Abstract
The role of magnetism in the biological functioning of hemoglobin has been debated since its discovery by Pauling and Coryell in 1936. The hemoglobin molecule contains four heme groups each having a porphyrin layer with a Fe ion at the center. Here, we present combined densityfunctional theory and quantum Monte Carlo calculations for an effective model of Fe in a heme cluster. In comparison with these calculations, we analyze the experimental data on human adult hemoglobin (HbA) from the magnetic susceptibility, Mössbauer and magnetic circular dichroism (MCD) measurements. In both the deoxygenated (deoxy) and the oxygenated (oxy) cases, we show that local magnetic moments develop in the porphyrin layer with antiferromagnetic coupling to the Fe moment. Our calculations reproduce the magnetic susceptibility measurements on deoxy and oxyHbA. For deoxyHbA, we show that the anomalous MCD signal in the UV region is an experimental evidence for the presence of antiferromagnetic Feporphyrin correlations. The functional properties of hemoglobin such as the binding of O_{2}, the Bohr effect and the cooperativity are explained based on the magnetic correlations. This analysis suggests that magnetism could be involved in the functioning of hemoglobin.
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Introduction
Pauling and Coryell showed that the magnetic susceptibility of deoxyHbA exhibits a Curietype (1/T) temperature dependence, while for oxyHbA it is weakly negative implying that the total spin S = 0 for the molecule^{1,2}. Mössbauer experiments^{3} are consistent with the view that Fe is in an S = 2 state in deoxyHbA, while its magnetic moment is found to be \(\lesssim 1\,{{\mu }}_{{\rm{B}}}\) in the oxy case. The magnetic circular dichroism (MCD) measurements^{4} find an anomalous line shape for the temperaturedependent MCD spectra in the UV region. The HbA molecule exhibits remarkable functional properties such as the cooperativity^{5,6,7} and the Bohr effect^{8,9,10,11}, which enhance its oxygen carrying capacity. When one of the four Fe ions in HbA combines an oxygen molecule, the other three Fe ions attract oxygens cooperatively. The Bohr effect denotes the characteristic of HbA through which the oxygen affinity depends on the pH of the medium. Despite many years of study, there still remains open questions on the nature of the spin and charge distributions and the role of magnetism in the functioning of HbA and other hemeproteins^{7,12,13}.
We study the electronic state of one heme group by using an effective multiorbital Anderson impurity model^{14,15}, which is described in the Methods section. The parameters of this model are obtained by the density functional theory^{16} (DFT) calculations, where we use the molecular coordinates determined by the Xray measurements^{17,18}. This way the stereochemical effects are included. In the DFT calculation of the Anderson model parameters, we use the Gaussian program^{19} with the BP86 energy functional (BP86)^{20,21} and the 6–31 G basis set. We then study this model with the quantum Monte Carlo (QMC) calculations by using the HirschFye algorithm^{22} while keeping all of the host states obtained by the DFT. However, the transverse component of the Hund’s coupling is not taken into account during the simulations. For the Coulomb interaction parameters we use U = 4 eV and J = 0.9 eV. This DFT + QMC approach is described in more detail in the Methods section and the Supplementary Information.
We note that the electronic state of the heme proteins and related molecules were previously studied by using the DFT^{23}, multiconfigurational methods^{24}, DFT + U^{25,26}, and by combining the DFT with the dynamical mean field theory (DMFT)^{27,28,29,30}. The DFT + DMFT method is similar to the DFT + QMC approach used here. However, there are differences. In the DFT + DMFT calculations the molecular coordinates are obtained computationally, while in the DFT + QMC approach we use those obtained by the Xray measurements. In particular, the DFT + DMFT calculations were used to study the charge and spin state of the Fe site. On the other hand, we study the magneticmoment formation in the host along with the Fe site, and also the Fehost magnetic correlations. In addition, we analyze the available experimental data on the magnetic correlations along with the DFT + QMC results, and investigate possible consequences for the biological functioning of HbA.
Figure 1(a) shows the molecular structure of deoxyHbA^{18}. We have performed our DFT + QMC calculations for the truncated clusters shown in Fig. 1(b,c).
Figure 2 shows DFT + QMC results for the magneticmoment density M(r) at temperature T = 150 K. In the deoxy case, we observe that the Fe site has a large up moment ≈4.6 μ_{B}. The neighboring nitrogen sites have smaller moments, while at the carbon sites M(r) points down. These down moments originate from the partiallyoccupied host states which consist of the C(2p_{z}) orbitals. Hence, antiferromagnetic correlations exist between the large Fe magnetic moment and the host moments spread out in the porphyrin layer in deoxyheme. In oxyheme, the Fe moment is reduced, but remains finite. In addition, the neighboring O_{2} and N sites have magnetic moments which are antiferromagnetically coupled to that of Fe. In the oxy case, the antiferromagnetic screening cloud is more tightly localized around the Fe site. We note that the realspace structure of the magnetic correlations found by DFT + QMC are different than those found by GGA + U^{26}. In the deoxy case, GGA + U does not obtain the antiferromagnetic Feporphyrin correlations. In the oxy case, GGA + U finds magnetic moment formation only at the Fe and the O_{2} sites. The DFT + QMC results show that, in the oxy case, magnetic moments also form in the porphyrin layer and in particular at the N sites neighboring Fe. Additional DFT + QMC data on the spin and charge distributions are presented in the Supplementary Information. The calculation of M(r) is also described in the Supplementary Information.
Figure 3(a) shows the temperature dependence of the total spin susceptibility χ_{t}. For the deoxyheme cluster, χ_{t} follows a nearlyperfect Curie Tdependence. The total effective magnetic moment M_{t} is ≈4.1 μ_{B} at T = 150 K, which is reduced from that of Fe due to the Feporphyrin antiferromagnetic correlations. In the oxy case, there are two temperature regimes separated by a crossover temperature T^{*} ≈ 300 K: In the highT regime, T > T^{*}, χ_{t} has a Curietype T dependence with an effective total moment ≈2.1 μ_{B}. In the lowT regime, T < T^{*}, χ_{t} decreases with decreasing T. When the QMC finiteΔτ effects are taken into account, this decrease becomes more rapid as shown in the Supplementary Fig. S4. For T < 300 K, the reduction of χ_{t} with respect to that of deoxyheme is mainly due to the collapse of the Fe magnetic moment because of the loss of the ferromagnetic correlations among the Fe(3d_{v}) orbitals. For T < 300 K, χ_{t} is reduced due to the suppression of magnetism at the Fermi level. This is seen in Fig. 3(b), which shows that the total moment M_{t} gets suppressed within ≈0.15 eV of the Fermi level as T decreases. In Supplementary Fig. S1(b), the total spin susceptibility χ_{t} is plotted as a function of μ, where the suppression of χ_{t} at the Fermi level is seen. In order to study the susceptibility as a function of the real frequency, it would be necessary to carry out a maximumentropy analytic continuation, which requires QMC data with very good statistics. Nevertheless, by simply plotting χ_{t} or M_{t} as a function of μ at the Fermi level, it is possible to obtain a characteristic energy scale for the suppression of magnetism. We attribute this suppression to the transfer of electrons from mainly the O_{2} to the Fe(3d_{v}) orbitals, in particular, to the socalled t_{2g} orbitals, v = xy, xz and yz. These are discussed further in the Supplementary Information.
Experimentally, the spin susceptibility of oxyHbA vanishes at room temperature^{1,31}, while we find that χ_{t} ≈ 150 \({\mu }_{{\rm{B}}}^{2}/\)eV per heme. It is possible that because of the various approximations, such as the neglect of the transverse component of the Hund’s coupling or the temperature dependence of the molecular coordinates and the use of the reduced hemeclusters, the DFT + QMC approach is underestimating the value of T^{*}. If this is indeed the case, then it will have consequences for the binding mechanism of O_{2} to Fe in heme and the Bohr effect: We note that, in general, charge transfer to the Fe(3d) orbitals would be energetically costly because of the large Coulomb repulsion over there. However, as seen in the Supplementary Information, in heme this is overcome by making use of the the upperHubbard level of the Fe(3d_{xy}) orbital. This is where the Fe(3d_{xy}) orbital becomes doubly occupied, and it is located very close to the Fermi level. This turns out to be a critical feature of the electronic structure of oxyheme, because when the chemical potential is away from this region, the magnetism is not suppressed. We think that, in order to minimize its energy, the system is developing magnetic moments and magnetic correlations by redistributing electrons. It is a real possibility that this gain in energy is responsible for the binding of O_{2} to heme. These suggest that the binding of O_{2} to Fe in heme has a magnetic origin. We note that the stereochemical effects^{6} are clearly important in O_{2} binding. We are proposing that the magnetism, which is controlled by the stereochemical effects, has the key role in the perfectly reversible binding of O_{2} to Fe in heme.
In the Bohr effect, the oxygen affinity of HbA is controlled by the hydrogen ion concentration in the red blood cells. We have seen above that the magnetism is suppressed within a narrow energy window at the Fermi level. Hence, the magnetic properties depend sensitively on the electron filling. Any changes which effectively moves the chemical potential away from this narrow region will affect the binding of O_{2}. This could be how the pH influences the oxygen affinity. However, we note that a more realistic modelling of the Bohr effect would also require taking into account how pH affects the complex interactions between the medium and the HbA molecule^{7}. Here, we consider only the initial and the final states of the O_{2} binding reaction. To the extent that the spin correlations dominate the O_{2} binding, we expect this discussion to be relevant for the actual process.
We note that the above suggested mechanism for the Feoxygen bonding may not be limited to heme, but may also be relevant for bonding in other compounds containing transition metals, for example the transitionmetal oxides. This type of bonding is more complicated than the usual covalent or ionic bondings because it involves the upperHubbard level of the 3d orbitals and the magnetic correlations.
An interesting feature which emerges from these calculations is the antiferromagnetic coupling between Fe and the porphyrin layer. Experimental evidence for this is provided by the MCD data on deoxyHbA in the UV region. The MCD intensity Δε(E) is the difference between the leftcircularly polarized (LCP) and the rightcircularly polarized (RCP) light absorption within an applied magnetic field parallel to the direction of light propagation. The MCD spectrum of deoxyHbA has a peak in the UV region at ≈3 eV which has an anomalous line shape and a 1/T temperaturedependence^{4}. The optical absorption ε(E) has a Tindependent peak at the same energy. It is known that the optical absorption at ≈3 eV is due to transitions from the bonding π to the antibonding π^{*} states which consist of the C(2p_{z}) orbitals of the porphyrin layer^{4}. The composition of the π and π^{*} states are illustrated in the Supplementary Fig. S6. The MCD spectrum of deoxyHbA in UV region is anomalous in the following sense: In the usual case, the Tdependent piece of the MCD spectrum first has a dip and then a peak as the frequency increases, whereas in deoxyHbA the MCD spectrum first has a peak and then a shallow dip. Here, we show that the anomalous MCD signal is caused by the orbitalselective optical transitions from the occupied π orbitals to two partially occupied π^{*} orbitals, which we label as \({\pi }_{1}^{\ast }\) and \({\pi }_{2}^{\ast }\). According to the DFT + QMC calculations, the \({\pi }_{1}^{\ast }\) state is nearly halffilled while the \({\pi }_{2}^{\ast }\) is nearly empty, which is discussed further in the Supplementary Information. We note that this mechanism for the anomalous MCD spectrum of deoxyHbA was originally proposed by Sharanov et al.^{32,33}. Here, we observe that the DFT + QMC data reproduces the experimental data in agreement with the previous predictions. This is experimental evidence for the Fehost antiferromagnetic correlations found in the DFT + QMC calculations.
As illustrated in Fig. 4(a), within an applied field B_{app} pointing up, the Fe spin will be polarized in the down direction. Meanwhile, the spin polarization of \({\pi }_{1}^{\ast }\) will be parallel to the field, because of the Fe\({\pi }_{1}^{\ast }\) antiferromagnetic correlations. Hence, during an LCP (RCP) optical transition from the π state, it will be energetically more favorable for the \({\pi }_{1}^{\ast }\) (\({\pi }_{2}^{\ast }\)) state to absorb a downspin (upspin) electron. These orbitalselective optical transitions are illustrated in Fig. 4(b,c). We have calculated the MCD spectrum due to these transitions as discussed in the Supplementary Information. Figure 4(d,e) present the experimental and the calculated MCD spectra, respectively. The inset of Fig. 4(d) illustrates the line shape normally expected for the Tdependent MCD spectrum^{34}. The anomalous MCD line shape of deoxyHbA seen in Fig. 4(d) had been attributed to a negative spinorbit coupling^{4}. We show that it instead originates from the antiferromagnetic coupling between the Fe(3d) and π^{*} states, as was previously suggested by Sharonov et al.^{32,33}. Because of this, \({\pi }_{1}^{\ast }\) and \({\pi }_{2}^{\ast }\) porphyrin states act as if they have negative gfactors. The agreement with the experimental data can be improved further by incorporating into the analysis the optical absorption data as shown in the Supplementary Information.
Next, we compare the DFT + QMC results with the experimental magnetic susceptibility measurements. Assuming independent heme groups, the magnetic susceptibility measurements yield an effective magnetic moment per heme \({M}_{{\rm{heme}}}^{{\rm{expt}}}=5.46\,{\mu }_{{\rm{B}}}\) in deoxyHbA^{1}. Since this is larger than the spin S = 2 value of 4.9 μ_{B}, Pauling and Coryell already pointed out the possibility of interheme ferromagnetic correlations^{1}. These imply that the average interheme magnetic correlation \(\langle {M}_{{\rm{heme}},1}^{z}{M}_{{\rm{heme}},2}^{z}\rangle \approx 1.9\,{\mu }_{{\rm{B}}}^{2}\) for deoxyHbA. On the other hand, the DFT + QMC calculations find M_{Fe} ≈ 4.6 μ_{B} and \({M}_{{\rm{heme}}}\approx 4.1\,{\mu }_{{\rm{B}}}\), which is reduced from M_{Fe} because of the antiferromagnetic Fehost coupling. This yields an average interheme magnetic correlation \(\langle {M}_{{\rm{heme}},1}^{z}{M}_{{\rm{heme}},2}^{z}\rangle \approx 4.3\,{\mu }_{{\rm{B}}}^{2}\) for deoxyHbA. Hence, the average interheme magnetic correlations are found to be stronger when the antiferromagnetic Fehost coupling is taken into account. From comparison with the experimental data, we thus find evidence for the existence of significant interheme ferromagnetic correlations.
The cooperativity of HbA enhances its O_{2} carrying capacity. It arises from the property that the binding rate of O_{2} to HbA depends on how many of the four heme groups in HbA are already oxygenated. For example, the binding of the first O_{2} to HbA occurs at a rate much slower than that of the last (fourth) O_{2} to bind. Clearly, any discussion of cooperativity requires a multiheme model. However, we will briefly comment on the implications of the DFT + QMC results. In particular, we suggest two possible scenarios on how the presence of interheme ferromagnetic correlations could lead to the cooperativity. In the first scenario, the first O_{2} to bind HbA needs to overcome the interheme ferromagnetic interactions, because it will break three of the interheme ferromagnetic bonds. Upon the binding of the first O_{2} to a heme group in HbA, the magnetic moment of that heme group vanishes. The following O_{2}’s will bind more easily because now there are fewer interheme ferromagnetic bonds to break. A second alternative scenario is based on the spin nonconservancy in the binding of O_{2} to heme. We note that while O_{2} is in a triplet state (S = 1) and deoxyheme is considered to be S = 2, the resultant oxyheme is in an S = 0 state, hence the spin is not conserved in the binding of O_{2} to a heme group. This nonconservancy of the total spin may be limiting the reaction rate for O_{2} binding^{23}. However, if there indeed exist sufficiently strong interheme ferromagnetic correlations, then spin transfer may be possible from one heme group to another within HbA. Hence, when the O_{2}’s bind cooperatively, the total spin can be conserved by interheme spin transfer. This could eliminate the limit on the O_{2} binding rate arising from the spin nonconservancy. According to this scenario, we expect the cooperative (simultaneous) binding of four O_{2}’s to HbA to occur faster than the binding of the first O_{2} to HbA. Clearly, these scenarios are only speculative ideas at the moment. It would be necessary to study multiheme models in order to test them.
We have shown that magnetic moment formation and magnetic correlations are key electronic properties of deoxy and oxyHbA. It is remarkable that these properties, which arise from the stronglycorrelated electrons, could play a role in the functioning of hemoglobin. We note that there are a large number of metalloproteins, metalloenzymes and other bioinorganic molecules containing transitionmetal centers^{35}. Hence, we suggest that the magnetic effects could have a general role in the functioning of bioinorganic molecules with a distinct place in the emerging field of quantum biology.
Methods
Effective Anderson impurity model for heme
We use an effective multiorbital Anderson impurity model^{14,15}, where the five Fe(3d) orbitals are taken as the impurity states and the remaining orbitals are treated as the host states, to describe the electronic properties of the deoxy and oxyheme clusters. The multiorbital Anderson Hamiltonian with the intra and interorbital Coulomb interactions is given by
where \({c}_{m\sigma }^{\dagger }\) (c_{mσ}) operator creates (annihilates) an electron in the m’th host state with spin σ, and \({d}_{\nu \sigma }^{\dagger }\) (d_{vσ}) is the creation (annihilation) operator for a localized electron with spin σ at the Fe(3d_{v}) orbital. The electron occupation operator for the Fe(3d_{v}) orbitals is \({n}_{\nu \sigma }={d}_{\nu \sigma }^{\dagger }{d}_{\nu \sigma }\). The energies of the host and the Fe(3d_{v}) states are ε_{m} and ε_{dv}, respectively. The hybridization matrix element between the m’th host state and the Fe(3d_{v}) orbital is V_{mv}. The intraorbital Coulomb repulsion is U, while U′ and U″ = U′ − J are the Coulomb interactions between two 3d electrons in different orbitals with antiparallel and parallel spins, respectively. Here, J is the ferromagnetic Hund’s coupling constant. In the case of a free atom, the relation U′ = U − 2J holds, which we also use here. The chemical potential μ is introduced because the QMC calculations are performed at finite temperatures in the grand canonical ensemble by using the HirschFye algorithm^{22}. At each value of the temperature, we adjust μ so that the cluster has the correct number of electrons, which is discussed in Supplementary Fig. S3. We obtain the values of ε_{m}, ε_{dv} and V_{mv} by the densityfunctional theory (DFT)^{16}. The DFT calculations are carried out by using the Gaussian program^{19} with the BP86 energy functional^{20,21} and the 6–31 G basis set with 483 basis functions for the deoxyheme cluster and 501 basis functions for the oxyheme cluster. Further information on this procedure is given in the Supplementary Information. For the interaction parameters we use U = 4 eV and J = 0.9 eV. A similar approach, which also uses a maximallylocalized singleparticle basis, was introduced by Ref. ^{36} for mapping the electronic state of molecular nanomagnets onto strongly correlated models.
We have performed the DFT + QMC calculations for clusters obtained by truncating the full deoxy and oxyheme molecular structures from the Protein Data Bank as described in the Supplementary Information. For understanding the functioning of HbA, the role of the stereochemical effects have been investigated. In particular, it has been emphasized that Fe moves by about 0.4 Å towards the porphyrin ring upon O_{2} binding^{17}. Since we are using coordinates determined by the Xray measurements, these stereochemical effects are already included in our model.
In Eq. (1), we include the longitudinal component of the Hund’s interaction, however the transverse component, which consists of the spinflip and the pairhopping terms, is not included, because it cannot be treated with the HirschFye algorithm. In addition, we neglect the temperature dependence of the molecular coordinates. In spite of these approximations, the DFT + QMC technique applied to this effective impurity model offers a realistic treatment for the electronic state of HbA. We note that these DFT + QMC calculations represent the only computational approach which yields agreement with the magnetic susceptibility, Mössbauer, and the MCD data on HbA at the same time.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Nuran Elmaci Irmak, Talat Yalcin, Acar Savaci and Tahir Cagin for comments. Support by the Turkish Scientific and Technical Research Council (TUBITAK grant number 113F242) is gratefully acknowledged. This work was supported by JST ERATO Grant No. JPMJER1402, JSPS GrantinAid for Scientific Research on Innovative Areas Grant No. JP26103005, and JSPS KAKENHI Grant Nos. JP16H04023 and JP17H02927. The numerical calculations reported here were performed at the TUBITAK ULAKBIM High Performance and Grid Computing Center (TRUBA resources).
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S. Mayda perfomed the QMC calculations, Z.K. performed the DFT calculations, N.B. and S. Maekawa wrote the paper together.
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Mayda, S., Kandemir, Z., Bulut, N. et al. Magnetic mechanism for the biological functioning of hemoglobin. Sci Rep 10, 8569 (2020). https://doi.org/10.1038/s4159802064364y
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DOI: https://doi.org/10.1038/s4159802064364y
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