Prepore Stability Controls Productive Folding of the BAM-independent Multimeric Outer Membrane Secretin PulD

Abstract

Members of a group of multimeric secretion pores that assemble independently of any known membrane-embedded insertase in Gram-negative bacteria fold into a prepore before membrane-insertion occurs. The mechanisms and the energetics that drive the folding of these proteins are poorly understood. Here, equilibrium unfolding and hydrogen/deuterium exchange monitored by mass spectrometry indicated that a loss of 4–5 kJ/mol/protomer in the N₃ domain that is peripheral to the membrane-spanning C domain in the dodecameric secretin PulD, the founding member of this class, prevents pore formation by destabilizing the prepore into a poorly structured dodecamer as visualized by electron microscopy. Formation of native PulD-multimers by mixing protomers that differ in N₃ domain stability, suggested that the N₃ domain forms a thermodynamic seal onto the prepore. This highlights the role of modest free energy changes in the folding of pre-integration forms of a hyperstable outer membrane complex and reveals a key driving force for assembly independently of the β-barrel assembly machinery.

Introduction

Outer membrane proteins (OMPs)⁴ allow Gram-negative bacteria to exchange nutrients and macromolecules with their environment. Classically, periplasmic broad-specificity chaperones deliver OMPs to the β-barrel assembly machinery (BAM) that catalyzes OMP OM insertion (1). In contrast, a group of large multimeric OMPs in which all protomers contribute part of their sequence to a single, shared transmembrane pore rely on the lipoprotein outer membrane localization (Lol) system for membrane targeting and on lipid-assisted self-insertion (2). Many OMPs that use this alternative assembly route are secretion portals for virulence factors or for the surface presentation of pili, and thus are potential targets for chemicals that abolish bacterial virulence.

The first OMP shown to assemble independently of BAM was the secretin PulD (3). PulD dodecamerizes to form the OM pore of the Klebsiella oxytoca type II secretion system for the secretion of pullulanase (PulA) (4). Mature full-length PulD (PulD^fl), comprising amino acids 28 to 660, forms a modular structure (Fig. 1A) with periplasmic flexible arms (N₀ to N₂) (5), a core structure that is referred to as an inverted cup-and-saucer (N₃ in the periplasm and the partly membrane-embedded C (Fig. 1B)) (6–8), and a domain S for binding to the fatty acylated pilotin PulS (9, 10), which delivers PulD to the OM (11) via the Lol system (12). The PulD core structure is hyperstable and resists denaturation by SDS, urea, and trypsin. The acquisition of these characteristics over time showed that a PulD variant lacking the flexible arms, PulD^{28–42/259–660} (where the superscript denotes the numbers of the included amino acids (Fig. 1A)), forms at least one multimeric intermediate (a prepore) prior to membrane insertion (13). The prepore was isolated by a Thr⁴⁷⁰ to Ile substitution in the C domain of both PulD^{28–42/259–660} and PulD^fl (14). Prepores are only SDS resistant and have a well structured cup but a collapsed outer chamber (14). Genetic dissection demonstrated that PulD cannot multimerize without N₃ (15) and PulD^{28–42/259–660}I322S, a PulD^{28–42/259–660} variant carrying an I322S substitution on the N₃ surface (Fig. 1B), is multimerization defective (16). These results suggest an important role for N₃ as a regulator for membrane insertion. To test this, the role of N₃ in PulD folding was investigated here by domain exchange of N₃ with the homologous N₁ and N₂ domains, by the effect of the I322S substitution on the structural, energetic, and dynamic properties of N₃ and by the formation of mixed multimers between unaltered PulD protomers and PulD protomers containing modifications that affect at least one folding step. The results provide the first insights into the folding energy landscape of an OMP of such large dimensions.

FIGURE 1.

PulD structure. A, domain structure of PulD^fl, amino acids delineating the subdomains are numbered and substitutions of interest are highlighted; B, structure of PulD^{28–42/259–598} obtained by cryo-EM reconstruction (left, MSD 2628 (8)) and the N₃ structure (modeled on an extramembranous domain of HOFQ (PDB 2Y3M (41)), highlighting the substitution I322S (purple). A model of the N₃-ring with 12-fold symmetry created using SymmDock (42) was fitted into the PulD^{28–42/259–598} structure using Chimera. OC, outer chamber (or “inverted saucer”); PV, periplasmic vestibule (or “inverted cup” open toward the periplasmic space; the rim of the cup is defined by N₃); C, averaged images of major classes obtained by TEM of the negatively stained PulD^{28–42/259–660} variants as indicated. The boxes are 25 nm wide.

Results

Evaluation of the Effect of the I322S Substitution on PulD^{28–42/259–660} Folding

Although PulD^{28–42/259–660}I322S appears multimerization defective upon in vitro synthesis when analyzed by SDS-PAGE (16), structural analysis by electron microscopy (EM) in Fig. 1C shows that it actually formed cup-and-saucer complexes when solubilized from liposomes with zwittergent 3–14 (ZW 3–14), a detergent typically used for structural studies of the PulD core (6). However, the outer chamber and the periplasmic vestibule were less well defined in PulD^{28–42/259–660}I322S than in unaltered PulD^{28–42/259–660} (Fig. 1C and supplemental Fig. S1), suggesting a folding defect. SDS-PAGE analysis of ZW 3–14-solubilized PulD^{28–42/259–660}I322S showed more abundant SDS-resistant multimers than in the absence of ZW 3–14 (Fig. 2A and Table 1). These observations were unlikely to be artifactual, as it was shown before that PulD^{28–42/259–660} cannot multimerize in the presence of ZW 3–14 alone (13). Although PulD^{28–42/259–660}I322S extracted from the liposomes with dodecylmaltoside (DDM), the detergent used to isolate the PulD^{28–42/259–660}T470I prepore (14), also migrated as monomers, it became in part SDS-resistant upon ZW 3–14 treatment (supplemental Fig. S2A). ZW 3–14 treatment did not induce SDS resistance when PulD^{28–42/259–660}I322S multimers were dissociated first by SDS or urea (supplemental Fig. S2, A and B), nor did it do so on other multimerization defective PulD^{28–42/259–660} variants (supplemental Fig. S2C), suggesting a specific effect of ZW 3–14 on PulD^{28–42/259–660}I322S multimers.

FIGURE 2.

SDS-PAGE analysis of the modifications made in the N domains on PulD multimerization. Immunoblot analysis after SDS-PAGE of in vitro synthesis in the presence of lecithin liposomes of (A) PulD^{28–42/259–660} variants containing the multimerization-defective I322S substitution, the multimerization neutral ΔL₂ deletion, or both; B, PulD^fl variants containing substitutions homologous to I322S in each of the homologous N domains; C, truncated PulD variants containing one of the homologous N domains replacing N₃ adjacent to the C domain and the homologous substitution to I322S where appropriate. Samples were solubilized in SDS-loading buffer (S), in zwittergent 3–14 (ZW) or dissociated in phenol as a total synthesis control (P). Equal amounts of the samples were migrated on the SDS-PAGE. Mo, monomer; Mu, multimer; fl, full-length; tr, 28–42/259–660; WT, unmodified mature PulD^fl; unaltered, PulD^tr without further modification. Numbers indicate the molecular weight of the markers and Mu^tr and Mu^fl between brackets; D, alignment of homologous N domains in PulD. The secondary structure elements correspond to those found in the model of N₃. The N domains were aligned using MAFFT version 7 with the highest gap penalty. The position of Ile³²² is highlighted by the asterisk and the deleted L₂-loop by the rectangle. The sequence similarity of N₃ with N₁ and N₂ is 36 and 42%, respectively, and sequence identity is 16 and 22%, respectively.

TABLE 1.

Quantification of PulD multimer formed upon in vitro synthesis in the presence of lecithin liposomes

The multimer fraction was determined by densitometry of the multimer and monomer bands on immunoblots after SDS-PAGE analysis as shown in Fig. 2. Errors represent standard deviations over 3 independent measurements.

		Without ZW 3–14	With ZW 3–14
PulD28-42/259–660	Unaltered	0.81 ± 0.23	0.79 ± 0.22
	I322S	0.12 ± 0.12	0.65 ± 0.28
	ΔL2	0.93 ± 0.10	0.94 ± 0.06
	ΔL2I322S	0.58 ± 0.28	0.70 ± 0.22
PulDfl	WT	0.83 ± 0.06	0.89 ± 0.05
	I322S	0.14 ± 0.10	0.48 ± 0.08
	L241S	0.84 ± 0.10	0.90 ± 0.08
	L168S	0.77 ± 0.15	0.82 ± 0.10
PulD28–42/188–260/342–660		0.41 ± 0.14	0.34 ± 0.15
PulD28–42/188–260/342–660	L241S	NSa	NS
PulD28–42/124–190/342–660		NS	NS

^a NS, not significant.

The Effect of the I322S Substitution Is Specific for the N-subdomain Adjacent to the C Domain

Introducing the I322S substitution into PulD^fl resulted in a similar effect as observed for PulD^{28–42/259–660}I322S: PulD^flI322S predominantly migrated as a monomer, but the amount of multimers increased when it was solubilized from the liposomes with ZW 3–14 prior to SDS-PAGE (Fig. 2B and Table 1). Although some SDS-resistant PulD^flI322S multimer was formed in vitro and in vivo, PulD^flI322S did not support efficient PulA secretion in the presence of all the Pul components in vivo (Fig. 3A). To validate that the secretion defect was not because of inefficient OM targeting of PulD^flI322S, the phage shock protein (Psp) response was measured by the amount of PspA produced upon PulD^flI322S production in the presence or absence of PulS. Typically, PspA levels are high when PulD^fl is suggested to insert into the inner membrane in the absence of PulS, whereas they are low when PulS efficiently delivers PulD^fl to the OM (11). In the presence of PulS, PulD^flI322S elicited a smaller Psp response than PulD^flWT, consistent with the smaller amount of PulD^flI322S multimers produced when compared with the amount of PulD^flWT multimers (Fig. 3B). Importantly, the amount of PspA produced in the presence of PulD^flWT or PulD^flI322S (but not PulS) was more than halved when either of the PulD^fl variants was produced in the presence of PulS, suggesting that both PulD^fl variants were targeted to the OM with equal efficiency (Fig. 3, A and B). Hence, the I322S substitution created a folding defect in vitro and in vivo.

FIGURE 3.

In vivo analysis of PulD^fl function and outer membrane targeting. A, PulA secretion by E. coli strain Pap105 producing a PulD^fl variant and all other type II secretion system components and PspA production in the presence PulS (as indicated). Secretion was normalized to the amount of PulA secreted by PulD^flWT, which typically reaches 70–75% (4). PspA levels were determined by densitometry of the bands shown in B. Errors represent standard deviations over 3 independent measurements; B, comparison of the small amount of PulD^fl multimers produced in the absence of PulS with the large amount produced when PulD^fl is efficiently targeted to the OM in the presence of PulS and the levels of PspA produced in each case as a measure for the phage shock response by PulD^fl assembly into the inner membrane (11). OmpF was used as a loading control. WT, PulD^fl; I322S, PulD^flI322S; ΔL₂, PulD^flΔL₂; ΔL₂I322S, PulD^flΔL₂I322S; PulDΔS, PulD with cleaved S-domain, which typically occurs in the absence of PulS; Mu, multimer; Mo, monomer. Numbers indicate the molecular weight of the markers and of Mu^fl between brackets.

Of the three domains that precede N₃ in PulD^fl (Fig. 1A), N₁ and N₂ are structurally homologous to N₃ (Fig. 2D, (17)), but neither are required for in vitro folding and in vivo assembly of PulD (18). Consistent with this, creating the homologous substitutions to I322S in N₁ and N₂ by the substitutions L168S and L241S, respectively, resulted into SDS-resistant multimers without ZW3–14 treatment (Fig. 2B and Table 1). Next, N₃ was replaced by N₂ or N₁, in PulD^{28–42/188–260/342–660} and PulD^{28–42/124–190/342–660}, respectively (Fig. 1A), to evaluate the effect of introducing L241S and L168S into these domains while positioned adjacent to C. Only PulD^{28–42/188–260/342–660} formed SDS-resistant multimers, albeit much less than PulD^{28–42/259–660} (Fig. 2C and Table 1). Introducing L241S into PulD^{28–42/188–260/342–660} yielded monomers upon SDS-PAGE, even with ZW 3–14 treatment (Fig. 2C). Thus, it appears that the substitution typified by I322S only induces misfolding when it is present in the N domain that directly precedes C.

Although homologous, replacing N₃ with N₂ and N₁ did not result in efficient PulD folding. A striking difference between the three domains is the length of the L₂-loop, which is shorter in N₂ than in N₃ and is only a few amino acids in N₁ (Fig. 2D). Deletion of L₂ in N₃ still allowed PulD^{28–42/259–660}ΔL₂ to form SDS-resistant multimers, excluding a role for L₂ in folding (Fig. 2A and Table 1). Surprisingly, introducing I322S in PulD^{28–42/259–660}ΔL₂ yielded SDS-resistant PulD^{28–42/259–660}ΔL₂I322S multimers (Fig. 2A and Table 1). PulD^{28–42/259–660}ΔL₂ was indistinguishable from PulD^{28–42/259–660}when examined by EM (Fig. 1C). PulD^{28–42/259–660}ΔL₂I322S exhibited shortened periplasmic vestibules near the rim of the cup, suggesting that the N₃ domains occupied less defined positions or were partially unstructured (Fig. 1C), a feature that it shared with PulD^{28–42/259–660}I322S (Fig. 1C). Both PulD^flΔL₂ and PulD^flΔL₂I322S were targeted to the OM in vivo and secreted PulA (Fig. 3). The relatively high amount of PspA produced in the presence of PulD^flΔL₂ and PulS is likely due to the increased susceptibility of the variant to proteolysis of the S domain (Fig. 3).

The I322S Substitution Destabilizes N₃

To determine the effect of the I322S substitution on N₃ structure and stability, N₃ domains were purified that carried none, one, or both of the I322S and ΔL₂ modifications. Circular dichroism (CD) spectroscopy indicated that the secondary structure of N₃ was unaffected by any of the modifications made (Fig. 4A). N₃ domains resisted complete unfolding upon thermal denaturation (Fig. 4, A and B). N₃I322S was the least thermostable with thermostability increasing in the order of N₃I322S < N₃ΔL₂I322S < N₃WT < N₃ΔL₂ (Fig. 4B).

FIGURE 4.

Stability of N₃ domains. A, CD spectra of N₃ (full black), N₃I322S (full purple), and N₃ΔL₂I322S (full green), and of thermally (dashed purple) and chemically (dotted purple) unfolded N₃I322S; B, thermal unfolding and C, chemical unfolding (open symbols) and refolding (filled symbols) of N₃WT (black squares), N₃I322S (purple circles), N₃ΔL₂ (cyan triangles), and N₃ΔL₂I322S (green triangles). Lines represent fits to a two-state equilibrium equation; D, comparison of local free energy changes (ΔΔG⁰_local) calculated from the three lowest H/DX rates with global free energy changes (ΔΔG⁰_global) obtained from chemical refolding of the entire domains (the contribution of 2 cis-Pro was subtracted, according to Ref. 43). The full line represents a linear fit to the data and the dotted line shows a linear correlation with a slope of 1. In A–C, MRE, mean residue ellipticity.

Unfolding in urea was reversible and approximated by global fitting to a two-state model with a shared m-value (2.33 ± 0.13 kJ/mol/M), resulting in a free energy of unfolding (ΔG⁰) of −18.82 ± 1.04 kJ/mol for N₃WT (Fig. 4C). The ΔΔG⁰ between N₃WT and N₃I322S was calculated to be 8.33 ± 1.19 kJ/mol, whereas deleting L₂ stabilized N₃ΔL₂I322S compared with N₃I322S by −4.69 ± 0.99 kJ/mol (Fig. 4C). N₃ΔL₂ was more stable than N₃WT (ΔΔG⁰ = −1.40 ± 1.56 kJ/mol) and unfolding was limited even at 8 m urea (Fig. 4C).

The I322S Substitution Increases N₃ Dynamics

Retention of the structure in the N₃ variants was confirmed by the exchange of backbone amide hydrogens with deuteriums measured by mass spectrometry (MS) (19, 20). Except for the expected loss of 13 fast-exchangeable amino acids in L₂, neither the L₂ deletion nor the I322S substitution affected the global hydrogen/deuterium exchange (H/DX) behavior of intact N₃ at early time points (supplemental Fig. S3A and Table S1). However, the number of slow exchangeable amide hydrogens was reduced by ∼30% in N₃ and in N₃ΔL₂ by the I322S substitution, whereas the number of intermediate exchangeable ones increased (supplemental Table S1), indicating a change in protein dynamics.

To locate which N₃ segments were affected by the I322S substitution, H/DX in N₃ domains was analyzed by MS after on-line pepsin digestion (Fig. 5A, supplemental Fig. S4A). Three N₃I322S segments showed increased deuterium uptake compared with those of N₃WT. The largest average uptake difference (17%) was observed in a segment represented by peptide Glu³³⁴-Leu³⁴⁰, which forms the C terminus of α-helix 2 (Fig. 5B) and is the most proximal structural element to the C domain in the PulD sequence (Figs. 1 and 2D). The two other segments (e.g. peptides Lys²⁷⁰-Leu²⁸² and Asp³⁰⁷-Asp³³²) had an uptake difference of 7% (Fig. 5A) and form a patch on the N₃ structure opposite to the C terminus of α-helix 2 (Fig. 5B). Introducing the I322S substitution in N₃ΔL₂ affected the same region of the N₃ domain, but uptake differences were smaller or disappeared (for peptide Ile³²²-Leu³³³) (Fig. 5, C–E, and supplemental Fig. S4B). Glu³³⁴-Leu³⁴⁰ remained the peptide with the largest uptake difference (9%), but kinetics closely resembled that of N₃WT (Fig. 5E, far right panel).

FIGURE 5.

Effects of the I322S substitution on the dynamics of N₃WT and N₃ΔL₂ probed by H/DX-MS. A, differential deuterium uptake map of N₃ in the presence or absence of the I322S substitution. Bars below the primary sequence correspond to unique peptides colored according to the deuterium uptake difference (in %, see scale bar) between N₃WT and N₃I322S, averaged over the entire H/DX-MS time course; B, structural homology model of N₃ highlighting the regions of increased deuterium uptake in the presence of the I322S substitution (see scale bar for color code); C, differential deuterium uptake map of N₃ΔL₂ in the presence or absence of the I322S substitution. The dashed lines indicate the L₂ deletion. Peptides are colored according to the deuterium uptake difference (in %, see scale bar) between N₃ΔL₂ and N₃ΔL₂I322S, averaged over the entire H/DX-MS time course; D, structural homology model of N₃ highlighting the regions with increased deuterium uptake in the presence of the I322S substitution (see scale bar for color code). The L₂ deletion is marked by the positions of Ser²⁹⁵ and Lys³⁰⁸. E, deuterium uptake curves for selected N₃-peptides in the segments showing increased deuterium uptake in the presence of the I322S substitution. The dotted black lines indicate the full deuterium uptake level measured for each peptide after 24 h incubation time at 60 °C. All H/DX-MS measurements were statistically validated using MEMHDX (supplemental Fig. S3, B and C).

H/DX coupled with NMR or MS can correlate local H/DX rates to the global ΔG⁰ of a protein or the global ΔΔG⁰ between a protein and its variants (21, 22). The peptides in Fig. 5E represent each of the segments that showed differential H/DX behavior upon the introduction of the I322S substitution. Excluding Ile³²²-Leu³³³, which is not directly comparable as it contains the site of the substitution, all peptides fitted best to a double exponential (Fig. 5E and supplemental Table S1). The slowest exchange rates were found in Glu³³⁴-Leu³⁴⁰, which also had the largest uptake difference, suggesting that the C terminus of α-helix 2 plays a role in maintaining N₃ stability. ΔΔG⁰_local were calculated from the three slowest exchangeable hydrogens in the peptides. Hence, for all N₃ domains the slow rates of Glu³³⁴-Leu³⁴⁰ were used, whereas the slow rates from the three peptides were used for N₃I322S as each peptide had one slowly exchanging hydrogen (supplemental Table S1). ΔΔG⁰_local correlated well with ΔΔG⁰_global obtained by equilibrium unfolding with a slope of 0.9 (Fig. 4D).

Folding Defects Are Overcome in Mixed PulD Multimers

How does N₃ energetics affect PulD multimer folding? Because multiple domains might need to interact to fold the PulD native multimer, it was assessed whether PulD^{28–42/259–660}I322S can form mixed multimers with wild-type PulD^fl (PulD^flWT) during co-synthesis. As shown before (13), co-synthesis of PulD^flWT and PulD^{28–42/259–660} produces mixed multimers containing from 0 up to 12 protomers from one form and, reciprocally, from 12 down to 0 of the other form. The mixed multimers form a regular ladder between 540 (a PulD^{28–42/259–660}dodecamer) and 825 kDa (a PulD^fl dodecamer) upon SDS-PAGE (Fig. 6A). The biochemical determinants of the native PulD multimer can be used to assign structural states to the mixed multimers: an SDS- and urea-resistant mixed multimer forms a membrane-inserted PulD^{28–42/259–660} pore (13), whereas an SDS-resistant but urea-sensitive mixed multimer forms a prepore (as typified by PulD^{28–42/259–660}T470I (14)). Trypsin resistance of the PulD pore is not useful in this case, because it trims all protomers to the same molecular mass. Ladders produced upon co-synthesis of PulD^flWT and PulD^{28–42/259–660} were SDS- and urea-resistant, indicative of native PulD structures in the mixed multimers (Fig. 6A and Table 2). In the presence of PulD^{28–42/259–660}I322S, SDS and urea resistance required six or more PulD^flWT protomers in the multimer (Fig. 6, A and B, and Table 2). Thus, native structures can form in the presence of the I322S substitution and suggest that either neighboring N₃WT domains correct the conformation of N₃I322S or that the N₃-ring is required to reach a critical stability. The distribution of protomers in a multimer is not easily determined (the N₀-N₂ domains in PulD^fl are not visible by EM). Hence, to distinguish between these possibilities, mixed multimers were produced between PulD^{28–42/259–660}I322S and PulD^fl variants that differ in N₃ stability. PulD^{28–42/259–660}I322S formed mixed multimers with all PulD^fl variants (Fig. 6A). Importantly, SDS- and urea-resistant mixed multimers containing PulD^flΔL₂ (which carries the most stable N₃ domain) could incorporate up to four extra PulD^{28–42/259–660}I322S protomers when compared with multimers containing PulD^flWT or PulD^flΔL₂I322S (Fig. 6A and Table 2). Consistent with the poor multimerization of PulD^{28–42/259–660}I322S and PulD^flI322S, SDS- and urea-resistant mixed PulD^{28–42/259–660}I322S/PulD^flI322S-multimers were low in abundance (Fig. 6A and Table 2). Hence, the results mirrored the stability and dynamics of the N₃ domains and support the idea that the stability in the N₃-ring needs to attain a critical value to enable PulD membrane insertion.

FIGURE 6.

Mixed multimer formation of PulD with its variants. SDS and urea resistance of mixed multimers formed in the presence of the same amount of one PulD^fl variant and one PulD^{28–42/259–660} variant. Equal amounts of the samples were migrated on the SDS-PAGE. Mu, multimer; Mo, monomer; fl, full-length; tr, 28–42/259–660; WT, unmodified mature PulD^fl; unaltered, PulD^{28–42/259–660} without further modification. Numbers indicate the molecular weight of the markers and of Mu^tr and Mu^fl between brackets.

TABLE 2.

Quantification of mixed PulD multimers formed upon in vitro synthesis in the presence of lecithin liposomes

The multimer fraction was determined by densitometry of the multimer and monomer bands on immunoblots after SDS-PAGE analysis as shown in Fig. 6. Errors represent standard deviations over 3 independent measurements.

		SDS resistant	Urea resistant
PulDfl	PulD28–42/259–660
WT	unaltered	0.90 ± 0.13	0.96 ± 0.03
WT	I322S	0.85 ± 0.15	0.82 ± 0.14
WT	T470I	0.80 ± 0.29	0.82 ± 0.15
WT	I322S/T470I	0.83 ± 0.24	0.71 ± 0.24
ΔL2	I322S	0.85 ± 0.24	0.86 ± 0.05
ΔL2I322S	I322S	0.87 ± 0.07	0.81 ± 0.06
I322S	I322S	0.32 ± 0.10	0.37 ± 0.10

As the ring formed by the 12 N₃ domains is structured in the PulD prepore (14), it was next investigated whether I322S substitution affected prepore formation. PulD^{28–42/259–660}I322S/T470I did not form SDS-resistant multimers and was not stabilized by ZW 3–14 (supplemental Fig. S2D). Consistent with PulD^{28–42/259–660}T470I forming SDS-resistant multimers (14), PulD^{28–42/259–660}T470I formed almost complete SDS-resistant ladders in mixed multimers with PulD^flWT (Fig. 6B and Table 2). However, whereas PulD^{28–42/259–660}T470I multimers dissociate in urea (14), mixed multimers with PulD^flWT were urea-resistant (Fig. 6B and Table 2). In contrast, like PulD^{28–42/259–660}I322S, PulD^{28–42/259–660}I322S/T470I formed SDS- and urea-resistant mixed multimers with PulD^fl when no more than six protomers contained a substitution (Fig. 6B and Table 2), confirming that the I322S substitution affected the formation of the SDS-resistant PulD-prepore.

Effect of the I322S Substitution on the PulD Folding Mechanism

To fold into the SDS-resistant prepore, the SDS-sensitive PulD^{28–42/259–660} multimer must overcome a transition barrier, defined by the Arrhenius activation energy (E_a). Failure to produce an SDS-resistant prepore in PulD^{28–42/259–660}I322S either results from an insurmountable E_a (which relates to the reaction rate constant k via k = Aexp(−E_a/RT), where A is a constant that depends on the collision frequency) or from destabilization of the SDS-resistant prepore to such an extent that the SDS-sensitive multimer is the more stable structure (supplemental Fig. S5, A–C). Rate constants for SDS-resistant PulD multimerization are measured by the intensities of the multimer bands over time upon SDS-PAGE (13). Although these are easily obtained for PulD^{28–42/259–660} and PulD^{28–42/259–660}T470I (0.14 ± 0.04 and 0.07 ± 0.05 min⁻¹, respectively), the rate constant is not easily determined for PulD^{28–42/259–660}I322S, which remains largely monomeric upon ZW 3–14 treatment in kinetic experiments (Fig. 7, inset, and supplemental Fig. S5, D–F). However, PulD^{28–42/259–660}I322S monomers decayed with a rate constant of 0.34 ± 0.14 min⁻¹ (Fig. 7, inset), providing an estimate for the multimerization rate constant. This places the rate constants for all PulD^{28–42/259–660} variants within 2-fold of each other, indicating that E_a would change by less than 2 kJ/mol/multimer. The effect of the I322S substitution on SDS-resistant prepore stability was also estimated via the ΔΔG⁰ = 8.33 kJ/mol between N₃ and N₃I322S. The sharp reduction in the amount of mixed multimers formed with PulD^flWT when the number of PulD^{28–42/259–660}I322S or PulD^{28–42/259–660}I322S/T470I protomers exceeded six (Fig. 6B), suggested that the maximal destabilization tolerated in the N₃-ring to produce a large amount of native multimers is ∼50 kJ/mol (≈6 × 8.33 kJ/mol). If this would be attributed to an increase in E_a, the mixed multimers would take years to fold (because k_unaltered/k_mixed = exp(−ΔE_a/RT) or k_T470I/k_mixed = exp(−ΔE_a/RT)). This is inconsistent with the abundant SDS and urea-resistant PulD-mixed multimers observed within 6 h. Thus, kinetic and thermodynamic measurements suggested that the I322S substitution must primarily destabilize the SDS-resistant prepore.

FIGURE 7.

Free energy diagram for PulD folding. Monomers (Mo) assemble into an EP before consolidation into an SDS-resistant LP and subsequent membrane insertion (NMu). RC, reaction coordinate. Numbers indicate the ΔΔG⁰ (in kJ/mol; see text for details). Schematic of the detergent-solubilized PulD structures (with the substitutions that arrest folding at a given step as indicated by the flat arrow) that correspond to each of the multimeric membrane-bound forms and the biochemical characteristics they share with it are also shown. Inset, kinetics of the most abundant PulD-population upon the introduction of substitutions (Mu or Mo) compared with PulD^{28–42/259–660} in 54 mm lecithin liposomes after 10 min synthesis. PulD^{28–42/259–660} is in black, PulD^{28–42/259–660}I322S in purple, and PulD^{28–42/259–660}T470I is in orange.

Discussion

Although each PulD protomer only shares parts of the C domain to form the dodecameric transmembrane pore and the gate that closes it (6), PulD will not assemble into such a structure in the absence of N₃ (15). The results presented here reveal why: N₃ provides thermodynamic stability to the obligate prepore that is formed prior to PulD membrane insertion. Substitution of Ile³²² into Ser in N₃ increases the conformational dynamics of structural elements in N₃, in particular in the α-helix that is next to the C domain and in the β-sheet opposite of this helix. Essential interactions between neighboring N₃ domains and/or between N₃ and the C domain might be disrupted in the context of the PulD dodecamer, but mixed multimers suggest that the global stability in the N₃-ring predominates. The ΔΔG⁰ = 4.69 kJ/mol between N₃I322S and N₃ΔL₂I322S indicates that decreasing the ΔG⁰ of the multimer with ∼56 kJ/mol (=12 × 4.69 kJ/mol) prevents the formation of the native PulD structure. The same idea emerges from the ΔΔG⁰ = 8.33 kJ/mol between N₃WT and N₃I322S and the observation that six PulD^flWT protomers are required to form a mixed multimer with PulD^{28–42/259–660}I322S (6 × 8.33 kJ/mol = 50 kJ/mol). This is the equivalent of few van der Waals contacts or a H-bond per monomer. We propose that the N₃-ring forms a seal onto the assembled C domains and must attain a critical stability to facilitate the conformational rearrangements in the C domain that drive pore formation (13), while preventing multimer disruption.

Insights into the folding of OMPs that multimerize into a shared transmembrane pore mainly come from the isolation of intermediates, but the kinetic and thermodynamic relationship with the native structure is mostly unknown (14, 23–25). Previous reports (13, 14, 26) and data obtained here help to define these relationships for PulD by characterizing intermediates that form in the presence of a lipid membrane in detail after detergent solubilization (Fig. 7). Combined, these data indicate that four structures reside in local energy minima along the reaction coordinate: the monomer (Mo), an SDS-sensitive early prepore (EP), the sealed SDS-resistant late prepore (LP), and the pore or native multimer (NMu) (Fig. 7). The largest driving force for PulD pore formation likely comes from membrane insertion of the transmembrane domain, consistent with the remarkable stability of NMu and the ease by which LP is dissociated by moderate urea concentrations (14). CD spectra and EM reconstructions support that PulD traverses the membrane by a β-barrel that is 9 nm in diameter (6, 13, 15). To create such a β-barrel, each PulD protomer should contribute at least 4 β-strands (27). With each β-strand contributing 10 kJ/mol to OMP stability (28), this amounts to an overall folding free energy of at least 480 kJ/mol (Fig. 7). Yet, the balance between EP and LP is attenuated by a modest ΔΔG⁰ of 50 kJ/mol/dodecamer. Furthermore, the low multimerization efficiency of PulD^{28–42/259–660}I322S, as determined by SDS-PAGE, suggests that the ΔΔG⁰ associated with the Mo to EP transition is of the same order of magnitude (=12 × 8.33 kJ/mol; Fig. 7).

Can this reflect the in vivo pathway? PulD belongs to an expanding group of BAM-independent OMPs that rely on lipid-assisted self-assembly once monomers are delivered to the OM via the Lol system, which in the case of PulD is mediated by the lipoprotein PulS (2, 3, 12, 26). This provides a rationale to study the membrane-associated steps of PulD assembly in a controlled in vitro environment in the presence of liposomes and to compare the results from in vitro studies to the in vivo pathway. Whereas PulS targets PulD to the OM in vivo, PulS accelerates PulD multimerization in vitro (26), providing a first driving force for PulD folding. The binding energy between PulD S domains and PulS was determined to be 40–50 kJ/mol (10). With the Mo to EP and EP to LP transitions each stabilizing the PulD multimer 50 kJ/mol (Fig. 7), the formation of these structures does not provide the energy to dissociate PulD/PulS heterodimers. Interestingly, dissociation is not required to form NMu in vivo (26) and PulS co-purifies with PulD from in vivo sources (29). PulS prevents proteolysis of PulD protomers in vivo (30), which could rationalize the necessity for a prolonged PulD/PulS association as PulD protomers join and break free from growing complexes until the correct conformation is found. Alternatively, the PulD/PulS heterodimer is required to retain PulD folding competence during multimerization. Hence, whereas the rough free energy landscape from Mo to EP only provides modest stability, it effectively lowers the entropy associated with PulD multimerization, providing a second driving force for PulD assembly. Lowering entropy ultimately facilitates rapid lipid-assisted membrane insertion, a third driving force in PulD assembly. In this context it is worth noting that a loss of 100 kJ/mol in a PulD^{28–42/259–660}I322S dodecamer relative to PulD^{28–42/259–660} is equivalent to the folding free energy of a small to medium-sized BAM-dependent OMP.

Multimeric OMPs (3, 14, 23–25, 31) and bacterial pore-forming toxins and eukaryotic-pore forming complexes (32) that form prepores typically contain at least one domain in the membrane periphery. Folding pathways are not easily obtained, because of the fast assembly rates and the complex environment that these proteins often require. However, multiple sequential pre-integration structures have been described for the bacterial toxins aerolysin (33) and perfringolysin O (34). A salt bridge between neighboring domains in the membrane periphery critically stabilizes an EP into an LP before NMu formation occurs in perfringolysin O, but removing it increases E_a only by 2 kJ/mol/monomer (34). Because a salt bridge is typically stronger, it is possible that it also acts to seal EP into LP. In conclusion, we propose that the energies associated with EP-to-LP transitions driven by domains in the membrane periphery will emerge as key driving forces for membrane insertion of all constitutive and non-constitutive multimeric complexes that assemble independently of a membrane-embedded insertase.

Experimental Procedures

Strains and Growth Conditions

Cloning and stress response experiments were performed in the Escherichia coli strain K-12 PAP105 (Δ(lac-pro) F′ (lacI^q1 ΔlacZM15 proAB⁺ Tn10)). E. coli strain MC4100 PAP7447 (F′ lacI^q pro⁺ Tn10)], with pulS, pulA, and pulC-O integrated into malPp and with a large deletion in pulD, was used for in vivo secretion (11). Strains were grown at 30 °C in LB medium with 100 μg/ml of ampicillin and/or 25 μg/ml of chloramphenicol (as appropriate). Expression of the pul genes was induced with 0.4% maltose.

Plasmid Construction and Site-directed Mutagenesis

Plasmids and primers used are given in supplemental Table S2. The plasmid encoding for the truncated version of PulD with the N₁ domain instead of the N₃ domain (PulD^{28–42/124–190/342–660}) was created in 3 steps. In step 1 (a) the DNA fragment from the SphI site on the pIVEX2.3 vector up to the DNA sequence encoding for PulD^fl amino acid 42 was amplified from pCHAP3716 with the primers ING318 and ING319; (b) the sequence encoding for the N₁ domain was amplified from pCHAP3731 using the primers ING320 and ING321; (c) the sequence encoding for the C and S domains was amplified from pCHAP3716 using the primers ING322 and ING323, where ING323 overlapped with the sequence of the EcoRI site on the pIVEX2.3 MCS vector. Step 2 involved mixing the purified PCR products at equimolar ratios to generate a DNA fragment that encoded for the entire PulD^{28–42/124–190/342–660} amino acid sequence between the SphI and EcoRI restriction sites using primers ING318 and ING323. In step 3 this fragment and pIVEX2.3 MCS were digested with the restriction enzymes SphI and EcoRI and ligated to yield pCHAP3359. The same procedure was used to generate pCHAP3357 that encoded for PulD^{28–42/188–260/342–660}, in which N₃ was replaced by N₂. The primer pairs ING318/ING324, ING325/ING326, ING327/ING323, and ING318/ING323 were used in PCR in step 1a, step 1b, step 1c, and step 2, respectively. pCHAP3358 that encodes PulD^flI322S was created by digesting A6 with AgeI and EcoRI and the short fragment carrying the mutation encoding for I322S was ligated into the long fragment of pCHAP3731 digested with the same enzymes. Plasmids encoding for PulD^flL168S (pCHAP3355) and PulD^flL241S (pCHAP3356) were obtained by site-directed mutagenesis of pCHAP3731 using the primer pairs L168SF/R and L241SF/R, respectively. The plasmid encoding PulD^{28–42/188–260/342–660}L241S (pCHAP3357) was obtained by site-directed mutagenesis of pCHAP3375 using the primers L241SF/R. To delete the sequence encoding for L₂ (amino acids 205 to 308) in N₃, the purified PCR product of pCHAP3716 or A6 and the phosphorylated primers ING349/ING350 were ligated to yield pCHAP3369 (encoding for PulD^{28–42/259–660}ΔL₂) or pCHAP3374 (encoding for PulD^{28–42/259–660}ΔL₂I322S), respectively. pCHAP3395 and pCHAP3396, encoding PulD^flΔL₂ and PulD^flΔL₂I322S, respectively, were obtained by ligating the short fragments from pCHAP3369 and pCHAP3374 digested with AgeI and EcoRI (encoding for the L₂-deletion and the I322S substitution, respectively) into pCHAP3731 digested with the same enzymes. The plasmid encoding PulD^{28–42/259–660}I322S/T470I (pCHAP3328) was obtained by exchanging the short fragment generated by AgeI and AflII digestion of pCHAP3727 (which contains the sequence for the T470I substitution) with that generated from A6 after digestion with the same enzymes (in which the large fragment contains the sequence with the I322S substitution). The sequences encoding for N₃WT, N₃I322S, N₃ΔL₂, and N₃ΔL₂I322S were obtained by PCR from the pCHAP3716, A6, pCHAP3369, and pCHAP3374 using primers ING347 and ING348. Digestion of the amplicons with NdeI and XmaI enabled ligation in-frame with the His₆ tag encoded on pIVEX2.3 MCS to give pCHAP3362, pCHAP3368, pCHAP3392, and pCHAP3393, respectively. To generate the pCHAP3378 that carries the pulD^flI322S gene with the signal sequence and the His₆ tag, the pulD^flI322S was excised from A6 using AgeI and BlpI and inserted into pCHAP3671 digested with the same enzymes. To insert pulD^flI322S in a moderate copy number vector, pCHAP3378 was digested with EcoRI and HindIII and the resulting fragment carrying pulD^flI322S was inserted into pSU18 digested with the same enzymes to yield pCHAP3389. Finally, as pCHAP3389 encodes for a PulD^flI322S variant with a His₆ tag, pCHAP3389 and pCHAP3635 were digested with EcoRI and HpaI and the fragment carrying pulD^flI322S was ligated into pCHAP3635 to yield pCHAP3394. The same procedure was followed to generate plasmids encoding for PulD^flΔL₂ and PulD^flΔL₂I322S, using pCHAP3369 and pCHAP3374 as parent plasmids to give pCHAP3390 and pCHAP3391, respectively.

Gel Electrophoresis and Immunoblotting

Proteins were separated by electrophoresis on a 4–10% SDS-polyacrylamide gel. Samples were mixed with SDS loading buffer (final concentration of 62.5 mm Tris-HCl, pH 6.8, 12.5% glycerol, 2% SDS). Following SDS-PAGE, the proteins were transferred onto a nitrocellulose sheet by semi-dry blotting. PulD variants, PspA and OmpF were detected using anti-PulD, anti-PspA, and anti-OmpF antibodies, respectively. Primary antibodies were detected by a secondary antibody coupled to horseradish peroxidase (GE Healthcare). Peroxidase activity was measured by the chemiluminescence produced by ECL2 (Pierce) and integrated on a Typhoon imager (GE Healthcare).

PspA Induction and Pullulanase Secretion

PspA induction was performed in the absence or presence of PulS (pCHAP585). Cultures were inoculated from overnight growths at D₆₀₀ = 0.15 and grown for an additional 6 h at 30 °C before cell harvesting. PspA production was analyzed by SDS-PAGE and immunoblotting of cells in SDS-loading buffer to a final concentration of 10 D₆₀₀/ml.

The same growth conditions were used to perform the secretion assay. Cells were grown to a D₆₀₀ = 1.0–1.5 and chilled on ice. The pullulanase assay was performed as described previously (35). The level of secretion is represented as the percentage of the total amount of pullulanase measured in unlysed cells.

In Vitro PulD Synthesis, Zwittergent 3–14 Solubilization, and Mixed Multimer Formation

In vitro PulD synthesis was done with the RTS100 E. coli HY kit (18). The reaction mixture was supplemented with lecithin liposomes at a final concentration of 2 mg/ml and incubated for 6 h at 30 °C. For ZW 3–14 treatment of PulD variants, part of the reaction mixture was mixed with a final concentration of 1.5% ZW 3–14 in 25 mm Tris, pH 7.2, and incubated for 1 h at room temperature. Controls were diluted with detergent-free buffer. To assess the effect of urea or DDM on the multimeric state of PulD^{28–42/259–660}I322S, liposomes, isolated from the in vitro synthesis reaction (16,100 × g for 15 min), were resuspended into 100 mm Tris, pH 7. 5, 500 mm NaCl and diluted 2-fold with 1, 2, 4, 6, or 8 m urea or 2% DDM. After incubation for 1 h at room temperature, a final concentration of 1.5% ZW 3–14 (and 25 mm Tris, pH 7.5, 125 mm NaCl) was added to all samples, except the controls, and incubated overnight. PulD multimers were dissociated by phenol where indicated (11). Formation of mixed multimers was achieved by supplementing the reaction mixture with 10 mg/ml of DNA in a 1:3 ratio of the plasmids carrying the pulD^fl and pulD^{28–42/259–660} genes, respectively, in the presence of lecithin liposomes. Liposomes were sedimented, resuspended, and divided into two parts. Liposomes were diluted in 25 mm Tris, pH 7.2, with or without 4 m urea (final) and incubated 1 h at room temperature. All samples were analyzed by SDS-PAGE and immunoblotting.

Folding Kinetics Followed by SDS Treatment

Folding kinetics of PulD^{28–42/259–660} variants were obtained by mixing aliquots of the synthesis reaction at the time points indicated with SDS-loading buffer to stop folding and incubated on ice before analysis by SDS-PAGE (13). Where kinetics relied solely on integration of the monomer, curves were corrected for linear degradation of the monomer. Curves were fitted to a single exponential equation in Origin 8.0.

Equilibrium Unfolding

N₃ variants were purified as described before (10), diluted to 0.2 mg/ml into 50 mm sodium phosphate, pH 8.0, and the appropriate amount of urea and incubated overnight to reach equilibrium. CD spectra were taken at 25 °C on an AVIV spectrometer between 190 and 260 nm (where possible) at a rate of 20 nm/min and a bandwidth of 1 nm in a 0.1-cm cuvette. Spectra were averaged over three measurements. For equilibrium unfolding experiments, 100 data points taken at 220 nm were averaged. The data were fitted to y = ((y_N + m_N[urea]) + (y_U + m_U[urea])exp(−(ΔG⁰_UN − M_UN[urea])/RT))/(1 + exp(−(ΔG⁰_UN-M_UN[urea])/RT)), with shared y_U, m_U, and M_UN values. Restraining the fit further using shared y_N and m_N values did not significantly alter the fitting parameters.

Transmission Electron Microscopy (TEM) and Image Processing

His-tagged PulD^{28–42/259–660} variants were extracted from liposomes by 3% ZW 3–14 in 50 mm Tris, pH 7.5, and 250 mm NaCl and purified using a cobalt affinity chromatography (Talon, Clontech). Solubilized PulD^{28–42/259–660} variants were bound to the resin for 1 h and washed with 5 column volumes of 50 mm Tris, pH 7.5, 250 mm NaCl, and 0.6% ZW 3–14 before elution in the same buffer supplemented with 5 mm EDTA. Eluted PulD^{28–42/259–660} variants were concentrated and frozen for EM analysis. His-tagged proteins were used as they increased the amount of particles that adsorbed perpendicular on the EM grid.

A 4-μl aliquot was adsorbed onto a glow discharged carbon film-coated copper EM-grid, washed with three droplets of pure water, and subsequently negatively stained with 2% (w/v) uranyl-acetate. The grids were imaged using a Philips CM10 TEM (FEI Company, Eindhoven, the Netherlands) operating at 80 kV. The images were recorded by the 2k × 2k side-mounted Veleta CCD camera (Olympus, Germany) at a magnification of 130,000. The pixel size at the sample level was 3.7 Å.

Image processing was done in the EMAN2 software package (36). The images were contrast transfer function corrected and particle projections were semi-automatically selected. e2refine2d was used to classify the particle projections and yielded reference-free class averages from a population of mixed, unaligned particle projections. The representative class averages with the best signal-to-noise ratio were selected and gathered in a gallery.

Sample Preparation for H/DX-MS

All samples were prepared in triplicate using a fully automated LEAP H/DX Pal robot (LEAP Technologies, Carrboro, NC). Prior to deuterium labeling, N₃ variants (15 μm in 50 mm HEPES buffer, pH 7.0) were equilibrated for 60 min at room temperature and placed at 4 °C in the protein compartment of the robot. H/DX was initiated by diluting 2 μl of each protein (30 pmol) with 48 μl of HEPES buffer made with 99.9% D₂O, pD 7.0, and at 20 °C. At time points between 10 s and 4 h, 30 μl of the labeling reaction was removed, quenched with 60 μl of ice-cold 1.5% formic acid to lower the pH to 2.5, and immediately analyzed by MS. Fully deuterated samples were prepared manually by 24 h incubation at 60 °C (final D₂O content of 96%), quenched as described above and frozen until LC-MS analysis.

H/DX-MS Data Acquisition

Quenched samples (10 pmol) were injected into a refrigerated Waters nanoACQUITY UPLC HDX system maintained at 0 °C (37). For global H/DX-analyses, samples were desalted for 2 min on a C4 trap column (VanGuard BEH C4 1.7 μm, Waters, Milford, MA) at 100 μl/min with 95% mobile phase A (0.15% formic acid, pH 2.5) and 5% mobile phase B (acetonitrile, 0.15% formic acid, pH 2.5). Proteins were eluted from the trap column to the mass spectrometer using a 2-min gradient of 5–90% mobile phase B at 100 μl/min. For local H/DX analyses, samples were passed across a Poroszyme-immobilized pepsin column held at 20 °C (2.1 × 30 mm, Applied Biosystems) and the resulting peptides were trapped for 2 min onto a C18 trap column (VanGuard BEH C18 1.7 μm, Waters) at 100 μl/min mobile phase A and 0 °C. Peptides were eluted from the trap column to an analytical C18 column (CSH C18, 130 Å, 1.7 μm, 1 × 100 mm, Waters) with a 6-min linear gradient of 8 to 40% mobile phase B at 40 μl/min. After each run, the pepsin column was cleaned with 0.8% formic acid, 5% acetonitrile, 1.5 m guanidinium chloride, pH 2.5. Blank injections were performed between each injection to confirm the absence of carryover.

Mass spectra were acquired in resolution and positive ion mode (m/z 50–2,000) on a Synapt G2-Si HDMS mass spectrometer (37) equipped with a standard electrospray ionization source. Mass accuracy was ensured by continuously infusing a Glu-1-Fibrinogen solution (100 fmol/μl in 50% acetonitrile) through the reference probe of the electrospray ionization source. Peptides were identified in non-deuterated samples by a combination of MS^E, data-dependent acquisition, and exact mass measurement. Peptide identifications were made by database searching in ProteinLynx Global server 3.0 (Waters) and each fragmentation spectrum was manually inspected for assignment confirmation.

H/DX-MS Data Processing

For global H/DX analyses, the centroid mass value of each protein was extracted in MassLynx 4.1 (Waters) and the amount of incorporated deuterium was determined from the mass difference between the deuterated and non-deuterated samples.

DynamX software 3.0 (Waters) was used to extract the centroid masses of all peptides selected for local H/DX analyses; only one charge state was considered per peptide. No adjustment was made for back-exchange and the results are reported as relative deuterium exchange levels expressed in either mass unit (Da) or relative fractional exchange. Relative fractional exchange data were obtained by dividing the measured deuterium uptake for each peptide by the maximum number of exchangeable backbone amide hydrogens that could be theoretically replaced into each peptide. This maximum number is the number of amino acid residues in the peptide minus the number of proline residues and the N terminus (38). All local H/DX-MS results were analyzed using MEMHDX (39).

Calculation of Thermodynamic Parameters from H/DX Rates

Under conditions where the native state is preferentially populated, the exchange rate, k_ex, is determined by the product of the internal rate of exchange for the unfolded protein, k_int, and the ratio of the rates for opening and closing of the protein fold, k_op and k_cl, respectively, k_ex = (k_op/k_cl)k_int, if k_cl ≫ k_int (40). If k_int is known or calculated, the free energy associated with opening of the protein fold is calculated from the equilibrium constant for opening, K_op = k_op/k_cl = k_ex/k_int via ΔG⁰_op = −RTlnK_op = −RTln(k_ex/k_int) (40). When two protein variants a and b are compared, k_int is the same except for the sites where modifications were made. Thus, when comparing peptides originating from two protein variants (excluding those containing modifications), the local difference in free energy between variants a and b is calculated via ΔΔG⁰_local = −RTln(k_ex-a/k_ex-b).

Author Contributions

G. H. M. H., A. P. P., and I. G. conceived the study; G. H. M. H., I. G., S. B., and M. C. worked on the investigation; analysis was done by G. H., S. B., I. G., and M. C.; software development by V. H.; resources were supplied by O. F., J. C.-R., A. P.P., and G. H. M. H.; and all authors contributed to writing.