Carbohydrates are crucial for living cells, playing myriads of functional roles that range from being structural or energy-storage devices to molecular labels that, through non-covalent interaction with proteins, impart exquisite selectivity in processes such as molecular trafficking and cellular recognition. The molecular bases that govern the recognition between carbohydrates and proteins have not been fully understood yet. In the present study, we have obtained a surface-area-based model for the formation heat capacity of protein–carbohydrate complexes, which includes separate terms for the contributions of the two molecular types. The carbohydrate model, which was calibrated using carbohydrate dissolution data, indicates that the heat capacity contribution of a given group surface depends on its position in the saccharide molecule, a picture that is consistent with previous experimental and theoretical studies showing that the high abundance of hydroxy groups in carbohydrates yields particular solvation properties. This model was used to estimate the carbohydrate's contribution in the formation of a protein–carbohydrate complex, which in turn was used to obtain the heat capacity change associated with the protein's binding site. The model is able to account for protein–carbohydrate complexes that cannot be explained using a previous model that only considered the overall contribution of polar and apolar groups, while allowing a more detailed dissection of the elementary contributions that give rise to the formation heat capacity effects of these adducts.

INTRODUCTION

Heat capacity effects have long been recognized as key to understanding the molecular principles that underpin non-covalent driven processes [14]. The absolute heat capacity of a system is determined by dispersion of the enthalpy distribution, which in turn is related to the number of vibrational, rotational and translational ‘soft’ modes susceptible to absorbing thermal energy. Changes in these soft modes are particularly large in processes that take place in aqueous solutions, due to the rearrangement of solvent molecules around solutes. The way the solvent restructures, and the consequent impact of this restructuring on the solution's heat capacity, is highly dependent on the solute's chemical nature. The analysis of a large body of thermodynamic data for the hydration of small organic compounds has shown an increase in heat capacity as a thermodynamic signature for the solvation of apolar groups, whereas the opposite effect is characteristic of polar groups. These distinct effects have been rationalized in terms of different restructuring responses of water molecules, depending on the surface polarity [5]. Hydrophobic solvation is thought to increase the number of energy reservoirs, by forming water–water hydrogen bonds at the first solvation layer that are stronger than at the bulk phase. In contrast, the stronger interactions occurring between solvating water molecules and polar groups prevent these interactions from being excited significantly at ordinary temperatures. Thus ΔCp (change in heat capacity) constitutes a powerful sensor for revealing the types and amounts of solute surfaces involved in processes such as protein folding and binding [6,7].

Carbohydrates have particular hydration characteristics which are crucial for dictating the structure and functionality of these biomolecules [810]. Hydroxy groups are thought to behave similarly to solvent water molecules. Nevertheless, the abundance and proximity of these groups in carbohydrates yield solvation characteristics, such as the total number of water molecules and the way they are shared inside the first hydration shell, which might depart significantly from individual hydroxy groups. In fact, heat capacities for saccharides in dilute aqueous solutions are significantly larger than values calculated from the sum of their individual chemical constituents [11]. Moreover, despite the highly polar character of carbohydrates, Raman scattering experiments have identified them as water-structure enhancers [12]. These properties must be relevant for determining how carbohydrates interact with proteins, imparting unique characteristics to the adducts they form [13,14]. Thus the study of carbohydrate and P–CH (protein–carbohydrate) complexes represents an excellent opportunity to expand our knowledge of the role that the solvent plays in biomolecular recognition processes. In the present study, a model for the heat capacity formation of P–CH complexes (ΔCpP-CH) was developed. This model performs better than an ad hoc model previously obtained [15], allowing a more detailed dissection of the elementary contributions that give rise to the formation heat capacity effects of these adducts.

THEORY

Surface-area-based models for the formation heat capacity of protein–carbohydrate complexes

Simple additivity models have proved to be very useful for estimating heat capacities from structural information for many classes of small non-ionic compounds. Among the different additivity schemes developed and calibrated with model compounds, those based on solvent-accessibility surface area changes (ΔA) became widely used in the analysis of conformational changes in macromolecules. Accessible surface area is a geometrical parameter considered to be proportional to the amount of water in contact with the solute. Therefore ΔA models allow us to take into account a partial exposition/burial of the interacting chemical groups. A widely used ΔA model for ΔCp considers this thermodynamic parameter as a simple function of the solvent-accessibility changes of polar (ΔApol) and apolar (ΔAap) areas [6]:

 
formula
(1)

Using thermodynamic and structural information for a database of seven P–CH complexes, García-Hernández et al. [15] derived an ad hoc parameterization of eqn (1) for these adducts (Table 1). According to this parameterization, both polar and apolar surfaces elicit a decreasing effect on heat capacity upon complex formation. This result is at variance with several results obtained previously for eqn (1), where typically a negative coefficient for polar groups was obtained (Table 1). Furthermore, the absolute contribution of apolar surfaces, although of the sign expected, is relatively small. The structural–energetic behaviour of P–CH complexes is in qualitative agreement with both experimental and theoretical studies, which indicate that carbohydrates bear unique solvation properties imparted by the large abundance of hydroxy groups mounted on the sugar-ring scaffold [11,12,16,17].

Table 1
Heat capacity parameterizations based on polar and apolar surface area changes

Values are means±S.D. and are in units of cal·K−1·mol−1·Å−2. S.D. is only given where available.

 Δcppol Δcpap System 
García-Hernández et al. [15−0.23±0.04 −0.07±0.03 Seven P–C complexes 
The present study −0.24±0.09 −0.06±0.09 14 P–C complexes 
Spolar et al. [18−0.14±0.04 −0.32±0.04 Liquid amides 
Murphy and Freire [19−0.27±0.03 −0.45±0.02 Dipeptide crystals 
Makhatadze and Privalov [20−0.21 −0.51 Protein unfolding 
Myers et al. [21−0.09±0.30 −0.28±0.12 Protein unfolding 
Robertson and Murphy [22−0.12±0.08 −0.16±0.05 Protein unfolding 
Madan and Sharp [23−0.17 −0.17 Nucleic acid fragments 
 Δcppol Δcpap System 
García-Hernández et al. [15−0.23±0.04 −0.07±0.03 Seven P–C complexes 
The present study −0.24±0.09 −0.06±0.09 14 P–C complexes 
Spolar et al. [18−0.14±0.04 −0.32±0.04 Liquid amides 
Murphy and Freire [19−0.27±0.03 −0.45±0.02 Dipeptide crystals 
Makhatadze and Privalov [20−0.21 −0.51 Protein unfolding 
Myers et al. [21−0.09±0.30 −0.28±0.12 Protein unfolding 
Robertson and Murphy [22−0.12±0.08 −0.16±0.05 Protein unfolding 
Madan and Sharp [23−0.17 −0.17 Nucleic acid fragments 

Table 2 shows the seven P–CH complexes originally used to calibrate eqn (1), along with another seven complexes of a known 3D (three-dimensional) structure for which ΔCpP-CH has been measured calorimetrically. Figure 1 shows ΔCpP-CH as a function of ΔApol and ΔAap. In this plot, eqn (1) was normalized by ΔAap to obtain a 2D (two-dimensional) representation of eqn (1). Thus the slope and the y-intercept correspond to the values of Δcppol and Δcpap respectively. It can be seen that the new P–CH complexes (numbers 8–14 in Figure 1) show the same trend as that seen for the original complexes (numbers 1–7), yielding Δcppol and Δcpap values close to those obtained originally with the reduced set of seven P–CH complexes (see the first and second data rows in Table 1). Nevertheless, it is also evident that for several of the new complexes, the dispersion with respect to the fitting line is wider. In particular, complexes 11, 12 and 13 show the widest dispersion in Figure 1. Since the binding heat capacities for these complexes seem to have been measured carefully (for instance, at least nine different temperatures were sampled to obtain ΔCpP-CH in each instance) [27,30,31], the divergences observed are presumably related to particular stereochemical features at their binding interfaces. Complex 11 is made of glucose and a CBM9 (family 9 carbohydrate binding module) from Thermotoga maritima xylanase 10A [27]. Notably, the complex of the same protein module binding cellobiose (complex 7) is clearly better explained by the model. A comparison between the two CBM9-complex crystal structures shows that glucose is bound with an orientation that deviates significantly from that of cellobiose [27], yielding quite different contact patterns. For instance, although the disaccharide buries more polar areas at the binding interface than the monosaccharide, the latter buries a larger amount of the secondary hydroxy's area. Complex 12 (fucose bound to a calcium-dependent tetrameric lectin from Chromobacterium violaceum) shows the widest dispersion in Figure 1. An inspection of the crystal structure reveals that none of the fucose's primary hydroxy groups makes contact with the protein's binding site, whereas the calcium ion is directly mediating the binding [30]. These features are not observed in any other complex in Table 2. Finally, complex 13 corresponds to cellopentaose bound to the CBM4 from Cellulomonas fimi β1-4-glucanase [31]. Surprisingly, the ΔCpP-CH value of this oligosaccharide complex is unusually small, falling in the range observed for monosaccharide complexes. In contrast, the complex made of an evolutionarily related CBM4 from T. maritima β1-3-glucanase and laminaripentaose (complex 14) shows the largest ΔCpP-CH value in Table 2.

Table 2
Heat capacity and surface area changes in protein–carbohydrate complexes

Values for –ΔCpP-CH are means±S.D. S.D. is only given where available. Hev-GlcNAc2: hevein-quitobiose (GlcNAcβ1-4GlcNAc), Hev-GlcNAc3: hevein-quitotriose (GlcNAcβ1-4GlcNAcβ1- 4GlcNAc), Lisoz-GlcNAc2: lysozyme-quitobiose, Lisoz-GlcNAc2: lysozyme-quitotriose, ConA-mMan: concanavalin A-methylmannose, ConA-Man2: concanavalin A-trimannoside core (Manα1-6[Manα1-3]Man), CBM9-Glc2: CBM9 from Thermotoga maritima xylanase 10A-cellobiose (Glcβ1-4Glc); ConA-mMan: concanavalin A-methylglucose, ConA-Man2: concanavalin A-mannobiose (Manα1-3Man), DGL-Man3: Dioclea grandiflora lectin-trimannoside core, CBM9-Glc: CBM9 from Thermotoga maritima xylanase 10A-glucose (Glc), CVIIL-mFuc:Chromobacterium violaceum lectin-methylfucose, CfCBM4-Glc5: CBM4 from Cellulomonas fimi β1-4-glucanase-cellopentaose (Glcβ1-4Glcβ1-4Glcβ1-4Glcβ1-4Glc), TmCBM4-Glc6: CBM4 from Thermotoga maritima β1-3-glucanase-laminarihexaose (Glcβ1-3Glcβ1-3Glcβ1-3Glcβ1-3Glc1-3Glc). Structure-based calculations of water-accessible surface areas were performed with the NACCESS program [32], using a probe radius of 1.4 Å and a slice width of 0.1 Å. Total changes in surface area (ΔAt) were estimated from the difference between the complex and the sum of free molecules. Polar area changes (ΔApol) were obtained from the change in accessible area of nitrogen and oxygen atoms, while the apolar area change (ΔAap) was computed from the contributions of carbon atoms.

 Hev-GlcNAc21 Hev-GlcNAc32 Lisoz-GlcNAc23 Lisoz-GlcNAc34 ConA-mMan 5 ConA-Man36 CBM9-Glc27 ConA-mGlc 8 ConA-Man29 DGL-Man310 CBM9-Glc 11 CVIIL-mFuc 12 CfCBM4-Glc513 TmCBM4-Glc614 
–ΔApol 162 237 276 404 162 364 239 145 246 364 208 157 341 407 
–ΔAap 313 345 309 384 186 245 301 185 205 253 175 152 522 490 
–ΔAt 475 582 545 788 348 609 540 330 451 617 383 309 863 897 
–ΔCpP-CH 64±6 83±8 83±5 119±3 52±11 109±5 67±2 37±5 67±2 96 34±16 80 50±10 172±14 
PDB − − 1lzbm 1lzb 5cna 1cvn 1i82 1gic 1i3h 1dgl 1i8a 2boi 1gu3 1gui 
Reference(s) [13[13[15[15[24,25[24,26[27[25[28[24,29[27[30[31[31
 Hev-GlcNAc21 Hev-GlcNAc32 Lisoz-GlcNAc23 Lisoz-GlcNAc34 ConA-mMan 5 ConA-Man36 CBM9-Glc27 ConA-mGlc 8 ConA-Man29 DGL-Man310 CBM9-Glc 11 CVIIL-mFuc 12 CfCBM4-Glc513 TmCBM4-Glc614 
–ΔApol 162 237 276 404 162 364 239 145 246 364 208 157 341 407 
–ΔAap 313 345 309 384 186 245 301 185 205 253 175 152 522 490 
–ΔAt 475 582 545 788 348 609 540 330 451 617 383 309 863 897 
–ΔCpP-CH 64±6 83±8 83±5 119±3 52±11 109±5 67±2 37±5 67±2 96 34±16 80 50±10 172±14 
PDB − − 1lzbm 1lzb 5cna 1cvn 1i82 1gic 1i3h 1dgl 1i8a 2boi 1gu3 1gui 
Reference(s) [13[13[15[15[24,25[24,26[27[25[28[24,29[27[30[31[31

Heat capacity changes for the formation of protein–carbohydrate complexes as a function of changes in polar (ΔApol) and apolar (ΔAap) surface areas

Figure 1
Heat capacity changes for the formation of protein–carbohydrate complexes as a function of changes in polar (ΔApol) and apolar (ΔAap) surface areas

Normalization of eqn (1) by ΔAap yields a straight-line model where ΔApolAap and ΔCp/ΔAap are the independent and dependent variables respectively and the slope and y-intercept correspond to Δcppol and Δcpap respectively.

Figure 1
Heat capacity changes for the formation of protein–carbohydrate complexes as a function of changes in polar (ΔApol) and apolar (ΔAap) surface areas

Normalization of eqn (1) by ΔAap yields a straight-line model where ΔApolAap and ΔCp/ΔAap are the independent and dependent variables respectively and the slope and y-intercept correspond to Δcppol and Δcpap respectively.

Overall, the above observations draw attention to the limited utility of the model in eqn (1), which only considers the global contributions of polar and apolar surface areas. An improved model is needed, which can more finely capture the molecular determinants of ΔCp in P–CH complexes. In the present study, we aimed to develop a refined model that considers separately the contributions from carbohydrate (ΔCpCH) and protein (ΔCpP) chemical groups:

 
formula
(2)

Where Δcpi and ΔAi are the specific heat capacity and surface area changes of group type i respectively.

The validity of this term separation follows from the evidence that, once corrected for protonation/deprotonation effects, heat capacity effects in the formation of P–CH complexes are mostly dictated by changes in the hydration extent of the interacting groups [15]. Following the same approach as that used with model compounds [33], the transfer of a sugar molecule from an aqueous medium to a solid (pure) state may be used to mimic the incorporation of free sugar into the highly ordered and packed environment of a P–CH interface [34]. Accordingly, dissolution data of carbohydrates may be taken advantage of to generate and calibrate a model for the ΔCpCH term in eqn (2). Once the ΔCpCH model has been parameterized, the carbohydrate contribution to the formation of a given P–CH adduct can be estimated by using the corresponding area changes of the ligand upon binding to its protein counterpart. This contribution can then be subtracted from the experimental ΔCpP-CH to obtain the heat capacity change associated with the protein's binding site.

Surface-area-based model for the dissolution heat capacity of carbohydrates

Table 3 shows a dataset of carbohydrates for which the dissolution ΔCp (ΔCpdiss) has been determined calorimetrically. ΔCpdiss values were obtained by subtracting the sugar's absolute heat capacity in the solid (anhydrous) state from its absolute heat capacity in an aqueous solution (at infinite dilution). The dataset is composed of 14 saccharides spanning a wide range of chemical characteristics, including furanoses, pyranoses, pentoses, hexoses, monosaccharide derivatives and lineal oligosaccharides. The availability of these data was crucial in the present study, as they allowed the inclusion of all the carbohydrate chemical groups present in the P–CH complex database (Table 2). Table 3 also shows surface area changes for carbohydrate dissolution, assuming that in the solid state the molecule is completely dehydrated, whereas in the aqueous medium it is fully exposed to the solvent.

Table 3
Heat capacity and accessible surface area changes for the dissolution of carbohydrates

Surface areas were classified according to the heteroatom type and the configurational location within the carbohydrate molecule in terms of the distance (number of covalent bonds) from the sugar ring (see Figure 2). Surface areas were calculated using the NACCESS program [32]. ΔCp values were determined at 25 °C.

  ΔCpdiss   ΔAi2 
 Carbohydrate (cal·mol−1·K−1·A−2ΔAap2ΔApol2C0 C1 C2,3 O0,1 O2 NCO Reference 
1. Arabinose (Ara) 24 115 156 115   156   [11
2. Fructose (Fru) 33 126 182 38 88  108 74  [11
3. Galactose (Gal) 31 116 188 75 41  147 41  [11
4. Glucose (Glc) 33 117 186 70 47  151 35  [11
5. Mannose (Man) 32 126 178 82 44  139 39  [11
6. Ribose (Rib) 22 118 152 118   152   [11
7. Xylose (Xyl) 29 114 158 114   158   [11
8. Sucrose (Suc) 51 191 258 81 110  175 83  [11
9. Maltotriose (Mtri) 55 277 357 190 87  248 109  [35
10. Maltotetrose (Mtet) 80 364 429 236 128  303 126  [35
11 Methyl-mannose (mMan) 48 187 144 66 46 75 109 35  
12. Methyl-glucose (mGlc) 44 200 141 69 48 83 105 36  
13. dGlc 35 150 137 103 47 101 36  
14. GlcNAc 36 188 182 62 46 80 101 38 43 
  ΔCpdiss   ΔAi2 
 Carbohydrate (cal·mol−1·K−1·A−2ΔAap2ΔApol2C0 C1 C2,3 O0,1 O2 NCO Reference 
1. Arabinose (Ara) 24 115 156 115   156   [11
2. Fructose (Fru) 33 126 182 38 88  108 74  [11
3. Galactose (Gal) 31 116 188 75 41  147 41  [11
4. Glucose (Glc) 33 117 186 70 47  151 35  [11
5. Mannose (Man) 32 126 178 82 44  139 39  [11
6. Ribose (Rib) 22 118 152 118   152   [11
7. Xylose (Xyl) 29 114 158 114   158   [11
8. Sucrose (Suc) 51 191 258 81 110  175 83  [11
9. Maltotriose (Mtri) 55 277 357 190 87  248 109  [35
10. Maltotetrose (Mtet) 80 364 429 236 128  303 126  [35
11 Methyl-mannose (mMan) 48 187 144 66 46 75 109 35  
12. Methyl-glucose (mGlc) 44 200 141 69 48 83 105 36  
13. dGlc 35 150 137 103 47 101 36  
14. GlcNAc 36 188 182 62 46 80 101 38 43 
*

Dr Aaron Rojas and Dr Luis A. Torres, Departamento de Química, CINVESTAV, IPN. Dr Ángeles Olvera and Dr Miguel Costas, Facultad de Química, UNAM, personal communication.

Since the heat capacity contribution of a given chemical surface may depend significantly on the particular environment in which it is immersed, the surface area-based ΔCpCH model was partitioned taking into account both the atom type (carbon, oxygen or nitrogen) and the distance of the atom from the sugar's ring (in terms of the number of covalent bonds). The number of covalent bonds separating a particular atom from the closest sugar ring atom is indicated by numerical subscripts, where 0 corresponds to atoms forming part of the ring skeleton. This classification yields nine different surface area types, as shown schematically in Figure 2. The areas of O0 and glycosidic O1 atoms are marginal in relation to the total polar area exposed by a carbohydrate molecule (∼5% and ∼1% respectively). So, to reduce the number of independent variables, O0 and O1 were gathered into a single group (O0,1). Furthermore, C2 and C3 atoms corresponding to methyl groups were summed up to form another group (C2,3), whereas atoms forming the amide group were also considered as a single group (NCO). Thus the specific model for ΔCpCH takes the form:

 
formula
(3)

Schematic representation of a disaccharide molecule showing the different sugar surface area types considered in the present study

Figure 2
Schematic representation of a disaccharide molecule showing the different sugar surface area types considered in the present study

Surface areas were classified according to the heteroatom type and the configurational location in terms of the distance (number of covalent bonds) from the sugar ring. Atoms forming the amide group were considered to form a single group (NCO). O0 and O1 surfaces were gathered into a single group (O0,1), whereas C2 and C3 surfaces (excepting C2 for the NCO group) were summed up to conform to another group (C2,3). See text for details.

Figure 2
Schematic representation of a disaccharide molecule showing the different sugar surface area types considered in the present study

Surface areas were classified according to the heteroatom type and the configurational location in terms of the distance (number of covalent bonds) from the sugar ring. Atoms forming the amide group were considered to form a single group (NCO). O0 and O1 surfaces were gathered into a single group (O0,1), whereas C2 and C3 surfaces (excepting C2 for the NCO group) were summed up to conform to another group (C2,3). See text for details.

The complete set of parameters for eqn (3) was obtained through three sequential steps. First, ΔcpC0, ΔcpC1, ΔcpO0,1 and ΔcpO2 were obtained from a multilinear regression fit of eqn (3) to data in Table 3, but excluding data for mMan, mGlc and GlcNAc. Secondly, ΔcpC2,3 was then obtained from the analysis of mMan and mGlc, as follows: using the respective surface area changes in these methyl derivatives, the theoretical contribution of C0, C1, O0,1 and O2 to the total ΔCpdiss was calculated. This quantity was subtracted from the experimental ΔCpdiss value to calculate the contribution of the methyl group (ΔCpC2,3). ΔcpC2,3 was then obtained by dividing ΔCpC2,3 by ΔAC2,3. Finally, ΔcpNCO was obtained in a similar way from the analysis of GlcNAc data. Table 4 shows the fitting values obtained through this sequential analysis. The associated errors given in this Table correspond to standard errors of the regression analysis.

Table 4
Heat capacity parameterization based on surface area changes for the dissolution of carbohydrates

Values are in units of cal·K−1·mol−1·Å−2.

 Value Error 
ΔcpC0 −0.09 0.05 
ΔcpC1 −0.36 0.10 
ΔcpC2,3 −0.23 0.04 
ΔcpO0,1 −0.10 0.03 
ΔcpO2 −0.15 0.09 
ΔcpNCO −0.19 − 
 Value Error 
ΔcpC0 −0.09 0.05 
ΔcpC1 −0.36 0.10 
ΔcpC2,3 −0.23 0.04 
ΔcpO0,1 −0.10 0.03 
ΔcpO2 −0.15 0.09 
ΔcpNCO −0.19 − 

As can be seen in Figure 3, the early ad hoc ΔCp model for P–CH complexes (Table 1) systematically overestimates ΔCpdiss of saccharides (solid symbols), with a mean excess [=(ΔCpcalc–ΔCpexp)/ΔCpexp] of 53±23%. This bias is not surprising, as the parameters are weighted by the contributions of both protein and carbohydrate chemical groups. In contrast, the ΔCpCH model (open symbols) satisfactorily accounts for the experimental ΔCpdiss values (R2=0.98). As anticipated [36], results in Table 4 indicate that the thermodynamic behaviour of a given surface atom type is influenced by the relative position of the atom in relation to the saccharide ring. The contribution is larger for carbon atoms located outside the ring (C1 and C2,3) than for those forming the ring (C0). Oxygen surfaces located two covalent bonds away from the ring (O2) show a negative value, whereas those closer to the ring (O0,1) have a small positive value. The amide group has a negative value, similar to that reported for liquid amides [18] (Table 1).

Calculated versus experimental ΔCp for the dissolution of carbohydrates

Figure 3
Calculated versus experimental ΔCp for the dissolution of carbohydrates

Solid symbols were obtained using parameters obtained from the analysis of the 14 P–CH complexes (the second row in Table 1). Open symbols correspond to calculated values obtained with eqn (3) and parameters in Table 4. The solid line represents the best fitting of a straight line.

Figure 3
Calculated versus experimental ΔCp for the dissolution of carbohydrates

Solid symbols were obtained using parameters obtained from the analysis of the 14 P–CH complexes (the second row in Table 1). Open symbols correspond to calculated values obtained with eqn (3) and parameters in Table 4. The solid line represents the best fitting of a straight line.

Heat capacity changes of non-saccharide hydroxylated compounds

It has been typical to assume that the heat capacity effect for the solvation of a given atom or chemical group surface is constant, regardless of the covalent or non-covalent connectivity. Nevertheless, the above results paint a different picture. To explore this issue in an alternative way, an analysis of simpler hydroxylated compounds was carried out. Table 5 shows transfer heat capacity data and accessible surface areas for seven n-alcohols, four diols with no contiguous hydroxy groups [i.e. HO-(CH2)n-OH, where n>2], and seven polyols with contiguous hydroxy groups (including 1,2-ethanediol and 1,2,3-propanetriol). Analysis of these dissolution data can be used to estimate the specific heat capacity contribution of hydroxy and carbon surfaces (ΔcpOH and ΔcpC respectively) in non-cyclic compounds with varying densities of hydroxy groups. For mono-hydroxylated and di-hydroxylated compounds with no contiguous hydroxy groups, ΔcpOH=–0.34 and ΔcpC=0.49 cal·mol−1·K−1·Å−2 (where 1Å=0.1 nm). In contrast, the analysis of polyols yields a positive value for ΔcpOH (=0.17 cal·mol−1·K−1·Å−2), and a significantly decreased value for ΔcpC (=0.13 cal·mol−1·K−1·Å−2). These results indicate that the elementary heat capacity contributions in hydroxylated compounds indeed depend on the density of hydroxy groups. As described in the Introduction section, this behaviour should be related to different characteristics of the solvent's organization. It is worth noting that Δcp values for O2 and C1 surfaces in carbohydrates (Table 4) are similar both in sign and magnitude to ΔcpOH and ΔcpC values in n-alcohols and diols. In contrast, Δcp values for O0,1 and C0 surfaces in carbohydrates are closer to ΔcpOH and ΔcpC values. These concordances suggest that the contribution of a given atom in a sugar molecule depends on its spatial position inside the molecule: the further an atom is from the sugar ring, the closer its solvation behaviour will be to that of simpler compounds.

Table 5
ΔCp and ΔA for the dissolution of simple hydroxylated compounds
Compound ΔCpdiss* (cal·K−1·mol−1·Å−2ΔAC† (Å2ΔAOH‡ (Å2
Methanol 27 81 43 
Ethanol 47 134 43 
1-Propanol 64 167 39 
1-Butanol 79 193 38 
1-Pentanol 96 219 41 
1-Hexanol 112 247 41 
Cyclohexanol 92 210 38 
1,3-Propanediol 40 139 76 
1,4-Butanediol 53 164 75 
1,5-Pentanediol 69 191 80 
1,6-Hexanediol 84 226 81 
1,2-Ethanediol 27 108 83 
1,2,3-Propanetriol 29 111 114 
Arabinitol 43 126 167 
Ribitol 43 120 170 
Xylitol 36 119 176 
Mannitol 53 132 201 
Sorbitol 45 126 203 
Compound ΔCpdiss* (cal·K−1·mol−1·Å−2ΔAC† (Å2ΔAOH‡ (Å2
Methanol 27 81 43 
Ethanol 47 134 43 
1-Propanol 64 167 39 
1-Butanol 79 193 38 
1-Pentanol 96 219 41 
1-Hexanol 112 247 41 
Cyclohexanol 92 210 38 
1,3-Propanediol 40 139 76 
1,4-Butanediol 53 164 75 
1,5-Pentanediol 69 191 80 
1,6-Hexanediol 84 226 81 
1,2-Ethanediol 27 108 83 
1,2,3-Propanetriol 29 111 114 
Arabinitol 43 126 167 
Ribitol 43 120 170 
Xylitol 36 119 176 
Mannitol 53 132 201 
Sorbitol 45 126 203 
*

Values were obtained by subtracting the absolute Cp of the solid from the absolute Cp of a dilute aqueous solution of the same compound. Absolute heat capacities for the solid state were estimated using eqn (4). Data for aqueous solutions were taken from [37,38].

Accessible surface area of carbons.

Accessible surface area of oxygen atoms in hydroxy groups.

Absolute heat capacity of carbohydrate crystals

Benson and Buss [39] found that absolute heat capacities of amino acid crystals can be described adequately in terms of atomic or covalent bond composition. For instance, the authors obtained the following atom-based model from the analysis of amino acid crystals:

 
formula
(4)

where Cpsolid is the crystal's absolute heat capacity, and Ni stands for the number of type i atoms present in the amino acid. Almost 40 years later, Gomez et al. [40] used the parameters of Benson and Buss to reproduce the absolute heat capacity of crystals of anhydrous globular proteins, finding excellent agreement between calculated and experimental values. Therefore it was concluded that Cpsolid (or primary heat capacity, using Gomez et al.'s terminology) is basically determined by the protein's atomic composition, whereas stereochemical contribution (including the particular network of intra and inter non-covalent bonds in the crystal) is negligible. Extending this argument, it would be expected that models such as that of eqn (4) would account for the Cpsolid of other kinds of compounds, including saccharide crystals. As shown in Figure 4, this is, in fact, the case (mean error=2±4%). This agreement is consistent with the picture that ΔCpdiss is not determined by the compound's covalent architecture itself, but by the way the organization of solvating water molecules is affected by the compound's stereochemistry.

Calculated versus experimental Cpsolid for different carbohydrates

Figure 4
Calculated versus experimental Cpsolid for different carbohydrates

Solid bars were obtained using eqn (4), whose parameters were obtained from the analysis of a dataset of amino acid crystals [39]. Open bars correspond to experimental Cpsolid values. Numbers in the X-axis stand for the list position of the corresponding carbohydrate in Table 3.

Figure 4
Calculated versus experimental Cpsolid for different carbohydrates

Solid bars were obtained using eqn (4), whose parameters were obtained from the analysis of a dataset of amino acid crystals [39]. Open bars correspond to experimental Cpsolid values. Numbers in the X-axis stand for the list position of the corresponding carbohydrate in Table 3.

Heat capacity model for carbohydrate-binding sites in proteins

Once having parameterized the ΔCpCH model, the carbohydrate contribution to the formation of each P–CH adduct was estimated using the corresponding ligand's area changes upon binding to its protein counterpart (Table 6). This contribution was then subtracted from the experimental ΔCpP-CH to obtain the heat capacity change associated with the protein's binding site desolvation:

 
formula
(5)
Table 6
ΔA and ΔCp of carbohydrates and proteins upon complex formation
 −ΔACH Hev-GlcNAc21 Hev-GlcNAc32 Lisoz-GlcNAc23 Lisoz-GlcNAc34 ConA-mMan5 ConA-Man36 CBM9-Glc27 ConA-mGlc8 ConA-Man29 DGL-Man310 CBM9-Glc11 CVIIL-mFuc12 CfCBM4-Glc513 TmCBM4-Glc614 Δcp (Makhatadze and Privalov)* 
Sugar O0,1 43 50 63 59 87 198 112 68 108 200 120 104 150 155  
 O2 18 55 66 35 35 46 36 55 36 36 49 88   
 NCO 47 67 52 69            
 C0 58 56 63 82 33 74 115 49 55 73 75 48 164 146  
 C1 30 30 42 28 46 67 56 47 59 68 27 36 89 106  
 C2,3 85 93 78 133 33  31  16       
 Total 266 314 357 437 234 374 329 230 277 377 258 204 452 495  
 −ΔCpCH 32 29 30 33 30 45 35 29 29 46 23 31 54 54  
                 
Protein −ΔAP                
 Cal 27 17 47 65 59 81 25 38 72 89 17 52 144 62 −0.53 
 Car 113 149 75 76 15 23 105 21 19 23 56  125 176 −0.29 
 Polar part of 
 Arg   12 12 12 25 10 18  15  −0.05 
 Asn   19 67 10 15 20 17 15 12  38 12 −0.23 
 Asp   18 47 32  10 35 29  21  −0.32 
 Cys               −0.93 
 Gln  14     13    12  37 63 −0.05 
 Glu 24 27          −0.12 
 His  16             −0.31 
 Lys              −0.38 
 Met               −0.93 
 Ser             −0.31 
 Thr     19   18     −0.30 
 Trp 20 20  32   12      33 −0.90 
 Tyr 22 23   27  12 29   21 −0.02 
 -CONH- 17 52 13 26 11 16 27  31 28 −0.39 
 Calcium            22   −0.39† 
 Total 209 268 188 351 114 235 211 100 174 240 125 105 411 402  
 −ΔCpP 32 54 53 86 18 64 32 38 50 11 40 −4 118  
 −ΔACH Hev-GlcNAc21 Hev-GlcNAc32 Lisoz-GlcNAc23 Lisoz-GlcNAc34 ConA-mMan5 ConA-Man36 CBM9-Glc27 ConA-mGlc8 ConA-Man29 DGL-Man310 CBM9-Glc11 CVIIL-mFuc12 CfCBM4-Glc513 TmCBM4-Glc614 Δcp (Makhatadze and Privalov)* 
Sugar O0,1 43 50 63 59 87 198 112 68 108 200 120 104 150 155  
 O2 18 55 66 35 35 46 36 55 36 36 49 88   
 NCO 47 67 52 69            
 C0 58 56 63 82 33 74 115 49 55 73 75 48 164 146  
 C1 30 30 42 28 46 67 56 47 59 68 27 36 89 106  
 C2,3 85 93 78 133 33  31  16       
 Total 266 314 357 437 234 374 329 230 277 377 258 204 452 495  
 −ΔCpCH 32 29 30 33 30 45 35 29 29 46 23 31 54 54  
                 
Protein −ΔAP                
 Cal 27 17 47 65 59 81 25 38 72 89 17 52 144 62 −0.53 
 Car 113 149 75 76 15 23 105 21 19 23 56  125 176 −0.29 
 Polar part of 
 Arg   12 12 12 25 10 18  15  −0.05 
 Asn   19 67 10 15 20 17 15 12  38 12 −0.23 
 Asp   18 47 32  10 35 29  21  −0.32 
 Cys               −0.93 
 Gln  14     13    12  37 63 −0.05 
 Glu 24 27          −0.12 
 His  16             −0.31 
 Lys              −0.38 
 Met               −0.93 
 Ser             −0.31 
 Thr     19   18     −0.30 
 Trp 20 20  32   12      33 −0.90 
 Tyr 22 23   27  12 29   21 −0.02 
 -CONH- 17 52 13 26 11 16 27  31 28 −0.39 
 Calcium            22   −0.39† 
 Total 209 268 188 351 114 235 211 100 174 240 125 105 411 402  
 −ΔCpP 32 54 53 86 18 64 32 38 50 11 40 −4 118  
*

Specific heat capacity coefficients (in units of cal·K−1·mol−1·Å−2) for protein chemical groups obtained by Privalov and Makhatadze [41] from the analysis of transference data of pertinent model compounds.

Hydration ΔCp for calcium was taken from Markus [42]. This value was corrected for volume effects by the method of Privalov and Makhatadze [41] and normalized by the total accessible surface area to obtain the specific heat capacity coefficient.

ΔCpCH and ΔCpP as long as ΔA values for the different surface types are shown in Table 6. In general, the two binding counterparts contribute unequally to the overall ΔCpP-CH. However, although ΔCpCH can be either significantly larger or smaller than ΔCpP, the carbohydrate tends to bury more surface area (on average 60% of ΔAt) than the protein.

Makhatadze and Privalov [41] developed a ΔCp model for protein folding reactions, in which a fine dissection of protein constituent groups was performed. It was used here to calculate ΔCpP in the formation of P–CH complexes. The Makhatadze and Privalov model includes terms for aliphatic and aromatic carbons (Cal and Car respectively), the peptide group (-CONH-) and the polar moiety of each of the side chains bearing a polar chemical group. The parameters were obtained from the analysis of heat capacity data for various organic model compounds used to mimic the different protein constituent groups. This model proved to estimate satisfactorily the absolute heat capacity of the native and unfolded states of a number of globular proteins [20]. The set of parameters for the Makhatadze and Privalov model, as long as the area changes for each of the proteins in Table 2, are shown in Table 6. Figure 5(A) compares ‘experimental’ ΔCpP values, obtained through eqn (4), with the estimates obtained from the Makhatadze and Privalov model [41]. The poor agreement observed may suggest that protein-binding sites for carbohydrates have also a distinct thermodynamic behaviour. Therefore an ad hoc model for these binding sites was calibrated using ΔCpP values. To keep a moderate number of independent variables, polar areas were separated only into neutral and charged polar areas (Pneu and Pchg respectively), whereas apolar areas, following the Makhatadze and Privalov scheme, were partitioned into aliphatic and aromatic carbon areas.

 
formula
(6)

As mentioned above, the CfCBM4–Glc5 complex (complex 13 in Table 2) has an unusually small ΔCpP-CH value (–50 cal·K−1·mol−1), comparable with those shown by monosaccharide complexes. In contrast, the evolutionarily related TmCBM4–Glc6 complex (complex 14) shows a much more negative ΔCpP-CH value. In spite of this large difference, the two complexes bury fairly similar amounts and kinds of areas at the binding interface (Table 6). Even though detailed structural and thermodynamic comparisons between these complexes have been performed, the bases for this dissimilar behaviour are still unclear [31]. According to our ΔCpCH model, ligands in complexes 13 and 14 contribute similar heat capacity changes, –54 cal·K−1·mol−1 (Table 6). Whereas for complex 14 this contribution represents about one-third of the total ΔCpP-CH, it exceeds the total change in complex 13. Therefore a positive ΔCpP value for CfCBM4 is predicted. In view of this anomalous result, complex 13 was excluded in parameterizing eqn (6).

Heat capacity changes for dehydration of the protein-binding site (A) and for formation of P–CH complexes (B)

Figure 5
Heat capacity changes for dehydration of the protein-binding site (A) and for formation of P–CH complexes (B)

In both panels, solid circles correspond to experimental values. In (A), open triangles represent values calculated with the Makhatadze and Privalov model (see Table 6), and solid triangles are values calculated with the ad hoc parameters for carbohydrate-binding sites in proteins (eqn 6 and Table 7). In (B), open squares correspond to values calculated with eqn (1) using parameters for P–CH complexes (the second row in Table 1), and solid triangles are values calculated with the extended P–CH's ΔCp model obtained in the present study (eqn 7).

Figure 5
Heat capacity changes for dehydration of the protein-binding site (A) and for formation of P–CH complexes (B)

In both panels, solid circles correspond to experimental values. In (A), open triangles represent values calculated with the Makhatadze and Privalov model (see Table 6), and solid triangles are values calculated with the ad hoc parameters for carbohydrate-binding sites in proteins (eqn 6 and Table 7). In (B), open squares correspond to values calculated with eqn (1) using parameters for P–CH complexes (the second row in Table 1), and solid triangles are values calculated with the extended P–CH's ΔCp model obtained in the present study (eqn 7).

Table 7 shows the parameterization obtained for eqn (6) from a multilinear regression analysis of ΔCpP and ΔAP values from Table 6. It is worth noting that both ΔcpCal and ΔcpCar values for carbohydrate-recognition protein sites are similar to those in other parameterizations (Table 1), with ΔcpCal being larger than ΔcpCar, in agreement with that obtained by Makhatadze and Privalov using model compounds [41]. Furthermore, the Δcppchg value is comparable with the specific values for an aspartic acid residue or lysine, although considerably more negative than those for glutamic or arginine. In contrast, the large positive value for Δcppneu differs significantly from most of the specific values for neutral polar areas in the Makhatadze and Privalov model, although it is smaller than the coefficient for the polar part of tryptophan. It is worth mentioning that we tried grouping neutral polar areas in different ways (for instance, separating the aromatic and/or amide surface from the rest of neutral polar surfaces), always obtaining positive values for the corresponding fitting coefficients.

Table 7
Specific heat capacity coefficients for surfaces on binding sites of carbohydrate-recognition proteins
 Value Error 
ΔcpCal −0.35 0.19 
ΔcpCar −0.16 0.09 
Δcppneu −0.43 0.16 
Δcppchg −0.27 0.36 
 Value Error 
ΔcpCal −0.35 0.19 
ΔcpCar −0.16 0.09 
Δcppneu −0.43 0.16 
Δcppchg −0.27 0.36 

Among polar areas forming part of carbohydrate-binding sites of proteins, neutral areas predominate over charged areas. Thus the positive sign of Δcppneu could help rationalize why the Makhatadze and Privalov model tends to underestimate ΔCpP. Nevertheless, other factors not considered hitherto could also be responsible for such a discrepancy. One of these factors could be the incorporation of water molecules into the P–CH interface. The transference of a bulk water molecule to a highly ordered protein environment is expected to decrease the heat capacity of the solution [4345]. To test whether this factor is significant in the formation of P–CH adducts, we performed a regression analysis, in which a term for interface water molecules was included in eqn (6): ΔCpw=ΔcpwNw, where Nw is the number of water molecules exchanged. By considering the crystallographic water molecules located within 3 Å (1 Å=0.1 nm) from both the protein and the carbohydrate, the analysis yielded Δcpw=–5.8±11.0 cal·K−1·mol−1. This value is close to the difference between the absolute heat capacities of liquid water and ice (approx. –8 cal·K−1·mol−1 at 25 °C). However, it has a large regression standard error associated. Furthermore, the inclusion of ΔCpw does not affect significantly the Δcppneu value (=0.37±0.24 cal·K−1·mol−1), whereas the quality of the fitting decreases slightly (R=0.93 with ΔCpw versus R=0.95 without ΔCpw).

Heat capacity model for protein–carbohydrate complexes

Using values in Tables 4 and 7 in eqn (2), the final parameterized model for protein–carbohydrate complexes derived here is:

 
formula
(7)

As shown in Figure 5(B), this new model performs better for reproducing experimental ΔCpP-CH than the original model, which considers only total polar and apolar areas. Furthermore, the refined model is able to explain quantitatively Δcppol and Δcpap values for the complete dataset of P–CH complexes in Table 2. By summing up the contributions of O0,1, O2, NCO, Pneu and Pchg surfaces, the total heat capacity contribution for polar groups (ΔCppol) can be obtained for each complex. The slope of ΔCppol versus ΔApol yields directly Δcppol (=0.16±0.04). The same approach yields Δcpap=0.22±0.03. The values obtained in this way are, within statistical uncertainty, the same as those obtained for eqn (1) using P–CH complexes in Table 2.

A co-linearity analysis of eqn (7) yielded values for VIF (variance inflation factor) smaller than 10 for all the independent variables, with exception of ΔAC0 and ΔAO0,1. Although this indicates a low co-linearity degree in the model, it is clear that ΔCpdiss data for additional (preferentially deoxy and other derivatized) carbohydrates are required in order to further strengthen the model statistically. In the meantime, we think that the co-linearity for ΔAC0 and ΔAO0,1 is tolerable due to the following reasons: (i) none has a VIF value larger than 30; and (ii) the exclusion of any of the saccharides in Table 2 does not yield significant variations in the regression parameters, showing therefore an acceptable stability in the estimations. Additionally, the model is able to account satisfactorily for ΔCpdiss of: (i) 2-deoxy-glucose, which exhibits a significant variation between ΔAC0 and ΔAO0,1 in relation to the other saccharides considered here and (ii) other sugars not included in Table 3, namely α, β and γ cyclodextrins, which are cyclic β1-4 oligosaccharides composed of six, seven and eight glucose residues respectively (results not shown).

DISCUSSION

Carbohydrates have stereochemical properties that impart unique thermodynamic behaviour to these molecules. Not surprisingly, P–CH complexes bearing unique thermodynamic properties have also evolved. Formation ΔG and ΔH, normalized per unit of interface surface area, are significantly larger for P–CH complexes than for other types of protein systems [14]. Furthermore, models obtained for protein folding or protein–protein binding processes perform poorly in accounting for ΔCpP-CH [15]. This situation makes it imperative to obtain ad hoc structural–energetic models for P–CH adducts. In the present study, we have derived a refined model for ΔCpP-CH, taking advantage of dissolution thermodynamic data of sugars. The use of transference data of model compounds has been recurrent in trying to elucidate the molecular determinants of the thermodynamic behaviour of macromolecules. Nevertheless, there is a question always lingering in these kinds of studies, regarding the appropriateness of extrapolating thermodynamic data derived from small compounds to macromolecular systems. As stated by Hedwig and Hinz [46], ‘…to ensure the best quantitative success of a group additivity scheme, it is preferable to choose model compounds that reflect as closely as possible the size, surface area, charge and hydrophobicity of the target moieties for which the thermodynamic properties are to be evaluated.’ Following this premise, the authors used data of oligopeptide molecules to mimic the properties of unfolded proteins, obtaining an additivity scheme that performed better than those calibrated using simpler model compounds. To a large extent, our ΔCpP-CH model achieves this reliability in stereochemical representation, as the transference data used to parameterize it correspond directly to one of the counterparts involved in the macromolecular complex studied.

According to the lines of reasoning exposed in the Introduction section of the present paper, the high positive ΔCp values observed for the hydration of saccharides indicate predominance of hydrophobic solvation effects. Nevertheless, the ΔCpCH model obtained here yields an alternative picture. Contrary to what has been observed for polar groups in most of the previous studies, hydroxy groups attached directly to carbon atoms forming the sugar ring increase the solution's heat capacity. Since secondary hydroxy groups are abundant in saccharide molecules, their positive contribution to the heat capacity is significant. Based on molecular simulations, Lemieux [16,36] proposed that hydroxy groups in sugars direct water molecules towards them, generating ‘gaps’ or ‘void’ spaces over carbon atoms forming part of the sugar ring, i.e. they should have a diminished heat capacity contribution due to an inefficient hydration. Therefore it would be expected that the solvation thermodynamics of carbon atoms forming the sugar ring would be altered in relation to the properties of ‘fully’ solvated carbon atoms. In agreement with this, the heat capacity contribution of the ring's carbon atoms (ΔcpC0) is significantly smaller than that of those located outside the ring (ΔcpC2,3). Furthermore, ΔcpC0 is smaller than the corresponding contribution of carbon atoms in simple hydroxylated compounds (ΔcpC) or smaller than the Δcpap value of any of the previous parameterizations for eqn (1) [1823]. A corollary of Lemieux's hypothesis is that the more distant a chemical group is from the sugar ring, the more ‘typical’ should its solvation be. This is what is seen in the ΔCCH model. Both primary hydroxy and acetamide groups show the characteristic decreasing heat capacity effect of polar groups, whereas the contribution of outer carbon surfaces falls within the range of values observed in simpler model compounds.

Surfaces at the binding sites of carbohydrate-recognition sites in proteins presented similar specific heat capacity contributions to those observed in model compounds and protein folding events. Nevertheless, polar-neutral surfaces are a notable exception, as they showed a large positive solvation heat capacity. Interestingly, Madan and Sharp [23] also obtained a positive value (0.17 cal·K−1·mol−1·Å−2) for polar groups in nucleic acids. The reason for this thermodynamic behaviour is still unclear, although it is presumably related to the hydration properties elicited by the spatial proximity of polar groups in these molecules.

Concluding remarks

P–CH complexes have unique properties that put them in a separate energetic–structural class among protein complexes. Accordingly, ad hoc structural–energetic correlations for these heterocomplexes are required. Previously, we obtained a ΔA-based model for heat capacity formation of P–CH adducts with preformed protein-binding sites. This model, containing a single term for each of the polar and apolar area changes, indicated that both kinds of surfaces diminish the heat capacity upon binding, with the apolar contribution being relatively small in comparison with other parameterizations obtained from protein folding and model compound studies. These results are in qualitative agreement with previous studies demonstrating that the hydration behaviour of carbohydrates is atypical. In the present study, a more sophisticated ΔCpP-CH model was developed, in which the contributions of the two binding counterparts were dissected. No previous parameterizations are able to account for desolvation heat capacity effects of carbohydrates or protein-binding sites. The model obtained for saccharides shows that, indeed, these molecules exhibit unique and rather complex solvation behaviours. The specific heat capacity of a given saccharide surface depends not only on the atom and group type, but also on its configurational location. The further away the surface is from the sugar ring, the closer is its solvation behaviour to simpler model compounds. Carbohydrate-binding sites of proteins, in particular neutral polar groups, seem also to show unique solvation properties. Explicit consideration of these stereochemical properties allows accounting for the energetics of a larger variety of P–CH complexes.

Although these results are encouraging, it is to be anticipated that the inclusion of other factors not considered in the present study might lead to an improvement of predicted P–CH heat capacities. For binding processes where solvation/desolvation effects are predominant, ΔA models are able to explain the observed ΔCp satisfactorily. Nevertheless, large differences between experimental and calculated ΔCp values have been observed for an increasing number of protein complexes [7,43]. In many of these instances, the occurrence of coupled equilibria such as exchange of counterions, structural water molecules, protonation/deprotonation of ionizable groups and large conformational dynamics changes has been established. These factors, which imply additional contributions to the simple change in solvent exposition of the interacting surfaces, have been named by Ladbury and Williams [47] as the extended interface. Clear evidence for non-local effects in P–CH complexes has appeared in the literature in the last few years [48]. As more quantitative information about these effects accumulates, it will be possible to elaborate more accurate semi-empirical models for these kinds of biomolecular adducts.

We thank Dr Miguel Costas for his critical review and valuable comments on this paper prior to submission.

Abbreviations

     
  • ΔA

    change in surface area

  •  
  • ΔCp

    change in heat capacity

  •  
  • CBM9

    family 9 carbohydrate binding module

  •  
  • 3D

    three-dimensional

  •  
  • P–CH complex

    protein–carbohydrate complex

  •  
  • VIF

    variance inflation factor

FUNDING

This work was supported by the CONACyT (Consejo Nacional de Ciencia y Tecnología) [grant numbers 47097 and 41328] and DGAPA [PAPIIT; IN204609-3]. E. A. C. received fellowships from the CONACyT and DGAPA.

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