Mathematical models describing oxygen binding by hemoglobin

Abstract

Despite the fact that the investigation of the structural and functional properties of hemoglobin dates back more than 150 years, the topic has not lost its relevance today. The most important component of these studies is the development of mathematical models that formalize and generalize the mechanisms determining the cooperative binding of ligands based on data on the structural and functional state of the protein. In this work, we review the mathematical relationships describing oxygen binding by hemoglobin, ranging from the classical Hüfner, Hill, and Adair equations to the Szabo-Karplus and tertiary two-state mathematical models based on the Monod-Wyman-Changeux and Koshland-Némethy-Filmer concepts. The generality of the considered equations as mathematical functions, bearing in their basis a power dependence, is demonstrated. The problems and possible solutions related to approximation of experimental data by the oxygenation equations with correlated fitting parameters are noted. Attention is paid to empirical equations, extended versions of the Hill equation, where the coefficient of cooperation is modulated by Gauss and Lorentz distributions as functions of partial oxygen pressure.

Introduction

Biological systems are inherently self-regulating, and one manifestation of this regulation is cooperativity. Despite the advances in the study of the structure of biological macromolecules, the mechanisms of cooperative binding of ligands to them are still largely unclear (Imai 1982; Mahadevi and Sastry 2016; Nagatomo et al. 2015; Pabis et al. 2018; Porter and Miller 2012; Stefan and Le Novere 2013). The emergence of new experimental data demands improvements in existing classical models, the development of new concepts, and the search for alternative interpretations (Cui and Karplus 2008; Edelstein 2014; Goutelle et al. 2008; Griffon et al. 1998; Gruber et al. 2019; Henry et al. 2002; Henry et al. 2021; Henry et al. 1997; Hofmeyr and Cornish-Bowden 1997; Horovitz and Mondal 2021; Lee and Karplus 1983; Prinz 2010; Rapp and Yifrach 2017; Szabo and Karplus 1972a; Weiss 1997). Hemoglobin has played a huge role in the understanding of cooperativity and allosteric interactions. A number of critical, reviewing, and retrospective publications have been devoted to this molecule (Eaton 2022; Eaton et al. 2007; Eaton et al. 1999; Edelstein 1975; Edsall 1972; Gell 2018; Giardina et al. 1995; Yuan et al. 2015).

Following the chronology of the events, we consider that the mathematical models are the most significant, in our opinion, in this context. The description of structural concepts in terms of the evolution of cooperativity and allosteric mechanisms in living systems is certainly one of the most interesting and important biological tasks, but because of its depth, it requires a separate discourse that is beyond the format of this review.

Specifically, in this work, we review the mathematical expressions that describe the binding of oxygen by hemoglobin, ranging from the classical Hüfner, Hill, and Adair equations to the Szabo-Karplus mathematical models and tertiary two-state models based on the concepts by Monod-Wyman-Changeux and Koshland and Némethy-Filmer.

We also consider our previously proposed empirical mathematical models of oxygenation based on the Hill equation, where the coefficient of cooperativity is modulated by the Gauss and Lorentz distributions as a function of oxygen partial pressure. These models, on the one hand, are based on modern concepts of molecular biophysics and, on the other hand, allow a better description of the experimental results in relation to its most famous prototype, the Hill equation.

First mathematical models of oxygen binding by hemoglobin

Based on the notion of the reversibility of a chemical reaction and its equilibrium, the law of mass action, Hüfner (1890) obtains probably the first mathematical equation describing oxygen binding by hemoglobin as a bimolecular chemical reaction. Although this equation could not satisfactorily approximate the experimental points of the S-shaped dissociation curve of oxyhemoglobin (ODC), it nevertheless served as the basis for the construction of other regression models of oxygenation.

It should be noted that the dependence of the form

$$\frac{y}{1-y}=kx\mathrm{or}y=\frac{\mathit{kx}}{1+kx},$$ 1

where y is the fraction of occupied binding centers, x here and hereafter is the concentration of the component to be bound (ligand, adsorbate, substrate, partial gas pressure, etc.), and k is the equilibrium constant also underlies the description of various physicochemical and biological regularities: the enzymatic Michaelis-Menten kinetics equation (Michaelis and Menten 1913), the Langmuir absorption isotherm (Langmuir 1916), the dependence of microbial growth rate on substrate concentration (Monod 1949), etc.

Hill (1910) proposes an empirical equation of oxygenation, where he introduces another parameter n, which can take a non-numerical value. Hill gives a physical meaning to this parameter, presenting it as an average value of the degree of aggregation of hemoglobin molecules.

In general terms, Hill’s equation can be represented as follows:

$$y=\frac{{(kx)}^n}{1+{(kx)}^n}=\frac{K{x}^n}{1+K{x}^n}.$$ 2

Hüfner’s equation (1) can be seen as a special case of the Hill equation with n = 1.

The simplicity of the treatment and fitting of the parameters K (macroscopic constant, also known as Hill constant, K_h) and n (“cooperativity indicator,” Hill coefficient, n_h, h), combined with the good approximating ability of this equation, allowed to expand the limits of its application, covering such fields of research as enzymology, pharmacology, regulation of gene transcription, etc. (Aramaki et al. 2011; Bordbar et al. 2004; Goutelle et al. 2008; Li et al. 1993; Srinivasan et al. 2020). It follows from Hill’s assumptions about possible aggregation that hemoglobin molecules within the aggregate must coordinate ligand binding in a certain way, i.e., the aggregate itself can be viewed as a system with several binding centers to which ligands are simultaneously attached.

Although the concept of “cooperativity” would be introduced into science somewhat later (Fowler and Kapitza 1929), the idea of aggregation of hemoglobin molecules would be developed in the studies of G. Adair, but in a different way (Adair et al. 1925).

Having established the fact that the hemoglobin molecule has four binding centers, considering the variant of consecutive oxygen attachment by closely located iron atoms, where the fully liganded state of the system should be the most stable, Adair proposed his equation for oxygenation (3):

$$y=\frac{K_{1}x+2·K_2{x}^2+3·K_3{x}^3+4·K_4{x}^4}{4·\left(1,+,K_1,x,+,K_2,x^2,+,K_3,x^3,+,K_4,x^4\right)}$$ 3

Adair was probably the first who tried to relate the phenomenological equation to the still unknown spatial structure of this protein molecule.

Subsequently, Klotz (1946) performed a deconvolution of the macroscopic K₁–K₄ constants in the Adair’s equation to the differing virtual (apparent) k₁–k₄ reaction equilibrium constants (Eq. (4), also known as the Adair-Klotz equation):

$$y=\frac{k_{1}x+2·k_1{k}_2{x}^2+3·k_1{k}_2{k}_3{x}^3+4·k_1{k}_2{k}_3{k}_4{x}^4}{4·\left(1,+,k_1,x,+,k_1,k_2,x^2,+,k_1,k_2,k_3,x^3,+,k_1,k_2,k_3,k_4,x^4\right)}.$$ 4

Like Hüfner’s Eq. (1), Adair’s equation (3) assumes a bimolecular reaction, the only difference being that ligand binding in Eq. (1) takes one step, while in Eq. (3) it takes four steps. For hemoglobin with four binding centers, the Hill equation with n = 4 should describe a pentamolecular reaction that is practically impossible. Therefore, the terms in this equation can be seen more as a result of differentiating a partition function of possible liganded states of the hemoglobin molecule (Nechipurenko 2014), where Ξ is the partition function:

$$\Xi \left(\mathrm{Ad}\right)=1+4·K_{1}x+6·K_2{x}^2+4·K_3{x}^3+K_4{x}^4.$$ 5

Later, Bernard (1960) also proposes an equation with a partition function of the form (6):

$$\Xi \left(\mathrm{Be}\right)=1+6·K_2{x}^2+K_4{x}^4.$$ 6

The equation describes the two-step oxygenation of hemoglobin (trimolecular reaction) by simultaneous addition of two oxygen molecules to the protein (n = 2 and 4, in contrast to Adair equation, where n = 1, 2, 3, and 4). Equations (1) and (2) with n = 1 and n = 4, respectively, as well as Eq. (5), can be considered as a special case of Eq. (3). At the same time, the Adair equation can also be represented as an approximating polynomial with non-integer values of power exponents.

It should be noted that the Adair equation provides greater agreement with the experiment relative to the Hill equation, not so much because of a more realistic view of oxygenation as a stepwise bimolecular reaction, but as a mathematical model that has a larger number of fitting parameters (four vs two). At the same time, finding the constants in the Adair equation is fraught with errors due to both measurement errors in the experimental data and the choice of method and initial optimization conditions (Imai 1990; Yonetani et al. 2002). Also, the resulting virtual reaction equilibrium constants of this equation are difficult to interpret since they can take close to zero or even negative values. It is also not clear from these constants how to find the value of hemoglobin half-saturation with oxygen p₅₀, which is easily defined in the Hill equation as K⁻ⁿ.

Pauling (1935) reinterprets the Adair equation from the position of biophysical chemistry, assuming that liganding proceeds with binding constant k, which is pH dependent but constant for each oxygenation step. An additional parameter a, characterizing the interaction of the two liganded gems, is introduced. Just like constant k, parameter a has a constant value at all stages of hemoglobin oxygenation.

Giving priority to the planar configuration of the hemes, assuming that they must be located as close to each other as possible for effective interaction, Pauling proposes an equation with a partition function of the form (7):

$$\Xi \left({\mathrm{P}}_{\mathrm{sq}}\right)=1+4·kx+\left(4,·,a,+,2\right)·k^2{x}^2+4·a^2{k}^3{x}^3+a^4{k}^4{x}^4.$$ 7

Pauling also considers a variant of arrangement of hemes in the form of a tetrahedron and the oxygenation equation corresponding to this spatial configuration. In this case, the partition function has the following form (8):

$$\Xi \left({\mathrm{P}}_{\mathrm{th}}\right)=1+4·kx+6·a{k}^2{x}^2+4·a^3{k}^3{x}^3+a^6{k}^4{x}^4.$$ 8

At the same time, Pauling rejected the scheme of arrangement of hemes in the form of a tetrahedron, which is close to the natural structure of hemoglobin, as he suggested that the emerging cooperation between the ligand binding centers requires the direct interaction of the latter.

In 1937, with the beginning of studies of the hemoglobin structure using an X-ray diffraction on protein crystals, M. Perutz opened a new stage in the study of oxygen binding properties of this biopolymer. It became clear that the hemoglobin molecule did not fit into the structural model based on a simple geometric order. Perutz (1970a, 1970b, 1978, 1979, 1989) established the spatial organization and investigated the conformational rearrangements of the hemoglobin macromolecule (Perutz et al. 1960), which stimulated the development of new structural and related mathematical models of oxygenation.

The interpretation of the parameter n in the Hill’s equation proposed earlier by Hill can now be considered not as an average of the variable number of hemoglobin molecules in its aggregates, but as a number of monomers in a macromolecule (tetramer), which corresponds to the number of oxygen binding centers established by Adair. In this case, the parameter introduced by Pauling characterizing the interaction of liganded hemes can be used to explain why the experimental non-integer value of n in the Hill equation is lower than the number of subunits of the macromolecule.

Classical mathematical models of hemoglobin oxygenation Monod-Wyman-Changeux and Koshland-Némethy-Filmer

Monod et al. (1965), based on the data from X-ray analysis of hemoglobin, on their own previous studies, and on studies by other scientific groups, propose a symmetric (consistent) model explaining the appearance of cooperativity in oligomeric proteins consisting of identical protomers (subunits). They suggested that each subunit can exist in two forms—relaxed (R) and tense (T).

The partition function of the ligand binding equation for the Monod-Wyman-Changeux (MWC) model appears as (9)

$$\Xi \left(\mathrm{MWC}\right)=L_0{\left(1,+,k_T,\mathrm{x}\right)}^n+{\left(1,+,k_R,\mathrm{x}\right)}^n,$$ 9

where L₀ is the equilibrium constant between the unliganded T and R forms of the quaternary structure of the protein [T₀]/[R₀], k_T and k_R are the equilibrium constants of ligand binding by the oligomer in its T and R form, respectively, and n is the number of ligands bound (for hemoglobin, equal to four).

Koshland Jr et al. (1966), considering/contemplating the data on the conformational rearrangement of hemoglobin molecules and the previously proposed structural concept of induced matching (Koshland Jr. 1958), developed a model of cooperative binding of ligands (known as “sequential model”) and analyzed various variants of the geometric construction of subunits in the oligomer composition. The symmetric model of allosteric regulation of MWC proteins can then be seen as a particular case of the Koshland-Némethy-Filmer (KNF) sequential model, just as the Hill model is a particular case of the Adair model (Newsholme and Start 1973).

The Koshland-Némethy-Filmer equation follows from the Pauling equation, despite some difference in the treatment of the mechanism of cooperativity (Koshland Jr et al. 1966). Therefore, the partition function for the tetrahedron configuration is similar to that of the Pauling equation (10):

$$\Xi \left({\mathrm{KNF}}_{\mathrm{th}}\right)=1+4·k_{\mathit{AB}}^3{k}_R{k}_{F}x+6·k_{\mathit{AB}}^4{k}_{\mathit{BB}}{k}_R^2{k}_F^2{x}^2+4·k_{\mathit{AB}}^3{k}_{\mathit{BB}}^3{k}_R^3{k}_F^3{x}^3+k_{\mathit{BB}}^6{k}_R^4{k}_F^4{x}^4.$$ 10

Here, k_R is the equilibrium constant between the inactive A and active B conformations of the [B]/[A] subunits, k_F is the equilibrium constant of ligand binding by the active B subunit, k_AB is the equilibrium constant ([AB][A])/([AA][B]), and k_BB is the equilibrium constant ([BB][A][A])/([AA][B][B]) with three possible variants of interacting subunits AA, AB, and BB.

It is evident that the Hill and MWC equations are similar in the sense that the Hill model assumes simultaneous ligand binding, while the MWC model considers simultaneous transition of all subunits from one conformational state to another. In addition, the MWC model, despite its opposition to KNF, has a common conceptual basis with it, realized, however, at different levels of the spatial organization of the oligomer with some difference in the interpretation of the mechanisms of conformational changes. In the MWC model, ligand binding induces the change at the quaternary structure level (or contributes to the change), while in the KNF model, the ligand-induced conformational rearrangement of the macromolecule takes place at the tertiary level, affecting the neighboring subunits. For this reason, the KNF model assumes a sequential transition of the oligomer from one conformational state to another, in contrast to MWC, which makes it possible to explain the negative cooperativity observed for some enzymes.

Mathematical models of oxygenation based on Monod-Wyman-Changeux equation

Perutz’s X-ray data with higher spatial resolution, nanosecond ligand binding kinetics data (Henry et al. 1997), and several other findings (Mozzarelli et al. 1991) made it possible to further develop the MWC model.

Ogata and McConnell (1972a, 1972b) propose an improved version of the MWC equation with five parameters, where α- and β-subunits are not equivalent to each other. This equation is also consistent with the concept of coordinated transition of tetramer subunits from one conformational state to another. Szabo and Karplus (1972a) make a refinement to the Ogata-McConnell equation by introducing additional parameters c^α and c^β, representing it with the following partition function (11):

$$\Xi \left(\mathrm{OMc},/,\mathrm{SK}\right)=L_0{\left(1,+,c^{\alpha },k_T^{\alpha },\mathrm{x}\right)}^2{\left(1,+,c^{\beta },k_T^{\beta },\mathrm{x}\right)}^2+{\left(1,+,k_R^{\alpha },\mathrm{x}\right)}^2{\left(1,+,k_R^{\beta },\mathrm{x}\right)}^2,$$ 11

where the equilibrium constants k_T and k_R for α- and β-subunits are separated and c^α and c^β are the allosteric constants for the hemoglobin deoxyform.

Furthermore, based on Perutz (1970a, 1970b), works related to the study of the role of H⁺ ions, OH⁻, subunit salt bridges, and 2,3-diphosphoglycerate (DPG) in the structural rearrangements of the hemoglobin molecule during its oxygenation, Szabo and Karplus propose a mathematical model of the Perutz stereochemical mechanism as an equation with six parameters, two variables, with the following partition function (Szabo and Karplus 1972a, 1972b) (12a–12c):

$$\Xi {\left(\mathrm{SK}\right)}_{{\mathrm{OH}}^{-}}={\Xi }_D\left(x,,,\mu \right)+{\Xi }_O\left(x,,,\mu \right),$$ 12a

$${\Xi }_D\left(x,,,\mu \right)=Q{S}^6{\left[1,+,\frac{\mu H^{\alpha }}{S}\right]}^2{\left[1,+,\frac{\mu H^{\beta }}{S}\right]}^2{\left\{1,+,\frac{1+\mu H^{\alpha }}{S^2\left(1,+,\frac{\mu H^{\alpha }}{S}\right)},k^{\alpha },x\right\}}^2{\left\{1,+,\frac{1+\mu H^{\beta }}{S^2{\left(1,+,\frac{\mu H^{\beta }}{S}\right)}^2},k^{\beta },x\right\}}^2,$$ 12b

$${\Xi }_O\left(x,,,\mu \right)={\left[1,+,\mu ,H^{\alpha }\right]}^2{\left[1,+,\frac{\mu H^{\beta }}{S}\right]}^2{\left\{1,+,k^{\alpha },x\right\}}^2{\left\{1,+,\frac{1+\mu H^{\beta }}{S\left(1,+,\frac{\mu H^{\beta }}{S}\right)},k^{\beta },x\right\}}^2,$$ 12c

where Ξ_D and Ξ_O are the partition functions of the deoxy- and oxy-states of the quaternary structure of hemoglobin; Q is the (hypothetical) equilibrium constant for the quaternary structure of the protein across the four inter-subunit salt bridges; S is the strength of the salt bridge (equal for all six bridges, including two intrasubunit bridges); H^α and H^β are hydroxyl binding constants for salt bridges originating from α- and β-subunits, respectively; k^α and k^β are ligand binding equilibrium constants of α- and β-subunit tetramer; and μ is the OH⁻ concentration.

It should be noted that at high values of the hydroxide ion concentration, the statistical sum for the Szabo-Karplus (SK) equation will be compatible with the classical Monod-Wyman-Changeux equation (9) (Eaton 2022).

Szabo and Karplus later proposed a mathematical model of hemoglobin oxygenation that takes into account the contribution of DPG (Szabo and Karplus 1976). Since this phosphate not only shifts the equilibrium between the deoxy- and oxyforms of the quaternary structure of the protein but also changes the oxygen affinity of the β-subunits of the deoxyform oligomer (Lindstrom and Ho 1972), the model obtained by the authors began to describe cooperativity, not only in the MWC views and formulations but also in KNF. This model postulates that each of the two conformations of the quaternary structure of hemoglobin has two conformations of the tertiary structure of its constituent subunits. Moreover, the subunit conformation variant is determined by the presence of a homotropic ligand. The mathematical model has nine parameters and two variables. The corresponding thermodynamic model relies on a partition function of the form (13a–13c):

$$\Xi {\left(\mathrm{SK}\right)}_{\mathrm{DPG}}={\Xi }_D\left(x,,,\upsilon \right)+{\Xi }_O\left(x,,,\upsilon \right),$$ 13a

$${\Xi }_D\left(x,,,\upsilon \right)=L_0\left[1,+,\upsilon ,P_D\right]{\left[1,+,c^{\alpha },k^{\alpha },x\right]}^2\left\{1,+,2,\left[\frac{1+\upsilon P_{D\prime }}{1+\upsilon P_D}\right],c^{\beta },k^{\beta },x,+,\left[\frac{1+\upsilon P_{D″}}{1+\upsilon P_D}\right],{\left(c^{\beta },k^{\beta },x\right)}^2\right\},$$ 13b

$${\Xi }_O\left(x,,,\upsilon \right)=\left[1,+,\upsilon ,P_O\right]{\left\{1,+,k^{\alpha },x\right\}}^2{\left\{1,+,k^{\beta },x\right\}}^2,$$ 13c

where L₀ is the equilibrium constant between the deoxy and oxy states of the quaternary structure of the protein (with refinement—also unliganded and DPG); P_D, P_D′, and P_D″ are the association constants of DPG with β-subunits for deoxyform quaternary structure with possible number of oxygen molecules as 0, 1, and 2, respectively; P_O is the binding constant of DPG to the quaternary oxyform (hypothetically); k^α and k^β are the equilibrium ligand binding constants for α- and β-subunit tetramers; c^α and c^β are allosteric constants for α- and β-subunits; and υ is the DPG concentration.

Having refined Perutz’s interpretation of the stereochemical mechanism, taking into account the results of studies on salt bridges of mutant hemoglobin with low affinity for oxygen (Anderson 1975), Lee and Karplus revised and generalized the previously proposed SK model (Lee and Karplus 1983). The equations have ten parameters and two variables, with the following partition functions (14a–14c):

$$\Xi {\left(\mathrm{SKL}\right)}_{{\mathrm{OH}}^{-}}={\Xi }_D\left(x,,,\mu \right)+{\Xi }_O\left(x,,,\mu \right),$$ 14a

$${\Xi }_D\left(x,,,\mu \right)=\frac{Q{S}^6}{r{\prime }^2}{\left[1,+,\frac{\mu H^{\alpha }}{S}\right]}^2{\left[1,+,\frac{\mu H^{\beta }}{S}\right]}^2{\left\{1,+,r^2,\frac{\left[1,+,\frac{\mu H^{\alpha }}{\mathit{rS}}\right]}{\left[1,+,\frac{\mu H^{\alpha }}{S}\right]},k_D^{\alpha },x\right\}}^2{\left\{1,+,r^2,\frac{\left[1,+,\frac{\mu H^{\beta }}{\mathit{rS}}\right]}{\left[1,+,\frac{\mu H^{\beta }}{S}\right]},k_D^{\beta },x\right\}}^2,$$ 14b

$${\Xi }_O\left(x,,,\mu \right)={\left[1,+,\mu ,H^{\alpha }\right]}^2{\left[1,+,\frac{\mu H^{\beta }}{r^{\prime }S}\right]}^2{\left\{1,+,k_O^{\alpha },x\right\}}^2{\left\{1,+,\frac{\left[1,+,\mu ,H^{\beta }\right]}{r^{\prime }S\left[1,+,\frac{\mu H^{\beta }}{r^{\prime }S}\right]},k_O^{\beta },x\right\}}^2,$$ 14c

where r and r′ are constants, in the expression rS and r′S determine the effective salt-bridge strength of the liganded tetramer deoxyformed chains (subunits) and the effective internal salt-bridge strength of the β-chain (β-subunit) unliganded oxyform, respectively.

Lee et al. (1988) also considered the constants of the Adair equation within the SKL mathematical model, noting that the constants k₁–k₄ depend on r and r′, but that k₁ and k₄ are the most accessible to experimental determination.

Brunori et al. (1986) proposed a model that incorporated the postulates of MWC and KNF. In this model, the subunits interact according to the principle of induced correspondence, forming a functional unit “сooperon” that exists in two of the possible states of the quaternary conformation of the oligomer. However, the assumption that the T conformation cooperatively binds oxygen has not been experimentally confirmed, which has led to a debate (Eaton 2022). The partition function for this equation with six parameters is as follows:

$$\Xi \left(\mathrm{Cooperon}\right)=L_0{\left[1,+,\left(k_T^{\alpha },+,k_T^{\beta }\right),\mathrm{x},+,{\delta }_T,k_T^{\alpha },k_T^{\beta },{\mathrm{x}}^2\right]}^2+{\left[1,+,\left(k_R^{\alpha },+,k_R^{\beta }\right),\mathrm{x},+,{\delta }_R,k_T^{\alpha },k_T^{\beta },{\mathrm{x}}^2\right]}^2,$$ 15

where δ_T and δ_R are the affinity increases to the binding of the second ligand by the αβ-dimer T and R forms of the tetramer, respectively.

Based on the experimental X-ray data analysis of hemoglobin single crystals grown in polyethylene glycol solutions (Mozzarelli et al. 1991), Henry et al. (2002) proposed a model based on the SK/SKL idea with the difference that the subunit conformation is no longer determined by the obligatory presence of a homotropic ligand. Thus, this model has two levels of two states: quaternary and tertiary (tertiary two-state), in contrast to the classical MWC with one level of two states (quaternary two-state). The mathematical model uses a total of 18 fitting parameters to fit the known kinetic and equilibrium data. The partition function assuming α=β has five fitting parameters:

$$\Xi {\left(\mathrm{TTS}\right)}_{\alpha =\beta }=\frac{L_0}{l_T^4}{\left[1,+,k_r,x,+,l_T,\left(1,+,k_t,x\right)\right]}^4+{\left[1,+,k_r,x,+,l_R,\left(1,+,k_t,x\right)\right]}^4,$$ 16

where l_T and l_R are the t/r ratio (t and r level states of the tertiary structure) of the unliganded subunits for the T and R states of the quaternary structure, respectively, and k_t and k_r are the equilibrium constants of ligand binding by the subunits in their t and r states.

It should be noted that despite the greater amount of evidence in favor of the MWC concept, both MWC and KNF models are phenomenological because they do not answer the question of how ligand binding gives the observed allosteric effect at the atomic level of detail (Cornish-Bowden 2014; Cui and Karplus 2008).

An interesting structural concept has been the idea of a concerted dissociative model, which describes the equilibrium of alternative quaternary structural ensembles whose architecture is determined by alternative conformations in the dissociated state (Jaffe and Lawrence 2012). The accumulation of experimental evidence supporting such a mechanism, its mathematical description, might allow a better understanding of the biological evolution of the cooperativity phenomenon.

The considered mathematical models of hemoglobin oxygenation have a common conceptual basis and, in our opinion, can be represented as a unified scheme of possible combinations of the structural state of a macromolecule. This state of hemoglobin can be described by three parameters: ligand binding (L-State), conformational state of the tertiary (T-State), and quaternary (Q-State) protein structures (Fig. 1).

Fig 1.

A unified scheme of possible combinations of the structural state of hemoglobin. Models: A TTS, B MWC, C KNF, D Adair, E Bernard, F Hill (n = 4), and G Hüfner. Legend: L-State (Ligand State), liganding state (E, Empty; B, Busy), number of vacant E0–E4 and occupied binding sites B0–B4; T-State (Tertiary State), subunit state (T-T, Tense; T-R, Relax), the number of subunits in the R- and T-state R0–R4 and T0–T4, respectively; Q-State (Quaternary State), tetramer state (Q-T, Tense; Q-R, Relax) and the number of such states R0–R1 and T0–T1 (i.e., R or T). The sum of E and B determines the number of ligand binding sites (4); the sum of T-R and T-T specifies the number of subunits (4), and the sum of Q-R and Q-T specifies the number of tetramers (1). The TTS model corresponds to 50 combinations of such states; MWC corresponds to a combination of 10 states (states T-R1T3–T-R3T1, as well as Q-R1T0/T-R0T4 and Q-R0T1/T-R4T0 are not provided by the model); KNF—5 (Q-State does not differ and is equal to 1, L- and T-State are conjugated, E4B0/R0T4, E3B1/R1T3, E2B2/R2T2, E1B3/R3T1, and E0B4/R4T0 are available); Adair’s model corresponds to a combination of 5 states (Q- and T-State are indistinguishable, L-states E4B0, E3B1, E2B2, E1B3, and E0B4 are available); Bernard’s model corresponds to a combination of 3 states (similar to Adair, states E4B0, E2B2, and E0B4); Hill—2 (for the hemoglobin tetramer, similar to the Bernard model, states E4B0 and E0B4); and Hüfner’s model corresponds to a combination of 2 states (similar to Hill, states E1B0 and E0B1)

Also, in the form of a diagram for each of the possible combinations of the structural state of hemoglobin, one can represent a set of microstates that are attained with varying degrees of probability (Fig. 2).

Fig 2.

A set of hemoglobin microstates for the E2B2/R2T2 combination. Legend: hemoglobin molecule described by L-, T-, and Q-State (left); one of the Q-States of the tetramer (here, Q-T, center); matrix of possible microstates for E2B2/R2T2 (right); green arrows represent intra- and intersubunit salt bridges; red arrow corresponds to β-subunit salt bridges with DPG

Limitations in applicability of mathematical models in solving inverse problems: possible solutions

Increasing the accuracy of the description of allosteric models tends to increase the parameters in the corresponding mathematical equations, which makes it difficult to solve inverse problems due to problems with the stability of such solutions. Therefore, the fundamental problem that hindered the use of the classical MWC and KNF models (and, of course, the solutions based on them) was that it was difficult to reliably determine the values of the parameters of these equations, because they correlated with each other and were sensitive to the method of data fitting. Because of this, experimental data are often fitted to the Hill equation, which provides an estimate of the measure of cooperativity but does not allow an understanding of its origin. Gruber et al. (2019) propose a solution to relate the Hill coefficient from the equation of the same name to the MWC model parameters, which improves robustness in finding the combination of physically relevant parameters such as L₀, k_T, and k_R.

A somewhat different approach in describing the experimental dissociation curves is proposed by Lavrinenko et al. (2022a, b, 2023). Given the importance and prevalence of the Hill equation in describing cooperativity in biochemistry, pharmacology, and molecular biology in general, it has been suggested that the Hill coefficient should be considered as a function of ligand concentration, in particular the partial pressure of oxygen (17):

$$y=\frac{x^h}{p_{50}^h+x^h},h=f(p).$$ 17

In Eq. (17), the Hill coefficient is modulated by the Gauss (17a) or Lorentz (17b) distribution:

$$h=\frac{h_{\max }-1}{exp{\left[ln,\left(p,/,p_{\max }\right),/,m\right]}^2}+1,$$ 17a

$$h=\frac{h_{\max }-1}{1+{\left[ln,\left(p,/,p_{\max }\right),/,m\right]}^2}+1,$$ 17b

where x is the ligand concentration (in this case, the partial pressure of oxygen pO₂), h_max is the maximum value of the Hill coefficient, p_max is the value of the ligand concentration (or pO₂) at which the maximum of the Hill coefficient is determined, and m is the factor determining the degree of expression of the maximum (or scale by x-axis).

Like the original Hill equation (Hill 1910), the Hill/G (17a) and Hill/L (17b) equations are empirically derived from the analysis of data from hemoglobin oxygenation curves. It should be noted that the Hill/L equation is not inferior to the Adair equation in the approximation of the oxyhemoglobin dissociation curve and has all the advantages of the classical Hill equation. Besides, from the four fitting parameters, it is possible to obtain additional indices (Fig. 3) allowing a more complete description, not only of the oxyhemoglobin dissociation curve but also of a number of similar functional dependences, such as kinetic curves for enzymatic reactions (Ricard and Noat 1985), pharmacology (Gesztelyi et al. 2012) and dose-response relationships (Chou 2011), ion transport (Lolkema and Slotboom 2015), and a number of other applications (Maguire et al. 2012).

Fig 3.

Relation between model fitting parameters and other derived from them, which describe the oxyhemoglobin dissociation curve (ODC). Legend: 1, curve representing Hill’s coefficient dependence respect to oxygen partial pressure; 2, oxyhemoglobin dissociation curve (ODC); pO₂, partial pressure of oxygen; SO₂, oxygen saturation; h, Hill coefficient; h_max, the maximum value of the Hill coefficient; p_max, oxygen partial pressure at which the Hill coefficient is maximal; SO₂, degree of hemoglobin saturation with oxygen at p_max; ∆pO₂ and ∆SO₂ correspond to the difference in oxygen partial pressures and the degree of hemoglobin oxygenation, respectively (Lavrinenko et al. 2023, with modifications)

The parameters h_max and p_max make it possible to find the highest value of the cooperativity coefficient and determine the oxygen partial pressure at which this coefficient is maximal. The p_max value corresponds to a certain point on the SO₂ ligand binding curve. This point characterizes the degree of saturation of hemoglobin with oxygen at the maximum value of h_max. Two points on the oxygenation curve with arguments p₅₀ and p_max make it possible to estimate the difference in oxygen partial pressures (∆pO₂). The corresponding projections on the y-axis (SO₂ for p_max and SO₂ for p₅₀) make it possible to determine the difference in hemoglobin oxygenation levels (∆SO₂). The parameters ∆pO₂ and ∆SO₂ are complementary and characterize the degree of deviation of p_max and SO₂ respect to p_max relative to the “symmetry” point—p₅₀. Approximation of the experimental ODC points by curves constructed according to the Hill/G and Hill/L equations showed a shift in h_max towards higher pO₂ values relative to p₅₀, which can probably be associated with the functional asymmetry of the tetramer subunits (Lavrinenko et al. 2022a, b).

Conclusion

The ability of hemoglobin to reversibly bind oxygen, which underlies the oxygen-transporting system of blood, as well as availability of hemoglobin and its easy isolation, allowed this protein to become one of the most important and well-studied objects of research. The S-shaped dependence of hemoglobin oxygen saturation on its partial pressure, discovered by Bohr (1885), opened a whole line of research related to the study of the cooperativity phenomenon in living systems, which further enhanced the interest in this hemoprotein. The deciphering by Perutz et al. (1960) of the spatial structure of hemoglobin, the cooperative allostery in proteins described by Monod et al. (1963), the revealed specific features of this biopolymer for different animal species and hemoglobin’s mutant forms, allowed to link its structure and functions. The prosthetic group of this protein can be considered as an analog of the active center of many enzymes, which made hemoglobin a unique model and “honorary enzyme” (Brunori 1999). All these basic facts and circumstances and a number of interesting discoveries even today determine the unflagging interest of biophysicists in this unique protein, strengthening its role and importance, as he most studied, but at the same time, to the end unexplored macromolecule and a unique model. A more detailed description of the structure, functions, genetics, taxonomy, and evolution of this protein is well beyond the scope of this brief review and deserves appropriate in-depth survey.

Author contribution

All authors contributed to the study conception and design. Conceptualization: Igor A. Lavrinenko; writing—original draft preparation: Igor A. Lavrinenko; writing—review and editing: Igor A. Lavrinenko, Yury D. Nechipurenko, Gennady A. Vashanov, and José L. Hernández Cáceres; visualization: Igor A. Lavrinenko; and supervision: Yury D. Nechipurenko

Funding

This work was supported by the Program of Fundamental Research in the Russian Federation for the 2021–2030 period (project No. 121052600299-1).

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Conflict of interest

The authors declare no competing interests.