Principles and equations for measuring and interpreting protein stability: From monomer to tetramer.

The ability to measure the thermodynamic stability of proteins with precision is important for both academic and applied research. Such measurements rely on mathematical models of the protein denaturation profile, i.e. the relation between a global protein signal, corresponding to the folding states in equilibrium, and the variable value of a denaturing agent, either heat or a chemical molecule, e.g. urea or guanidinium hydrochloride. In turn, such models rely on a handful of physical laws: the laws of mass action and conservation, the law that relates the protein signal and concentration, and the one that relates stability and denaturant value. So far, equations have been derived mainly for the denaturation profiles of homomeric proteins. Here, we review the underlying basic physical laws and show in detail how to derive model equations for the unfolding equilibria of homomeric or heteromeric proteins up to trimers and potentially tetramers, with or without folding intermediates, and give full demonstrations. We show that such equations cannot be derived for pentamers or higher oligomers except in special degenerate cases. We expand the method to signals that do not correspond to extensive protein properties. We review and expand methods for uncovering hidden intermediates of unfolding. Finally, we review methods for comparing and interpreting the thermodynamic parameters that derive from stability measurements for cognate wild-type and mutant proteins. This work should provide a robust theoretical basis for measuring the stability of complex proteins.

Direct and indirect interactions in the recognition between a cross-neutralizing antibody and the four serotypes of dengue virus.

Dengue fever is the most important vector-borne viral disease. Four serotypes of dengue virus, DENV1 to DENV4, coexist. Secondary infection by a different serotype is a risk factor for severe dengue. Monoclonal antibody mAb4E11 neutralizes the four serotypes of DENV with varying efficacies by recognizing an epitope located within domain-III (ED3) of the viral envelope (E) protein. To better understand the cross-reactivities between mAb4E11 and the four serotypes of DENV, we constructed mutations in both Fab4E11 fragment and ED3, and we searched for indirect interactions in the crystal structures of the four complexes. According to the serotype, 7 to 12 interactions are mediated by one water molecule, 1 to 10 by two water molecules, and several of these interactions are conserved between serotypes. Most interfacial water molecules make hydrogen bonds with both antibody and antigen. Some residues or atomic groups are engaged in both direct and water-mediated interactions. The doubly-indirect interactions are more numerous in the complex of lowest affinity. The third complementarity determining region of the light chain (L-CDR3) of mAb4E11 does not contact ED3. The structures and double-mutant thermodynamic cycles showed that the effects of (hyper)-mutations in L-CDR3 on affinity were caused by conformational changes and indirect interactions with ED3 through other CDRs. Exchanges of residues between ED3 serotypes showed that their effects on affinity were context dependent. Thus, conformational changes, structural context, and indirect interactions should be included when studying cross-reactivity between antibodies and different serotypes of viral antigens for a better design of diagnostics, vaccine, and therapeutic tools against DENV and other Flaviviruses.

Cross-reactivities between human IgMs and the four serotypes of dengue virus as probed with artificial homodimers of domain-III from the envelope proteins.

BACKGROUND: Dengue fever is the most important vector-borne viral disease. Four serotypes of dengue virus, DENV1 to DENV4, coexist. Infection by one serotype elicits long-lasting immunity to that serotype but not the other three. Subsequent infection by a different serotype is a risk factor for severe dengue. Domain III (ED3) of the viral envelope protein interacts with cell receptors and contains epitopes recognized by neutralizing antibodies. We determined the serotype specificity and cross-reactivity of human IgMs directed against ED3 by using a well-characterized collection of 90 DENV-infected and 89 DENV-uninfected human serums. METHODS: The recognitions between the four serotypes of ED3 and the serums were assayed with an IgM antibody-capture ELISA (MAC-ELISA) and artificial homodimeric antigens. The results were analyzed with Receiving Operator Characteristic (ROC) curves. RESULTS: The DENV-infected serums contained IgMs that reacted with one or several ED3 serotypes. The discrimination by ED3 between serums infected by the homotypic DENV and uninfected serums varied with the serotype in the decreasing order DENV1 > DENV2 > DENV3 > DENV4. The ED3 domain of DENV1 gave the highest discrimination between DENV-infected and DENV-uninfected serums, whatever the infecting serotype, and thus behaved like a universal ED3 domain for the detection of IgMs against DENV. Some ED3 serotypes discriminated between IgMs directed against the homotypic and heterotypic DENVs. The patterns of cross-reactivities and discriminations varied with the serotype. CONCLUSIONS: The results should help better understand the IgM immune response and protection against DENV since ED3 is widely used as an antigen in diagnostic assays and an immunogen in vaccine candidates.

In-vitro and in-vivo analysis of the production of the Bordetella type three secretion system effector A in Bordetella pertussis, Bordetella parapertussis and Bordetella bronchiseptica.

Bordetella pertussis, Bordetella parapertussis, and Bordetella bronchiseptica are three closely related pathogens. They all possess the gene coding for the Bordetella type three secretion system effector A (bteA) toxin that became a focus of interest since it was demonstrated that B. pertussis Japanese non-vaccine-type isolates produce BteA unlike vaccine-type isolates. We thus explored the in-vitro production of BteA in B. pertussis isolates collected in France during periods of different vaccine policy as well as in B. parapertussis and B. bronchiseptica isolates. We also analyzed the in-vivo induction of anti-BteA antibodies after infection with different isolates of the three species. We produced a recombinant His6-tagged BteA (rBteA) protein. Specific rBteA polyclonal serum was prepared which enabled us to screen Bordetella isolates for in-vitro BteA production: 99.0% (293/296) of tested B. pertussis isolates, including French vaccine strains, and 97.5% (79/81) of B. bronchiseptica isolates produced BteA in-vitro but only the latter was capable of inducing an in-vivo immune response. No in-vitro or in-vivo production of BteA was detected by any of the B. parapertussis isolates tested.

Thermodynamic stability of domain III from the envelope protein of flaviviruses and its improvement by molecular design.

The Flavivirus genus includes widespread and severe human pathogens like the four serotypes of dengue virus (DENV1 to DENV4), yellow fever virus, Japanese encephalitis virus and West Nile virus. Domain III (ED3) of the viral envelope protein interacts with cell receptors and contains epitopes recognized by virus neutralizing antibodies. Its structural, antigenic and immunogenic properties have been thoroughly studied contrary to its physico-chemical properties. Here, the ED3 domains of the above pathogenic flaviviruses were produced in the periplasm of Escherichia coli. Their thermodynamic stabilities were measured and compared in experiments of unfolding equilibriums, induced with chemicals or heat and monitored through protein fluorescence. A designed ED3 domain, with the consensus sequence of DENV strains from all serotypes, was highly stable. The low stability of the ED3 domain from DENV3 was increased by three changes of residues in the protein core without affecting its reactivity towards DENV-infected human serums. Additional changes showed that the stability of ED3 varied with the DENV3 genotype. The T(m) of ED3 was higher than 69°C for all the tested viruses and reached 86°C for the consensus ED3. The latter, deprived of its disulfide bond by mutations, was predominantly unfolded at 20°C. These results will help better understand and design the properties of ED3 for its use as diagnostic, vaccine or therapeutic tools.

The folded and disordered domains of human ribosomal protein SA have both idiosyncratic and shared functions as membrane receptors.

The human RPSA [ribosomal protein SA; also known as LamR1(laminin receptor 1)] belongs to the ribosome but is also a membrane receptor for laminin, growth factors, prion, pathogens and the anticarcinogen EGCG (epigallocatechin-gallate). It contributes to the crossing of the blood-brain barrier by neurotropic viruses and bacteria, and is a biomarker of metastasis. RPSA includes an N-terminal domain, which is folded and homologous to the prokaryotic RPS2, and a C-terminal extension, which is intrinsically disordered and conserved in vertebrates. We used recombinant derivatives of RPSA and its N- and C-domains to quantify its interactions with ligands by in-vitro immunochemical and spectrofluorimetric methods. Both N- and C-domains bound laminin with K(D) (dissociation constants) of 300 nM. Heparin bound only to the N-domain and competed for binding to laminin with the negatively charged C-domain, which therefore mimicked heparin. EGCG bound only to the N-domain with a K(D) of 100 nM. Domain 3 of the envelope protein from yellow fever virus and serotypes-1 and -2 of dengue virus bound preferentially to the C-domain whereas that from West Nile virus bound only to the N-domain. Our quantitative in-vitro approach should help clarify the mechanisms of action of RPSA, and ultimately fight against cancer and infectious agents.

Multiple folding states and disorder of ribosomal protein SA, a membrane receptor for laminin, anticarcinogens, and pathogens.

The human ribosomal protein SA (RPSA) is a multilocus protein, present in most cellular compartments. It is a multifunctional protein, which belongs to the ribosome but is also a membrane receptor for laminin, growth factors, prion, pathogenic microorganisms, toxins, and the anticarcinogen epigallocatechin gallate. It contributes to the crossing of the blood-brain barrier by neurotropic viruses and bacteria and is used as a biomarker of metastasis. RPSA includes an N-terminal domain, which is homologous to the prokaryotic ribosomal proteins S2, and a C-terminal extension, which is conserved in vertebrates. The structure of its N-domain has been determined from crystals grown at 17 °C. The structure of its C-domain remains unknown. We produced in Escherichia coli and purified the full-length RPSA and its N- and C-domains. We characterized the folding states of these recombinant proteins mainly by methods of fluorescence and circular dichroism spectrometry, in association with quantitative analyses of their unfolding equilibria, induced with heat or urea. The necessary equations were derived from first principles. The results showed that the N-domain unfolded according to a three-state equilibrium. The monomeric intermediate was predominant at the body temperature of 37 °C. It also existed in the full-length RPSA and bound ANS, a small fluorescent molecule. The C-domain was in an intrinsically disordered state. The recombinant N- and C-domains weakly interacted together. These results indicated a high plasticity of RPSA, which could be important for its multiple cellular localizations and functional interactions.

Mechanism of dengue virus broad cross-neutralization by a monoclonal antibody.

The dengue virus (DENV) complex is composed of four distinct but serologically related flaviviruses, which together cause the present-day most important emerging viral disease. Although DENV infection induces lifelong immunity against viruses of the same serotype, the antibodies raised appear to contribute to severe disease in cases of heterotypic infections. Understanding the mechanisms of DENV neutralization by antibodies is, therefore, crucial for the design of vaccines that simultaneously protect against all four viruses. Here, we report a comparative, high-resolution crystallographic analysis of an "A-strand" murine monoclonal antibody, Mab 4E11, in complex with its target domain of the envelope protein from the four DENVs. Mab 4E11 is capable of neutralizing all four serotypes, and our study reveals the determinants of this cross-reactivity. The structures also highlight the mechanism by which A-strand Mabs disrupt the architecture of the mature virion, inducing premature fusion loop exposure and concomitant particle inactivation.

Reagentless fluorescent biosensors from artificial families of antigen binding proteins.

Antibodies and artificial families of antigen binding proteins (AgBP) are constituted by a connected set of hypervariable (or randomized) residue positions, supported by a constant polypeptide backbone. The residues that form the binding site for a given antigen, are selected among the hypervariable residues. We showed that it is possible to transform any AgBP of these families into a reagentless fluorescent biosensor, specific of the target antigen, simply by coupling a solvatochromic fluorophore to one of the hypervariable residues that have little or no importance for the interaction with the antigen, after changing this residue into cysteine by mutagenesis. We validated this approach with a DARPin (Designed Ankyrin Repeat Protein) and a Nanofitin (also known as Affitin) with high success rates. Reagentless fluorescent biosensors recognize their antigen in an immediate, quantitative, selective and specific way, without any manipulation of the sample to analyze or addition of reagent.

Recombinant antibodies specific for the Plasmodium falciparum histidine-rich protein 2.

Early diagnosis and appropriate treatment are key elements of malaria control programs in endemic areas. A major step forward in recent years has been the production and use of rapid diagnostic tests (RDTs) in settings where microscopy is impracticable. Many current RDTs target the Plasmodium falciparum histidine-rich protein 2 (PfHRP2) released in the plasma of infected individuals. These RDTs have had an indisputably positive effect on malaria management, but still present several limitations, including the poor characterization of the commercial monoclonal antibodies (mAbs) used for PfHRP2 detection, variable sensitivity and specificity, and high costs. RDT use is further limited by impaired stability caused by temperature fluctuations during transport and uncontrolled storage in field-based facilities. To circumvent such drawbacks, an alternative could be the development of well-characterized, stabilized recombinant antibodies, with high binding affinity and specificity. Here, we report the characterization of the cDNA sequences encoding the Fab fragment of F1110 and F1546, two novels anti-PfHRP2 mAbs. FabF1546 was produced in the Escherichia coli periplasm. Its properties of binding to the parasite and to a recombinant PfHRP-2 antigen were similar to those of the parental mAb. As the affinity and stability of recombinant antibodies can be improved by protein engineering, our results open a novel approach for the development of an improved RDT for malaria diagnosis.

Knowledge-based design of reagentless fluorescent biosensors from a designed ankyrin repeat protein.

Designed ankyrin repeat proteins (DARPins) can be selected from combinatorial libraries to bind any target antigen. They show high levels of recombinant expression, solubility and stability, and contain no cysteine residue. The possibility of obtaining, from any DARPin and at high yields, fluorescent conjugates which respond to the binding of the antigen by a variation of fluorescence, would have numerous applications in micro- and nano-analytical sciences. This possibility was explored with Off7, a DARPin directed against the maltose binding protein (MalE) from Escherichia coli, with known crystal structure of the complex. Eight residues of Off7, whose solvent accessible surface area varies on association with the antigen but which are not in direct contact with the antigen, were individually mutated into cysteine and then chemically coupled with a fluorophore. The conjugates were ranked according to their relative sensitivities. All of them showed an increase in their fluorescence intensity on antigen binding by >1.7-fold. The best conjugate retained the same affinity as the parental DARPin. Its signal increased linearly and specifically with the concentration of antigen, up to 15-fold in buffer and 3-fold in serum when fully saturated, the difference being mainly due to the absorption of light by serum. Its lower limit of detection was equal to 0.3 nM with a standard spectrofluorometer. Titrations with potassium iodide indicated that the fluorescence variation was due to a shielding of the fluorescent group from the solvent by the antigen. These results suggest rules for the design of reagentless fluorescent biosensors from any DARPin.

Crystal structure of a nucleocapsid-like nucleoprotein-RNA complex of respiratory syncytial virus.

The respiratory syncytial virus (RSV) is an important human pathogen, yet neither a vaccine nor effective therapies are available to treat infection. To help elucidate the replication mechanism of this RNA virus, we determined the three-dimensional (3D) crystal structure at 3.3 A resolution of a decameric, annular ribonucleoprotein complex of the RSV nucleoprotein (N) bound to RNA. This complex mimics one turn of the viral helical nucleocapsid complex, which serves as template for viral RNA synthesis. The RNA wraps around the protein ring, with seven nucleotides contacting each N subunit, alternating rows of four and three stacked bases that are exposed and buried within a protein groove, respectively. Combined with electron microscopy data, this structure provides a detailed model for the RSV nucleocapsid, in which the bases are accessible for readout by the viral polymerase. Furthermore, the nucleoprotein structure highlights possible key sites for drug targeting.

NKp44 receptor mediates interaction of the envelope glycoproteins from the West Nile and dengue viruses with NK cells.

Dengue virus (DV) and West Nile virus (WNV) have become a global concern due to their widespread distribution and their ability to cause a variety of human diseases. Antiviral immune defenses involve NK cells. In the present study, we investigated the interaction between NK cells and these two flaviviruses. We show that the NK-activating receptor NKp44 is involved in virally mediated NK activation through direct interaction with the flavivirus envelope protein. Recombinant NKp44 directly binds to purified DV and WNV envelope proteins and specifically to domain III of WNV envelope protein; it also binds to WNV virus-like particles. These WNV-virus-like particles and WNV-domain III of WNV envelope protein directly bind NK cells expressing high levels of NKp44. Functionally, interaction of NK cells with infective and inactivated WNV results in NKp44-mediated NK degranulation. Finally, WNV infection of cells results in increased binding of rNKp44 that is specifically inhibited by anti-WNV serum. WNV-infected target cells induce IFN-gamma secretion and augmented lysis by NKp44-expressing primary NK cells that are blocked by anti-NKp44 Abs. Our findings show that triggering of NK cells by flavivirus is mediated by interaction of NKp44 with the flavivirus envelope protein.

Germline humanization of a non-human primate antibody that neutralizes the anthrax toxin, by in vitro and in silico engineering.

Fab 35PA83 is an antibody fragment of non-human primate origin that neutralizes the anthrax lethal toxin. Human antibodies are usually preferred when clinical use is envisioned, even though their framework regions (FR) may carry mutations introduced during affinity maturation. These hypermutations can be immunogenic and therefore FR that are encoded by human germline genes, encountered in IgMs and thus part of the "self" proteins, are preferable. Accordingly, the proportion of FR residues in 35PA83 that were encoded by human V and J germline genes, i.e. the germinality index (GI) of 35PA83, was increased in a multistep cumulative approach. In a first step, the FR1 and FR4 residues of 35PA83 were changed simultaneously into their counterparts coded by 35PA83's closest human germline genes, without prior modelling. The resulting derivative of 35PA83 had the same affinity as its parental Fab. In a second step, the 3D structures of this first 35PA83 derivative, carrying the same type of residue changes but in the FR2 and FR3 regions, were modelled in silico from sequences. Some of the changes in FR2 or FR3 modified the predicted peptide backbone. The changes that did not seem to alter the structure were introduced simultaneously in the Fab by an in vitro method and resulted in a loss of reactivity, which could however be fully restored by a single point mutation. The final 35PA83 derivative had a GI higher than that of a fully human Fab, which had neutralization properties similar to 35PA83 and which was used as a benchmark in this study.

Improvement of an antibody neutralizing the anthrax toxin by simultaneous mutagenesis of its six hypervariable loops.

The enhancement of antibody affinity by mutagenesis targeting only complementarity determining regions has the advantage of respecting the framework regions, which are important for tolerance if clinical use is envisaged. Here, starting from a Fab (antigen-binding fragment; 35PA(83)) capable of neutralizing the lethal toxin of anthrax and having an affinity of 3.4 nM for its antigen, a phage-displayed library of variants where all six complementarity determining regions (73 positions) were targeted for mutagenesis was built. This library contained 5 x 10(8) variants, and each variant carried four mutations on average. The library was first panned with adsorbed antigen and washes of increasing stringency. It was then screened in parallel with either small concentrations of soluble biotinylated antigen or adsorbed antigen and long elution times in the presence of soluble antigen. The stringencies of both selections were pushed as far as possible. Compared with 35PA(83), the best selected clone had an affinity enhanced 19-fold, to 180 pM, and its 50% inhibitory concentration was decreased by 40%. The results of the two selection methods were compared, and the generality of these methods was considered.

Pediatric measles vaccine expressing a dengue antigen induces durable serotype-specific neutralizing antibodies to dengue virus.

Dengue disease is an increasing global health problem that threatens one-third of the world's population. Despite decades of efforts, no licensed vaccine against dengue is available. With the aim to develop an affordable vaccine that could be used in young populations living in tropical areas, we evaluated a new strategy based on the expression of a minimal dengue antigen by a vector derived from pediatric live-attenuated Schwarz measles vaccine (MV). As a proof-of-concept, we inserted into the MV vector a sequence encoding a minimal combined dengue antigen composed of the envelope domain III (EDIII) fused to the ectodomain of the membrane protein (ectoM) from DV serotype-1. Immunization of mice susceptible to MV resulted in a long-term production of DV1 serotype-specific neutralizing antibodies. The presence of ectoM was critical to the immunogenicity of inserted EDIII. The adjuvant capacity of ectoM correlated with its ability to promote the maturation of dendritic cells and the secretion of proinflammatory and antiviral cytokines and chemokines involved in adaptive immunity. The protective efficacy of this vaccine should be studied in non-human primates. A combined measles-dengue vaccine might provide a one-shot approach to immunize children against both diseases where they co-exist.

Mapping to completeness and transplantation of a group-specific, discontinuous, neutralizing epitope in the envelope protein of dengue virus.

Dengue is caused by a taxonomic group of four viruses, dengue virus types 1-4 (DENV1-DENV4). A molecular understanding of the antibody-mediated protection against this disease is critical to design safe vaccines and therapeutics. Here, the energetic epitope of antibody mAb4E11, which neutralizes the four serotypes of DENV but no other flavivirus, and binds domain 3 (ED3) of their envelope glycoprotein, was characterized. Alanine-scanning mutagenesis of the ED3 domain from serotype DENV1 was performed and the affinities between the mutant domains and the Fab fragment of mAb4E11 were measured. The epitope residues (307-312, 387, 389 and 391) were at the edges of two distinct beta-sheets. Four residues constituted hot spots of binding energy. They were aliphatic and contributed to form a hydrophobic pocket (Leu308, Leu389), or were positively charged (Lys307, Lys310). They may bind the diversity residues of mAb4E11, H-Trp96-Glu97. Remarkably, cyclic residues occupy and block the hydrophobic pocket in all unrelated flaviviruses. Transplanting the epitope from the ED3 domain of DENV into those of other flaviviruses restored affinity. The epitope straddles residues of ED3 that are involved in virulence, e.g. Asn/Asp390. These results define the epitope of mAb4E11 as an antigenic signature of the DENV group and suggest mechanisms for its neutralization potency.

Improving the stability of an antibody variable fragment by a combination of knowledge-based approaches: validation and mechanisms.

Numerous approaches have been described to obtain variable fragments of antibodies (Fv or scFv) that are sufficiently stable for their applications. Here, we combined several knowledge-based methods to increase the stability of pre-existing scFvs by design. Firstly, the consensus sequence approach was used in a non-stringent way to predict a large basic set of potentially stabilizing mutations. These mutations were then prioritized by other methods of design, mainly the formation of additional hydrogen bonds, an increase in the hydrophilicity of solvent exposed residues, and previously described mutations in other antibodies. We validated this combined method with antibody mAbD1.3, directed against lysozyme. Fourteen potentially stabilizing mutations were designed and introduced into scFvD1.3 by site-directed mutagenesis, either individually or in combinations. We characterized the effects of the mutations on the thermodynamic stability of scFvD1.3 by experiments of unfolding with urea, monitored by spectrofluorometry, and tested the additivity of their effects by double-mutant cycles. We also quantified the individual contributions of the resistance to denaturation ([urea](1/2)) and cooperativity of unfolding (m) to the variations of stability and the energy of coupling between mutations by a novel approach. Most mutations (75%) were stabilizing and none was destabilizing. The progressive recombination of the mutations into the same molecule of scFvD1.3 showed that their effects were mostly additive or synergistic, provided a large overall increase in protein stability (9.1 kcal/mol), and resulted in a highly stable scFvD1.3 derivative. The mechanisms of the mutations and of their combinations involved variations in the resistance to denaturation, cooperativity of unfolding, and likely residual structures of the denatured state, which was constrained by two disulfide bonds. This combined method should be applicable to any recombinant antibody fragment, through a single step of mutagenesis.

Diversity and junction residues as hotspots of binding energy in an antibody neutralizing the dengue virus.

Dengue is a re-emerging viral disease, affecting approx. 100 million individuals annually. The monoclonal antibody mAb4E11 neutralizes the four serotypes of the dengue virus, but not other flaviviruses. Its epitope is included within the highly immunogenic domain 3 of the envelope glycoprotein E. To understand the favorable properties of recognition between mAb4E11 and the virus, we recreated the genetic events that led to mAb4E11 during an immune response and performed an alanine scanning mutagenesis of its third hypervariable loops (H-CDR3 and L-CDR3). The affinities between 16 mutant Fab fragments and the viral antigen (serotype 1) were measured by a competition ELISA in solution and their kinetics of interaction by surface plasmon resonance. The diversity and junction residues of mAb4E11 (D segment; V(H)-D, D-J(H) and V(L)-J(L) junctions) constituted major hotspots of interaction energy. Two residues from the D segment (H-Trp96 and H-Glu97) provided > 85% of the free energy of interaction and were highly accessible to the solvent in a three-dimensional model of mAb4E11. Changes of residues (L-Arg90 and L-Pro95) that statistically do not participate in the contacts between antibodies and antigens but determine the structure of L-CDR3, decreased the affinity between mAb4E11 and its antigen. Changes of L-Pro95 and other neutral residues strongly decreased the rate of association, possibly by perturbing the topology of the electrostatic field of the antibody. These data will help to improve the properties of mAb4E11 for therapeutic applications and map its epitope precisely.

Quantitative measurement of protein stability from unfolding equilibria monitored with the fluorescence maximum wavelength.

The fluorescence of tryptophan is used as a signal to monitor the unfolding of proteins, in particular the intensity of fluorescence and the wavelength of its maximum lambda(max). The law of the signal is linear with respect to the concentrations of the reactants for the intensity but not for lambda(max). Consequently, the stability of a protein and its variation upon mutation cannot be deduced directly from measurements made with lambda(max). Here, we established a rigorous law of the signal for lambda(max). We then compared the stability DeltaG(H(2)O) and coefficient of cooperativity m for a two-state equilibrium of unfolding, monitored with lambda(max), when the rigorous and empirical linear laws of the signal are applied. The corrective terms involve the curvature of the emission spectra at their lambda(max) and can be determined experimentally. The rigorous and empirical values of the cooperativity coefficient m are equal within the experimental error for this parameter. In contrast, the rigorous and empirical values of the stability DeltaG(H(2)O) generally differ. However, they are equal within the experimental error if the curvatures of the spectra for the native and unfolded states are identical. We validated this analysis experimentally using domain 3 of the envelope glycoprotein of the dengue virus and the single-chain variable fragment (scFv) of antibody mAbD1.3, directed against lysozyme.