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1 Unité de Repliement et Modélisation des Protéines (CNRS URA 2185) and
2 Unité de Bio-Informatique Structural (CNRS URA 2185), Institut Pasteur, Paris Cedex, France
Reprint requests to: Arnaud Blondel, Unité de Repliement et Modélisation des Protéines, Institut Pasteur, F-75724, Paris Cedex 15, France; e-mail: ablondel{at}pasteur.fr; fax: 33-1-40-61-30-43.
(RECEIVED July 31, 2003; FINAL REVISION September 17, 2003; ACCEPTED September 19, 2003)
Supplemental material: See www.proteinscience.org
3 Present address: Laboratory for Structural Biology, Center for Advanced Research in Biotechnology, University of Maryland Biotechnology Institute, 9600 Gudelsky Drive, Rockville, MD 20850, USA. 
Article and publication are at http://www.proteinscience.org/cgi/doi/10.1110/ps.03309504.
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Abstract |
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 TOP Abstract IntroductionResultsDiscussionMaterials and methodsElectronic supplemental materialReferences |
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Keywords: Association interface; equilibrium dissociation constant; titrimetry; multiple equilibria; data modeling
Abbreviations: R67 DHFR, dihydrofolate reductase from resistance plasmid R67 • (nX)2 and (nX)4, R67 DHFR homodimer or homotetramer bearing point mutation at position n leading to residue X • (nX;mZ)2 and (nX;mZ)4, R67 DHFR homodimer or homotetramer bearing two point mutations at position n and m • (WT)4, wild-type R67 DHFR homotetramer • (nX)(mZ), heterodimers • (nX)2:(mZ)2, heterocomplex of homodimers • [(nX)(mZ)]2, homocomplex of heterodimers • (nX)2 + (mZ)2, heterocomplex of unspecified symmetry, or pair of complementing mutants • NADPH, nicotinamide adenine dinucleotide phosphate • THF, 5,6,7,8-tetrahydrofolate • DHF, 7,8-dihydrofolate • Tris, Tris [hydroxymethyl]aminomethane • MES, 2-[N-Morpholino]ethanesulfonic acid • MTA buffer, 100 mM Tris, 50 mM MES, 50 mM acetic acid • DTT, 1,4-dithio-DL-threitol • TCA, trichloro acetic acid • IEF PAGE, isoelectrofocalization polyacrylamide gel electrophoresis • U-Model, complex model of association yielding a U-shaped iso-fluorescence titration curve • Us-Model, simplified U-model yielding a U-shaped curve • V-Model, simple model of association yielding a V-shaped iso-fluorescence titration curve
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Introduction |
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TOPAbstractIntroductionResultsDiscussionMaterials and methodsElectronic supplemental materialReferences |
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The R67 plasmid encoded DHFR is a homotetrameric protein of 33.7 kD. Each protomer consists of 78 amino acid residues (Smith et al. 1979). The protein is active as a tetramer but has no or very little activity in the dimeric form (Nichols et al. 1993). It is not sensitive to trimethoprim in contrast with the chromosomal DHFR, and thus, confers antibiotic resistance to bacteria (Huovinen et al. 1986). The X-ray crystal structures of the dimeric and tetrameric forms were solved (Matthews et al. 1986; Narayana et al. 1995), and revealed that the tetramer is a doubly isologous dimer of dimers. Equilibrium denaturation experiments showed that the dimeric form is very stable. Consequently monomeric species are rare (
G30°C 
13.9 kcal/mole or KD 100 pM; Reece et al. 1991). The histidines 62 of one dimer form hydrogen bonds with the serines 59 of the other dimer, forming four such bonds per tetramer. The protonation of the histidines at low pH brakes these bonds and causes the pH dependence of the equilibrium between dimers and tetramers (Nichols et al. 1993).
The interactions involving His 62 and Ser 59 were investigated by site-directed mutagenesis. The construction, production, and characterization of proteins bearing single mutations were described previously (Dam et al. 2000; S59[A/C], H62[A/C/D/E/F/G/I/K/L/M/N/Q]). Most of those proteins were unable to confer trimethoprim resistance to bacteria plated on Petri dishes. They had no or very low enzymatic activity in vitro, and were all dimeric, as shown by sedimentation/diffusion equilibrium. However, mixing those dimeric proteins together restored some activity. The complementing pairs were identified and some of them had a strong activity: (59A)2 + (62L)2, (59A)2 + (62C)2, (59A)2 + (62F)2, (59A)2 + (62M)2, and (59A)2 + (62N)2. The heterotetramer (59A)2:(62L)2 was studied in detail in a previous work, and its dissociation constant as a function of pH was determined with a method especially developed for that purpose: the iso-fluorescence titration (Dam et al. 2000). The analysis of the results was performed according to a standard model that was based on the assumption that dimers did not dissociate into monomers. That model, thereafter called the "V-model" (V for the shape of the curve observed in the iso-fluorescence titration), was fully satisfactory, and provided reproducible results (deviation on KD generally below 20%).
In the present work, four other heterocomplexes yielding significant complementation, were characterized in the unprotonated forms: (59A)2 + (62[C/F/M/N])2 at pH 8.0. In contrast to what was observed previously for (59A)2:(62L)2, the iso-fluorescence titration data of pairs (59A)2 + (62[F/M/N])2 slightly, but consistently, deviated from the predictions of the V-model. To explain the data for these maverick pairs, the formation of monomers and then heterodimers was proposed and investigated. A more comprehensive model of association taking other reactions into consideration was presented (U-model). Limits for the equilibrium constants of these other reactions were determined experimentally for each mutated protein. To determine a confidence interval for KD (X2 + Z2 
X2:Z2), the U-model was fit to the iso-fluorescence titration data with the constants of the other reactions preset to combinations of values systematically covering values within their respective experimental limits. A simplification of the U-model, the Us-model, was derived by assuming a negligible concentration of monomer and equipartition of protomers among heterodimers and heterotetramers of relevant symmetries. It was found that the data for each pair of mutants could be fit with either the V- or the Us-model, and that the prediction thus made was in agreement with that of the U-model. The applicability of V- or Us-model to each pair of complementing mutants was shown to depend on the kinetics of protomer redistribution among dimers. Thus, it appeared necessary to determine the reaction mechanisms for each complex prior to a rigorous analysis of the determinants of the association.
Examples from the literature show that redistribution of protomer can be important for biological functions and cellular response to external signals: pathological variants of hemoglobin or transthyretin were associated with a modified rate of protomer redistribution (McDonald et al. 1987; Schneider et al. 2001); the arrangement of subunits might control the final cell location of human nucleotide diphosphate kinase (e.g., Gilles et al. 1991), or modify the function of the eukaryotic proteasome (e.g., DeMartino and Slaughter 1999). The methods developed to solve the complex U-model proved to be rather flexible and efficient, and could be useful in the study of a vast variety of such systems.
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Results |
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TOPAbstractIntroductionResultsDiscussionMaterials and methodsElectronic supplemental materialReferences |
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and in the Supplemental Material (Fig. S1). The data were first analyzed according to the V-model, which assumes an association between stable dimers X2 and Z2 into a tetramer X2:Z2 (Materials and Methods). The fits with this simple model was apparently satisfactory. Yet, a slight but consistent deviation between the experimental data and the theoretical curve was observed for the pairs, (59A)2 + (62F)2, (59A)2 + (62M)2, and (59A)2 + (62N)2 (Figs. 1B,S1). The systematic bias in the shape of the data distribution suggested that the model was not fully adapted for these pairs of mutants and had to be revised. A new model was therefore sought.
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X2:Z2 should be the only significant reactions (Supplemental Material, Fig. S2), and the V-model should apply. Because the V-model could not account for the data, the assumption does not hold, and heterodimer formation had to be considered despite the very low dissociation constant of the dimers inferred from Reece et al. (1991). Confirming that hypothesis, heterodimers could be detected by IEF PAGE: The mutants could be separated by their isoelectric point and a distinct band appeared between that of the two initial proteins when they were mixed prior to electrophoresis (Supplemental Material, Fig. S3). This band could not be due to tetramers because at pH = pI (between 4 and 5) they cannot form (KD > 300 µM and 130 mM, respectively; see Dam et al. 2000). Those data also showed that dimers were sufficiently stable in that pH range to overcome the electrophoretic separating force.
Evidence of heterodimers were also obtained at neutral pH in solution, that is, under the conditions of the association experiments. Indeed, the double mutant (59A;62L)4, which can normally form active tetramers (Dam et al. 2000), was inhibited by dimeric inactive mutants that appeared to sequester the active mutant protomers in inactive heterodimers (Supplemental Material, Fig. S4A). As expected, the higher the concentration of inactive mutant, the stronger the inhibition (Supplemental Material, Fig. S4B).
Elaboration of a relevant model
Two steps can be identified in the formation of the wild-type R67 DHFR homotetramer. First, monomers associate tightly into dimers through one protomer interface (dimerization interface, inset of Fig. 2A) with a KD of about 100 pM (Reece et al. 1991). Second, dimers associate into tetramers through two protomer interfaces (tetramerization interface, inset of Fig. 2A) with a KD of about 10–50 nM (Nichols et al. 1993).
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All the mutations studied here were introduced in the tetramerization interface and no homotetramer (X4 and Z4) could be detected by analytical centrifugation (Dam et al. 2000). Thus, tetramer with identical protomer interacting through the tetramerization interface should be in negligible amount and the model could be simplified accordingly. The concentrations of X3Z and XZ3 should also be small, but to a lesser extent (KX3Z > 20 µM; Supplemental Material Fig. S5).
The reactions of the simplified model are shown in Figure 2A
. The mathematical solution is derived in Materials and Methods (equations 7–14, and comments), and was called the U-model (for the shape of the function, see Fig. 1B). This model bears seven equilibrium constants (KX2, KZ2, KXZ (
), KX2Z2 (=KD), K(XZ)2 (=4 • ß • KD), KX3Z, and KXZ3; 
and ß are constants expressing the relative propensity to form different symmetry related oligomers) and eight fluorescence signals (FX2, FZ2, F, FX, FZ, FXZ, FX3Z, and FXZ3).
The number of parameters was too large to fit them simultaneously. Therefore, a confidence interval was determined for each of the parameters, except for KX2Z2(= KD ) and F, which were fit, and ß, which could be treated independently. Indeed, ß only appears in the equation giving the total concentration of tetramers, T = [X2:Z2] + [(XZ)2] = [X2] • [Z2] • (1 + 2/ß)/KD, and the difference of fluorescence between the two tetramers is hidden by the definition of F (equations 11 and 14). Kinetic data presented below showed that ß was between 1/3 and infinity. was between 0 (no heterodimers) and 1 (equipartition of the protomers); KX3Z and KXZ3 were above 20 µM (Supplemental Material); FX2 and FZ2 were measured; FX and FZ, were between 0 and (FX2 + FZ2)/4 (from Reece et al. 1991), FX3Z and FXZ3 were bracketed generously between 0 and 2 • F; and finally, FXZ, which should be zero, was set within ±10% of the average fluorescence of the homodimers (see Discussion). An upper limit for KX2 and KZ2 was determined experimentally (see below).
The V-model has two parameters (F, KD), and is a limit case of the U-model. A model bearing the same parameters but accounting for formation of heterodimers was derived and called the Us-model (Fig. 2B). It is based on the same assumptions as that of the V-model: KX2, KZ2, KXZ, 
and 
are zero and ß is 1. In the Us-model, is 1, corresponding to equipartition of protomers among the dimers, instead of 0, no heterodimers, in the V-model. The three models were solved differently, as described in Materials and Methods.
Monomer–dimer equilibrium by sedimentation/diffusion centrifugation
The dissociation constant for the monomer–dimer equilibrium was studied by sedimentation/diffusion equilibrium in an analytical ultracentrifuge. Recording the absorbance at 230 nm and using an initial concentration of 4 µM yielded a stable signal while lower concentrations proved unsatisfactory (the total protomer concentration in the useful part of the gradient was between 70 nM and 10 µM). Recording the gradients for each sample after equilibration at different speeds (23,000, 25,000, 28,000, 30,000, and 50,000 rpm) and fitting them simultaneously with a Fortran program adapted for that purpose (example shown in Fig. 3) allowed to make a robust analysis of the association (see Materials and Methods). The mass and the partial specific volume (
) used in the analysis were deduced from the amino acid sequence.
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, which has a strong impact on the analysis, was set to a low value (Supplemental Material, Table S1). The quality of the data appeared to allow a sufficient discrimination to exclude dissociation constants above 12 nM (when was set so that KDdim 10 nM, confidence interval on KDdim 5%; Supplemental Material, Table S1).
Data analysis
The V-, Us-, and U-models were applied to the experimental data and solved numerically as described in Materials and Methods. The V- and Us- models bore only two parameters, KD and F, and thus, could be solved conventionally as described previously (Dam et al. 2000; method-4, see Materials and Methods).
By contrast, the number of parameters of the U-model was too large to allow their accurate determination with the iso-fluorescence data alone. Therefore, the U-model was fit to the data by adjusting KD and F and presetting the other parameters in a systematic grid search to values spanning their respective confidence intervals (above and Supplemental Material, Fig. S6). Thus, for each mutant, each of the three pair of experimental curves (high + low concentration) were fit simultaneously as exposed in Materials and Methods for each set of preset parameters. The three independent KD and F thus obtained were then averaged and the 
2 for the six curves summed for each set of preset parameter. The lower part of the distribution of averaged KD /summed 2 for all the combinations of preset parameters defined the confidence interval for KD (Table 1; Supplemental Material, Fig. S6). The predictions for ß in [1/3,
[ were deduced from that with ß = 1 by the formula: KD(ß) = (ß + 2)/(ß(1 + 2)) • KD (1).
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= 1, and FXZ = 0, as expected (Table 1 and Supplemental Material, Fig. S6B,C). The three parameters, FXZ, , and KD, were strongly coupled in the fit, and other combinations also yielded good fits. These other combinations had no specific reason to occur, considering the distance of the mutations from the dimerization interface. They could result from the large number of parameters of the model (13) in the presence of experimental uncertainties, which opens new areas to reach minima that could be artificial. Therefore, the values obtained with FXZ = 0 were shown in Table 1 for reference. Nevertheless, all possible combination were considered to determine the confidence interval which was asymmetric: 
and 
(Table 1). The Us-model was in good agreement with the U-model (KD = 0.36 µM), while the V-model was not (KD = 0.77 µM; Table 1).
The results for the other mutants are summarized in Table 1, and the observed differences are described now.
59A + 62L
In striking contrast to 59A + 62F, the optimal was clearly 0, indicating that no significant redistribution of the protomers occurred. As the KD of this pair was only a few time larger than the upper limit for KX2 or KZ2, the preset parameters had a stronger impact on the analysis than for the other pairs. However, the parameters involving heterodimers (FXZ, KX3Z, KXZ3, FX3Z, FXZ3, and ß) had no influence, because was zero. Any deviation of the preset parameters from 0 led to a reduction of the deduced KD. The upper bound of the confidence interval of the U-model was raised so that it covered the confidence interval of the V-model, yielding 0.031 ± 0.012 µM, in relatively good but not exact agreement with the V-model.
59A + 62C
The optimal 2 was obtained at = 0.3–0.4, implying that there was a partial redistribution of the protomers among dimers. The deduced KD was in surprisingly good agreement with that of the V-model. This was in contrast with the results obtained with the other pairs of mutants (except for 59A + 62L) for which the U- and Us-model gave similar results. The confidence interval was 
and 
.
59A + 62M
The 2 was slightly lower with FXZ > 0 than with FXZ = 0. This deviation from F XZ = 0 is likely to be due to the fact that the system is underdetermined as discussed above, and thus, the most probable value for the KD should be 0.99 µM. However, the possibility for FXZ > 0 had to be integrated, and resulted in an asymmetric confidence interval: 
and 
.
59A + 62N
The KD range of value for this pair has been shown previously to be less favorable for the iso-fluorescence method (Dam et al. 2000). As a result, the distribution of predicted KD was comparatively wider than for the other pairs. At the same time, the effect of was less acute ( in 0.5–1.0 gave similar 2 and KD values). Here again, the confidence interval was asymmetric: 
and 
.
The confidence intervals determined for the U-model (Table 1) were much wider and cautious than the usual 68% or 90% confidence interval based on a Gaussian convolution of probability, but still acceptably narrow.
For pairs (59A)2 + (62[L/C])2, the prediction of the V- and U-models were similar, and correlatively, was small and the free energy uncertainties with the U-model limited (0.25 kcal/mole). On the contrary, for pairs (59A)2 + (62[F/M/N])2, the Us- and the U-models gave similar predictions; was close to 1, and the uncertainties were slightly larger (0.5 kcal/mole for (59A)2 + (62[F/M])2 and 1 kcal/mole for (59A)2 + (62N)2.
Why are (59A)2 + (62[L/C])2 different? Can they not form heterodimers?
It was intriguing that the V-model fit the data for the two pairs (59A)2 + (62[L/C])2 while the Us-model fit the data of the other pairs, suggesting that the former pairs could not form heterodimers. However, heterodimer (59A)(62L) could be seen by IEF PAGE, but formed slowly (Supplemental Material, Fig. S3B). This suggested that those pairs did not have time to form heterodimers under the experimental conditions of the iso-fluorescence titration.
To test this hypothesis, mixtures of (59A)2 and (62L)2 prepared in the usual conditions were then either incubated at 30°C for 20 h or denatured and renatured prior to the usual incubation for 1–2 h at 20°C. These treatments affected the quality of the data. However, the Us-model fit better those data than the V-model (Supplemental Material, Annex 1). Thus, the differences between distinct pairs of mutants could be explained by the rate of protomer redistribution. This suggested that all pairs had the same mechanism of association: a rapid association (
) followed by a slow redistribution of the protomers (
).
This mechanism could be tested because the V-model predicts a lager proportion of tetramers than the U-/Us-model for the same association constant (indeed, the lager number of species in the U-/Us-model corresponds to a partial dilution which, by compensation, lead to the prediction of a lower KD for the same data; see Table 1). Hence, after mixing, the fluorescence signal should first follow a rapid decrease and then a slow increase—opposite directions for pair (59A)2 + (62N)2. This was confirmed by manual or stopped flow fluorescence kinetics for pairs (59A)2 + (62[F/M])2.
Manual mixing kinetics did not show the rapid association of homodimers, but the signal at trecord = 0 was lower than that of the initial solutions. Although the slow phase could not be isolated accurately (exponential fit given as an indication, Fig. 4A), the signal increase corresponded to 3.5% of the total signal for (59A)2 + (62F)2—similar value for (59A)2 + (62M)2. By contrast, the signal was stable over 1000 sec for (59A)2 + (62L)2, suggesting a kinetics of protomer redistribution much longer than the time of the experiment. The amplitude of the slow phase for (59A)2 + (62F)2 and (59A)2 + (62M)2 was in agreement with the predictions of the V- and Us-models with the respective KD (see comment on equation 3). This suggested that ß should indeed be close to 1. This evaluation of ß was possible because the detection spectra were the same than for the iso-fluorescence titrations.
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The kinetics for the pairs (59A)2 + (62[F,M,N])2 showed two phases with amplitudes of opposite signs. This would not occur if the formation of species (XZ)2 was much stronger than that of X2:Z2 (i.e., ß was small). Calculations comparing the V- and the U-models in the conditions of the experiments showed that values of ß below 1/3 would result in a further association rather than in a partial dissociation. Therefore, ß had to be between 1/3 and infinity—with ß = , [(XZ)2] = 0, equation 9.
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Discussion |
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TOPAbstractIntroductionResultsDiscussionMaterials and methodsElectronic supplemental materialReferences |
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X2:Z2) was attempted in the context of multiple equilibria between various oligomeric species. The results showed that the existence of homologue but not identical subunits, as it is often the case in biological systems, has an impact on the stability of the protein assembly. The quantification of a single microscopic reaction was possible, although not easy, despite the global complexity of the system. This fundamental information is essential for the refinement of our understanding of protein stability, as it can serves to benchmark molecular modeling methods through free energy calculations, for example. This study would have been simpler with a dimer instead of a tetramer. However, the lever effect of the mutations on the KD was much larger with a high degree of oligomerization. Hence, none of the mutants formed tetramers, and they all appeared as dimers. This paved the way to the combinatorial search of complementing mutation pairs and to the determination of the dissociation constant with high precision by the iso-fluorescence titration method (Dam et al. 2000). Conversely, with a dimer, the residual concentration of homodimeric mutant would have "polluted" the combinatorial search and the measurement of the heteroassociation.
The number of parameters of the U-model was too large to fit them all. Thus, intervals enclosing them were determined. The confidence intervals for KD was then determined by intervals combination (
+ 
( + i) and -( - i)), much larger than that of the usual Gaussian probability convolution (
). Despite this and the experimental difficulty due to the strength of the interactions (nM to µM), the sensitivity of the iso-fluorescence titration method allowed the determination of cautious and reasonably narrow confidence intervals [except for (59A)2 + (62N)2 due to experimental conditions].
The use of pairs of complementing mutations turned an homo-oligomer into an hetero-oligomer. Some of the consequences of this fact were used in this work, as, for instance, the possibility to perform titrations. Some other interesting consequences will now be discussed.
Effect at distance of the mutations on the formation of dimers
The mutations were located far from the dimerization interface, and thus, should have almost no effect on dimerization (KX2 or KZ2). However, kinetic data suggested the contrary. Analytical ultracentrifugation gave an upper limit for the dissociation constants, but failed to give exact values because they were too low. Their determination, for instance, by urea denaturation, was not pursued because the upper limit proved to be sufficient in the application of the U-model.
Another aspect that is difficult to test is the effect of the mutations on the relative propensity to form homo- versus heterodimers (factor in equation 8). Because the mutations were located in the tetramerization interface, they should not affect the dimerization directly, but the monomer stability first, and then, as a consequence, dimerization. These assumptions imply that the relative affinity should not be affected in the absence of kinetic limitation ( 1, Supplemental Material, Annex 2).
Effect of mutations on the kinetics of protomer redistribution
The mutations could affect the kinetic of protomer redistribution through two mechanisms. First, they can affect the strength of tetramerization, thus, the dimer concentration, and finally the speed of protomer redistribution. Second, as they might destabilize the monomers and introduce new intermediate species they might increase the dimer dissociation constant and accelerate the redistribution among dimers. Is it possible to determine from the existing data which mechanism prevails?
The wild-type protein and the heterotetramer (59A)2:(62L)2, which associate with a strong affinity (KD 30–40 nM) have a slow exchange kinetics [tested by catalytic inactivation with (62D)2, not shown]. Conversely, weaker complexes, (59A)2 + (62[F/M/N])2, displayed faster subunit redistribution. However, (59A)2:(62C)2 did not reach protomer equipartition, although it formed weaker tetramers than (59A)2 + (62F)2, which undergoes fast protomer redistribution.
Thus, tetramerization is not the only factor influencing the kinetics of protomer redistribution, and the protomer stability should also be important. Thus, the mutations did not only affect some local interactions but also had a global effect on the system and its molecular mechanisms.
Relative stability of the X2:Z2 and the (XZ)2 forms
The two reactions X2 + Z2 
X2:Z2 and 2(XZ) (XZ)2 (with the symmetry where X faces Z in the tetramerization interface) create the same sets of interactions. Therefore, they should have the same strength and ß should be close to 1 (equation 9). However, second-order effects could affect ß because the protomers have different relative positions within X2:Z2 and (XZ)2. However, these effects should be small for two reasons. First, long-range interactions (electrostatic or dipole) are not modified (the mutations do not change the charges at the experimental pH). Second, the mutations should not distort the dimer structure because they are not in the hydrophobic core or the monomer–monomer interface.
Yet, it is difficult to measure the equilibrium dissociation constant of the two individual tetrameric species. The only available data were from the kinetics of association. Although it bore large uncertainties, the amplitude of the slow phase was in agreement with the predictions of the V- and Us-models (Results, Fig. 4). This suggested that ß should indeed be close to 1. More crudely, the mere fact that there were two phases of opposite amplitude gave the boundaries for the value of ß (see Results).
Relative specific fluorescence signals of the various species or reactions
This aspect was also difficult to test directly because it is impossible to isolate individual species from the others. Reactions yielding symmetry related products, for example, X2 + Z2 2XZ or X2:Z2 (XZ)2, should induce negligible signal variations resulting from second-order effects at distances of about 20 Å. In the second example, the formulation of the U-model was such that potential differences would have no effect on the KD determination.
Intrinsic and quantitative differences between V-, Us-, and U- models
The V-model is a particular case of the U-model. Correspondingly, (59A)2 + (62L)2 was associated with the V-model, but could also undergo slow protomer exchange and then followed the Us-model. Indeed, the same physics apply to all the pairs of mutants, but their kinetics of protomer redistribution differed (e.g., Results, Fig. 4).
The V- and Us-models both have two adjustable parameters only (KD and F). Thus, when they apply, they are more robust than the U-model, which bears 13 independent parameters. Correlatively, the 2 of the V- or Us-models can be meaningfully compared while the comparison of either 
or 
with 
is more questionable.
The predictions of the V-, Us-, and U-models were not strikingly different (see Table 1). However, the using a relevant model should give more precise results, especially in difficult cases.
It is noteworthy that the U-model made KD prediction in agreement with that of the two-parameters model giving the lowest 2 (V- or Us-model, depending on the pair of mutant). Therefore, the latter two models appeared sufficient to analyze the iso-fluorescence data. The pairs seemed to either form heterodimers rapidly (minutes, while incubation was 1 or 2 h) or to take much longer (days). In principle, a pair having an intermediate rate of exchange would have to be analyzed with the U-model. However, this principle seemed contradicted by pair (59A)2 + (62C)2 for which the V- and U-models predictions were in good agreement, although it appeared to have an intermediate rate of exchange ( = 0.3–0.4, 2 ratio close to 1, Table 1).
Mutation effects and dissociation constant
All measures were performed at pH 8.0, where histidines are mostly unprotonated. Thus, the molecules were in a well-defined state, the association was not affected, and meaningful comparisons should be possible.
Molecular models of the heterotetramers (59A)2 + (62[C/F/L/M/N])2 were built as previously described (Pelosi et al. 1999; PDB: 1VIE
[PDB]
) to gain some structural insight on the mutations effect. Because pair (59A)2 + (62A)2 was almost unable to associate (Dam et al. 2000) although no steric hindrance were introduced, new interactions had to be made in the complementing pairs. Indeed, the histidine Cß, C
, N
1, C2 atoms were replaced in the molecular models of the mutants. The hydrophobic environment was favorable for (62[C/F/L/M]), but less for (62N), in agreement with the KD measures. An unfavorable hydrophobic crevice was created for 62[C/L/M/N], where the C
1 and N2 atoms were not replaced. By contrast, the whole space was filled by the phenylalanine ring in (59A)2:(62F)2.
This latter point was paradoxical, because (59A)2 + (62F)2 made a significantly weaker association than (59A)2 + (62L)2, while the latter had almost the same affinity as (WT)4. Conformational changes in (62[F/L])2 displacing the equilibria could explain this (see above and Supplemental Material, Annex 2).
The effect of mutations was clearly dependent on the context in which they were introduced. First, as a consequence of their selection by complementation they could not be additive (two positive free energy contributions yielding an almost zero total contribution). Similarly, the example of (59A;62L)4 show that the simple extrapolation of the effects observed on (59A)2:(62L)2 does not hold (two almost zero-free energy contributions yield a positive contribution; Dam et al. 2000). The latter example suggest that water plays an essential and a strongly context dependent role at this interface. Therefore, a detailed molecular model and a comprehensive thermodynamics formalism seems necessary to account for the experimental data. This is far beyond the scope of this work, but makes the present results all the more interesting as a benchmark to control theoretical methods. More comprehensive simulations taking the effect of the solvent and that of the mutations on the dimer into account would be necessary to fully interpret the experimental facts. Such simulations are underway in the laboratory and will be reported elsewhere.
Impact of protomer redistribution and relation to physiologic systems
In this work, protomer redistribution had to be considered to allow a rigorous study. It can also be important for biological phenomena as exemplified by a pathological variant of transthyretin, which was associated with a slower rate of protomer redistribution (Schneider et al. 2001). The S- and N-Baltimore variants of hemoglobin were associated with faster and slower rates of protomer redistribution, respectively, in a detailed study of the multiple equilibria (McDonald et al. 1987). More generally, when subunit isoforms exist, protomer exchanges can have an impact on the apparent affinity constants as well as on the protein function. For example, human nucleotide diphosphate kinase, which consists of random association of two kinds of protomers, A and B (A6, A5B, . . ., AB5, B6; Gilles et al. 1991) probably undergoes subunits redistribution that may control its final location within the cell. Similarly, the eukaryotic proteasome (700 kD) is arranged as four heptameric rings formed by two sets of seven -type subunits and seven ß-type subunits (DeMartino and Slaughter 1999). The exact arrangement of subunits in such complex molecules is thought to modify the function, and could be controlled by protomer exchange. Thus, the kinetics of such exchange can be of primary importance to understand the cellular response to external signals. The resolution of a complex model like the U-model might be useful in the study of such systems. The approach that was used to solve the U-model proved to be rather flexible and efficient, and it could be applied on systems following a vast variety of mechanisms.
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-Tryptophane Synthase. The two solutions were mixed in different proportions: 
mL of dimer X2 and (1 - ) mL of dimer Z2, with varying from 0 to 1 by 0.1 increments. Dimer X2 was (62C)2, (62F)2, (62M)2, or (62N)2 and Z2 was (59A)2. After a specified incubation time, the fluorescence of each mixture was continuously recorded for 30 sec at 20°C, pH 8.0 using a Perkin-Elmer LS-5B spectrofluorimeter and averaged. The excitation wavelength was 297 nm (2.5 nm excitation slit). Emission was detected at 350 nm (5 nm emission slit) for all proteins [except for (59A)2 + (62N)2, em = 324 nm]. Two sets of mixtures, one set at low and the other at high protein concentration, were prepared in triplicates for the determination of each KD value.
Analysis of the iso-fluorescence titrations
Data were analyzed according to three distinct models (V-, Us-, and U- models). First, a standard association equilibrium model (the "V-model") was used (Dam et al. 2000; Fig. 2B):
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and the fluorescence was calculated as (Dam et al. 2000):
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where dC (=CZ2 - CX2) is the difference between the concentrations of the species Z2 (CZ2) and X2 (CX2) in the initial solutions. FX and FZ are the fluorescence signals of the initial solutions of X2 and Z2, respectively, and F is the specific fluorescence signal due to the association reaction. CX2 and CZ2 were determined from FX, FZ and the specific fluorescence signal of each species.
Second, a slightly more complex model (the "Us-model," Fig. 2B) was used. This model took the presence of hybrid dimers into account, but supposed that the monomers were in negligible amounts, that X2:Z2 and (XZ)2 were the only possible tetramers and, that the symmetry did not influence the association strength if it allowed the same interactions [i.e., the two tetrameric species X2:Z2 and (XZ)2, or the three dimeric species X2, Z2, and (XZ)2]. Because the dissociation constant of an heterodimerization is half that of an homodimerization involving the same interactions (Dam et al. 2000, see Discussion), the monomers–dimers equilibria can be written as:
 | (1a) |
 | (1b) |
 | (1c) |
When KDdim, the homodimer dissociation constant, is almost zero, equation 1 reduces to:
 | (2) |
Similarly, the dissociation constants for the different dimer–tetramer equilibria involving the same interactions could be written as:
 | (3a) |
 | (3b) |
KD in equation 3a is the same as that in the V-model, and corresponds to the heterodimerization of homodimers. Factor 4 in equation 3b is entropic, and is decomposed in two factors 2. The first one is homologous to the factor 1/2 in equation 1c (see comment above). The second one is due to the fact that they are two possible orientations for the reaction X2 + Z2 X2:Z2, and only one for the reaction 2 • XZ (XZ)2. In agreement with the expected equipartition for protomers X and Z randomly distributed among the authorized tetramers, the combination of equations 2 and 3 predicts the same concentrations for X2:Z2 and (XZ)2. The total concentration of tetramers (T = [X2:Z2] + [(XZ)2]) is related to the concentration of heterodimer XZ by combination of equations 2 and 3:
 | (4) |
The concentration of heterodimer XZ is the root of the polynomials given by the conservation of mass and the above relations:
 | (5) |
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The root was determined numerically. Equation 4 gave the concentration of tetramers as a function of , T(), and equation 6 gave the fluorescence signal F() as a function of the mixing parameter :
 | (6) |
Finally, the U-model followed the scheme shown in Figure 2A, and made no assumption except that X4 or Z4 could not be formed, and consequently, (XZ)2 was assembled with X facing Z in the tetramerization interface. The model was expressed as a pair of functions, F1 and F2, to avoid the use of a single polynomial of high degree and complexity as well as possible ambiguities among the roots. F1 and F2 were expressed as a function of [X2] and [Z2] to produce the Us- and V-model as limit cases. The concentrations of the other species were deduced from the equilibrium constants:
 | (7) |
A factor was introduced to integrate possible structural effect on XZ:
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 | (8) |
A factor ß was introduced to discriminate the structural effects on the two legitimate tetramers, K(XZ)2 = 4 • ß • KX2Z2 (with KX2Z2 = KD):
 | (9) |
The concentrations of illegitimate tetramers were given by:
 | (10) |
The mass conservation gave the functions (F1,F2) as:
 | (11a) |
 | (11b) |
where XT and ZT are the total concentrations of X and Z protomers, respectively. The concentrations of the individual species were substituted using equations 7–10 and yielded F1([X2],[Z2]) and F2([X2],[Z2]). The root of the system of equation 11 was determined recursively by the Newton formula:
 | (12) |
where Mn is the derivative matrix of (F1,F2) with respect to ([X2],[Z2]) at step n:
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To avoid numerical singularities, the matrix inversion was written by simplification of the following expression:
 | (13) |
The convergence of equation 12 proved to be very fast, and the quantity F12 + F22 was usually brought below 10-60 (µM)2 after only five to six iterations. The fluorescence was then given by the following relation:
 | (14) |
FX2 and FZ2 were measured from concentrated solution of pure species where monomers are in negligible amount. Because the two legitimate tetramers are in constant relative amount (equation 9), FX2Z2 and FXZ2 were merged in the quantity,
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which was one of the fitted parameters of the model. The other FS were determined from the following quantities, which were sampled within a fixed interval (see Results): FX = (F(X) - 
FX2), FZ = (F(Z) - FZ2), FXZ = (FXZ - FX2 - FZ2), FX3Z = (FX3Z - FX2 - FXZ), and FXZ3 = (FXZ3 - FXZ - FZ2). The specific florescence of monomeric species are written F(X) and F(Z) to distinguish them from FX and FZ, the florescence of the initial solutions.
Finally, the experimental data were analyzed with the V-, the Us-, or the U-model, using the fitting procedure described previously (Dam et al. 2000: method-4: fitting each pair of titration so that F is optimal for the high concentration data and KD is optimal for the low concentration data; that is, 
and 
).
Sedimentation/diffusion equilibrium
The monomer–dimer equilibrium of the modified proteins was characterized by sedimentation/diffusion equilibrium experiments (S/D–E) performed at 20°C with a Beckman XLA ultracentrifuge. Optical path cells (12 mm) with standard double-sector aluminum centerpieces were used in an An60Ti rotor. Proteins, dialyzed against 50 mM Tris-HCl at pH 8 with 2 mM ß-mercaptoethanol, were centrifuged at various speeds and equilibrated after each increment of speed until perfect superposition of three consecutive scans (0–23 krpm: ~20 h, 23–25, 25–28; 28–30 krpm: ~12 h, and 30–50 krpm: 6 h). The solvent density was determined with a pychnometer (
= 1.009 mg/mL). The data were analyzed by global fitting of all the data sets for each homodimer and by minimization of the function 2:
 | (15) |
where Aij is the absorption at radius xj recorded at velocity 
i, xai and xbi are, respectively, the lower and upper limit for the use of the data in the analysis, and Ai(x) is given by the function:
 | (16) |
where Bi is the baseline at velocity i, x0i is a reference radius (chosen at the bottom of the useful part of the gradient for numerical precision), M1 is the mass of the monomer, n is the oligomerization order (here n = 2), KA is the association constant in (absorbance units)1-n, A0i is the absorption of the monomer at x0i, and Hi is the thermodynamic quantity given by:
 | (17) |
where R is the perfect gas constant. To make the fitting more robust, the quantity 
, the total absorbance at x = x0i was fit, and A0i was deduced by numerical root finding. Bi, xai, and xbi were the same for all i, except with mutation H62N because the baseline showed an regular drift during the data collection (~0.08 cm-1 in ~60 h). Thus, the fitting allowed one baseline, five absorbance values, and one dissociation constant to vary (for H62N, five baselines varied independently). Finally, KD was deduced from KA by the relation:
 | (18) |
where 1 and n = n • 1 are the molar extinction coefficients for the monomer and the n-mer, respectively.
Fluorescence kinetics
Kinetics were recorded at 20°C for 1000 sec with a thermostated Perkin-Elmer LS-5B spectrofluorimeter. The excitation wavelength was 297 nm (excitation slit at 2.5 nm) and emission was measured at 350 nm (emission slit at 5 nm). (59A)2 at 2.4 µM (dimers) was mixed manually with (62F)2 or (62L)2 at 2 µM (dimers) in MTA polybuffer pH 8.0 with 5 mM ß-mercaptoethanol and 0.08 mg/mL -Tryptophan Synthase.
Fast kinetic measurements were performed with a SFM-3 stopped-flow module with large (18 mL) syringes (Bio-Logic) thermostated at 20°C and coupled to a Bio-Logic optical bench with a 35 µL F-15 fluorescence cell (1.5 x 1.5 mm). Excitation was at 297 nm. Emission was detected through a high-pass filter (LG-350-F: > 350 nm). Protein concentration was 2 µM (dimers) so the absorbance was under 0.05 to minimize the internal filter effects. The dead time was 5 msec. The kinetics were recorded for 500 sec with a sampling interval and a filtering time constant of 500 msec. Four kinetics were accumulated and averaged for analysis.
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The Supplemental Material contains:
Figure S1A,B, similar to Figure 1 A,B with titration curves for other pairs of mutations; Figure S2, possible reactions without heterodimers; Figure S3A,B, IEF PAGE showing the formation of heterodimers and the effect of incubation time and temperature; Figure S4A,B, inhibition by an third-party mutant showing the formation of heterodimers; Figure S5, identification of the relevant reactions among possible ones; Figure S6, illustration of data analysis; Table S1, effect of V values on the fitting of analytical ultracentrifugation data, robustness assessment; Annex 1, effect of the incubation conditions on the association regime of pair (59A)2 + (62L)2 showing that protomer redistribution, although very slow, can occur for that pair; Annex 2, formalization of the effect of the protomer stability on the association showing that there should be no symmetry propensity.
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Acknowledgments |
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The publication costs of this article were defrayed in part by payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 USC section 1734 solely to indicate this fact.
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References |
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DeMartino, G.N. and Slaughter, C.A. 1999. The proteasome, a novel protease regulated by multiple mechanisms. J. Biol. Chem. 274: 22123–22126.
Gilles, A.M., Presecan, E., Vonica, A., and Lascu, I. 1991. Nucleoside diphosphate kinase from human erythrocytes. Structural characterization of the two polypeptide chains responsible for heterogeneity of the hexameric enzyme. J. B
