Exploring the terminal region of the proton pathway in the bacterial nitric oxide reductase

The c-type nitric oxide reductase (cNOR) from Paracoccus (P.) denitrificans is an integral membrane protein that catalyzes NO reduction; 2NO + 2e⁻ + 2H⁺ → N₂O + H₂O. It is also capable of catalyzing the reduction of oxygen to water, albeit more slowly than NO reduction. cNORs are divergent members of the heme-copper oxidase superfamily (HCuOs) which reduce NO, do not pump protons, and the reaction they catalyse is non-electrogenic. All known cNORs have been shown to have five conserved glutamates (E) in the catalytic subunit, by P. denitrificans numbering, the E122, E125, E198, E202 and E267. The E122 and E125 are presumed to face the periplasm and the E198, E202 and E267 are located in the interior of the membrane, close to the catalytic site. We recently showed that the E122 and E125 define the entry point of the proton pathway leading from the periplasm into the active site [U. Flock, F.H. Thorndycroft, A.D. Matorin, D.J. Richardson, N.J. Watmough, P. Ädelroth, J. Biol. Chem. 283 (2008) 3839-3845]. Here we present results from the reaction between fully reduced NOR and oxygen on the alanine variants of the E198, E202 and E267. The initial binding of O₂ to the active site was unaffected by these mutations. In contrast, proton uptake to the bound O₂ was significantly inhibited in both the E198A and E267A variants, whilst the E202A NOR behaved essentially as wildtype. We propose that the E198 and E267 are involved in terminating the proton pathway in the region close to the active site in NOR.     

A pathway for protons in nitric oxide reductase from Paracoccus denitrificans

Nitric oxide reductase (NOR) from P. denitrificans is a membrane-bound protein complex that catalyses the reduction of NO to N₂O (2NO + 2e⁻ + 2H⁺ → N₂O + H₂O) as part of the denitrification process. Even though NO reduction is a highly exergonic reaction, and NOR belongs to the superfamily of O₂-reducing, proton-pumping heme-copper oxidases (HCuOs), previous measurements have indicated that the reaction catalyzed by NOR is non-electrogenic, i.e. not contributing to the proton electrochemical gradient. Since electrons are provided by donors in the periplasm, this non-electrogenicity implies that the substrate protons are also taken up from the periplasm. Here, using direct measurements in liposome-reconstituted NOR during reduction of both NO and the alternative substrate O₂, we demonstrate that protons are indeed consumed from the ‘outside’. First, multiple turnover reduction of O₂ resulted in an increase in pH on the outside of the NOR-vesicles. Second, comparison of electrical potential generation in NOR-liposomes during oxidation of the reduced enzyme by either NO or O₂ shows that the proton transfer signals are very similar for the two substrates proving the usefulness of O₂ as a model substrate for these studies. Last, optical measurements during single-turnover oxidation by O₂ show electron transfer coupled to proton uptake from outside the NOR-liposomes with a τ = 15 ms, similar to results obtained for net proton uptake in solubilised NOR [U. Flock, N.J. Watmough, P. Ädelroth, Electron/proton coupling in bacterial nitric oxide reductase during reduction of oxygen, Biochemistry 44 (2005) 10711-10719]. NOR must thus contain a proton transfer pathway leading from the periplasmic surface into the active site. Using homology modeling with the structures of HCuOs as templates, we constructed a 3D model of the NorB catalytic subunit from P. denitrificans in order to search for such a pathway. A plausible pathway, consisting of conserved protonatable residues, is suggested.     

A pathway for protons in nitric oxide reductase from Paracoccus denitrificans

Nitric oxide reductase (NOR) from P. denitrificans is a membrane-bound protein complex that catalyses the reduction of NO to N(2)O (2NO+2e(-)+2H(+)-->N(2)O+H(2)O) as part of the denitrification process. Even though NO reduction is a highly exergonic reaction, and NOR belongs to the superfamily of O(2)-reducing, proton-pumping heme-copper oxidases (HCuOs), previous measurements have indicated that the reaction catalyzed by NOR is non-electrogenic, i.e. not contributing to the proton electrochemical gradient. Since electrons are provided by donors in the periplasm, this non-electrogenicity implies that the substrate protons are also taken up from the periplasm. Here, using direct measurements in liposome-reconstituted NOR during reduction of both NO and the alternative substrate O(2), we demonstrate that protons are indeed consumed from the 'outside'. First, multiple turnover reduction of O(2) resulted in an increase in pH on the outside of the NOR-vesicles. Second, comparison of electrical potential generation in NOR-liposomes during oxidation of the reduced enzyme by either NO or O(2) shows that the proton transfer signals are very similar for the two substrates proving the usefulness of O(2) as a model substrate for these studies. Last, optical measurements during single-turnover oxidation by O(2) show electron transfer coupled to proton uptake from outside the NOR-liposomes with a tau=15 ms, similar to results obtained for net proton uptake in solubilised NOR [U. Flock, N.J. Watmough, P. Adelroth, Electron/proton coupling in bacterial nitric oxide reductase during reduction of oxygen, Biochemistry 44 (2005) 10711-10719]. NOR must thus contain a proton transfer pathway leading from the periplasmic surface into the active site. Using homology modeling with the structures of HCuOs as templates, we constructed a 3D model of the NorB catalytic subunit from P. denitrificans in order to search for such a pathway. A plausible pathway, consisting of conserved protonatable residues, is suggested.     

Transmembrane proton translocation by cytochrome c oxidase

Respiratory heme-copper oxidases are integral membrane proteins that catalyze the reduction of molecular oxygen to water using electrons donated by either quinol (quinol oxidases) or cytochrome c (cytochrome c oxidases, CcOs). Even though the X-ray crystal structures of several heme-copper oxidases and results from functional studies have provided significant insights into the mechanisms of O₂-reduction and, electron and proton transfer, the design of the proton-pumping machinery is not known. Here, we summarize the current knowledge on the identity of the structural elements involved in proton transfer in CcO. Furthermore, we discuss the order and timing of electron-transfer reactions in CcO during O₂ reduction and how these reactions might be energetically coupled to proton pumping across the membrane.     

Functional proton transfer pathways in the heme-copper oxidase superfamily

Heme-copper oxidases (HCuOs) terminate the respiratory chain in mitochondria and most bacteria. They are transmembrane proteins that catalyse the reduction of oxygen and use the liberated free energy to maintain a proton-motive force across the membrane. The HCuO superfamily has been divided into the oxygen-reducing A-, B- and C-type oxidases as well as the bacterial NO reductases (NOR), catalysing the reduction of NO in the denitrification process. Proton transfer to the catalytic site in the mitochondrial-like A family occurs through two well-defined pathways termed the D- and K-pathways. The B, C, and NOR families differ in the pathways as well as the mechanisms for proton transfer to the active site and across the membrane. Recent structural and functional investigations, focussing on proton transfer in the B, C and NOR families will be discussed in this review.     

Influence of inhibitors of alternative respiration pathway and oxygen on growth and proton secretion

A possible involvement of the alternative oxidase pathway in proton translocation was investigated. Net H⁺ efflux- and elongation-rates were simultaneously and continuously measured by means of a pH-stat and an angular position transducer. Disulfiram, an inhibitor of the alternative path, reduces the IAA- and Fusicoccin-induced as well as endogenous proton secretion and growth. Fusicoccin-induced H⁺ secretion is very sensitive to reduced oxygen concentration values far apart from the K_m of cytochrome oxidase. The sensitivity of non stimulated proton secretion to reduction of oxygen concentration depends on the age of plant material. It is proposed that more than one system is responsible for proton translocation across the plasmalemma. One of them has a high sensitivity to reduced oxygen concentration which is within the same range of the high K_m value of the alternative oxidase.     

The proton pump of heme-copper oxidases.

Proton pumping heme-copper oxidases represent the terminal, energy-transfer enzymes of respiratory chains in prokaryotes and eukaryotes. The Cu_B-heme a₃ (or heme o) binuclear center, associated with the largest subunit I of cytochrome c and quinol oxidases, is directly involved in the coupling between dioxygen reduction and proton pumping. The role of the other subunits is less clear. The following aspects will be covered in this paper:i) the efficiency of coupling in the mitochondrial aa₃ cytochrome c oxidase. In particular, the effect of respiratory rate and protonmotive force on the H⁺/e^? stoichiometry and the role of subunit IV; ii) mutational analysis of the aa₃ quinol oxidase of Bacillus subtilis addressed to the role of subunit III, subunit IV and specific residues in subunit I; iii) possible models of the protonmotive catalytic cycle at the binuclear center. The observations available suggest that H⁺/e^?coupling is based on the combination of protonmotive redox catalysis at the binuclear center and co-operative proton transfer in the protein.     

Proton polarizability and proton transfer in histidine-phosphate hydrogen bonds as a function of cations present: ir investigations

(L -His)_n- dihydrogen phosphate systems are studied by ir spectroscopy in the presence of various cations and as a function of the degree of hydration. Ir continua indicate that (I) OH … N ? O^?…H⁺N (IIR) hydrogen bonds are formed and that these bonds show high proton polarizability, which increases from the Li⁺ to the K⁺ system. In the K⁺?system, His-P_i-P_i chains are formed, showing particularly high proton polarizability due to collective proton motion within both hydrogen bonds. The OH N ? O^?…H^?N equilibria are determined from ir bands. With the Li⁺ system, 55% of the protons are present at the histidine residues; this percentage is smaller with the Na⁺ system (41%), and amounts to only 32% with the K⁺ system. With the increasing degree of hydration the removal of the degeneracy of ν_as?PO^2?₃ vanishes, indicating loosening of the cations from the phosphates. Nevertheless, the hydrogen bond acceptor O atom becomes more negative; a shift of the equilibrium to the right is observed in the OH… N ? O^?…H⁺N bond. This is explained by the strong interaction of the dipole of the hydrogen bonds with the water molecules. All these results show that protons can be shifted easily in these hydrogen bonds due to their high proton polarizability. The transfer equilibria can be controlled easily by local electrical fields. In addition, these results may be of significance when phosphates interact with proteins.     

Inter‐subunit interaction of gastric H+,K+‐ATPase prevents reverse reaction of the transport cycle

The gastric H⁺,K⁺‐ATPase is an ATP‐driven proton pump responsible for generating a million‐fold proton gradient across the gastric membrane. We present the structure of gastric H⁺,K⁺‐ATPase at 6.5 Å resolution as determined by electron crystallography of two‐dimensional crystals. The structure shows the catalytic α‐subunit and the non‐catalytic β‐subunit in a pseudo‐E₂P conformation. Different from Na⁺,K⁺‐ATPase, the N‐terminal tail of the β‐subunit is in direct contact with the phosphorylation domain of the α‐subunit. This interaction may hold the phosphorylation domain in place, thus stabilizing the enzyme conformation and preventing the reverse reaction of the transport cycle. Indeed, truncation of the β‐subunit N‐terminus allowed the reverse reaction to occur. These results suggest that the β‐subunit N‐terminus prevents the reverse reaction from E₂P to E₁P, which is likely to be relevant for the generation of a large H⁺ gradient in vivo situation.     

Intramolecular proton transfer in adenine imino tautomers

The electronic structural impact on intramolecular proton transfer in the cis- and trans-imino N7 and N9 tautomers of adenine (A) has been studied quantum mechanically, using density functional theory (B3LYP/TZVP, SAOP/TZ2P, LB94/TZ2P) and Green function (OVGF/TZVP) models. It is found that proton transfer does not significantly change isotropic properties but has profound impact on electron distributions of the species through anisotropic properties. The relative energies with respect to the canonical A tautomer (amino-9H), ΔE, for imino 7Hcis, imino 7Htrans, imino 9Hcis and imino 9Htrans are calculated as 16.15, 16.43, 18.46 and 13.80 kcal mol^? 1 (B3LYP/TZVP model) and some minor changes in perimeters of the purine ring is also observed. The Hirshfeld atomic charges indicate that whether a proton attached to N₍₇₎ or N₍₉₎ causes a significant local charge redistribution. However, these charges are insensitive to cis–trans proton transfer. Condensed Fukui function reveals N₍₁₀₎ and C₍₈₎ as the most electrophilic reactive site among N- and C-atom sites, respectively. We also found that proton transfer significantly alters in-plane σ orbitals, rather than out of plane π orbitals including the frontier orbital 6a″. Moreover, orbital based responses to various proton transfers are presented: the orbital 29a′ (HOMO-1) is a signature orbital differentiating all the four tautomers. Orbital 27a′ is a site (N₍₇₎ and N₍₉₎) sensitive orbital, whereas orbital 22a′ is only sensitive to proton orientation on the imino group = N–H.     

Oxygen and proton pathways in cytochrome c oxidase

Cytochrome c oxidase is a redox-driven proton pump, which couples the reduction of oxygen to water to the translocation of protons across the membrane. The recently solved x-ray structures of cytochrome c oxidase permit molecular dynamics simulations of the underlying transport processes. To eventually establish the proton pump mechanism, we investigate the transport of the substrates, oxygen and protons, through the enzyme. Molecular dynamics simulations of oxygen diffusion through the protein reveal a well-defined pathway to the oxygen-binding site starting at a hydrophobic cavity near the membrane-exposed surface of subunit I, close to the interface to subunit III. A large number of water sites are predicted within the protein, which could play an essential role for the transfer of protons in cytochrome c oxidase. The water molecules form two channels along which protons can enter from the cytoplasmic (matrix) side of the protein and reach the binuclear center. A possible pumping mechanism is proposed that involves a shuttling motion of a glutamic acid side chain, which could then transfer a proton to a propionate group of heme α₃. Proteins 30:100–107, 1998. © 1998 Wiley-Liss, Inc.     

Cytochrome oxidase: pathways for electron tunneling and proton transfer

 Electrons from cytochrome c, the substrate of cytochrome oxidase, a redox-linked proton pump, are accepted by Cu_A in subunit II. From there they are transferred to the proton pumping machinery in subunit I, cytochrome a and cytochrome a ₃–Cu_B. The reduction of the latter site, which is the dioxygen reducing unit, is coupled to proton uptake. Dioxygen reduction involves a peroxide and a ferryl ion intermediate, and it is the transition between these and back to the resting oxidized enzyme that are coupled to proton pumping. The X-ray structures suggest electron–transfer pathways that can account for the observed rates provided that the reorganization energies are small. They also reveal two proton-transfer pathways, and mutagenesis experiments have shown that one is used for proton uptake during the initial reduction of cytochrome a ₃–Cu_B, whereas the other mediates transfer of the pumped protons. Received: 23 March 1998 / Accepted: 11 May 1998     

A structured model for the kinetics of fungal amylase production

Summary The cell is considered to be divided into nucleic acids, proteinaceous material and storage compounds. The enzyme is believed to be constitutive but repressed by the rate of catabolism. A structured model is developed to describe the growth, amylase production and dissolved oxygen profile in the batch culture ofAspergillus oryzae.Nomenclature CA Conc. of enzyme SKB units/m³ - C_D,C_E,C_G,C_O,C_S Conc. of D.E.G. mass, oxygen, substrate kg/m³ - C_O* Mean oxygen conc. in gas-liquid interface kg/m³ - C_X Total cell conc. kg/m³ - D D-mass (proteins) kg - E E-mass (storage carbon) kg - G G-mass (necleic acids) kg - K_E Rate of usage of E-mass kg/m³/h - K_EO K_E with oxygen limitation kg/m³/h - Q Active fraction of promotor genes - - R_D,R_E,R_G Rate of production of D,E,G-mass kg/m³/h - R_O,R_S Rate of usage of oxygen, substrate kg/m³/h - R_DO,R_EO,R_GO R_D,R_E and R_G with oxygen limitation kg/m³/h - S Substrate (carbon) concentration kg/m³ - t Time h - t_o Time at which C_S = 0 h - K₁,K₂,K₃,K₂₄,K₂₅ Stoichiometric constants - - K₄,K₅,K₆,K₇ Rate constants h^–1 - K₈-K₁₂,K₁₈,K₂₀-K₂₃ Michaelis-Menten constants kg/m³ - K₂₆,K₂₆ Absorption coefficient, K₂₆ at C_X = 0 h^–1 - K₂₇ Empirical constant kg/m³ - K₁₅ Rate of enzyme formation SKB units/kg - K₁₆,K₁₇ Equilibrium constants m³/kg - K₁₉ Decay constant for mRNA h^–1     

On the role of subunit III in proton translocation in cytochromec oxidase

Mammalian mitochondrial cytochromec oxidase catalyzes the transfer of electrons from ferrocytochromec to molecular oxygen in the respiratory chain, while conserving the energy released during its electron transfer reactions by the vectorial movement of protons across the inner membrane of the mitochondrion. The protein domain that translocates the protons across the membrane is currently unknown. Recent research efforts have investigated the role of one of the transmembrane subunits of the enzyme (III,M _r 29,884) in the vectorial proton translocation reaction. The data that favor subunit III as integral in vectorial proton translocation as well as the data that support a more peripheral role for subunit III in proton translocation are reviewed. Possible experimental approaches to clarify this issue are presented and a general model discussed.     

The H channel is not a proton transfer path in yeast cytochrome c oxidase

Cytochrome c oxidases (CcOs) in the respiratory chains of mitochondria and bacteria are primary consumers of molecular oxygen, converting it to water with the concomitant pumping of protons across the membrane to establish a proton electrochemical gradient. Despite a relatively well understood proton pumping mechanism of bacterial CcOs, the role of the H channel in mitochondrial forms of CcO remains debated. Here, we used site-directed mutagenesis to modify a central residue of the lower span of the H channel, Q413, in the genetically tractable yeast Saccharomyces cerevisiae. Exchange of Q413 to several different amino acids showed no effect on rates and efficiencies of respiratory cell growth, and redox potential measurements indicated minimal electrostatic interaction between the 413 locus and the nearest redox active component heme a. These findings clearly exclude a primary role of this section of the H channel in proton pumping in yeast CcO. In agreement with the experimental data, atomistic molecular dynamics simulations and continuum electrostatic calculations on wildtype and mutant yeast CcOs highlight potential bottlenecks in proton transfer through this route. Our data highlight the preference for neutral residues in the 413 locus, precluding sufficient hydration for formation of a proton conducting wire.     

The KC Channel in the cbb3-Type Respiratory Oxygen Reductase from Rhodobacter capsulatus Is Required for Both Chemical and Pumped Protons

The heme-copper superfamily of proton-pumping respiratory oxygen reductases are classified into three families (A, B, and C families) based on structural and phylogenetic analyses. Most studies have focused on the A family, which includes the eukaryotic mitochondrial cytochrome c oxidase as well as many bacterial homologues. Members of the C family, also called the cbb₃-type oxygen reductases, are found only in prokaryotes and are of particular interest because of their presence in a number of human pathogens. All of the heme-copper oxygen reductases require proton-conducting channels to convey chemical protons to the active site for water formation and to convey pumped protons across the membrane. Previous work indicated that there is only one proton-conducting input channel (the K^C channel) present in the cbb₃-type oxygen reductases, which, if correct, must be utilized by both chemical protons and pumped protons. In this work, the effects of mutations in the K^C channel of the cbb₃-type oxygen reductase from Rhodobacter capsulatus were investigated by expressing the mutants in a strain lacking other respiratory oxygen reductases. Proton pumping was evaluated by using intact cells, and catalytic oxygen reductase activity was measured in isolated membranes. Two mutations, N346M and Y374F, severely reduced catalytic activity, presumably by blocking the chemical protons required at the active site. One mutation, T272A, resulted in a substantially lower proton-pumping stoichiometry but did not inhibit oxygen reductase activity. These are the first experimental data in support of the postulate that pumped protons are taken up from the bacterial cytoplasm through the K^C channel.     

Combined DFT and electrostatics study of the proton pumping mechanism in cytochrome c oxidase

Cytochrome c oxidase is a redox-driven proton pump which converts atmospheric oxygen to water and couples the oxygen reduction reaction to the creation of a membrane proton gradient. The structure of the enzyme has been solved; however, the mechanism of proton pumping is still poorly understood. Recent calculations from this group indicate that one of the histidine ligands of enzyme's Cu_B center, His291, may play the role of the pumping element. In this paper, we report on the results of calculations that combined first principles DFT and continuum electrostatics to evaluate the energetics of the key energy generating step of the model—the transfer of the chemical proton to the binuclear center of the enzyme, where the hydroxyl group is converted to water, and the concerted expulsion of the proton from δ-nitrogen of His291 ligand of Cu_B center. We show that the energy generated in this step is sufficient to push a proton against an electrochemical membrane gradient of about 200 mV. We have also re-calculated the pK_a of His291 for an extended model in which the whole Fe_a3-Cu_B center with their ligands is treated by DFT. Two different DFT functionals (B3LYP and PBE0), and various dielectric models of the protein have been used in an attempt to estimate potential errors of the calculations. Although current methods of calculations do not allow unambiguous predictions of energetics in proteins within few pK_a units, as required in this case, the present calculation provides further support for the proposed His291 model of CcO pump and makes a specific prediction that could be targeted in the experimental test.     

Electron and proton transfer in the M. smegmatis III2IV2 supercomplex

The M. smegmatis respiratory III₂IV₂ supercomplex consists of a complex III (CIII) dimer flanked on each side by a complex IV (CIV) monomer, electronically connected by a di-heme cyt. cc subunit of CIII. The supercomplex displays a quinol oxidation?oxygen reduction activity of ~90 e^?/s. In the current work we have investigated the kinetics of electron and proton transfer upon reaction of the reduced supercomplex with molecular oxygen. The data show that, as with canonical CIV, oxidation of reduced CIV at pH 7 occurs in three resolved components with time constants ~30 μs, 100 μs and 4 ms, associated with the formation of the so-called peroxy (P), ferryl (F) and oxidized (O) intermediates, respectively. Electron transfer from cyt. cc to the primary electron acceptor of CIV, Cu_A, displays a time constant of ≤100 μs, while re-reduction of cyt. cc by heme b occurs with a time constant of ~4 ms. In contrast to canonical CIV, neither the P → F nor the F → O reactions are pH dependent, but the P → F reaction displays a H/D kinetic isotope effect of ~3. Proton uptake through the D pathway in CIV displays a single time constant of ~4 ms, i.e. a factor of ~40 slower than with canonical CIV. The slowed proton uptake kinetics and absence of pH dependence are attributed to binding of a loop from the QcrB subunit of CIII at the D proton pathway of CIV. Hence, the data suggest that function of CIV is modulated by way of supramolecular interactions with CIII.     

Proton transfer in catalysis and the role of proton shuttles in carbonic anhydrase

The undisputed role of His64 in proton transfer during catalysis by carbonic anhydrases in the α class has raised questions concerning the details of its mechanism. The highly conserved residues Tyr7, Asn62, and Asn67 in the active-site cavity function to fine tune the properties of proton transfer by human carbonic anhydrase II (HCA II). For example, hydrophobic residues at these positions favor an inward orientation of His64 and a low pK_a for its imidazole side chain. It appears that the predominant manner in which this fine tuning is achieved in rate constants for proton transfer is through the difference in pK_a between His64 and the zinc-bound solvent molecule. Other properties of the active-site cavity, such as inward and outward conformers of His64, appear associated with the change in ΔpK_a; however, there is no strong evidence to date that the inward and outward orientations of His64 are in themselves requirements for facile proton transfer in carbonic anhydrase.     

The effects of the oxygen transfer coefficient and substrate concentration on the xylose fermentation by Debaryomyces hansenii

Relevant production of xylitol by Debaryomyces hansenii requires semiaerobic conditions since in aerobic conditions the accumulated reduced adenine dinucleotide coenzyme is fully reoxidized leading to the conversion of xylitol into xylulose. For oxygen transfer coefficient values from 0.24 to 1.88 min^-1, in shake flasks experiments, biomass formation increased proportionally to the aeration rate as shown in the oxygen transfer coefficient and xylose concentration isoresponse contours. The metabolic products under study, xylitol and ethanol were mainly growth associated. However, for oxygen transfer coefficient above 0.5 min^-1 higher initial xylose concentration stimulated the rate of production of xylitol. This fact was less evident for ethanol production. The direct relationship between increased biomass and products formation rate, indicated that the experimental domain in respect to the aeration rate was below the threshold level before the decreasing in metabolic production rates reported in literature for xylose-fermenting yeasts. The fact that ethanol was produced, albeit in low levels, throughout the experimental design indicated that the semiaerobic conditions were always attained. Debaryomyces hansenii showed to be an important xylitol producer exhibiting a xylitol/ethanol ratio above four and a carbon conversion of 54% for xylitol.Abbreviations K_La oxygen transfer coefficient - DO(T) dissolved oxygen (tension) - OUR oxygen uptake rate - NAD(H) oxidised (reduced) nicotinamide adenine dinucleotide - NADP(H) oxidised (reduced) nicotinamide adenine dinucleotide phosphate - CRC catabolic reduction charge - C oxygen concentration in the culture medium - C^* oxygen concentration at saturation conditions - Y_i response from experiment i - parameters of the polynomial model - x experimental factor level (coded units) - R² coefficient of multiple determination - t time