5 1.07 0.91 0.19 ) 0.13 15.1

174 E. Alexov (Eur. J. Biochem. 271) � FEBS 2003

optimumpH thatwere not in the database, and therefore arenot modeled in the paper. However, an additional four wellstudied proteins were used to benchmark the method inbroad pH range and to compare the effect of mutations.

Free energy and net charge of unfolded state

The unfolded state is modeled as a chain of noninteractingamino acids (the possibility of residual interactions in theunfolded state is discussed at the end of the discussionsection). Thus, the free energy of ionizable groups (pH-dependent free energy) is calculated as [31]:

DGunf ¼ �kT lnðZunfÞ

¼ �kTXNi�1

lnf1þ exp½�2:3cðiÞðpH� pKsolðiÞÞ�g

ð1Þ

where k is the Boltzmann constant, T is the temperature inKelvin degrees,N is the number of ionizable groups, c(i) is 1for bases, )1 for acids, pKsol(i) is the standard pKa value insolution of group �i � (e.g., [47]), pH is the pH of the solutionand N is the number of ionizable residues. Zunf is thepartition function of unfolded state and DGunf is the freeenergy of unfolded state. The reference state of zero freeenergy is defined as state of all groups in their neutral forms[31].

The net charge is calculated using the standard formulathat comes from Henderson–Hasselbalch equation:

qunf ¼XNi¼1

10�cðiÞðpH�pKsolðiÞÞ

1þ 10�cðiÞðpH�pKsolðiÞÞcðiÞ ð2Þ

where c(i) ¼ )1 or +1 in the case of acid or base,respectively.

Free energy and net charge of the folded state

The pH-dependent free energy of the folded state iscalculated using the 3D structure of proteins listed inTable 1. The 3D structure comprises N ionizable groups(the same number as in the unfolded state) and L polargroups. Each of them might have several alternative side-chain rotamers [50], or alternative polar proton positions[47]. In addition, ionizable groups are either ionized orneutral. All these alternatives are called �conformers�, beingionizational and positional conformers. There is no a prioriinformation to indicate which conformer is most likely toexist at certain conditions of, for example, pH and saltconcentration. Each microstate is comprised of one con-former per residue. The Monte Carlo method was used toestimate the probability of microstates. This procedureis called multi-conformation continuum electrostatics (MCCE) and it is described inmore details elsewhere [25,47,50].Abrief summary of the MCCE method is provided in a latersection.

To find the free energy one should calculate thepartition function for each of the proteins. Thus, oneshould construct all possible combinations of conformers.Because of the very large number of conformers (most of

the cases more than 1000), the Monte Carlo method(Metropolis algorithm [51]) is used to find the probabilityof the microstates [20,47,50,52]. However, to construct thepartition function one should know all microstate energiesand to sum them up as exponents. Each microstateenergy should be taken only once, which induces extralevel of complexity. A special procedure is designed thatcollects the lowest microstate energies and that assuresthat each microstate is taken only once [50]. A microstatewas considered to be unique if its energy differs by morethan 0.001 kT from the energies of all previouslygenerated states. A much more stringent procedure thatcompares the microstate composition would requiresignificant computation time and therefore was notimplemented. This results in a function that estimatesthe partition function. This effective partition functionwill not have the states with high energy (they are rejectedby the Metropolis algorithm), but they have negligibleeffect [53]. In addition, the constructed partition functionmay not have all low energy microstates, because givenmicrostate may not be generated in the Monte Carlosampling or because two or more distinctive microstatesmay have identical or very similar energies. Bearing inmind all these possibilities, the effective partition function(Zfol) is calculated as [50]:

Zfol ¼XXfol

n¼1

expð�DGfoln =kTÞ ð3Þ

where DGfoln is the energy of the microstate �n� and Xfol is the

number of microstates collected in Monte Carlo procedure.Then the free energy of ionizable and polar groups in foldedstate is:

DGfol ¼ �kT lnðZfolÞ ð4Þ

The occupancy of each conformer (qfoli ) [52] is calculatedin the Metropolis algorithm and then used to calculate thenet charge of the folded state:

qfol ¼XMi¼1

qfoli cðiÞ ð5Þ

M is the total number of conformers. [Note that c(i)¼ 0 fornon ionizable conformers.]

Free energy of folding

The pH-dependent free energy of folding is calculated as adifference between the free energy of folded and unfoldedstates:

DDGfolding ¼ DGfol � DGunf ð6Þ

An alternative formula of calculating the pH dependenceof the free energy of folding is [19,31,54,55]:

DDGfolding ¼ 2:3kT

ZpH2

pH1

DqdpH ð7Þ

where, pH1 and pH2 determine the pH interval andDq is thechange of the net charge of the protein from unfolded tofolded state.

� FEBS 2003 Calculating pH of maximal protein stability (Eur. J. Biochem. 271) 175

Computational method: MCCE method

The basic principles of the method have been describedelsewhere [47,50]. The MCCE [25] method allows us to findthe equilibrated conformation and ionization states ofprotein side chains, buried waters, ions, and ligands. Themethod uses multiple preselected choices for atomic posi-tions and ionization states for many selected side chains andligands. Then, electrostatic and nonelectrostatic energiesare calculated, providing look-up tables of conformer self-energies and conformer–conformer pairwise interactions.Protein microstates are then constructed by choosing oneconformer for each side chain and ligand. Monte Carlosampling then uses each microstate energy to find eachconformer’s probability.

Thus, the MCCE procedure is divided into three stages:(a) selection of residues and generation of conformers; (b)calculation of energies and (c) Monte Carlo sampling.

Selection of residues. The amino acids that are involved instrong electrostatic interactions (magnitude > 3.5 kT) areselected. They will be provided with extra side-chainrotamers to reduce the effects of possible imperfections ofcrystal structures. The reason is that a small change in theirposition might cause a significant change in the pairwiseinteractions [56]. The threshold of 3.5 kT is chosen based onextensive modeling of structures and fitting to experiment-ally determined quantities [25]. The selection is made bycalculating the electrostatic interactions using the ori-ginal PDB [57] structure. The alternative side chains forthese selected residues are built using a standard library ofrotamers [58] and by adding an extra side chain positionusing a procedure developed in the Honig’s laboratory [59].The backbone is kept rigid. Then the original structure andalternative side chains were provided with hydrogen atoms.Polar protons of the side chains are assigned by satisfying allhydrogen acceptors and avoiding all hydrogen donors [25].Thus, every polar side chain and neutral forms of acids havealternative polar proton positions.

Calculation of energies. The alternative side chains andpolar proton positions determine the conformationalspace for a particular structure, and they are called�conformers�. The next step is to compute the energies ofeach conformer and to store them into look-up tables.Because of conformation flexibility, the energy is nolonger only electrostatic in origin, but also has nonelec-trostatic component [47,50].

Electrostatic energies are calculated by DelPhi [60,61],using the PARSE [62] charge and radii set. Internaldielectric constant is 4 [63], while the solution dielectricconstant is taken to be 80. The molecular surface isgenerated with a water probe of radius 1.4 A [64]. Ionicstrength is 0.15 M and the linear Poisson–Boltzmannequation is used. Focusing technique [65] was employed toachieve a grid resolution of about two grids per Angstrom.TheM calculations, whereM is the number of conformers,produce a vector of length M for reaction field energyDGrxn,i and an MxM array of the pairwise interactionsbetween all possible conformers DGij

el. In addition, eachconformer has pairwise electrostatic interactions with thebackbone resulting in a vector of length M DGpol,i. The

magnitude of the strong pairwise and backbone interactionsis altered as described in [56]. Such a correction wasshown to improve significantly the accuracy of the calcu-lated pKas [25].

Having alternative side chains and polar hydrogenpositions requires nonelectrostatic energy to be taken intoaccount too. This energy is a constant in calculations thatuse a �rigid� protein structure (and therefore should not becalculated), but in MCCE plays important role discrim-inating alternative positional conformers. The non-electrostatic interactions for each conformer are thetorsion energy, a self-energy term which is independentof the position of all other residues in the protein, andthe pairwise Lennard–Jones interactions, both with por-tions of the protein that are held rigid, and withconformers of side chains that have different allowed posi-tions [25,47,50].

Thus, the microstate �n� pH-dependent free energy offolded state is [20,21,47,50]:

DGfoln ¼

XMi¼1

�2:3kTdnðiÞ½cðiÞðpH� pKsolðiÞÞ þ DpKintÞðiÞ�

þXMj¼iþ1

dnðiÞdnðjÞðGijel þ Gij

nonelÞ�;

DpKintðiÞ ¼ DpKsolvðiÞ þ DpKdipðiÞ þ DpKnonelðiÞð8Þ

where dn(i) is 1 if ith conformer is present in the nthmicrostate, M is the total number of conformers, DpKint(i)is the electrostatic and non electrostatic permanent energycontribution to the energy of conformer �i� (note that it doesnot contain interactions with polar groups), c(i) is 1 forbases, )1 for acids, and 0 for neutral groups,DpKsolv(i) is thechange of solvation energy of group �i�, DpKdip(i) is theelectrostatic interactions with permanent charges,DpKnonel(i) is the nonelectrostatic energy with the rigid partof protein, Gij

el and Gijnonel are the pairwise electrostatic and

non electrostatic interactions, respectively, between con-former �i� and �j �.

Monte Carlo sampling. The Monte Carlo algorithm isused to estimate the occupancy (the probability) of eachconformer at given pH. The convergence is consideredsuccessful if the average fluctuation of the occupancy issmaller than 0.01 [25]. The pHwhere the net charge of giventitratable group is 0.5 is pK½. To adopt a commonnomenclature, pK½ will be referred as pKa throughout thetext.

Optimum pH, isoelectric point (pI) and bases/acids ratio

The experimental pH of maximal stability for each of theproteins listed in Table 1 is taken from the websiteBRENDA [49]. The database does not always provide asingle number for the optimum pH. If given protein isreported to be stable in a range of pHs, then the optimumpH is taken to be the middle of the pH range.

The optimum pH in the numerical calculation is deter-mined as pH at which the free energy of folding hasminimum. In the case that the free energy of folding has a


minimum in a pH interval, the optimum pH is the middle ofthe interval. The calculations were carried out in steps ofDpH ¼ 1. Thus, the computational resolution of determin-ing the pH optimum was 0.5 pH units.

The calculated and experimental pH intervals were notcompared, because in many cases BRENDA databaseprovides only the pH of optimal stability. In addition, inmost cases the experimental pH interval of stability given inthe BRENDA database does not provide information forthe free energy change that the protein can tolerate and stillbe stable. Therefore it cannot be compared with thenumerical results which provide only the pH dependenceof the folding free energy. Some proteins may tolerate afree energy change of 10 kcalÆmol)1 and still be stable, whileothers became unstable upon a change of only a fewkcalÆmol)1.

The calculated isoelectric point (pI) is the pH at whichthe net charge of folded state is equal to zero. There ispractically no experimental data for the pI of the proteinslisted in Table 1. The net charge at optimum pH is thecalculated net charge of the folded protein at pHoptimum. Base/acid ratio was calculated by counting allAsp and Glu residues as acids and all Arg, Lys and Hisresidues as bases. In some cases, one or more acidic and/or His residues was calculated to be neutral at a particularpH optimum, but they were still counted. The reason forthis was to avoid the bias of the 3D structure and tocalculate the base/acid ratio purely from the sequence.The given residue is counted as 66% buried if itssolvent accessible surface (SAS) is one-third of the SASin solution. Averaged intrinsic pK shifts were calculatedas

1

N

XNi¼1

ðpKintðiÞ � pKsolðiÞÞ

and the averaged pKas shift as

1

N

XNi¼1

ðpKaðiÞ � pKsolðiÞÞ

Thus, a negative pK shift corresponds to conditions suchthat the protein stabilizes acids and destabilizes bases andvice versa. Arginines were not included in the calculationsbecause their pKas are calculated in many cases to beoutside the calculated pH range.

Results

Origin of optimum pH

The paper reports the pH dependence of the free energy offolding. Despite the differences among the calculatedproteins, the results show that the pH-dependence profileof the free energy of folding is approximately bell-shapedand has a minimum at a certain pH, referred to through thepaper as the optimum pH.

To better understand the origin of the optimum pH, aparticular case will be considered in details. Figure 1Ashows the free energies of cathepsin B calculated in pHrange 0–14. Three energies were computed: the free energyof the unfolded state (bottom line), the free energy of thefolded state (middle line) and the free energy of folding (topcurve). For the sake of convenience the free energies of thefolded state and folding are scaled by an additive constantsso to have the same magnitude as the free energy of theunfolded state at the pH of the extreme value (in this casepH ¼ 5). It improves the resolution of the graph withoutchanging its interpretation, because the energies contain anundetermined constant (hydrophobic interactions, entropychange, van der Waals interactions and other pH-inde-pendent energies).

Free energy of unfolded state. It can be seen (Fig. 1A) thatthe free energy of the unfolded state has a maximum valueat pH ¼ 5 and it rapidly decreases at low and high pHs.Such a behavior can be easily understood given equation 1.At low pH, the pKsol of all acidic groups is higher than thecurrent pH and thus they contribute negligible to thepartition function. In contrast, all basic groups contributesignificantly to the partition function. As the pH decreases,their contribution increases, making the free energy morenegative. At medium pHs, all ionizable groups are ionized(except His and Tyr), but their effect on the free energy isquite small, because their pKsol are close to the pH. Thisresults in a maximum of the free energy corresponding tothe least favorable state. At high pHs, the situation isreversed: all acidic groups have a major contribution to thepartition function, while bases add very little. Thus, the freeenergy profile of the unfolded state is always a smooth curve(bell-shaped) with amaximum at a certain pH. The shape ofthe curve and the position of the maximum depend entirelyupon the amino acid composition.

Fig. 1. Cathepsin B pH-dependent properties.

(A) Free energy; (B) net charge.


Free energy of folded state. The free energy of the foldedstate behaves in a similar manner, but it changes less withthe pH (Fig. 1A). Note that it has maximum at pH ¼ 6.The major difference occurs at low and high pHs where freeenergy of the folded state does not decrease as fast as for theunfolded state. The 3D structure adds to the microstateenergy (Eqn 8) and to the partition function several newenergy terms )DpKint(i) (that originates in part from thedesolvation energy) and pairwise interactions Gij (a detaileddiscussion on the effect of desolvation and pairwise energieson the stability is given in [31]). If these two termscompensate each other, then Eqn 8 might be thought toreassemble the microstate energy formula of the unfoldedstate, Eqn 1. But there is an important difference: the aminoacids are coupled through the pairwise interactions. Thepairwise energies are a function of the ionization states.Thus, the de-ionization of a given group will cancel itspairwise interaction energies with the rest of the protein.

The effect of the coupling can be easily understood at theextremes of pH. Consider a very low pH such that the pKasof all acidic groups are higher than the current pH. At suchpH all acids will be fully protonated and thus the bases(having their own desolvation penalty) will be left withoutfavorable interactions. Thus the energy of the folded statewill be less favorable (because of the desolvation energy andthe lack on favorable interactions) than the energy ofunfolded state.

Free energy of folding. The pH dependence of the freeenergy of folding results from the difference of the abovefree energies (Fig. 1A). It always will have a minimum atcertain pH (in principle it might have more than oneminimum). This minimum may or may not coincide withthe pH where the unfolded free energy has maximum. Thefolding free energy always has a bell shape, and it isunfavorable at low and high pHs as compared to the freeenergy at optimum pH.

Net charge. An alternative way of addressing the samequestion is to compute the net charge of the protein(Fig. 1B). One can see that at the extremes of pH, theprotein is highly charged. At low pH it has a huge netpositive charge and at high pH a huge net negative charge.A straightforward conclusion could be made that acidic/basic denaturation is caused by the repulsion forces amongcharges with the same type. However all these positivecharges at low pH exist also at medium pH, where theproteins are stable. The thing that is missing at low pH andcauses acid denaturation is the favorable interactions withnegatively charged groups. At low pH, bases are left withoutthe support of acids, and they have to pay an energy penaltyfor their desolvation and unfavorable pairwise energiesamong themselves.

Equation 7 provides an additional tool for determiningthe optimum pH. At the optimum pH, the curve of foldingfree energy must have an extremum, i.e. the curve mustinvert its pH behavior. At pH lower than the optimum pH,the free energy of folding should decrease with increasingthe pH, then it should have a minimum at pH equal to theoptimum pH, and then it should increase with furtherincrease of the pH. Such behavior corresponds to a negativenet charge difference between the folded and unfolded state

at pH smaller than the optimum pH. As pH increases, thenet charge difference should get smaller, and at the optimumpH, it should be zero. Further increase of the pH (above theoptimum pH) should make the net charge difference apositive number. One can see in Fig. 1B that the net chargeof folding follows such pattern and is zero at pH ¼ 5, wherethe free energy of folding has a minimum.

General analysis of the optimum pH

Comparison to experimental data. Although this paperfocuses on the pH of maximal stability, it is useful tocompare the calculated pH dependence of the folding freeenergy on a set of proteins subjected to extensive experi-mental measurements. Figure 2 plots the calculated andexperimental pH dependence of the free energy of folding.The experimental data is taken from Fersht [66,67],Robertson [68] and Pace [10]. One can see that thecalculated pH-dependent free energy agrees well withthe experimental data. The most important conclusion forthe aims of the paper is that the calculated pH dependenceprofile of the free energy of folding is similar to that of theexperiment. The only exception is ribonuclease A wherethe calculated pH optimum is 8 while the experiment findsthe best stability at pH ¼ 6. It should be noted that thecalculated results are similar to the results reported byElcock [33] and Zhou [36] in cases of idealized unfoldedstate. From the works of the above authors, as well as fromKarshikoff laboratory [34], one can see that the residualinteractions in unfolded state do not affect the pH optimumin majority of the studied cases.

An additional possibility for comparison is offered by themutant data. Table 2 shows the stability change of barnasecaused by mutations of charged residues. The calculatednumbers are the pKa shifts (in respect to the standard pKsol)of each of these ionizable residues. Thus, the energy of themutant residue is not taken into account in the numericalcalculations. Even under such simplification, the calculatednumbers are 0.84 kcalÆmol)1 rmsd from the experiment.

Figure 3 compares the calculated optimum pH vs.experimental optimum pH for 28 proteins listed in Table 1.One can see that calculated values are in good agreementwith experimental data. The slope of the fitting line is 0.93and Pearson correlation coefficient is 0.86. The rmsdbetween calculated and experimentally determined opti-mum pHs is 0.73. The optimum pH ranges from 2 to 9 (4–9experimentally) which provides a broad range of pHs to becompared.

The origin of the optimum pH. The position of theoptimum pH depends on the amino acid composition andon the organization of the amino acids within the 3Dstructure. To find which of these two factors dominates weplotted the calculated optimum pH of the free energy offolding vs. the pH at which the free energy of unfolded statehas maximum (Fig. 4). The free energy of folding resultsfrom the difference of the free energy of folded and unfoldedstates. Thus, if the last two energies have the same pHdependence, the free energy of folding will be pH independ-ent. If both the free energy of unfolded and of folded statehave similar shape andmaximumat the same pH, thenmostlikely the optimum pHwill also be at this pH. If the curve of


the free energy of the folded state is steeper at basic pHs (orflatter at acidic pHs) compared to the free energy of theunfolded state, then the difference, i.e. the free energy offolding will have optimum pH shifted to the right pH scale.Such a phenomenon will occur if the protein stabilizes acids.Then the optimum pH will be higher than the pH ofmaximal free energy of unfolded state (points above the

Table 2. Experimental and calculated effect of single mutants on the

stability of barnase.

Mutant Experiment (kcalÆmol)1) Calculation (kcalÆmol)1)

D12A ) 0.95 ) 1.83

R69S, R69M ) 2.67, ) 2.24 ) 1.9

D75N ) 4.51 ) 2.92

R83Q ) 2.23 ) 4.07

D93N ) 4.17 ) 4.27

R110A ) 0.45 ) 2.17

Fig. 2. The calculated pH dependence of the

free energy of folding (solid line) and experi-

mental data (d). The ionic strength was

selected to match experimental conditions:

barnase (I ¼ 50 mM), OMKTY3

(I ¼ 10 mM), CI2 (I ¼ 50 mM) and ribonuc-

lease A (I ¼ 30 mM).

Fig. 3. The calculated optimum pH vs. the experimental optimum pH.

The figure shows only 27 data points, because the calculated and

experimental data for 1b4u and 1qt1 overlap.

Fig. 4. The calculated optimum pH vs. the pH of maximal free energy of

unfolded state. Only 19 points can be seen in the figure, because of an

overlap, but all 28 points are taken into account in the calculation of

the correlation coefficient.


diagonal). If the protein stabilizes bases (or destabilizesacids), then the optimum pH is lower than the pH ofmaximum of the free energy of unfolded state (point belowthe diagonal). The points lying on the diagonal representcases for which the amino acid sequence dominates indetermining the optimum pH. The points below thediagonal show proteins with pH optimum lower than thepH of maximum of the free energy of unfolded state. Thepoints offset from the diagonal manifest the importance ofthe 3D structure. In each case where the 3D structure causesa shift of the solution pKa of ionizable groups, the stabilitychanges [31,69]. If protein favors the charges, then thestability increases. From 28 proteins studied in the paper,nine lie on the main diagonal (tolerance 0.5pK units), while19 are offset by more than of 0.5pK units. Thus, in 32% ofthe cases the amino acid composition is the dominant factordetermining the optimum pH and in 68% of the cases, the3D structure does.

To check for possible correlation between the optimumpH and the pK shifts in respect to the standard pKsol, theywere plotted in Fig. 5. Two pK shifts were calculated:intrinsic pK which does not account for the interactionswith ionizable and polar groups, and pKa shift whichreflects the total energy change from solution to the proteinfor each ionizable group. In both cases the correlation withpH optimum exists, although the correlation coefficients arenot very good. A positive pK shift corresponds to pK ofacids and bases bigger that ofmodel compounds and thus toelectrostatic environment that disfavors acids and favorsbases. The most acidic enzymes were found to use thisstrategy to lower their optimum pH (see themost right handside of the Fig. 5). The most basic enzymes induce slightpositive shift of the intrinsic pK, but adding the pairwiseinteractions turns the pK shift to a negative number. Theenzymes between these two extremes do not induce large pKshift on average.

It is well known that the pH dependence of the freeenergy is an integral of the net charge difference betweenfolded and unfolded states over a particular pH interval(Equation 7) [31,55,70]. A negative net charge differencecorresponds to a negative change of the free energy (the freeenergy gets more favorable as pH increases). Thus, if an acidhas a pKa lower than the standard pKsol, it will titrate atlower pH in the folded state compared to unfolded. As aresult, such a group will contribute to the net chargedifference by a negative number. Conversely, a positive netcharge difference corresponds to a positive free energychange, i.e. to a less favorable free energy of folding. This

corresponds to pKas higher than the standard pKsol. Atoptimum pH the net charge difference should be zero. Atvery low and at very high pHs, the free energy of folding isunfavorable, because either bases or acids are left withoutthe support of the contra partners. Between these twoextremes, the free energy of folding must have a minimum.Starting from very low pH to high pH, the first severalionization events will be the deprotonation of acids. Becausethese few acids are in the environment of the positivepotential of bases, they have pKas lower than of unfoldedstate and thus, the net charge difference between folded andunfolded states will be negative. Thus, the free energy offolding will decrease. If the protein does not support theacids, then the rest of acids will have pKas higher than thatof the unfolded state. This results to a positive net chargedifference between the folded and unfolded state andincreases the free energy of folding. Thus, the optimumpH will be at low pH. Conversely, if the protein favors theacids, then most of them will have pKas lower than ofunfolded state and the net charge difference between foldedand unfolded states will be negative. Thus, the free energy offolding will keep decreasing with increasing pH. This willresult in optimum pH shifted to higher pHs.

The optimum pH is not uniquely determined by the ratioof basic to acidic groups. Figure 6A demonstrates thatenzymes with quite different bases to acids ratio have similaroptimum pH and that proteins with similar bases to acidsratio function at completely different pHs. At the same time,the trend is clearly seen. The proteins that function at lowpH have fewer bases (low base to acid ratio), while theenzyme working at high pH have more bases than acids (seealso Table 2). The Pearson correlation coefficient is lessthan 0.4, which demonstrates that the base/acid ratio is notthe most important factor in determining the optimum pH.However, restricting the counting to buried amino acidsonly, one finds much better correlation (Fig. 6B). Thisimprovement suggests that the pH optimum is mostlydetermined by the buried charged groups, but the correla-tion is still weak.

The effect of the net charge on the stability of theproteins is demonstrated in Fig. 7A,B, where the optimumpH is plotted against the calculated isoelectric point (pI)and the net charge at optimum pH. At the isoelectricpoint the net charge of the protein is zero, i.e. there areequal number negative and positive charges. The graphshows that there is no correlation (Pearson coeffi-cient ¼ 0.09) between the isoelectric point and the opti-mum pH. At the same time, the correlation between the

Fig. 5. The experimental optimum pH vs. the

averaged pK shifts. (A)Averaged intrinsic pKa;

(B) averaged pKas shift.


optimum pH and the net charge of folded state is notneglectable. The signal is weak, but there is a cleartendency for proteins with acidic optimum pH to bepositively charged and for proteins with basic optimumpH to carry negative net charge. There are only a fewproteins which do not have net charge at optimum pH.

Discussion

The study has shown that the pH of maximal stability canbe calculated using the 3D structure of proteins. Twenty-eight different proteins were studied, most of them withundetectable sequence and structural similarity. The opti-mum pH varies from very acidic pH to very basic pH. Sucha diversity provided a good test for the computationalmethod (MCCE) used in the study. Relatively goodagreement with the experimental data was achieved result-ing to correlation of 0.85 and rmsd ¼ 0.73. At the sametime, as indicated in Fig. 3, there are three proteins withcalculated optimum pH of about 1.5 pK units offset fromthe experimental value (see Table 1). The reason for such adiscrepancy could be conformation changes that are notincluded in the model. In addition, all calculations werecarried out at physiological salt concentration (I ¼ 0.15 M),while the experimental conditions of measuring the opti-mum pH in many cases are not available. This may or maynot be a source of significant error, because although the saltconcentration strongly affects the pKa values in proteins[71,72] and in model compounds [73], it may not necessaryaffect the optimum pH [74]. At the same time, it isinteresting to point out that the average rmsd of calculated

to experimental pH optimum is 0.73, which is similar andslightly better than the average rmsd of pKas calculations[25].

Two major factors determine the optimum pH, aminoacid composition and 3D structure of the proteins. Therelative importance of these two factors varies among theproteins. To test our conclusions, two proteins that havedifferent optimumpH (acidic and basic) and are structurallysuperimposable will be discussed below.

Figure 8A shows a structural alignment of acida-amylase (pdb code 2aaa) and xylose isomerase (pdb code1qt1). The first protein has acidic optimum pH (calculatedoptimum pH ¼ 4, experimental optimum pH ¼ 4.9), whilethe second has basic optimum pH (calculated and experi-mental optimum pH ¼ 8). The core structures of theproteins are well aligned (rmsd ¼ 5.0 A and PSD ¼ 1.47[75]). The part of the sequence alignment generated from thestructural superimposition is shown in Fig. 8B. The posi-tions that correspond to Arg or Lys residues in the xyloseisomerase sequence and are aligned to nonbasic groups inacid a-amylase sequence are highlighted. One can see that31 basic groups of xylose isomerase sequence are replacedby negative, polar or neutral groups in acid a-amylasesequence. There are only a few examples of the oppositecase that are not shown in the figure. This results to base/acid ratio of 0.51 for acid a-amylase and 0.84 for xyloseisomerase. This difference in the amino acid compositionresults in a different pH dependence of the free energy of theunfolded state and thus demonstrates the effect of the aminoacid composition on the optimum pH. From a structuralpoint of view it is interesting to mention that most of the


calculated isoelectric point (A) and the net

charge at pH optimum (B).


ratio of bases/acids. Twenty-seven data points

can be seen, because of the overlap between

1qtl and 1b4u. (A) All amino acids; (B) buried

amino acids.


extra basic groups within the xylose isomerase structure arenot within the extra loop regions, but rather within the corestructure (see Fig. 8A). This confirms the observation(Fig. 7B) that buried groups affect the optimum pH andan enzyme that has acidic optimum pH has low acid/base

ratio. It remains to be shown that this is a general behaviorof all enzymes operating at low pH.

Three-dimensional structure of the protein plays an evenmore significant role than the sequence composition on theoptimum pH (68% of the cases in this work). The ability of

Fig. 8. Alignment of acid alpha-amylase

(2aaa.pdb) and xylose isomerase (1qt1.pdb).

(A) Structural and sequence alignments are

carried out with GRASP2 [79]. Structural

alignment in ribbon representation: acid

amylase backbone is shown in green and

xylose isomerase in blue. The red patches

show the positions of substitution of Arg/Lys

to negative, polar or neutral groups from

xylose isomerase to acid amylase (see Fig. 8B).

(B) Sequence alignment from the structural

superimposition: highlighted are the positions

at which Arg/Lys in the xylose isomerase

sequence are aligned to acid, polar or neutral

group in acid a-amylase sequence.


the proteins to reduce the bias of the amino acid sequencecomposition was shown by comparing the isoelectric point,the net charge and the optimum pH. It was shown that formost proteins the optimumpHdoes not coincidewith the pIand that the protein is most stable when it caries net charge.This was demonstrated experimentally by engineering thesurface charges of ribonuclease Sa [12]. Increasing the netcharge of the molecule does not change its pH of maximalstability, but changes the isoelectric point and increasessolubility [12].

Another strategy used to reduce the bias from the aminoacid composition is to change pKas of ionizable groups inthe protein. If protein favors the negative charges on acidicgroups, then the optimum pH is shifted towards high pH ascompared to the pH at which unfolded free energy hasmaximumand vice versa (Fig. 5). The same is valid for basicgroups but the effect is less noticeable simply because theirpKas are too high (except for histidines). It should beemphasized that one should distinguish between the ampli-tude of the free energy of folding and optimum pH. Asdiscussed in previous papers [31,69], the stabilization ofionizable groups by the protein always increases proteinstability.

It should be emphasized that this paper does not makean attempt to calculate the all of the details of pHdependence of the free energy of denaturation. This willrequire an appropriate model of the unfolded state [7,66],which is believed to be compact and native-like. (Inaddition, the denaturated state may not be the same inthermal, urea or guanidine denaturation experiments [10].)The modeling of the unfolded state would eventuallyrequire molecular dynamic runs [33] or some assumptionsof the organization of the amino acids in unfolded state[34,36] or even an experimental determination of the pKasin model compounds [35,73]. Our goal was to computethe pH at which the free energy of folding has minimum.It was shown in the literature that while the shape of thepH-dependence curve is sensitive to the model of theunfolded state, the optimum pH does not dependsignificantly on it [33–36].

The success of the modeling of the pH dependent freeenergy of folding critically depends of the accuracy of thecalculated pKas of the ionizable groups. Recent bench-marks of MCCE on 166 titratable groups resulted to anrmsd 0.83 pK as compared to the experimentally deter-mined pKas [25]. It was demonstrated that increasing theinternal dielectric constant to 20 makes the results slightlyworse, because a significant part of the protein dielectricresponse is captured explicitly in the MCCE methodology.Using a high dielectric constant and allowing explicit rear-rangement of protein dipoles would result to a doublecounting of the same effect. Thus, MCCE employs a lowinternal dielectric constant of 4 and no attempts were madeto study the sensitivity of the results against different valuesof the dielectric constant. Other parameters that were nottested include the charge set [76], the choice of molecularsurface (van der Waals surface vs. molecular surface)[56,77,78] and the effect of energy minimization of PDBstructures [26]. These will require a separate study. Inaddition, it should be noted that the relatively popular �null�method (a method that assumes that pKas of the protein asthe same as in model compounds) will not work in this case,

because it will result in pH-independent free energy offolding.

Despite of several failures, the presented methodologycan predict the optimum pHwith reasonable accuracy. Thisinformation can be used to identify a possible cellularcompartment or body organ where the protein mayfunction. Obviously a protein with a very basic optimumpH cannot be stable in the stomach or in the liposome. Onecan combine such information with information from othersources to achieve better functional prediction. In thepostgenomic era, when many proteins are crystallized andtheir structures determined, the challenge is to find theirputative function. In such a task, any seed of information isvaluable.

Acknowledgements

The author thanks Barry Honig for many inspirational discussions and

for the support during the work. We thank Trevor Siggers and Therese

Mitros for reading the manuscript and for the useful suggestions. This

work was supported by NIH grant GM-30518.

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Documents

Numerical calculations of the pH of maximal protein stability: The effect of the sequence composition and three-dimensional structure