Statistical Mechanical Model for pH-Induced Protein Folding

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Statistical Mechanical Model for pH-Induced Protein Folding: Application to Apomyoglobin Takuya Mizukami, Yosuke Sakuma, and Kosuke Maki J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b06936 • Publication Date (Web): 04 Aug 2016 Downloaded from http://pubs.acs.org on August 18, 2016

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Statistical Mechanical Model for pH-Induced Protein Folding: Application to Apomyoglobin

Takuya Mizukami1†, Yosuke Sakuma1 and Kosuke Maki1* 1

Graduate School of Science, Nagoya University, Furo-cho, Chikusa, Nagoya, Aichi 464-8602, Japan †

Current Address: Fox Chase Cancer Center, Philadelphia, PA 19111, USA

E-mail: [email protected] (Takuya Mizukami) [email protected] (Yosuke Sakuma) [email protected] (Kosuke Maki)

*Corresponding author: Kosuke Maki Address: Furo-Cho, Chikusa, Nagoya, Aichi 464-8602 Japan E-mail: [email protected] Phone: +81-52-789-2434 Fax: +81-52-789-2879

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ABSTRACT Despite the major role of pH in protein folding and stability, a quantitative understanding of the pH-induced protein folding mechanism remains elusive. Two conventional models, the Monod-Wyman-Changeux and the Linderstrøm-Lang smeared charge models have been used to analyze the formation/disruption of specific native structures and fluctuating non-native states, respectively. However, there are only a few models that could represent overall kinetic events of folding/unfolding independent of the properties of relevant molecular species, which has hampered efforts to systematically analyze pH-induced folding. Here, we constructed a statistical mechanical model that incorporates the protonation mechanism of conventional models along with a combined manual search and least-squares fitting procedure, which was used to investigate the folding of horse apomyoglobin over a wide pH range (2.2−6.7) with a time window of ~40 μs to ~100 s, using continuous-/stopped-flow fluorescence at 8°C. Quantitative analysis assuming a five-state sequential scheme indicated that 1) pH-induced folding/unfolding is represented by both specific binding and Coulombic interactions; 2) kinetic folding/unfolding intermediates share kinetic mechanisms with the equilibrium intermediate, indicating their equivalence; and 3) native-like properties are acquired successively during folding by intermediates and in transition states. This model could also be applied to a variety of association/dissociation processes.


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INTRODUCTION Numerous biological processes, including enzymatic catalysis, proton pumping and protein folding, are regulated by pH, mainly via protonation or deprotonation. In particular, protein folding and stability are markedly influenced by pH, with denaturation occurring at extreme values, making it highly useful for investigating the mechanisms of protein folding. Compared to folding/unfolding induced by denaturants, pH-induced folding/unfolding is more biologically relevant, and the underlying mechanism is in theory more straightforward, with denaturation linked to the binding of protons to ionizable groups.1,2 The protonation state of ionizable residues such as His, Asp, and Glu, as well as the chain termini, is a major factor determining specific native structures via the formation of hydrogen bonds and salt bridges. In the case of apomyoglobin (apoMb; Figure 1), specific protonation of His24 stabilizes a partially unfolded state (pH 4 intermediate or M-state) relative to the native state (N) at pH~4, leading to accumulation of the intermediate at equilibrium.3 Protonation is favored in an intermediate owing to the normal pKa of His24, but is largely prohibited in N because His24 is buried and forms a hydrogen bond, thereby exhibiting a lower pKa. The denaturation of native apoMb is interpreted by the Monod-Wyman-Changeux (MWC) model, which attributes the pH dependence of stability to the binding of protons, interpreted as ligands, to specific ionizable groups.3 Upon further acidification, the pH 4 intermediate transitions to the acid-unfolded state (U) at pH ~2. This may be better interpreted via an alternative scenario, whereby the intermediate expands to release destabilizing energy arising from Coulombic repulsion between positively charged residues, such as Arg, Lys, and protonated His (His+), whose charge is cancelled by negatively charged residues (Asp and Glu) under physiological conditions, leading to unfolding.4 In contrast to specific native structures, charged residues are obscured in non-native states including pH 4 intermediate owing to its fluctuating characteristics and the virtual absence of specific side-chain contacts, for which the Linderstrøm-Lang smeared charge (LL) model is appropriate.4 The mechanism underlying pH-induced folding remains poorly understood despite its importance in determining protein stability, folding, and function owing to the lack of a comprehensive model. In



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previous studies, especially from a theoretical point of view, a few models have been proposed to predict pKa values of ionizable groups in proteins based on the statistical mechanics, and applied to pH-induced equilibrium unfolding. 5-7 However, the models were not applied to kinetics of folding/unfolding of proteins exhibiting multi-state folding/unfolding probably owing to the complex folding/unfolding behavior. To understand the physical chemical mechanisms of pH-induced folding kinetics and unfolding equilibrium of various globular proteins, we require a model that represents kinetics and equilibrium of conformational change irrespective of the conformational properties of relevant species.

Figure 1. (A, B) Ribbon diagram of horse skeletal muscle myoglobin on a crystallographic structure (PDB: 1AZI) and (C) sequence alignment of horse and sperm whale proteins (sequences obtained from Uniprot database). (A) Basic and acidic residues are shown in blue and red, respectively. His residues (except His36) are shown in cyan. Side chains of His24, His36, His119, Trp7, and Trp14 are shown in the stick model. (B) The native state of myoglobin contains eight helices (A-H) while the native state of apoMb exhibits seven helices (A–E, G and H; red, pink and orange). The intermediate of apoMb exhibits helical structure in the A-, G-, H- (red) and part of B- (pink) helix regions (A(B)GH domain). (C) Horse and sperm whale proteins are highly homologous in sequence to each other. The figure was prepared using PyMol (Schrödinger, New York, NY, USA).


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Myoglobin is a 153-residue heme protein whose native structure consists of eight α helices (A– H helices) that adopt a globin fold (Figure 1). ApoMb, the apo form of myoglobin with the heme group removed, forms a compact structure similar to that of myoglobin, differing only in terms of the F helix, the N terminus of the G helix, and some loop regions, which appear largely disordered. Previous studies of horse skeletal muscle apoMb (h-apoMb) and sperm whale apoMb (sw-apoMb) have reported that apoMb is folded via a sequential mechanism involving at least two compact intermediates8,9 that transiently accumulate during folding under strongly native conditions. They have ~30% larger gyration radii (Rg) than N and an overall globular shape, while exhibiting ~40% of the native secondary structure in the case of h-apoMb.8 In sw-apoMb, amide protons are more protected from hydrogen exchange in the A-, G-, and H-helices and part of the B-helix of N than in the other regions (Figure 1B), forming as early as ~0.4 ms, indicating that native-like helical backbone structures are formed in these regions (A(B)GH domain) in kinetic intermediates.10 This is also predicted for h-apoMb due to its high sequence homology to sw-apoMb (Figure 1C). As described above, apoMb also populates equilibrium intermediates under various denaturing conditions such as moderately acidic pH, moderate urea concentrations, and acidic pH in the presence of salt.11-13 Among these, the pH 4 intermediate is the best-characterized intermediate. Although the pH 4 intermediate of apoMb consists of two forms, Ia and Ib, or I1 and I2, that differ in terms of stability against pH and urea,9 here we consider only the ensemble consisting of both forms. The pH 4 intermediate of sw-apoMb exhibited a pattern of protection from hydrogen exchange similar to that of kinetic intermediates, suggesting that they have similar structural properties.14,15 In addition, an unfolding intermediate was found to transiently accumulate early in unfolding at pH 2.7.16 ApoMb involves two Trp residues (Trp7 and Trp14) in the A helix (Figure 1A), which are located at the interfaces of A-/H-helices and A-/E-helices, respectively. The pH 4 intermediate (M state) emits Trp fluorescence more intensely than in N and U due to the hydrophobic environments sequestered from the solvent. In contrast, Trp fluorescence in N and U is relatively quenched by



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specific contact with quencher residues and exposure to the solvent, respectively. Thus, Trp fluorescence is a sensitive probe for monitoring tertiary structures of apoMb although the two Trp residues are located close to the N-terminus.

Although a series of intermediates has been identified and characterized in previous studies,3,4,8-16 their relationships and roles in the overall folding/unfolding reaction remain poorly understood. For example, the energetics of relevant species, which includes transition states (TS), as a function of pH are unclear. More specifically, it is unknown whether the pH 4 intermediate is thermodynamically equivalent to kinetic folding and unfolding intermediates, since these states are formed under completely different conditions and via different mechanisms.14,15 The folding and unfolding intermediates accumulate only transiently under strongly native and acidic conditions, respectively, whereas the pH 4 intermediate is stably populated under moderately denaturing conditions; this raises the question of whether pH 4 and kinetic intermediates are formed by shared or distinct kinetic mechanisms during folding. These long-standing questions arise mainly from the difficulty in quantitatively investigating folding/unfolding reactions from early to late stages over a wide pH range. A systematic analysis of pH-induced folding with sub-millisecond temporal resolution using appropriate models can potentially answer these questions.

To this end, we developed a statistical mechanical model of the kinetic steps of pH-induced protein folding that is independent of the conformational properties of the molecular species involved. We assumed two kinds of interactions based on protonation/deprotonation mechanisms of the MWC and LL models; that is, the binding of protons to specific ionizable groups and Coulombic interactions between charged residues, respectively. Our model is a statistical mechanical version of these conventional models and incorporates them as limiting cases at the thermodynamic level. The model allows estimation of expected values and fluctuation (variance) of not only pKa of protonation sites (the MWC model) but also apparent radii (rap), and hence the net charge (according to the


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Debye-Hückel theory used in the LL model) of relevant states, including the TS. We then measured the folding/unfolding kinetics over a pH range from 2.0 to 6.7 and a time window of ~40 µs to 100 s by combining continuous-flow (CF) and stopped-flow (SF) fluorescence methods along with unfolding equilibrium at 8°C. We concluded the following from a quantitative analysis of the results using the model, assuming a five-state sequential scheme. First, both energetics terms corresponding to the MWC and LL models are required to represent pH-induced folding/unfolding reactions. Interconversions between partially folded species accompany expansion/contraction to release destabilizing energy from Coulombic interactions (corresponding to the LL model) between bound protons, whereas specific proton binding (corresponding to the MWC model) plays a major role only in the unfolding of a native-like intermediate via disruption of contact at His24. Second, kinetic folding and unfolding intermediates share the folding mechanisms with pH 4 intermediate, indicating that the three intermediates are a single molecular species despite differences in the conditions and mechanisms of their formation. Third, native-like properties were successively acquired during the formation of not only intermediates, as in the case of other proteins, but also TS in the assembly of a specific native structure, albeit with potential transient formation of non-native local structures during this process. We further proposed the applicability of the model to systems other than pH-induced folding/unfolding.

EXPERIMENTAL METHODS Chemicals All chemicals were either specially prepared or of guaranteed reagent grade. Lyophilized horse skeletal muscle myoglobin was purchased from Sigma (St. Louis, MO, USA). The heme group was removed from myoglobin according to previously described methods; 17 the h-apoMb thus obtained was lyophilized and stored at −20°C.18 H-apoMb concentration was determined spectrophotometrically using an extinction coefficient of ε280 = 14,300 M−1 cm−1.8 All measurements were performed in 12 mM sodium citrate at pH 2.2–7.1 or in HCl at pH 2.0. Solutions were passed



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through a membrane filter (pore size: 0.20 or 0.45 µm) before measurements were performed.

Kinetic refolding/unfolding measurements and fitting of kinetic traces Protein stock solutions for kinetic measurements contained HCl at pH 2.0, 2 mM sodium citrate at pH 4.0, or 2 mM sodium citrate at pH 6.0. Refolding and unfolding reactions were initiated by two-fold dilution of the stock solution with appropriate sodium citrate buffers to give the final solution conditions of 12 mM sodium citrate at pH 2–7. Final protein concentrations were 20–40 µM for CF and 5 µM for SF measurements. Changes in Trp fluorescence over time were recorded using a long-pass filter (50% transmittance at 305nm) by excitation at 295 nm. CF and SF measurements covered time windows of ~40–980 µs and ~3.7 ms–100 s, respectively, and were performed at 8°C with cooled air and circulating water, respectively. The dead times of CF and SF devices were 42–63 µs (depending on in-house-constructed mixers that were used in CF experiments) and 3.7 ms, respectively, as calibrated by the quenching of N-acetyl-L-tryptophanamide fluorescence by N-bromosuccinimide.19 Kinetic traces obtained from CF and SF measurements under matching conditions were combined into a single trace to cover a time window ranging from 42–63 µs to ~100 s, and the traces were fitted by nonlinear least-squares fitting to a single or multi-exponential function using IGOR software (Wavemetrics, Lake Oswego, OR, USA) as: Fobs (t ) = Feq + ∑ i Fi e− λi t

(1)

where Fobs(t) and Feq are the fluorescence intensity at a reaction time t and after reactions were fully equilibrated, respectively, whereas Fi and λi are the amplitude and apparent rate of the i-th phase, respectively.

Estimation of the optimized values of physical parameters by a combined manual search and least-square fitting using the Monte Carlo method The parameters were manually varied to reproduce the pH dependence of apparent rates and fluorescence intensities by means of standard numerical methods for solving the system of linear


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differential equations that describe the kinetic scheme using IGOR software (Wavemetrics, Lake Oswego, OR, USA). The manual search values of the physical parameters were optimized by least-square fitting using the Monte Carlo method as described in SI Results S3.

RESULTS AND DISCUSSION Statistical mechanical model of pH-induced protein folding/unfolding The MWC model describes protonation as the binding of protons, which assumed to be independent in most cases, to specific ionizable groups.1,20,21 This model is usually only applied to the denaturation of N and fully native-like states, because it assumes proton binding to specific sites in the well-defined tertiary structure that are often lost in fluctuating non-native states. On the other hand, the LL model assumes that unfavorable energy arising from Coulombic repulsion between charges on bound protons is released by expansion of the protein when the charge exceeds a critical value, resulting in conversion to more unfolded species.4,22 Furthermore, the charge is assumed to be evenly spread over the surface of a spherical protein that reflects the average globular shape of fluctuating transient structures in the non-native state without specific side chain contacts.

A statistical mechanical model of protonation/deprotonation that incorporates both MWC and LL models at the thermodynamic limit was developed (SI Results S1.1 for details) to investigate kinetic steps of pH-induced protein folding. In this model, we assumed N(H) protonation sites with association energy uk,ρ at the ρ-th site (ρ = 1,…, N(H)) in the state k (an arbitrary state relevant to folding) in a protein molecule. Charges located at these sites also functioned as sources of Coulombic interaction (see below). The resultant electrostatic potential was assumed so that charges at these sites were uniformly spread over the surface of a spherical molecule with a charge-averaged apparent radius in state k, rap,k. This approximation allowed the electrostatic potential, w(rap,k), of a proton bound to the surface of a protein to be estimated using the Debye-Hückel theory as:



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 1 e2 1  2w ( rap,k ) = −   4π kBTε aq  rap,k rap,k + D 

(2)

for a protein with a unit net charge, e, where εaq and D are the permittivity of water (80.4ε0, ε0 is the vacuum permittivity) and Debye length, respectively, kB is the Boltzmann constant, and T is the absolute temperature (note that the potential energy was normalized by kBT). The simple representation of Coulombic interactions essentially by using a single parameter, rap,k, allowed us to analyze complex folding properties of multi-state folding proteins. Thus, the potential energy Ek with respect to protonation/deprotonation of a protein molecule in state k is represented as: Ek =

∑ρ

N

(H )

=1

p k( H, ρ) u k , ρ + 2 k B T

∑ρ

N

(H )

=1

(

p k( H, ρ) Z m in +

∑σ

>ρ

)

p k( H,σ ) w ( ra p , k

)

(3)

(Specific binding term) + (Coulombic interaction term) where ρ and σ indicate protonation sites (ρ, σ = 1,…, N(H)), Zmin is the minimum net charge of the molecule in the pH range of interest (< 7 in this study), and uk,ρ is the association energy of a proton at the ρ-th site as described above. pk,ρ(H) is the proton occupancy at the ρ-th site and hence, equal to 0 and 1 for unprotonated and protonated states, respectively. This potential energy was obtained by extracting the degree of freedom of protonation from the whole protein system by assuming that protonation and other factors such as hydrophobic effect are independent of each other (eqs S1.2 and S1.14). In eq 3, the first and second terms represent the total energy on specific proton binding and the total electrostatic potential energy between the charges on the protein, respectively, which are hereafter referred to as the specific binding and the Coulombic interaction terms, respectively. The former and latter terms correspond to the MWC and LL models, respectively, and serve as limiting cases at the thermodynamic level (SI Results S1.2 and S1.3). The grand partition function, Ξk, of the protein molecule system at the chemical potential of proton µH is represented as follows:

(

)

 1  N (H) Ξ k = ∑ { X } exp  − E k − ∑ ρ =1 pk( H, ρ) µ H  k  k BT 

The summation of Boltzmann factors runs over all possible combinations of pk,ρ(H) (ρ = 1,…, N(H))({Xk}). Eqs 2−4 allow us to estimate physical properties of the protein system including the


(4)

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energetic contributions of specific binding and Coulombic interaction terms for each kinetic step (see below).

These equations were further transformed to another formalism represented by physical parameters such as pKa and pH that were appropriate for dealing with the experimental results. 5,7 Chemical potential was represented as µH = µH0− 2.30 kBT pH, where µH0 is the chemical potential at a reference state, and the pKa value of the ρ-th site (pKa,k,ρ) is defined as pKa,k,ρ = (µH0-uk,ρ)/2.30kBT in the absence of a net charge, where the protonation site is "half-occupied" (eq S1.21 in SI Results S1.1). By substituting the above equations into eqs 2−4, Ξk is reduced as:

(

  D Z min + ∑ σ > ρ p k(H,σ) N (H) e2 (H)   Ξ k = ∑ { X } exp − ∑ ρ =1 pk , ρ 2.30 ( pH − pK a , k , ρ ) +  k  4π k BT ε aq rap,k ( rap,k + D )  

)     

(5)

Based on the transition state theory, the activation free energy upon conversion from state k to state l via TS, ∆Gk‡, is related to the ratio of the partition functions of TS to state k, as follows:

 Ξ Ξ‡ ( pH 6)  ∆Gk ‡ = ∆Gk ‡ ( pH 6) − RT log  ‡   Ξk Ξk ( pH 6) 

(6)

where ∆Gk‡(pH 6) is the reference activation free energy at pH 6 and R is the gas constant. The elementary rate constants for crossing the TS, kkl, are represented by eq 7:

 ∆G   ∆G ‡ (pH 6)   Ξ‡ Ξ‡ (pH 6)  0  Ξ‡ Ξ‡ (pH 6)  kkl = A0 exp  − k ‡  = A0 exp  − k  = kkl    RT RT Ξ Ξ (pH 6)     k k   Ξk Ξk (pH 6) 

(7)

where A0 is the pre-exponential factor with an arbitrary value of 106 s−1,23 and kkl0, which is defined as k kl0 ≡ A0 exp ( −∆ Gk ‡ (pH 6) / RT ) , is the elementary rate constant at pH 6.0.

pH dependence of refolding and unfolding reactions of h-apoMb The refolding and unfolding kinetics of h-apoMb were measured by monitoring Trp fluorescence (excitation at 295 nm) at various initial and final pH values and 8°C. Refolding was induced by a pH



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jump, either from pH 2.0 (in HCl) to a value ranging from 2.2 to 6.2, or from pH 4.0 (in 2 mM sodium citrate) to a value ranging from 4.0 to 6.6. Unfolding reactions were induced by a pH jump either from pH 4.0 (in 2 mM sodium citrate) to a value ranging from 2.6 to 4.0, or from pH 6.0 (in 2 mM sodium citrate) to a value ranging from 2.8 to 6.7. The initial states of the refolding and unfolding reactions were the acid-unfolded state (U) (pH 2.0) and the native state (N) (pH 6.0), respectively, in addition to an equilibrium (pH 4) intermediate (pH 4.0) for both reactions. Kinetic traces obtained from CF and SF experiments under matching conditions were combined to cover a time window from ~40 µs to 100 s. The dead times of CF and SF measurements were 42–63 µs and 3.7 ms, respectively. Traces of refolding and unfolding reactions (Figure 2) were fitted by nonlinear least-squares fitting to eq 1, where we used the index of phase, i, to identify refolding and unfolding reactions, with Ri and Ui representing the i-th phase of refolding and unfolding reactions, respectively. The number of phases was dependent on the initial and final conditions (Table 1). The fluorescence intensities were relaxed to essentially the same value after refolding and unfolding reactions were fully equilibrated under matching conditions, which confirms the reversibility of the refolding/unfolding under the conditions studied (Figure 2D). In addition to the observed folding/unfolding phases, a slow minor phase (phase D; apparent rate: λD~1 s−1, amplitude: FD) was observed to arise from the association/dissociation of oligomeric species,13,18 most likely dimeric species, since it was detected under most conditions, including upon dilution of the protein solution at a constant pH of 4.0. This was confirmed by measuring dependence of protein concentration on refolding, including dilution of the protein solution at pH 4.0 (SI Result S2 and Figure S1). Thus, phase D was not considered further in this study. 16,24

Table 1: Fitting functions used for refolding/unfolding reactions at initial and final conditions. Reaction

Initial pH

Refolding

2.0

1 phasea pH 2.9–4.8

2 phasesa pH 5.0–6.2


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4.0

pH 5.0–6.6

-

6.0

pH 3.6/4.6–6.7

pH 2.6–3.4/3.8–4.5

4.0

pH 2.6–3.1

-

Unfolding a

1 phase: a single-exponential function, 2 phases: a double-exponential function. Phase D arising

from the association/dissociation of oligomeric species was excluded.

When refolding was initiated at pH 2.0 (U), the kinetics were biphasic, consisting of a faster rising phase (phase R1) and a slower decreasing phase (phase R2) at pH > 5.0, whereas only a single phase R1 was observed at pH 2.9–4.8 (Table 1 and Figure 2A). When refolding was initiated at pH 4.0 (pH 4 intermediate), a single declining phase was observed at pH 5.0–6.6 (Figure 2C) whose apparent rate and amplitude were identical to those of phase R2 under matching conditions (Figure 2D), and which was therefore assigned to phase R2 irrespective of initial conditions (pH 2.0/4.0). On the other hand, the kinetics of unfolding from pH 6.0 (N) were complex (Table 1 and Figure 2B). At pH 2.6–4.5 with an exception at pH 3.6, the components of the phases were dependent on pH; however, the kinetics were biphasic, consisting of a rising phase (phase U2) and a slower decreasing phase (phase U3) at pH 2.6–3.8, and of a faster rising phase (phase U1) and phase U2 at pH 3.8–4.5. On the other hand, only phase U2 was observed at pH 3.6 and 4.6–6.7. When unfolding was initiated at pH 4.0 (pH 4 intermediate), a single decreasing phase was observed at pH 2.6–3.1 (Figure 2C) whose apparent rate and amplitude were very similar to those of phase U3 (Figure 2D), and which was therefore assigned to phase U3 irrespective of initial conditions (pH 4.0/6.0). There were no detectable kinetics under other conditions.



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Figure 2. Kinetic traces of h-apoMb (A) refolding from pH 2.0, (B) unfolding from pH 6.0, and (C) refolding and unfolding from pH 4.0 measured by CF and SF fluorescence methods at representative pH and 8°C. Final pH values are indicated in each panel. (D) Representative kinetic traces of refolding at pH 5.9 from pH 2.0 and 4.0, and of unfolding at pH 2.9 from pH 4.0 and 6.0, respectively. Black solid lines were obtained by nonlinear least-squares fitting of data. Colored dashed lines were obtained by quantitative modeling using the parameters shown in Tables 3 and S1. Arrows denote the fluorescence intensity at pH 2.0 (reference value).


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Figure 3. (A) pH dependence of apparent rates of refolding/unfolding obtained in CF and SF experiments. Triangles and inverted triangles: λR1 (orange) and λR2 (red) for refolding from pH 2.0;

λU1 (cyan), λU2 (blue), and λU3 (purple) for unfolding from pH 6.0; and λR2 (green) and λU3(yellow) for refolding and unfolding from pH 4.0. Triangles and inverted triangles indicate the rising and decreasing phases, respectively. Black solid lines represent apparent rates predicted by the quantitative modeling. (B) pH dependence of apparent and elementary rate constants of refolding/unfolding predicted by the quantitative modeling. Black and blue lines represent apparent and elementary rate constants, respectively. (C–E) Cumulative amplitudes of refolding/unfolding from (C) pH 4.0, (D) 2.0, and (E) 6.0, defined as F0R2 = Feq + FR2 and F0R1 = Feq + FR2 + FR1 for the refolding reaction and F0U3 = Feq + FU3, F0U2 = Feq + FU3 + FU2, and F0U1 = Feq + FU3 + FU2 + FU1 for the unfolding reaction, where Feq and Fi are the fluorescence intensity at equilibrium and the



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conventional amplitude of the i-th phase, respectively. Circles: (C) F0R2 (green) and F0U3 (yellow) and Feq (black) for refolding/unfolding from pH 4.0; (D) F0R1 (orange), F0R2 (red) and Feq (black) for refolding from pH 2.0; and (E) F0U1(cyan), F0U2(blue), F0U3 (purple) and Feq (black) for unfolding from pH 6.0. (C–E) Dashed lines represent fluorescence intensities of U (red), I (orange), M (green), N' (cyan), and N (blue), predicted by the quantitative modeling. Colored solid lines represent cumulative amplitudes predicted by quantitative modeling. Arrows represent direction of fluorescence change of each phase.

The pH dependence of the apparent rates of refolding and unfolding is shown in Figure 3A (red and blue symbols for refolding and unfolding, respectively, from pH 2.0 and 6.0 and green and yellow symbols for refolding and unfolding, respectively, from pH 4.0). The apparent rate of phase R1 (λR1) was almost independent of pH (~1 × 104 s−1) above 5.0, and smoothly decreased to ~1 × 103 s−1 when pH was reduced to 2.9, which was connected to the apparent rate of phase U3 (λU3) at pH 3.4−2.6. The apparent rate of phase R2 (λR2) reached a minimum value at pH~6 (~4 s−1), and increased with decreases in pH before disappearing at pH 5.0. λR2 was smoothly connected to the apparent rate of phase U2 (λU2), with the overlap region at pH 5.0−5.7, which reached~1 × 104 s−1 at pH 2.6. The detection of the slower phase R2 only under native conditions (pH > 5.0) confirmed it as the rate-limiting step of the overall refolding reaction, which roughly corresponded to the formation of the native/native-like species, whereas the faster phase R1 corresponded to the formation of an intermediate (M) (see below). λR1 and λU3 overlapped under matching conditions as did λR2 and λU2, indicating that each set of phases arose from the same kinetic step, because the apparent rates were independent of the initial conditions of the reactions. At pH~3.5, phases U2 and U3 showed similar rates due to the crossover, making it difficult to distinguish between them. Phase U1 (λU1 ~5–9 × 103 s−1) was detected at pH 3.8–4.5, with the rate being too large below pH 3.8 and the amplitude being too small above pH 4.5, which corresponded to the formation of the third, native-like intermediate N'.


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Figures 3C–E show the pH dependence of cumulative amplitudes of refolding and unfolding (panel C: refolding and unfolding from pH 4.0; panel D: refolding from pH 2.0; and panel E: unfolding from pH 6.0) in addition to Feq. The cumulative amplitudes were defined as F0R2 = Feq + FR2 and F0R1 = Feq + FR2 + FR1 for the refolding reaction and F0U3 = Feq + FU3, F0U2 = Feq + FU3 + FU2, and F0U1 = Feq + FU3 + FU2 + FU1 for the unfolding reaction, which yielded fluorescence intensities extrapolated to zero time of the reactions in the absence of preceding phases. The pH dependence of conventional amplitudes thus obtained is also shown in Figure S1. The pH dependence of Feq, the equilibrium unfolding transition curve, exhibited two transition regions at pH ~3.5 and ~5.5.13 Under equilibrium conditions, N and U were the major species at pH > ~5.5 and pH < ~3.5, respectively, while the pH 4 intermediate is predominantly populated between the transition regions at pH ~4−5, where Feq approximates the fluorescence intensity of the equilibrium state.

Figure 3C shows the pH dependence of F0R2 and F0U3, which closely approximated the fluorescence intensity of the pH 4 intermediate since it represented the initial state of refolding and unfolding reactions from pH 4.0. In fact, they linearly merged with Feq when the pH approached the transition regions. Thus, F0R2 and F0U3 constituted the baseline of the pH 4 intermediate, although F0U3 was resolved to a lesser extent. This was the case for refolding from pH 2.0 (U) (Figure 3D). F0R2 closely approximated the fluorescence intensity of a kinetic intermediate, M, which formed in phase R1 to accumulate at equilibrium at pH 2.9–4.8 (as a mixture of U and M at pH < ~4) without phase R2 and to transiently accumulate before the slow conversion to N in phase R2 at pH > 5.0, since λR1>>λR2. In addition, the pH dependence of phase R2 (λR2 and F0R2) was independent of the initial conditions (pH 2.0/4.0) (Figure 3A for λR2 and Figures 3C and D for F0R2), which indicated that M was involved in the pH 4 intermediate. On the other hand, the pH dependence of F0R1, which represented the fluorescence intensity extrapolated to zero time of the refolding observed by CF measurements, exhibited a sigmoidal feature that deviated from the linear baseline of U (i.e.,



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fluorescence intensity expected for U) above pH 3.5, indicating the accumulation of an additional kinetic intermediate (I) within the dead time of the CF measurements (i.e., burst phase or phase R0). Hence, phase R1 arose from the conversion of I to M. In unfolding from pH 6.0 (N) (Figure 3E), M formed in phase U2 due to the overlap of λR1 (I→M conversion) and λU3 (M→I⇄U conversion) to accumulate at equilibrium at pH 3.6–6.7 (as a mixture of N and M at pH > ~5) without phase U3 and to transiently accumulate as an unfolding intermediate before conversion to U in phase U3 at pH < 3.4. Therefore, M represented refolding and unfolding intermediates under native and unfolding conditions, respectively, and was involved in the pH 4 equilibrium intermediate. On the other hand, F0U2 deviated from the baseline of N below pH ~ 5; this was fully accounted for by the conventional amplitude, FU1, at pH 3.8–4.5, because F0U1 was at the baseline of N. Phase U1 was detected as the burst phase whose fluorescence intensity was represented by F0U2 at pH < 3.8, since it appeared too rapidly for CF to be measured. Therefore, phase U1 along with F0U2 below pH 3.8 was attributed to the accumulation of a third intermediate, N'. Therefore, phases U1 and U2 are assigned to the N→N' and N'→M conversions, respectively. Phases U2 and R2 arose from the same kinetic step; thus, phase R2 corresponded to the M→N' conversion.

Based on the above considerations and previous studies on pH-induced folding of h- and sw-apoMb,8,16,25 we assumed a five-state sequential model (Scheme 1), which is also fully consistent with urea-induced refolding/unfolding at pH 6.0,26 to reproduce the pH-induced refolding/unfolding kinetics of h-apoMb. The need for five states was also confirmed in terms of the rate equation. The four experimentally observed kinetic phases (phases R1/U3, R2/U2, and U1 along with phase R0 as a burst phase) corresponded to four non-zero eigenvalues of the rate equation of the refolding/unfolding kinetics, and therefore the number of the relevant states was predicted to be at least five. In addition, the sequential scheme was confirmed by pulsed hydrogen exchange experiments combined with nuclear magnetic resonance (NMR) and mass spectrometry analysis of apoMb, which revealed that each species accumulated in a sequential order along the reaction. The


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TSs in this scheme were numbered in order from U to N. For example, TS1 is the transition state in the U⇄I interconversion. (Scheme 1)

Practical application and evaluation of the statistical mechanical model to h-apoMb folding We applied our model (eqs 2−7) to h-apoMb folding based on the results obtained above and the folding scheme. In practice, the pH dependence of the apparent rates and cumulative amplitudes (Figure 3), and hence, the kinetic traces (Figure 2), were reproduced by elementary rate constants calculated based on eqs 5−7 and Scheme 1 along with the fluorescence intensities of the five species. The parameters used for the h-apoMb system (Zmin and N(H)) are listed in Table 2. The Debye length was 11.4 Å in 12 mM sodium citrate. The protonation sites were limited to those with normal pKa < ~7, i.e., carboxyl groups of Asp and Glu and imidazole groups of His (except for His24 and His36, see below), because we focused on conformational changes occurring under neutral-to-acidic conditions. Arg and Lys with normal pKa > ~7 were assumed to always be protonated. It was also assumed to be the case for His36 in all states because of its extremely high pKa value (> 8).3 The pKa of His24 (pKaHis24) was taken as an independent physical parameter and constrained at < 4 in N because of the low pKa> kMN' (Figure 3B; see below). Thus, both kIM and kMI were almost pH independent, which followed from the observation that pKa and rap remained unchanged in I, TS2, and M. Because M possesses a native-like A(B)GH domain,14 possible conformational changes occurring in the I ⇄ M interconversion were likely to represent further organization within the A(B)GH domain, including recruitment of the G helix and part of the B helices (see below).

N' ⇄ N interconversion F0U2, including the unfolding burst phase, approximated the unfolding transition curve from N to N' based on the pre-equilibrium occurring in phase U1, which was characterized by the pre-equilibrium constants (kNN'/kN'N). The transition observed at pH ~3.5–4.5 was accounted for by the difference in pKaHis (5.90 ± 0.02 and 5.78 ± 0.01) and rap (23.3 ± 0.1Å and 18.2 Å) between N' and N.8 By selecting these parameters in TS4, λU1~kNN'+kN'N was also reproduced as a function of pH, confirming that phase U1 arose from the N→N' conversion (Table 3and Figure 3A, B). Phase U1 became undetectable as an observable phase below pH 3.6 because λU1 was too large for CF


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measurements. The reverse N'→N conversion was undetectable under refolding conditions because the corresponding kinetic barrier was lower than that of the preceding rate-limiting step (the M→N' conversion). The quantitative analysis of unfolding kinetics revealed that N' differed from typical hidden intermediates because it was sufficiently stable to be transiently populated; N', which was only 1.6 kcal/mol less stable than N at pH 6.0, became more stable than N below pH 4.3. N' exhibited a native-like structure because pKa and rap, as well as Trp fluorescence, were closer to those of N than to those of M (Tables 3 and S1).

A recent NMR study of sw-apoMb using R2 relaxation dispersion revealed a native-like intermediate (I1) located between N and molten-globule (MG) states at moderately acidic pH (4.75– 5.5) and 35°C.27 I1 is an on-pathway, high-energy intermediate with more native-like properties than MG, such as low pKaHis24. The I1 and MG states likely corresponded to N' and M, respectively, in this study. The largest chemical shift change in I1 was found at the C terminus of the E helix, where the potential quenchers Lys77 and Lys79 were in contact with Trp14 and Trp7, respectively, in N; the small increase in fluorescence during the N→N' conversion may have resulted from this structural change (Table S1). However, the rate of the interconversion was estimated to be up to two orders of magnitude smaller than that of the N ⇄ N' interconversion estimated in this study. Therefore, although sw-apoMb is generally more stable than h-apoMb, the difference in their rate constants is large and this difference must be explained in future studies.

M ⇄ N' interconversion Overall refolding and unfolding reactions were rate-limited by phases R2 and U2, respectively, above pH 3.6. These phases arose from the M ⇄ N' interconversion, as reported in previous studies of sw-apoMb (N rather than N' for sw-apoMb).16,33λR2 ~ kIM/(kIM + kMI) × kMN' was reduced to kMN' at pH ~6–7 (kIM>>kMI); therefore, the weak pH dependence of λR2~ kMN' was attributed to a slight decrease in pKaHis and pKaHis24 (without a change in rap) during the M → TS3 conversion (Table 3),



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which resulted in the formation of N' with a moderate rate.27 λU2~kNN'/(kNN'+kN'N)×kN'M was reduced to kN'M below pH ~3.5, considering kNN' >> kN'N. The pH dependence of kN'M was attributed to changes in both pKaHis24 (1.99 ± 0.01 to 5.83 ± 0.04) and rap (23.3 ± 0.1 to 23.6 ± 0.1Å) during the N'→TS3 conversion, which were responsible for the increase in λU2 upon lowering of pH from ~6 to ~2. The normalization of pKaHis24 in TS3 mainly accounted for the pH dependent behavior of λU2 (N'→M conversion) at pH ~4–6, as reported in a previous study of sw-apoMb using the MWC model.33 Our model further indicated that an increase in rap from N' to TS3 induced a continuous increase in λU2 at pH values below ~4 because of the Coulombic interaction energy that apparently reduced pKaHis24 in N'. In addition, the steeper pH dependence of λU2 at pH ~4–6 than at pH ~2–4 was accounted for by kN'N >> kNN' at pH above ~4.

Protonation/deprotonation of carboxyl groups in refolding/unfolding As described above, pKacarboxyl was 3.6 (~0.4 lower than the typical value) in U, as reported for other proteins in their denatured state.36,37 Even after the fitting, this value was unchanged (< 0.05 units) in all states except N', TS4, and N, reflecting a dominant contribution by the Coulombic interaction term compared with the specific binding term, to the stability of these non-native states, as both imidazole and carboxyl groups exhibited near-normal pKa values. The near-normal values of pKacarboxyl in non-native states were consistent with previous reports on charged-site mutations of sw-apoMb,4,38 where a majority of charge deletion substitutions had little effect on stability of the pH 4 intermediate. In contrast, a decrease in pKacarboxyl from the normal value by 0.1–0.2, reflecting contribution by the specific binding term, would also be consistent with destabilization caused by a charge deletion substitution of Asp20 and Arg118, which has been demonstrated to form an ion pair in the native myoglobin with cyanoheme. In addition, it may contribute to the stabilization of native-like characteristics of N'.

Proton binding/dissociation vs. conformational changes


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pH-induced folding/unfolding is an inherently competitive reaction between conformational change and proton binding/dissociation. We assumed that protonation/deprotonation occurred more rapidly than the interconversion between relevant states. If rates of protonation/deprotonation were slower than rates of conformational change, protonation and not conformational change would limit overall reaction rates. This was especially relevant to early folding events, because the estimated elementary rate constant was as large as ~104 s−1. In recent 13C NMR relaxation experiments on the B1 domain of protein G, protonation rates were found to be higher than the fastest interconversion rates, validating the assumption made in our analysis.39

pH-Induced unfolding equilibrium reproduced by kinetic parameters To further test the validity of our quantitative modeling, pH-induced equilibrium unfolding was analyzed according to results obtained from the quantitative analysis (SI Results S4 and Figure S4). The fraction populated was calculated for each of five states as a function of pH using the obtained parameters (Figure S4B). Using the fractions as known parameters, equilibrium unfolding transition curves monitored by Trp fluorescence at 301–449 nm were fitted by linear least-squares fitting. The fitted curves closely reproduced the unfolding transition curves (Figure S4C). The fluorescence emission spectra of the relevant species at equilibrium were in good agreement with those obtained by urea-induced unfolding equilibrium under the same conditions (pH 6.0 and 8°C) (Figure S4D).26 In addition, the integrated fluorescence intensity of each species estimated by the equilibrium measurements was similar to that obtained by quantitative analysis (Table S1), which validated our model.

Contribution of the specific binding and Coulombic interaction terms in the h-apoMb folding In previous studies, partial folding/unfolding reactions such as the unfolding equilibrium of the pH 4 intermediate and the late stages of folding were selected from overall reactions to investigate the mechanisms of pH-induced folding of apoMb using either the MWC 3,33 or the LL 4,38 model. Here,



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we employed a statistical mechanical model that incorporates both models. It was therefore necessary to confirm that both specific binding and Coulombic interaction terms were required to obtain consistent mechanisms of pH-induced h-apoMb folding/unfolding reactions. For this purpose, we fitted the data obtained in this study to either the MWC or LL model derived from the statistical mechanical model, assuming the five-state sequential scheme (Scheme 1) (SI Results S1.2 and S1.3). The pH dependence of apparent rates and cumulative amplitudes of folding/unfolding were well reproduced by both models (Figures S5and S6). However, the parameters obtained with the two models were inconsistent with previous results (Tables S3 and S4). The MWC model had two His residues with low pKa values ( kMN', respectively, resulting in the stable population of M (Figures 3A, B and 4A). In contrast, I→M and M→N' conversions were preferred at pH > ~5.5 and N'→M and M ⇄ I → U at pH < ~3, resulting in transient accumulation of M as folding and unfolding intermediates, respectively. This provides evidence that M and the pH 4 intermediate are a single molecular species, and hence share a kinetic mechanism. The unfolding intermediate M did not accumulate above pH 3.6 due to a switch in the rate-limiting step in the unfolding reaction. The rate-limiting step was the N' → M conversion at pH above ~3.5, which made the M → I conversion undetectable, whereas below pH ~3.5 the rate-limiting step was the M ⇄ I → U conversion, leading to accumulation of M as an unfolding intermediate. This was observed for sw-apoMb in the previous study based on apparent rates < ~1×103 s−1,16 and has been demonstrated in the present study by the detection of relevant kinetic steps with the time resolution extended to ~40 µs.

Our previous study of h-apoMb folding induced by urea found that the folding was consistent with the five-state sequential scheme (Scheme 1) with kinetic behavior and structural properties of each intermediate similar to those obtained in this study. In fact, the relevant species share common Gibbs free energies (including TS) and almost equivalent fluorescence intensities at pH 6.0 and 0 M urea at 8°C (Table 3 and S1). This suggests similarity in the mechanisms between urea- and pH-induced folding reactions. Quantitative analysis of urea-induced folding revealed a change in solvent-accessible surface area (SASA) relative to N (normalized to the difference in SASA between N and U), whereas quantitative analysis of pH-induced folding revealed rap for each species. Assuming similarity between urea- and pH-induced folding, we established a correlation between normalized SASA and rap for molecular species relevant to folding, including the TS (Figure5). Both parameters were reduced from U to N, confirming the successive accretion of native-like properties in the folding of h-apoMb. For example, the reduction in SASA with no change in rap, occurring in



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the I → M conversion was consistent with local organization within the A(B)GH domain.40,41 The reduction in the net charge from U to N was also consistent with the behavior of the dimension. A remarkable finding is that the TS also exhibited this trend. Although energetically unfavorable, the polypeptide chain became more compact than, or was at least similar in dimension to, the preceding state. This is consistent with results of the previous Φ-value analysis of sw-apoMb, which revealed that the Φ value increased from the intermediate (M) to the TS of the rate-limiting step for residues

located in the majority of helical regions.12

Figure 5. Relationship between relative SASA and rap. Previously reported SASA values 26 are plotted as a function of rap. The size of each symbol represents the relative stability of the state at pH 6.0 and 8°C in terms of Gibbs free energy; the Gibbs free energy scale is shown above the figure. A schematic structure of the observable species and the net charge of each species are also shown.

Application of the statistical mechanical model to other systems Salt (anion)-induced folding of h-apoMb


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As a test for the robustness of the model in systems other than the one used in this study, i.e., pH-induced folding of h-apoMb, we applied the model to salt (anion)-induced folding of h-apoMb. H-apoMb forms a partially unfolded intermediate in the presence of a high salt concentration such as > ~50 mM NaCl under strongly acidic conditions (Msalt).42 The salt-induced folding is thought to occur by reduction of the unfolding electrostatic repulsion in U via shielding of the net charge of protein molecules by the ions. The shielding changes the Debye length and effective dimension of protein molecules, which is expected to perturb the Coulombic interaction term but to have a much lower effect on the specific binding term at a constant pH of 2.0. Thus, salt-induced folding is independent of protonation/deprotonation. We found that the model reproduced the salt-induced folding kinetics of h-apoMb at 50 mM and 200 mM NaCl initiated in the absence of NaCl at pH 2.0 and 8°C, albeit with some discrepancy relative to the kinetic traces experimentally obtained, with a slight modification to the electrostatic potential energy of a bound proton, 2w(rap,k), in eq 2 as follows:

2w( rap,k , reff,k , C ) =

 1  e2 1 −  , 4π kBTε aq  rap,k reff,k + D ( C ) 

(8)

where C is the anion (Cl−) concentration, and reff,k is the radius of an effective anion-exclusive (anion-impenetrable) sphere of h-apoMb in state k (with the other parameters the same as those of eq 2 and 5). The Debye length, D (C), was represented explicitly as a function of the anion concentration, C, because the folding kinetics were measured at two salt concentrations. The effective radius, reff,k, was introduced by assuming that the solvent containing sodium and chloride ions was allowed to penetrate into the “inside” of the sphere with the apparent radius, rap,k, which characterized the dimension of the protein molecule in state k. Thus, reff was equal to rap in N because of the well-packed specific structure of N, whereas it was smaller than rap in partially or fully unfolded states that possessed unstructured/disordered regions. Eq 8 was derived in SI Results S1.4 and Figure S7A. Polypeptide chains were indicated to be contracted by the reduction of electrostatic repulsion, which corresponded to the reduction in the Coulombic interaction term (the corresponding



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exponential term, i.e., the probability, was increased) based on the model, in both pH- and salt-induced folding. However, the mechanisms of the reduction were different between the two types of folding. The number of protonated sites was reduced at pH > 2.0 in pH-induced refolding, whereas the Debye length and the effective dimension in the electrostatic potential energy were reduced due to the shielding-effects by the ions, compared with those in the absence of salt, in salt-induced refolding. Figure S7 shows that the kinetic traces were reasonably reproduced the modified model assuming a three-state scheme U ⇄ Isalt ⇄ Msalt, where Isalt is a kinetic intermediate accumulating within the dead time of the CF measurements. Considering the quality of the modeling, it was possible that additional intermediates accumulated, especially within the dead time of the CF device. Although a detailed analysis of the mechanism of salt-induced folding is required for a more appropriate model (which is beyond the scope of this study), the successful application of the model to another system indicates that it is sufficiently robust to be applied to a wider range of systems other than the pH-induced folding of h-apoMb. In the analysis of the salt-induced folding, the folding scheme and the fluorescence properties remained unchanged (U ⇄ Isalt ⇄ Msalt vs. U ⇄ I ⇄ M (pH 4 intermediate)), which would suggest that these two types of folding, i.e., that induced by pH and salt, share a common mechanism.

Prediction of charge distribution as a function of pH by using the optimized physical parameters. Another example of the application of the model is demonstrated in Figure S8. The pH dependence of the charge distributions of h-apoMb was predicted based on the model and optimized physical parameters obtained in this study. Because the mass of a protein molecule (m) is almost independent of the charge state (Z), the charge distribution of the protein is estimated by electrospray ionization mass spectrometry (as a function of m/Z). The predicted charge distribution reproduced the majority of the features of the charge distributions obtained by mass spectrometry at pH < 5.43 The charge states with a maximum population were reproduced within three charge units of the values experimentally obtained, and the charge distributions were also reproduced for higher charge states


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of the distribution in the pH region (Figures S8B−F). In addition, the pH with the highest population was also reproduced for charge state with Z > +12 (Figure S8A). Additional minor charge distributions found at lower charge states were not accounted for by the model. Some of the discrepancies might arise from experimental details of the mass spectrometry because the mass spectra were indicated to exhibit some variation in the spectral features probably due to the difference in instrumentation and solvent conditions.43,44 The semi-quantitative predictions of the charge distributions of h-apoMb further confirmed the applicability of the model to other systems, and demonstrated that physical parameters obtained from kinetic experiments along with quantitative analysis can explain a majority of the experimental results obtained by an independent methodologies. The results exhibited the advantage of the statistical mechanics-based model in that distributions and thus fluctuations of observables can be obtained in addition to expected values. In this case, the charge distributions were directly obtained from the grand canonical distribution of the h-apoMb system used in this study. Recently it has been recognized that the fluctuations of biomolecules including proteins play important roles in their functions, but the methodologies used, such as NMR and single-molecule experiments, were associated with limitations. The statistical mechanical model provided us with a means to investigate the fluctuations through kinetic experiments.

Modeling of the equilibrium and kinetic properties of a two-state folding protein. We applied the model to pH-induced folding/unfolding of a protein system other than h-apoMb for further testing of the model. For this purpose, we chose chymotrypsin inhibitor 2 (CI2; 64 residues) as a model protein, as it is a two-state folding protein. 45 Simple two-state proteins require fewer physical parameters to represent the folding compared with h-apoMb. In addition, the parameters required to predict the pH dependence of stability and folding/unfolding rates are available in the literature. 46-48 These properties make CI2 an ideal example to demonstrate the versatility of the



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model. In fact, pH-induced unfolding equilibrium of CI2 has been modeled by statistical mechanical models in previous studies. 6

The parameters used for the CI2 system (Zmin and N(H)) are listed in Table S5. The reference parameters taken from the literature are i) the Gibbs free energy of TS and N relative to U at pH 6.349, ii) Rg of U, TS and N 50, and iii) the apparent pKa (pKAP) of all carboxyl groups in the four Asps and six Glus in N and the averaged pKAP of Asp and Glu carboxyl groups in U at 50 mM and 200 mM NaCl and 25°C (the pKAP of Asp23 was not available for 200 mM NaCl)46. Wild-type CI2 has no histidine. The pH of the reference states was 6.3 (except for the analysis of the R48H variant; see below), and rap values were approximated by the Rg values. The Gibbs free energy at pH 6.3 and Rg values of U, TS, and N are listed in Table S6. Although the pKa values in the absence of Coulombic interactions were required for modeling, the values available in the literature were the apparent values (pKAP) at 50 mM and 200 mM NaCl. Thus, we started the modeling by converting the pKAP in N and U at 50 mM NaCl into the corresponding pKa values by using the model.

The procedure for the conversion was straightforward and used to predict the salt dependence of the pKAP values below. The detailed procedure is described in SI Results S1.5. In brief, the pKa at the protonation sites (eq S1.21 in SI Results S1.5) was numerically searched so that each protonation site was half occupied, i.e., the proton occupancy was equal to 0.5, at 50 mM NaCl and at the pH of pKAP of the site (eq S1.49). Once the pKa values are appropriately estimated, we could predict the salt dependence of the pKAP by using the pKa values thus obtained as known parameters (eq S1.49). After conversion of the pKAP of N at 50 mM NaCl into the corresponding pKa (Table S6), we in turn estimated the pKAP of N at 200 mM NaCl based on the converted pKa using the model, and compared these values with the pKAP experimentally obtained in a previous study. 46 Figure S10 shows the correlation between the predicted pKAP (pKAP,Calc) and experimentally obtained pKAP (pKAP,Exp) at 200 mM NaCl.46 Seven of the nine pKAP,Calc values were in good agreement with pKAP,Exp with an


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accuracy of ±0.08 units. The two outliers were the values for Glu14 and Glu15, where pKAP,Calc < pKAP,Exp by -0.53 and -0.74, respectively, which suggested that the model underestimated the electrostatic interactions of these residues. This discrepancy can be explained by the fact that these residues are adjacent to each other. Although the charge was assumed to be evenly spread over the surface of the protein molecule in the model, Glu14 and Glu15 would be located too close to each other for the charge to be evenly distributed, which would result in much stronger electrostatic interactions than those assumed in the model. Nevertheless, the results suggest that the model could predict pKAP with an accuracy of ±0.08 units if the protonation sites are at least a few residues distant from each other. Because the salt dependence of pKAP,Calc was determined only by the Coulombic interaction term in the model, the reproducibility of pKAP,Exp determined by the model indicates that the electrostatic interactions are fully accurately approximated by the Coulombic interaction term in the range of NaCl concentrations used here. In addition, the results indicate that the Coulombic interaction term (the LL model) is required even for N in addition to the specific binding term (the MWC model).

A previous study indicated the pKAP of CI2 in U at 50 mM NaCl had to be slightly lower (~0.3) than the normal value to reproduce the pH dependence of the charge states in the MWC model. 46 The pKAP values of Asp and Glu in U thus estimated were 3.6 and 4.0, respectively, at 50 mM NaCl. The corresponding pKa values were obtained from the above pKAP values in the same procedure used for calculation of the pKa values in N, which resulted in values of 4.09 and 4.35 for Asp and Glu, respectively. The estimated pKa values for U were more similar to the normal values compared with the pKAP values for U reported previously. 46 The pKa and pKAP values for U are listed in Table S6. Considering that the conversion mainly corresponds to the removal of the contribution of the Coulombic interaction term from the total energy of protonation/deprotonation, the slightly lower-than-normal values of pKAP for U might be attributed not only to the residual structure, as indicated previously,46 but also to the charge effects on protonation/deprotonation.



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Figure S11A-D shows the pH dependence of the stability in terms of the Gibbs free energy and the folding/unfolding rates of CI2 predicted by using the model based on a two-state scheme (U ⇄ N). The physical parameters estimated by the quantitative modeling are listed in Table S6. Throughout the modeling described below, the Gibbs free energy and Rg of U, TS, and N and the pKa values in U were constrained (boldface in Table S6). We used the manual search procedure because the results for the equilibrium and kinetics were taken from the literature. 46-48 It should be noted that CI2 is not fully unfolded even under the acidic (pH ~1) condition. 46 The pH dependence of the Gibbs free energy difference between N and U and the apparent rate were fully reproduced by the model (eqs 2-7) even with each pKa in N constrained, which indicates that the model well approximates the protonation/deprotonation mechanisms. Very similar results were obtained when the pKa values for U, TS, and N were averaged to the corresponding averaged values (pKa,ave) with the pKa,ave for U constrained and the pKa,ave for TS and N varied. This indicates that the averaged pKa values for N and TS are appropriately estimated even without detailed knowledge of the pKa value at each protonation site. The experiments were performed in the presence of salt (50 mM and 200 mM NaCl). 46 Thus, the binding/penetration of ions may affect folding, although it is likely that the pH-induced folding is predominantly controlled by the protonation/deprotonation rather than by the binding/penetration of ions, especially for CI2, which is much smaller than h-apoMb. Thus, the change in the Debye lengths fully accounts for the effects of salt. Therefore, we also modeled the equilibrium and kinetic behavior using the modified model (eqs 3-8), again, although a more appropriate model is required to investigate the effects of the binding/penetration of the ions in more detail. The quality of the fit was slightly better than that using the original model, especially for the equilibrium unfolding. 46 However, the fitting generally improved when a parameter (reff) was added. Furthermore, reff ~ 0.9 × rap indicates that the effects of the binding/penetration of the ions are, if any, very small and that salt effects are fully accounted for by the Debye length for CI2 folding. In fact, previous studies indicated that TS was fairly compact with a loosely formed hydrophobic core, 49


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which may prevent ions from easily penetrating into the interior of the protein molecule. In addition, it was also indicated that electrostatic interactions were less developed in TS than the overall structural formation. 48 Because the results were similar regardless of whether the pKa values in N were constrained or varied, and whether the original or the modified model was used, our further analyses utilized the physical parameters obtained by the original model with the pKa in N constrained, because this model is the simplest and the parameters are the most unambiguous.

To investigate the energetics of CI2 folding, the stability relative to U was calculated for TS and N in terms of Gibbs free energy at pH 2.0 and 6.3 in the presence of 50 mM and 200 mM NaCl along with the contribution of the specific binding and the Coulombic interaction terms to protein stability at pH 2.0 relative to the intrinsic stability at pH 6.3, which was determined by calculating the Gibbs free energy of either of the two terms (Figure S11E and F). The procedure for the calculation was essentially identical to that for h-apoMb. The contributions of the specific binding and Coulombic interaction terms to the destabilization were comparable to each other in N, whereas the Coulombic interaction term played a major role in the destabilization of TS. The salt dependence of pKAP 46 indicates the contribution of the Coulombic interaction term in N as described in the case of h-apoMb, whereas the abnormal pKa values of the carboxyl groups in N (Table S6) indicate the contribution of the specific binding term. As described above, TS was indicated to be compact with a loose hydrophobic core, with favorable electrostatic interactions developed to lesser extent, 48 which suggests that large fraction of the protons is nonspecifically bound on the compact protein molecule, and thus the source of destabilization is the Coulombic repulsion. This is consistent with the dominant contribution of the Coulombic interaction term and virtual absence of the contribution of the specific binding term. In support of this conclusion, the pH dependence of the charge state in N, TS, and U (Figure S11G and H), which also reproduced that reported in a previous study, 46 showed that the charge states of TS and U were similar to each other (with < 1 unit charge difference). There was no qualitative difference in the contributions of the two terms between 50 mM and 200 mM



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NaCl (Fgiure S11E and F). These results again indicate that both the specific binding (the MWC model) and Coulombic interaction (the LL model) terms are required to represent pH-induced folding.

To establish the applicability of the model, it is critical to determine whether it can predict the effects of amino acid substitutions on folding. Among CI2 variants, the R48H variant folds faster than the wild type only at pH > ~6, with the rate increased at higher pH (four-fold faster at pH ~8). 48 A similar variant, the R48F variant, folds >10 fold faster than the wild type, even at pH 6.3 51. The mechanism underlying the accelerated folding kinetics was interpreted as follows: the replacement of Arg48 by Phe eliminated unfavorable charge interactions with neighboring Arg46 and increased favorable interactions with nearby hydrophobic residues. 51 Accordingly, the accelerated folding rate of the R48H variant at high pH is attributed to the neutralization of the histidine residue due to deprotonation at pH > pKa. We modeled the folding rate of the R48H variant as a function of pH by replacing Arg by His, where in practice the net charge was decreased by one unit charge (Arg was assumed to be always protonated in this study), and incorporating the pKa of the histidine residue (pKa values for U, TS, and N were 7.0, 6.4, and 6.4, respectively). A remarkable point of the analysis was that we used the Gibbs free energies of wild type CI2, and not the R48H variant, at pH 2.0 as the reference value. Wild-type CI2 and the R48H variant are considered to be almost identical in terms of the protonation state at pH 2.0 because the side chain of residue 48 is fully protonated at this pH regardless of whether the residue is Arg or His. Figure S11I shows that the folding rate increased at pH values above ~6, which reproduced the results obtained experimentally. 48 This indicates that our model can predict the effects of amino acid substitutions on the pH dependence of folding if the substitutions are represented by the change in protonation sites and if there is an appropriate reference state where the change caused by the replacements is cancelled. The successful application of the model to multiple proteins in the present study indicates that it is sufficiently robust to be applied to a wide range of systems.


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Further implications for the application of the model to systems other than protein folding. To further apply the model to other systems, it is worth considering the underlying assumptions used to analyze pH-induced folding, which were i) that specific binding energy of protons and/or Coulombic repulsion energy between charged residues varied according to the relevant states; and ii) that these terms were represented by physical parameters such as pKa and rap. We found that the above prerequisites held for a much wider variety of association/dissociation processes than protonation/deprotonation when the term “proton” was replaced by “ligands”, “partner proteins”, or “target nucleic acids”, although we originally aimed to construct a statistical mechanical version of the combined MWC and LL models. The model can be applied when the potential energy terms of association are appropriately represented by physical parameters that vary according to the reaction. Furthermore, other physical quantities such as total charge could be obtained as expected values along with the fluctuations as described above if they are represented by the statistical mechanics, given that the grand partition function is known. Figure S9 shows the pH dependence of the expected values and fluctuations of the net charge of h-apoMb in U, M, and N as examples. Therefore, the model is applicable to a wide variety of association/dissociation processes and can potentially provide various physical quantities. It should be noted that some parameters, such as pKa and rap in U in this study, must be constrained as reference values to uniquely determine the physical parameters.

A particularly interesting topic is the binding and folding mechanisms of intrinsically disordered proteins (IDPs), which are important for cellular functions.52,53 An IDP is disordered in isolation even under physiological conditions, and gains an ordered structure upon binding to partner proteins as part of a complex. In this study, partner proteins corresponded to protons. Upon coupled folding and binding, the IDP’s dimension and dissociation constant are likely to be reduced, which would satisfy the prerequisites of the model. In fact, an intrinsically disordered variant of staphylococcal nuclease folds to a native-like state upon binding to the substrate analog prAp



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(although prAp is not a protein but rather an oligonucleotide), with a corresponding decrease in the dissociation constant.54 The IDP’s affinity for the partner protein and its dimension can be estimated for each relevant species, including the corresponding TS, by measuring the kinetics of coupled folding and binding as a function of partner protein concentration (rather than pH as in this study).55,56 More practically, the effects of intrinsic charges of proteins could be estimated as Coulombic interaction terms without measuring salt dependence of the kinetics parameters. In addition, detailed investigations of the folding/binding kinetics would provide insight into the contributions of the induced-folding and conformational selection models in the coupled folding and binding of IDP.

CONCLUSIONS We constructed a statistical mechanical model of the protonation/deprotonation of proteins along with the combined manual search and least-squares fitting procedure to investigate pH-induced folding/unfolding irrespective of the structural characteristics of the relevant molecular species. We applied the model to the analysis of h-apoMb folding over a wide pH range (2.2 to 6.7) with a time window of ~40 µs to ~100 s. A quantitative analysis of results obtained using the model and assuming a five-state sequential scheme revealed the following. First, both energetic terms corresponding to the MWC and LL models were required to represent pH-induced folding/unfolding reactions. Coulombic interactions were responsible for some of the interconversions between partially folded species, whereas specific proton binding played a direct role only in the folding/unfolding of native/native-like states. Second, kinetic folding and unfolding intermediates shared a folding mechanism with the pH 4 intermediate, and hence, these three intermediates constitute a single molecular species. Third, native-like properties were successively recruited upon formation of not only intermediates, as revealed for many proteins including apoMb, but also transition states to achieve a specific native structure, albeit presumably with transient formation of a small amount of non-native local structure. We also suggest that a variety of


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association/dissociation-driven events including protonation/deprotonation as well as folding kinetics and associated fluctuations could be analyzed using the statistical mechanical model.

SUPPORTING INFORMATION Supplementary results: Derivation of the statistical mechanical model and its relationship to the Monod-Wyman-Changeux (MWC) and Linderstrøm-Lang smeared charge (LL) models (SI Results S1; Figures S1, S5, and S6; and Tables S1, S3 and S4); protein concentration dependence of kinetic parameters (SI Results S2 and Figure S1); physical parameter optimization and error estimation (SI Results S3; Figures S2 and S3; and Table S2); pH-induced unfolding equilibrium reproduced by kinetic parameters (SI Results S4 and Figure S4); Modification to the model for investigating salt-induced folding and the results (SI Results S1 and Figure S7); predicted charge distribution of h-apoMb as a function of pH (Figure S8); predicted expected values and fluctuations of h-apoMb as a function of pH (Figure S9); correlation between the pKAP values at 200 mM NaCl obtained experimentally (pKAP,Exp) and those predicted by the model based on the values at 50 mM NaCl (pKAP,Calc) for chymotrypsin inhibitor 2 (SI Results S1 and Figure S10); pH dependence of the stability and folding of CI2 at 25°C predicted by the model including the prediction of mutation effects (Figure S11, Tables S5, and S6).

ACKNOWLEDGMENTS We thank Drs. Heinrich Roder, Ming Xu, Marc Jamin, Kunihiro Kuwajima, Satoshi Takahashi, Takahisa Yamato and Takanori Uzawa for the helpful discussions. We thank the Center for Gene Research, Nagoya University for performing circular dichroism measurements and the Technical Center of Nagoya University for constructing continuous-flow mixers. This study was supported by Grants-in-Aid for Scientific Research (C) from the Japan Society for the Promotion of Science (JSPS; project numbers 24570181 and 20570153), Toyoaki Scholarship Foundation, and Astellas



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Foundation for Research on Metabolic Disorders to KM. TM was supported by the Nagoya University Program for Leading Graduate Schools Integrative Graduate Education and Research Program in Green Natural Sciences, the Nagoya University International Academic Exchange Scholarship for Overseas Study Program 2011, and in part by the JSPS Institutional Program for Young Researcher Overseas Visits.

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TABLE OF CONTENTS GRAPHIC


Statistical Mechanical Model for pH-Induced Protein Folding

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