Clifford W Fong Eigenenergy
email: cwfong@internode.on.net

The antioxidant properties of Edaravone analogs and Amyloid β induced oxidative stress and neurotoxicity in Alzheimer's disease -a quantum mechanical study

Keywords: Alzheimer's disease, Amyloid beta protein, Edaravone, Edaravone analogs, oxidative stress, quantum mechanics

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The antioxidant properties of Edaravone analogs and Amyloid β induced oxidative stress and neurotoxicity in

Alzheimer's disease -a quantum mechanical study

Introduction

Amyloid beta protein (A ) is a neurotoxic substance, which can damages nerve cells by oxidative stress, alters calcium homeostasis, and activates a variety of proapoptotic pathways, all of which are involved in the pathogenesis of Alzheimer's disease (AD). AD is a neurodegenerative disease, and one factor involved in AD is oxidative stress. Edaravone (Edv) is a free radical scavenger, which can reduce the damage caused by oxidative stress. Aβ can promote the generation of reactive oxygen species (ROS), which have an important role in oxidative damage to neurons. Aβ is a metalloprotein that can directly generate free radicals or bind with metal ions (Fe 3+ , Cu 2+ ) to generate OH• from H 2 O 2 or other ROS. Hydrogen peroxide is thought to be involved in the formation of the many oxidative products that form in the Alzheimer-affected brain. Brain tissue is particularly susceptible to oxidative stress due to its high aerobic metabolic activity and high lipid content. Because of the high lipid content of the brain, lipid peroxidation in which OH• abstract hydrogen from polyunsaturated fatty acids (PUFA) forming alkyl radicals is a common process. [START_REF] Mc Rosales Hernández | Involvement of Free Radicals in the Development and Progression of Alzheimer's Disease[END_REF][START_REF] Smith | The redox chemistry of the Alzheimer's disease amyloid β peptide[END_REF][START_REF] Da Butterfield | Oxidative Stress, Amyloid-Peptide, and Altered Key Molecular Pathways in the Pathogenesis and Progression of Alzheimer's Disease[END_REF] Edv can modulate Aβ-induced oxidative stress and neurotoxicity. The accumulation of Aβ and the subsequent aggregation into amyloid plaques in the brain are thought to initiate the AD pathogenesis. Edv can interact with Aβ and can inhibit Aβ aggregation and disaggregate preformed Aβ fibrils, indicating that Edv is a scavenger for both ROS and Aβ. Edaravone is thought to interfere with Aβ aggregation to form fibrils. Edaravone can improve cognitive deficits before and after the onset of Aβ deposition. [START_REF] Ss Jiao | Edaravone alleviates Alzheimer's disease-type pathologies and cognitive deficits[END_REF][START_REF] Zhang | Edaravone ameliorates oxidative damage associated with A 25-35 treatment in PC12 cells[END_REF][START_REF] He | Inhibitory Effects of Edaravone in -Amyloid-Induced Neurotoxicity in Rats[END_REF][START_REF] Ye Shen | Effects of Edaravone on Amyloidβ Precursor Protein Processing in SY5Y-APP695 Cells[END_REF][START_REF] Cheignon | Oxidative stress and the amyloid beta peptide in Alzheimer's disease[END_REF] Edv can reduce A toxicity by interfering with hydrolysis sites of the amyloid precursor protein (APP), or increasing the activity of -secretase, and reducing the hydrolysis of -secretase upon APP. The effect of Edv on APP processing, showed that Edv treatment for 24 h down-regulated β-amyloid (Aβ) production in a dose-dependent manner. Edv played an important role in regulating Aβ production by enhancing the non-amyloidogenic pathway and inhibiting the amyloidogenic pathway. [START_REF] Ye Shen | Effects of Edaravone on Amyloidβ Precursor Protein Processing in SY5Y-APP695 Cells[END_REF] Aβ are peptides of 36-43 amino acids that are precursors of the amyloid plaques found in the brains of people with AD. The peptides derive from the amyloid precursor protein (APP), a transmembrane glycoprotein, which is cleaved by beta secretase and gamma secretase to yield Aβ. Aβ molecules when released can aggregate to form flexible soluble oligomers and fibrils which may exist in several forms. These subsequently can react with metals or with microglial cells to produce a large amounts of ROS. Amyloid beta-peptide of 42 amino acid residues (Aβ 1- 42 ) is the main source of free radicals in patients with AD. [START_REF] Mc Rosales Hernández | Involvement of Free Radicals in the Development and Progression of Alzheimer's Disease[END_REF][START_REF] Smith | The redox chemistry of the Alzheimer's disease amyloid β peptide[END_REF][START_REF] Da Butterfield | Oxidative Stress, Amyloid-Peptide, and Altered Key Molecular Pathways in the Pathogenesis and Progression of Alzheimer's Disease[END_REF] Aβ [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] (GSNKGAIIGLM) has been used as a convenient alternative to Aβ 1-42 in AD investigations as the smaller peptide mimics several of the toxicological and oxidative stress properties of the native full-length peptide. It has been found that very rapid assembly leading to stable insoluble aggregates of Aβ 25-35 occur at pH 7.4, not pH 3.0 in water. [START_REF] Millucci | Rapid aggregation and assembly in aqueous solution of Aβ (25-35) peptide[END_REF] It is known that Aβ 25-35 is more rapidly toxic and causes more oxidative damage than the parent Aβ 1-42. The Cterminal methionine Aβ [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] has been identified as the cause of the enhanced reactivity of Aβ 25-35 [START_REF] Da Butterfield | Oxidative Stress, Amyloid-Peptide, and Altered Key Molecular Pathways in the Pathogenesis and Progression of Alzheimer's Disease[END_REF][START_REF] Varadarajan | Different mechanisms of oxidative stress and neurotoxicity for Alzheimer's A beta(1-42) and A beta(25-35)[END_REF][START_REF] Schöneich | Free Radical Reactions of Methionine in Peptides: Mechanisms Relevant to β-Amyloid Oxidation and Alzheimer's Disease[END_REF][START_REF] Fourré | Structural and Topological Studies of Methionine Radical Cations in Dipeptides: Electron Sharing in Two-Center Three-Electron Bonds[END_REF][START_REF] Me Clementi | Ab(31-35) and Ab(25-35) fragments of amyloid beta-protein induce cellular death through apoptotic signals: Role of the redox state of methionine-35[END_REF] The activity of the amyloid beta protein in AD may be related to modulation of membrane lipid peroxidation. The effect of Aβ 25-35 on lipid peroxidation was examined in liposomes enriched with polyunsaturated fatty acids. The activity of Aβ 25-35 was compared to that of Aβ [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] in which methionine was replaced with leucine. The relative antioxidant activities of the native Aβ [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] was greater than the methionine substituted Aβ 25-35. Aβ 25-35 inhibits lipid peroxidation at low concentrations in a dose-and time-dependent manner as a result of physicochemical intercalations with the membrane lipid bilayer. [START_REF] Walter | Alzheimer's disease amyloid beta peptide 25-35 inhibits lipid peroxidation as a result of its membrane interactions[END_REF][START_REF] Da Butterfield | Evidence that amyloid beta-peptideinduced lipid peroxidation and its sequelae in Alzheimer's disease brain contribute to neuronal death[END_REF][START_REF] Fabiani | Alzheimer's Disease as a Membrane Disorder: Spatial Cross-Talk Among Beta-Amyloid Peptides, Nicotinic Acetylcholine Receptors and Lipid Rafts[END_REF] Aβ 1-42 oligomers must be inserted into the lipid bilayer of brain cells to initiate lipid peroxidation. Lipid peroxidation is thought to be the first damaging event to occur with Aβ 1-42 in the AD brain. L-methionine 35 (particularly the sulfuranyl radical of L-met) of Aβ 1-42 plays a major role in lipid peroxidation. [START_REF] Da Butterfield | Oxidative Stress, Amyloid-Peptide, and Altered Key Molecular Pathways in the Pathogenesis and Progression of Alzheimer's Disease[END_REF] We have recently studied the lipid peroxidation of cell membranes and brain protection for cerebral ischemia by Edv and Edv analogs. The in vitro and in vivo study was consistent with a mechanism that is dominated by (a) the penetration of the drug into the cell membrane which requires initially desolvation of the drug as it moves from the bulk aqueous environment into the lipid bilayer, then the inherent lipophilicity of the drug as penetrates deeper into the more hydrophobic centre of the bilayer, and (b) the reactivity of the drug in the oxidized regions of the lipid bilayer, which can be assessed by the HOMO-LUMO energy gap. It is step (a) that is slow and rate determining. Once the drug penetrates into the lipid bilayer, reaction with allylic radical PUFAs will be rapid. [START_REF] Fong | Mechanism, structure activity relationships of lipid peroxidation of cell membranes and brain protection for cerebral ischemia by Edaravone and Edaravone analogs -a quantum mechanical study[END_REF] The mechanism by which Edv and its analogs exerts its antioxidative or free radical scavenging behaviour is considered to be that shown in Figure 1. [START_REF] Fong | Mechanism, structure activity relationships of lipid peroxidation of cell membranes and brain protection for cerebral ischemia by Edaravone and Edaravone analogs -a quantum mechanical study[END_REF][START_REF] Watanabe | How is edaravone effective against acute ischemic stroke and amyotrophic lateral sclerosis?[END_REF][START_REF] Ahmadinejad | Molecular Mechanisms behind Free Radical Scavengers Function against Oxidative Stress[END_REF][START_REF] Niki | Lipid peroxidation: Mechanisms, inhibition, and biological effects[END_REF] The single electron transfer (SET) reactions from the neutral and anionic forms of 27 Edv derivatives to 11 free radicals (including the stable free radical 2,2-diphenyl-1-picrylhydrazyl, DPPH) has been studied under physiological conditions and most of them were excellent free radical scavengers via the SET mechanism. [START_REF] Pérez-González | On the Outstanding Antioxidant Capacity of Edaravone Derivatives through Single Electron Transfer Reactions[END_REF] The anionic form of Edv has been predicted to react 8.6 times faster than its neutral form towards the OH• [START_REF] Pérez-González | OH Radical Scavenging Activity of Edaravone: Mechanism and Kinetics[END_REF]. The dominant view is that the anionic form of Edv and its analogs is the major species responsible for the free radical scavenging mechanism towards a wide range of free radicals. [START_REF] Fong | Mechanism, structure activity relationships of lipid peroxidation of cell membranes and brain protection for cerebral ischemia by Edaravone and Edaravone analogs -a quantum mechanical study[END_REF][START_REF] Watanabe | How is edaravone effective against acute ischemic stroke and amyotrophic lateral sclerosis?[END_REF][START_REF] Ahmadinejad | Molecular Mechanisms behind Free Radical Scavengers Function against Oxidative Stress[END_REF][START_REF] Niki | Lipid peroxidation: Mechanisms, inhibition, and biological effects[END_REF][START_REF] Pérez-González | On the Outstanding Antioxidant Capacity of Edaravone Derivatives through Single Electron Transfer Reactions[END_REF][START_REF] Pérez-González | OH Radical Scavenging Activity of Edaravone: Mechanism and Kinetics[END_REF] A recent study of the reaction of a wide range of Edv analogs with DPPH [START_REF] Bk Sarojini | DPPH Scavenging Assay of Novel 1,3-disubstituted-1H-pyrazol-5-ols and their in silico Studies on Some Proteins Involved in Alzheimer's Disease Signaling Cascade[END_REF] allows an examination of properties than drive the free radical scavenging ability of Edv. Hydrogen transfer is more preferential than an electron transfer for reaction of Edaravone with DPPH radical. [START_REF] Borges | A combined experimental and theoretical approach for radical scavenging activity of edaravone and its related derivatives[END_REF] Combined with our previous study of the lipid peroxidation of cell membranes by Edv and Edv analogs, a detailed examination of free radical scavenging properties of Edv and its analogs can be made, as shown in Figure 1, where X can be taken to be a PUFA allylic radical involved in lipid peroxidation. The reaction between Edaravone and the hydroxyl radical or DPPH is known to result in the formation of BPOH and OPB as reaction products. [START_REF] Watanabe | How is edaravone effective against acute ischemic stroke and amyotrophic lateral sclerosis?[END_REF] Sarojini [START_REF] Bk Sarojini | DPPH Scavenging Assay of Novel 1,3-disubstituted-1H-pyrazol-5-ols and their in silico Studies on Some Proteins Involved in Alzheimer's Disease Signaling Cascade[END_REF] has also studied the docking binding energy behavior of a number of Edv analogs with the Amyloid Precursor Protein (APP) which is the precursor to the A , Aβ 1-42 and Aβ [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] proteins which are neurotoxic substances which play an important role in the pathogenesis of AD. A can directly generate free radicals or bind with metal ions (Fe 3+ , Cu 2+ ) to generate hydroxyl radicals from H 2 O 2 or other ROS via the Fenton reaction. [START_REF] Mc Rosales Hernández | Involvement of Free Radicals in the Development and Progression of Alzheimer's Disease[END_REF][START_REF] Smith | The redox chemistry of the Alzheimer's disease amyloid β peptide[END_REF][START_REF] Da Butterfield | Oxidative Stress, Amyloid-Peptide, and Altered Key Molecular Pathways in the Pathogenesis and Progression of Alzheimer's Disease[END_REF] Aβ [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] has been used as a convenient alternative in AD investigations since the smaller peptide mimics several of the toxicological and oxidative stress properties of the native full-length peptides A and Aβ 1-42 . It is known that Aβ 25-35 is more rapidly toxic and causes more oxidative damage than the parent Aβ 1-42. The redox properties of the C-terminal methionine of Aβ 25-35 has been identified as the cause of the enhanced reactivity of Aβ 25-35 [START_REF] Varadarajan | Different mechanisms of oxidative stress and neurotoxicity for Alzheimer's A beta(1-42) and A beta(25-35)[END_REF][START_REF] Schöneich | Free Radical Reactions of Methionine in Peptides: Mechanisms Relevant to β-Amyloid Oxidation and Alzheimer's Disease[END_REF][START_REF] Fourré | Structural and Topological Studies of Methionine Radical Cations in Dipeptides: Electron Sharing in Two-Center Three-Electron Bonds[END_REF][START_REF] Me Clementi | Ab(31-35) and Ab(25-35) fragments of amyloid beta-protein induce cellular death through apoptotic signals: Role of the redox state of methionine-35[END_REF] Since the Aβ 25-35 protein is an integral part of APP, this well defined protein can serve as a model component of Edv binding with APP for which binding energies are available for a wide range of Edv analogs from Sarojini's study.

Edv reduces the generation of A toxicity involve interfering with hydrolysis sites of APP, increasing the activity of -secretase, and reducing the hydrolysis of -secretase upon APP. [START_REF] Ye Shen | Effects of Edaravone on Amyloidβ Precursor Protein Processing in SY5Y-APP695 Cells[END_REF] Edv also reduces the oxidative damage caused by A 25-35 in PC12 cells. [START_REF] Zhang | Edaravone ameliorates oxidative damage associated with A 25-35 treatment in PC12 cells[END_REF] In a beta amyloid A -induced rat model of AD, Edv can inhibit the neurotoxic effect of A toxicity in a concentration dependant manner, suggesting that Edv may be a potential complementary treatment strategy for AD. [START_REF] He | Inhibitory Effects of Edaravone in -Amyloid-Induced Neurotoxicity in Rats[END_REF] Study objectives (a) Examine the free radical scavenging ability of Edv and its analogs towards the stable free radical DPPH and compare such results to the behavior of Edv and analogs with lipid peroxidation. (b) Examine the quantum chemical reactivity properties of the Aβ 25-35 protein and Edv to elucidate the factors which drive the binding interactions between the protein and Edv.

Results (1) Free radical scavenging of DPPH by Edaravone and analogs

We have previously described a model that has been shown to apply to a wide range of drug transport, binding, metabolic and cytotoxicity properties of cells and tumours (Equation 1). The model is based on establishing linear free energy relationships between the four drug properties and various biological processes. Equation 1 has been previously applied to passive and facilitated diffusion of a wide range of drugs crossing the blood brain barrier, [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF] the active competitive transport of tyrosine kinase inhibitors by the hOCT3, OATP1A2 and OCT1 transporters, and cyclin-dependent kinase inhibitors and HIV-1 protease inhibitors. The model also applies to PARP inhibitors, the anti-bacterial and anti-malarial properties of fluoroquinolones, and active organic anion transporter drug membrane transport, and some competitive statin-CYP enzyme binding processes. There is strong independent evidence from the literature that ΔG desolvation , ΔG lipophilicity , the dipole moment and molecular volume are good inherent indicators of the transport or binding ability of drugs. [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF][START_REF] Fong | Predicting PARP inhibitory activity -A novel quantum mechanical based model[END_REF][START_REF] Fong | A novel predictive model for the anti-bacterial, anti-malarial and hERG cardiac QT prolongation properties of fluoroquinolones[END_REF][START_REF] Fong | Statins in therapy: Cellular transport, side effects, drug-drug interactions and cytotoxicity -the unrecognized role of lactones[END_REF][START_REF] Fong | Drug discovery model using molecular orbital computations: tyrosine kinase inhibitors[END_REF] Equation 1: Transport or Binding or Cytotoxicity = ΔG desolv,CDS + ΔG lipo,CDS + Dipole Moment + Molecular Volume

Eq 1 uses the free energy of water desolvation (ΔG desolv,CDS ) and the lipophilicity free energy (ΔG lipo,CDS ) where CDS represents the non-electrostatic first solvation shell solvent properties. CDS may be a better approximation of the cybotactic environment around the drug approaching or within the protein receptor pocket, or the cell membrane surface or the surface of a drug transporter, than the bulk water environment outside the receptor pocket or cell membrane surface. The CDS includes dispersion, cavitation, and covalent components of hydrogen bonding, hydrophobic effects. Desolvation of water from the drug (ΔG desolv,CDS ) before binding in the receptor pocket is required, and hydrophobic interactions between the drug and protein (ΔG lipo,CDS ) is a positive contribution to binding. The lipophilicity ΔG lipo,CDS is calculated from the solvation energy in n-octane or n-octanol. In some biological processes, where biological reduction may be occurring, and the influence of molecular volume is small, the reduction potential (electron affinity) has been included in place of the molecular volume. In other processes, the influence of some of the independent variables is small and can be eliminated to focus on the major determinants of biological activity.

We have recently used this model to develop a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues of tirapazamine and the effect on the extravascular penetration of tirapazamine into tumours. [START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF] It was found that the multiparameter model of the diffusion, antiproliferative assays IC 50 and aerobic and hypoxic clonogenic assays for a wide range of neutral and radical anion forms of tirapazamine (TPZ) analogues showed: (a) extravascular diffusion is governed by the desolvation, lipophilicity, dipole moment and molecular volume, similar to passive and facilitated permeation through the blood brain barrier and other cellular membranes, (b) hypoxic assay properties of the TPZ analogues showed dependencies on the electron affinity, as well as lipophilicity and dipole moment and desolvation, similar to other biological processes involving permeation of cellular membranes, including nuclear membranes, (c) aerobic assay properties were dependent on the almost exclusively on the electron affinity, consistent with electron transfer involving free radicals being the dominant species.

The model (eq 2) has also been recently applied to triple negative breast and ovarian cancers where transient and stable free radicals are involved in the cytotoxic oxidative stress processes. The electron affinity of the various drugs, along with the water desolvation, lipophilicity and dipole moment, has been shown to be an important predictor of cytotoxic efficacy. [START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF] In our recent study of ORAC and CellROX free radical anticancer drugs and oxidative stress in colorectal cancer cells [START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF] we found that eq 2 was also applicable.

The antioxidant capacity (or chemical reactivity) of the drugs can be assessed by the HOMO-LUMO energy gap, since we have recently shown that this gap is linearly related to the LD 50 toxicity of various drugs which are involved in oxidative stress processes in some cancers. [START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF] Eq 2 can then be represented by substituting the electron affinity by the HOMO-LUMO energy gap for analogs where no radical species is formed. Equation 2.

Oxidative Stress or Oxidative Properties = ΔG desolv,CDS + ΔG lipo,CDS + Dipole Moment + Electron Affinity (or HOMO-LUMO gap) Evaluation methodology

Given the successful transport, binding or cytotoxicity model as described by eq 1 and 2 for wide range of drugs, the method used was to calculate the molecular parameters for the DPPH scavenging activity for 17 Edv analogs (isolated in the enol form) in Table 1, at 25,50,75 μg/mL of Edv and Edv analogs taken from Sarojini [START_REF] Bk Sarojini | DPPH Scavenging Assay of Novel 1,3-disubstituted-1H-pyrazol-5-ols and their in silico Studies on Some Proteins Involved in Alzheimer's Disease Signaling Cascade[END_REF]: (1) the free energy of water desolvation (ΔG desolv,CDS ), ( 2) the lipophilicity free energy (ΔG lipo,CDS ) in n-octane, (3) the dipole moment in water, (4) the molecular volume in water, and ( 5) or the HOMO-LUMO energy gap in water. These values are shown in Table 1.

Stepwise multiple regression was then applied to seek out which drug molecular properties had the largest effect on the DPPH scavenging activity for Edv and Edv analogs shown in Table 1.

Stepwise multiple regression analysis showed that eqs 3(a), 3(b) and 3(c Eq 5 shows as expected that water desolvation and lipophilicity of the Edv analogs are significant factors affecting the binding energy, and the HOMO-LUMO energy gap is a major determinant.

)
Since the HOMO-LUMO gap of the Edv analog ligands is a major determinant of the binding energy, the HOMO-LUMO gap of the APP, or the Aβ 25-35 protein as a proxy for the APP, should correlate with the values for the Edv analogs.

Rudberg [40] has calculated the HOMO-LUMO gap of a wide range of proteins to be within the range of 0.1-5.8 eV in vacuo or 2-8 eV in water at the B3LYP level, and was fairly insensitive to the basis set employed (3-21G was adequate). It was noted that the gap was somewhat dependent on the functional employed, with the HF functional giving anomalously high values. Lever [START_REF] Lever | Electrostatic considerations affecting the calculated HOMO-LUMO gap in protein molecules[END_REF] has calculated the HOMO-LUMO gap of a range of proteins (ranging from 456-2386 atoms) in water to be 2.6-3.7 eV using the PBE functional. Beratan [START_REF] Dn Beratan | DNA: insulator or wire?[END_REF] has determined that the HOMO-LUMO gap in proteins is ca 5-7eV, with the HOMO of typical proteins being about -7 eV.

Fukushima [START_REF] Fukushima | An insight into the general relationship between the three dimensional structures of enzymes and their electronic wave functions: Implication for the prediction of functional sites of enzymes[END_REF] has found that the molecular orbitals of 112 enzymes fell into three groups according to the degree of electron delocalization: (a) highly localized orbitals, (b) highly delocalized orbitals whose electrons are distributed over almost the whole molecule, and (c) moderately delocalized orbitals. The MOs belonging to Group A are located near the HOMO-LUMO band gap, and thereby include the frontier orbitals of a given enzyme. The frontier orbitals of the 112 enzymes have a high propensity to be co-located with the known functional sites, especially when the enzymes are hydrated.

An examination of the frontier orbitals of the 128 conformers of the protein ubiquitin indicated that their localization is not affected by the protein dynamics. There was some variation in the energy levels of the HOMOs and LUMOs for the various conformers as expected using a semiempirical method, PM6. [START_REF] Pichierri | Insights into the interplay between electronic structure and protein dynamics: the case of ubiquitin[END_REF] Pang [START_REF] Pang | Two Rules on the Protein-Ligand Interaction[END_REF] has calculated the full electronic structure of a protein along with its hydrophobic pocket with perturbation theory, and concluded that interaction only occurs between the lowest unoccupied molecular orbitals (LUMOs) of a protein and the highest occupied molecular orbital (HOMO) of its ligand, not between the HOMOs of a protein and the LUMO of its ligand. However this conclusion is based on the assumption that the HOMO-LUMO gap in proteins is very large, ca. 11 eV which is not always true. [40][START_REF] Lever | Electrostatic considerations affecting the calculated HOMO-LUMO gap in protein molecules[END_REF][START_REF] Dn Beratan | DNA: insulator or wire?[END_REF] However it is clear that where the HOMO and LUMO of the ligand and the protein are close together, binding is facilitated as the complex is stabilized more than when the gap is large.

Discussion

Free radical scavenging of DPPH• by Edaravone and analogs compared to lipid peroxidation by Edaravone and analogs

The scavenging behavior by the Edv analogs of the stable radical DPPH occurs mainly by hydrogen atom transfer. Eq 3(a), 3(b) and 3(c) show the scavenging activity is directly proportional to the negative of the HOMO-LUMO gaps of the Edv analogs. There was no dependence on ΔG desolv,CDS, ΔG lipo,CDS, Dipole Moment, or Molecular Volume as expected for a simple reaction between two unencumbered species in water. However we have previously showed that IC 50 inhibition of lipid peroxidation for Edv and 35 analogs is well described by eq 4 which shows that lipid peroxidation is dependent on the ΔG desolv,CDS, ΔG lipo,CDS, Dipole Moment and HOMO-LUMO gap reflecting the water desolvation required for the inhibitors to enter a lipid environment described by ΔG lipo,CDS. In this hydrophobic environment, reactivity is determined by the HOMO-LUMO gap as well as the dipole interaction.

These equations describe the antioxidant capacity of Edv and a wide range of closely related analogs and that the HOMO-LUMO gaps of these compounds are good ab initio predictors of antioxidant activity. Lipid peroxidation is a known element of the pathogenesis of Alzheimer's disease. The HOMO, LUMO and HOMO-LUMO gaps of Edv, its tautomers and the anionic form of Edv and DPPH• shown in Table 3 indicate the relative reactivities of the various species.

Clearly the anionic form of Edv is more reactive than the enol form of Edv.

Quantum mechanical study of Aβ 25-35 protein and Edaravone

Since the HOMO-LUMO gap is strongly involved in antioxidant reactivity of Edv and a range of analogs with a simple stable free radical like DPPH• and in the more relevant lipid peroxidation of rat brain homogenate, it was convenient to examine the HOMO-LUMO gap of the Aβ 25-35 protein and Edv to obtain direct insight into the likely redox interactions.

Eq 5 shows the docking binding energy of 10 Edv analogs with Amyloid Precursor Protein (APP) is dependent on the water desolvation, lipophilicity and HOMO-LUMO gap of the 10 Edv analogs. The Amyloid Precursor Protein (APP) is the precursor to the A , Aβ Table 3 shows the HOMO, LUMO and HOMO-LUMO values for the Aβ 25-35 protein with Edv and the Edv anion which is known to be the more potent electron scavenger form, and the DPPH• radical.

Since it is known that high levels of dietary methionine is implicated in AD [START_REF] Tapia-Rojas | Is L-methionine a trigger factor for Alzheimer's-like neurodegeneration?: Changes in Aβ oligomers, tau phosphorylation, synaptic proteins, Wnt signaling and behavioral impairment in wild-type mice[END_REF] and there is ample evidence that the redox behaviour of methionine in Aβ, Aβ 1-42 and Aβ 25-35 [START_REF] Da Butterfield | Oxidative Stress, Amyloid-Peptide, and Altered Key Molecular Pathways in the Pathogenesis and Progression of Alzheimer's Disease[END_REF][START_REF] Varadarajan | Different mechanisms of oxidative stress and neurotoxicity for Alzheimer's A beta(1-42) and A beta(25-35)[END_REF][START_REF] Schöneich | Free Radical Reactions of Methionine in Peptides: Mechanisms Relevant to β-Amyloid Oxidation and Alzheimer's Disease[END_REF][START_REF] Fourré | Structural and Topological Studies of Methionine Radical Cations in Dipeptides: Electron Sharing in Two-Center Three-Electron Bonds[END_REF][START_REF] Me Clementi | Ab(31-35) and Ab(25-35) fragments of amyloid beta-protein induce cellular death through apoptotic signals: Role of the redox state of methionine-35[END_REF] is very important in β-amyloid oxidative stress and neurotoxicity. The HOMO and LUMO of Aβ 25- 35 in various conformations is examined in Table 3.

The Aβ 25-35 peptide is more rapidly toxic and causes more oxidative damage than its parent Aβ 1- 42 . The mechanism of action of these peptides is thought to be caused by the single C-terminal methionine. When the next amino acid in the sequence of Aβ 1-42 (valine) is appended to Aβ 25-35 , the resultant peptide, Aβ 25-36 in which methionine is no longer C-terminal, is neither toxic to cultured neurons nor does it cause oxidative damage. [START_REF] Da Butterfield | Oxidative Stress, Amyloid-Peptide, and Altered Key Molecular Pathways in the Pathogenesis and Progression of Alzheimer's Disease[END_REF][START_REF] Varadarajan | Different mechanisms of oxidative stress and neurotoxicity for Alzheimer's A beta(1-42) and A beta(25-35)[END_REF] The methionine radical cation in Aβ is thought to be critical in neurotoxicity and oxidative stress, and that the methionine radical cation in peptides can be stabilized through bond formation with either the oxygen or the nitrogen atoms of adjacent peptide bonds. [START_REF] Schöneich | Free Radical Reactions of Methionine in Peptides: Mechanisms Relevant to β-Amyloid Oxidation and Alzheimer's Disease[END_REF] The sulfur-centered radical cation could complex with oxygen, nitrogen, or other sulfur atoms from a neighboring residue or from the peptidic bond, forming an intramolecular S∴X two-center three-electron bond (X = S, N, O) [START_REF] Fourré | Structural and Topological Studies of Methionine Radical Cations in Dipeptides: Electron Sharing in Two-Center Three-Electron Bonds[END_REF] Evidence for these conjectures can be seen in Figures 2 and3 which shows the various conformations and configurations of Aβ 25-35 , and a truncated Aβ 25-35 which focuses on the Cterminal L-methionine at a more detailed level of theory.

Since Aβ 25-35 is known to be more neuotoxic than its parent Aβ 1-42 which is attributed to the Cterminal L-methionine, the formation of the rotated conformation of Aβ [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] shown in Figure 2(3) allows the formation of a cation radical which can engage in 3 electron bonding overlap between the S + atom and the N atom of methionine similar to that shown in Figure 2 (right). The water optimized S-N bonding length is 2.23Å indicating that this arrangement favours S---N bonding interaction rather than S---O interaction (3.63Å). It has been previously suggested that a three centred three electron bonding arrangement as shown in Figure 3 (right) can stabilize the cation radical, but this study suggests that S---N interaction is the dominant stabilizing interaction for the methionine moiety of Aβ [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] . In particular it is noteworthy that this bonding arrangement is a LUMO, not the usual HOMO which resides on methionine in most conformations and reactive configurations of Aβ 25-35 as illustrated in Table 3. This molecular arrangement would be critical in the potential interactions between Aβ 25-35 and other reactive species involved in oxidative stress or protein-protein interactions. The energy levels of the HOMO, LUMO and HOMO-LUMO gap for this cation radical closely match, for example, that for the Edv anion radical (see Table 3), with binding between the HOMO of Edv anion and the LUMO of the cation radical Aβ 25-35 on L-met35, and concomitantly between the HOMO of the cation radical Aβ 25-35 with the LUMO of the EDV anion. These levels more closely match each other than the alternative Edv keto or enol tautomers shown in Table 3. It is noted that the rotated cation radical conformation of Aβ 25-35 is overall less stable than the non-rotated conformation by 24.5 kcal/mol (total electronic energy basis in water), but this reactive conformation with the LUMO located on the L-methionine that appears to be the basis for the enhanced activity of the C-terminal Lmethionine of Aβ 25-35 in AD pathogenesis. An estimate of the degree of charge delocalization in the rotated cation radical conformation (Figure 2(4)) relative to the non-rotated conformation (Figure 2(5)) is seen in Table 3 where the former has an electrostatic charge of +0.118 on S and -0.249 on N compared to the values of +0.419 and -0.685 respectively, ie the charge on S decreases by 0.301 au in the rotated cation radical conformation.

It has been previously observed that when the next amino acid in the sequence of Aβ 1-42 (ie valine) is appended to Aβ 25-35 , the resultant peptide, Aβ 25-36 , (where methionine is no longer Cterminal), is neither toxic to cultured neurons nor does it cause oxidative damage. [START_REF] Varadarajan | Different mechanisms of oxidative stress and neurotoxicity for Alzheimer's A beta(1-42) and A beta(25-35)[END_REF] The reason for this is that the addition of valine in the Aβ 25-36 protein precludes the methionine35 from rotating into a conformation where the S can approach the N of methionine which it can do in Aβ 25-35 (see Figure 2(4)). That is, S is sterically hindered from approaching the N atom of methionine 35 when valine is present in Aβ 25-36 .

Table 3 lists various species of Aβ 25-35 which may be involved in redox reactions with other species, such as the radical, anion, cation and corresponding rotated conformations. The anion species where the N of methionine is deprotonated is particularly reactive as evidenced by the smaller HOMO-LUMO gaps. The HOMO is located on the L-methionine and the LUMO on the L-alanyl-L-isoleucyl-L-isoleucylglycyl region in all configurations except that of the cation radical forms where both the HOMO and LUMO are located on the L-lysylglycyl-L-alanyl-Lisoleucyl-L-isoleucylglycyl region for the non-rotated form (see Figure 2(5)) and the rotated form (see Figure 2(4)). Since the binding between Aβ 25-35 and other redox species is governed by the HOMO and LUMO interactions, these locations and their relative energy levels will decide reactive outcomes in AD pathogenesis.

The redox state of methionine-35 in Aβ 31-35 and Aβ 25-35 peptides primarily determines the toxicity in isolated rat brain mitochondria. The peptides are predominantly monomeric forms and are strongly neurotoxic, having the ability to enter within the cells, determining mitochondrial damage and trigger of apoptotic signals. The oxidation of methionine-35 to the sulphoxide is known to reduce toxicity to neurons and mitochondria, as well as reducing the solubility of the Aβ 31-35 and Aβ 25-35 peptides. A decreased or inhibited activity of methionine sulfoxide reductase is found in AD brains, so a cell protective mechanism which prevents the reduction of the oxidised (sulphoxide) methionine-35 residue back to the toxic Aβ 31-35 and Aβ 25-35 peptides may be possible. [START_REF] Me Clementi | Ab(31-35) and Ab(25-35) fragments of amyloid beta-protein induce cellular death through apoptotic signals: Role of the redox state of methionine-35[END_REF] There is a strong negative correlation between oxidative damage and Aβ deposition in AD brain. It is generally thought that drugs that target oxidative pathways can be effective against AD. [START_REF] Smith | The redox chemistry of the Alzheimer's disease amyloid β peptide[END_REF] The HOMO and LUMO energy levels (and the HOMO-LUMO gap) are inherent determinants of reactivity of proteins such as Aβ 25-35 , so drugs which can bind with Aβ 25-35 will have HOMO and LUMO close to those in Aβ 25-35 (HOMO Donor àLUMO Acceptor ). Table 3 shows the HOMO and LUMO of various reactive species of the Aβ 25-35 protein and the anti-AD drug Edv in different tautomeric forms and the reactive anionic form. It can be seen why Edv and especially the anionic form of Edv is an effective AD drug, based on the Aβ 25-35 being a proxy for AD pathogenesis, particularly because the methionine in the rotated conformation of the cation radical is a LUMO which can accept electrons from binding or interacting redox species. The anionic form of Edv has a HOMO of -4.48eV compared to the unrotated / unconstrained configurations of Aβ 25-35 which have HOMOs of -6.3 to -6.8eV (excepting the anionic species)

indicating that Edv HOMO is the donor to the Aβ 25-35 LUMO (L-alanyl-L-isoleucyl-Lisoleucylglycyl) acceptor. However for the rotated cation radical form of Aβ 25-35 interacting with the Edv anionic species, the Aβ 25-35 LUMO is located on the methionine moiety.

The anionic form of methionine (loss of H + from the N-H position) in the rotated conformation of the Aβ 25-35 protein is very reactive with a HOMO-LUMO gap (1.99eV), with the HOMO -3.65eV (located on the methionine) and LUMO -1.66eV. This configuration allows highly reactive redox binding or interacting with various species in AD pathogenesis which is opposite to that available to the rotated cation radical species of the Aβ 25-35 protein in redox processes. The anionic (on methionine) form of Aβ 25-35 is predicted to be a very reactive intermediate whereby the HOMO on methionine could be a donor binding species to other acceptor species involved in AD pathogenesis.

Conclusions

This study has examined the free radical scavenging ability of Edaravone and its analogs towards the stable free radical DPPH and compares such results to the behavior of Edv and analogs with lipid peroxidation. The scavenging activity is directly proportional to the negative of the HOMO-LUMO gaps of the Edv analogs. We have previously shown that inhibition of lipid peroxidation for Edaravone and 35 analogs is dependent on the water desolvation (ΔG desolv,CDS ) , lipophilicity (ΔG lipo,CDS ), Dipole Moment and the negative of the HOMO-LUMO gap. Lipid peroxidation is a known element of the pathogenesis of Alzheimer's disease.

This study has also examined the quantum chemical reactivity properties of the Aβ 25-35 protein and Edaravone to elucidate the factors which drive the binding interactions between the protein and Edaravone. Edaravone and its analogs are known to interact with the Amyloid Precursor Protein (APP) which is the precursor to the A , Aβ The studies of the Aβ 25-35 protein, Edaravone, and the binding energies of the Edaravone analogs to the Amyloid Precursor Protein are all consistent in showing that the HOMO, LUMO and HOMO-LUMO energy gaps are inherent predictors of how such molecular properties are important to Alzheimer's pathogenesis.

Experimental Methods

Calculations were carried out using the Gaussian 09 package. Energy optimizations were at the DFT/B3LYP/6-31G(d,p) (6d, 7f) level of theory for all atoms in water. Selected optimizations at the DFT/B3LYP/6-311+G(d,p) (6d, 7f) level of theory gave very similar results to those at the lower level. Optimized structures were checked to ensure energy minima were located, with no imaginary frequencies. Energy calculations were conducted at the DFT/B3LYP/6-31G(d,p) (6d, 7f) for neutral compounds and DFT/B3LYP/6-31+G(d) (6d, 7f) level of theory for anions with optimized geometries in water, using the IEFPCM/SMD solvent model. With the 6-31G* basis set, the SMD model achieves mean unsigned errors of 0.6 -1.0 kcal/mol in the solvation free energies of tested neutrals and mean unsigned errors of 4 kcal/mol on average for ions. [START_REF] Av Marenich | Universal Solvation Model Based on Solute Electron Density and on a Continuum Model of the Solvent Defined by the Bulk Dielectric Constant and Atomic Surface Tensions[END_REF] The 6-31G** basis set has been used to calculate absolute free energies of solvation and compare these data with experimental results for more than 500 neutral and charged compounds. The calculated values were in good agreement with experimental results across a wide range of compounds. [START_REF] Rayne | Accuracy of computational solvation free energies for neutral and ionic compounds: Dependence on level of theory and solvent model[END_REF][START_REF] Rizzo | Estimation of Absolute Free Energies of Hydration Using Continuum Methods: Accuracy of Partial Charge Models and Optimization of Nonpolar Contributions[END_REF] Adding diffuse functions to the 6-31G* basis set (ie 6-31+G**) had no significant effect on the solvation energies with a difference of less than 1% observed in solvents, which is within the literature error range for the IEFPCM/SMD solvent model. HOMO and LUMO calculations included both delocalized and localized orbitals (NBO).

Quantum mechanical study of the Aβ 25-35 protein optimization and energy calculations were conducted at the DFT/B3LYP/6-31G(d,p) and DFT/B3LYP/3-21G with the empirical dispersion = GD3BJ correction in water. Both SMD and CPCM solvation models were used with similar results. ESP atomic charges were calculated using the CHELPG method. Some studies involved the truncation of 3 amino acids at the opposite end to the methionine moiety as shown in Figure 2 at the DFT/B3LYP/6-31G(d,p) level with the empirical dispersion = GD3BJ correction in water to more closely focus on the electronic properties of the methionine moiety of Aβ 25-35 .

It is noted that high computational accuracy for each species in different environments is not the focus of this study, but comparative differences between various species is the aim of the study.

DPPH assay, structures of Edv analogs (enol form) in Table 1

1 -

 1 [START_REF] Dn Beratan | DNA: insulator or wire?[END_REF] and Aβ[START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[END_REF][START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[END_REF][START_REF] Fong | The role of free radicals in the effectiveness of anti-cancer chemotherapy in hypoxic ovarian cells and tumours[END_REF][START_REF] Fong | Free radicals in chemotherapy induced cytotoxicity and oxidative stress in triple negative breast and ovarian cancers under hypoxic and normoxic conditions[END_REF][START_REF] Fong | Role of stable free radicals in conjugated antioxidant and cytotoxicity treatment of triple negative breast cancer[END_REF][START_REF] Fong | Toxicology of platinum anticancer drugs: oxidative stress and antioxidant effect of stable free radical Pt-nitroxides[END_REF][START_REF] Fong | The effect of desolvation on the binding of inhibitors to HIV-1 protease and cyclin-dependent kinases: Causes of resistance[END_REF][START_REF] Fong | Physiology of ionophore transport of potassium and sodium ions across cell membranes: Valinomycin and 18-Crown-6 Ether[END_REF][START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] proteins which are neurotoxic substances that play important roles in the pathogenesis of AD. It is known that Aβ 25-35 is more rapidly toxic and causes more oxidative damage than the parent Aβ 1-42. The redox properties of the C-terminal methionine of Aβ 25-35 has been identified as the cause of the enhanced reactivity of Aβ 25-35[START_REF] Da Butterfield | Oxidative Stress, Amyloid-Peptide, and Altered Key Molecular Pathways in the Pathogenesis and Progression of Alzheimer's Disease[END_REF][START_REF] Varadarajan | Different mechanisms of oxidative stress and neurotoxicity for Alzheimer's A beta(1-42) and A beta(25-35)[END_REF][START_REF] Schöneich | Free Radical Reactions of Methionine in Peptides: Mechanisms Relevant to β-Amyloid Oxidation and Alzheimer's Disease[END_REF][START_REF] Fourré | Structural and Topological Studies of Methionine Radical Cations in Dipeptides: Electron Sharing in Two-Center Three-Electron Bonds[END_REF][START_REF] Me Clementi | Ab(31-35) and Ab(25-35) fragments of amyloid beta-protein induce cellular death through apoptotic signals: Role of the redox state of methionine-35[END_REF] Since the Aβ 25-35 protein is an integral part of APP, this well defined protein can serve as a model component of Edv binding with APP for which binding energies are available for a wide range of Edv analogs.

  1-42 and Aβ 25-35 proteins which are neurotoxic substances which play an important role in the pathogenesis of Alzheimer's disease. Aβ 25-35 has been used as a convenient proxy in AD investigations since the smaller peptide mimics several of the toxicological and oxidative stress properties of the native full-length peptides A and Aβ 1- 42 . It is known that Aβ 25-35 is more rapidly toxic and causes more oxidative damage than the parent Aβ 1-42. The redox properties of the C-terminal methionine of Aβ 25-35 has been previously identified as the cause of the enhanced AD reactivity of Aβ 25-35 .It is shown that the docking binding energy of 10 Edaravone analogs with Amyloid Precursor Protein (APP) is dependent on the water desolvation, lipophilicity and the negative of the HOMO-LUMO gap of the 10 Edv analogs.This study has also shown that a rotated cation radical conformation of Aβ 25-35 (which is overall less stable than the non-rotated conformation by 24.5 kcal/mol), has a reactive conformation with the LUMO located on the L-methionine that appears to be the basis for the enhanced activity of the C-terminal L-methionine of Aβ 25-35 in AD pathogenesis. The relative energies of the HOMO, LUMO and the HOMO-LUMO energy gaps of the Aβ 25-35 protein in relation to those of Edaravone dictate the reactivity between the Edaravone and the Aβ 25-35 protein. The rotated cation radical Aβ 25-35 conformation is unique in that the LUMO resides on the methionine moiety, whereas other conformations such as the radical, anionic and non-rotated conformations has the HOMO residing on the methionine. This arrangement allows 3 electron bonding overlap between the S + atom and the N atom of methionine. As such, this cation radical bonding arrangement can confer unique redox properties on the methionine of Aβ 25-35 protein which has strong implications for Alzheimer's pathogenesis.

[ 23 ]

 23 are shown below: Left structure: R = 3a (4fluoro-phenyl), 3b (2,4-difluorophenyl), 3c (4methyl-phenyl), 3d (2,4dichlorophenyl), 3e (4-methoxy-phenyl), 3f (2-pyridyl), 3g (6-trifluoromethyl-2 pyridyl), 3h (methyl), 3i (R=H), 3j (2-methyl-phenyl) Right structure: R = 3k (4-fluoro-phenyl), 3l (4-methyl-phenyl), 3m (2,4 difluoro-phenyl), 3n (methyl) Left structure: R = 3o (methyl), 3p (phenyl), 3q (2,6-difluoro-phenyl) Left structure: R = 3r (phenyl), 3s (4-fluoro-phenyl) Structure: R = 3t (4-fluoro-phenyl), 3u (methyl) Edaravone

Figure 1 .

 1 Figure 1. Free radical scavenging reactions of Edaravone with DPPH and X• where X• represents an allylic polyunsaturated fatty acid (PUFA) involved in lipid peroxidation.

Figure 2 .Figure 3 .

 23 Figure 2. (1) Aβ 25-35 , (2) Truncated Aβ 25-35 , (3) Rotated Conformation Truncated Aβ 25-35 and expanded view of C-terminal L-methionine, (4) Rotated Conformation Truncated Aβ 25-35 Cation Radical, (5) Truncated Aβ 25-35 Cation Radical

LUMO gap + 128.50

  were the best descriptions when tested against ΔG desolv,CDS, ΔG lipo,CDS, Dipole Moment, Molecular Volume, or the HOMO-LUMO energy gap. It was also clear that a number of analogs 3A, 3D, 3E, 3Q, 3L were consistent statistical outliers in all correlations.

	Where R 2 = 0.213, SEE = 15.66, SE(HOMO-LUMO) = 5.40, F=4.319, Significance=0.054
	Equations 3(a), 3(b) and 3 (c) for the scavenging of the stable free radical DPPH can be
	compared for the previously determined relationships found for lipid peroxidation by Edv and its
	analogs, as shown in eq 4.
	Eq 4 Inhibition of lipid peroxidation IC 50 for Edv and 35 analogs from [17] was:
	IC 50 = 4.19ΔG desolv,CDS + 2.20ΔG lipo,CDS -2.39Dipole Moment -2.95(HOMO-LUMO) + 72.16
	Where R 2 = 0.401, SEE = 7.73, SE(ΔG desolvCDS ) = 1.24, SE(ΔG lipoCDS ) = 1.53, SE(Dipole Moment) =0.92, SE(HOMO-LUMO) =
	2.50, F=5.20, Significance=0.0025
	(2)

Eq 3(a) DPPH scavenging activity for 18 Edv analogs in Table

1

, excluding 3A, 3D, 3E, 3Q, at 25 μg/mL: Activity % = -16.05 HOMO-LUMO gap + 128.

[START_REF] Pang | Two Rules on the Protein-Ligand Interaction[END_REF]

Where R 2 = 0.356, SEE = 13.194, SE(HOMO-LUMO) = 5.40, F=8.827, Significance=0.009 Eq 3(b) DPPH scavenging activity for 18 Edv analogs in Table

1

, excluding 3A, 3D, 3E, 3Q, at 50 μg/mL: Activity % = -12.73 HOMO-LUMO gap + 120.

[START_REF] Av Marenich | Universal Solvation Model Based on Solute Electron Density and on a Continuum Model of the Solvent Defined by the Bulk Dielectric Constant and Atomic Surface Tensions[END_REF]

Where R 2 = 0.356, SEE = 6.424, SE(HOMO-LUMO) = 5.40, F=3.928, Significance=0.065 Eq 3(c) DPPH scavenging activity for 18 Edv analogs in Table

1

, excluding 3A, 3D, 3E, 3Q, at 75 μg/mL: Activity % = -13.328 HOMO-

Quantum mechanical study of Aβ 25-35 protein and Edaravone

  Application of the general eq 2 to the docking binding energy studies of Edv analogs with APP (Table2) taken from Sarojini[START_REF] Bk Sarojini | DPPH Scavenging Assay of Novel 1,3-disubstituted-1H-pyrazol-5-ols and their in silico Studies on Some Proteins Involved in Alzheimer's Disease Signaling Cascade[END_REF] gives eq 5.

	Eq 5. Binding energy of 10 Edv analogs with Amyloid Precursor Protein (APP) in Table 2:
	Binding Energy = 0.26ΔG desolv,CDS + 0.44ΔG lipo,CDS -0.98(HOMO-LUMO) + 0.755
	Where R 2 = 0.885, SEE = 0.184, SE(ΔG desolvCDS ) = 0.07, SE(ΔG lipoCDS ) = 0.08, SE(HOMO-LUMO) = 0.28, F=11.81,
	Significance=0.006

Table 1 . DPPH scavenging assay for Edaravone and analogs at three concentration levels

 1

	(ref 23) with molecular properties of analogs
					HOMO-			Dipole	Molec
					LUMO	ΔG desolv,CDS	ΔG lipo,CDS	Moment	Volume
		25μg/mL	50μg/mL	75μg/mL	Gap eV	kcal/mol	kcal/mol	D	cm 3 /mol
	3A	80.99	84.12	85.98	4.25	-5.34	-7.44	7.73	209
	3B	69.21	71.32	79.95	4.38	-6.16	-6.73	7.88	232

Footnotes: See experimental methods for structures of Edv analogs

Table 2 . Docking binding energies of Edaravone analogs with Amyloid Precursor Protein, APP

 2

, (ref 23) with

molecular properties of analogs Binding Energy HOMO- LUMO Gap eV ΔG desolv,CDS kcal/mol ΔG lipo,CDS kcal/mol Dipole Moment D Molec Volume cm 3 /mol

  Footnotes: See experimental methods for structures of Edv analogs; Edv analogs 3 series are enol tautomers, and 4 series analogs are keto tautomers

	3F	-7.87	4.34	-3.22	-7.95	8.99	188
	3G	-7.82	4.35	-5.39	-6.99	11.92	220
	3B	-8.28	4.38	-6.16	-6.73	7.88	232
	3D	-8.91	4.4	-5.03	-8.94	8.48	206
	3A	-7.92	4.25	-5.34	-7.44	7.73	209
	4A	-7.62	3.83	-5.33	-7.7	6.74	183
	4B	-8.04	3.94	-5.92	-7.12	5.78	213
	4D	-8.51	4	-4.97	-9.24	7.12	210
	4F	-7.78	3.98	-3.29	-8.26	4.4	226
	4g	-7.84	4.03	-5.53	-7.33	6.96	223

Table 3 . HOMO, LUMO, HOMO-LUMO gaps for Aβ 25-35 protein in water plus ESP atomic charges on L-Methionine of Aβ 25-35 protein and Edaravone

 3

		HOMO	LUMO	HOMO-	HOMO	LUMO	ESP	ESP
		eV	eV	LUMO	Site	Site	S Methionine	N Methionine
				Gap eV
	Aβ 25-35	-6.28	-1.68	4.6


Generalized spacetime metric of a static black hole

Keywords: Generalized spacetime metric, static black hole, Lagrangian approach, classical solar system tests, capture cross section

published or not. The documents may come

Generalized spacetime metric of a static black hole

Mischa Moerkamp

Introduction

The Schwarzschild metric, the solution of the Einstein field equations in general relativity theory (GRT) that deals with a spherically symmetric and static universe containing a singularity, is given by:

ds 2 = 1 - r s r c 2 dt 2 -1 - r s r -1
dr 2 -r 2 dϕ 2 (Schwarzschild metric)

where r s = 2GM/c 2 is the Schwarzschild radius, r is the radial coordinate and ϕ is the azimuthal angle (we focus on the spacetime metric in the equatorial plane). Light propagates at zero eigentime: ds 2 = 0, so the coordinate speed of light in the Schwarzschild metric depends on the direction of the light ray: the coordinate speed of light is anisotropic. The Schwarzschild metric features an event horizon at which the metric becomes singular: as light gets closer to the event horizon at r = r s , the coordinate speed of light slows down. The spacetime metric of a static BH can be generalized by introducing three scalar functions f [r], g[r] and h[r]: ds 2 = f c 2 dt 2 -gdr 2 -hr 2 dϕ 2 (Generalized spacetime metric)

Note that the dtdr-term, which also obeys the spherical symmetry condition, has been neglected for simplicity's sake. The scalar functions can be written as series expansions:

f = 1+ ∞ j=1 f j r s r j g = 1+ ∞ j=1 g j r s r j h = 1 + ∞ j=1 h j r s r j
The refractive index of this spacetime metric is:

n[r, û] = h f + (û • r) 2 g -h f
where r = r/|r| denotes the unit vector that points in the direction of the position r with respect to the static BH and û is the direction of the light ray. Both the speed and the direction of light change in accordance with Fermat's principle. The photon sphere (or innermost circular orbit) can be obtained by solving the following equation for r:

d(r h/f ) dr = 0
The photon sphere in the Schwarzschild metric has a radius of r γ = 3r s /2.

The principle of least action reads:

δ ds = 0 and by changing variables from s to t, we can identify the Lagrangian of a test body of mass m:

L = -mc 2 f - 1 c 2 (g ṙ2 + hr 2 φ2
) where the overdot represents differentiation with respect to t. The identity g = h corresponds to an isotropic spacetime metric, in which the coordinate speed of light does not depend on the direction of the light ray. In 1921, H. A. Wilson introduced such an isotropic spacetime metric featuring the scalar refractive index: n = 1+r s /r. [START_REF] Wilson | An Electromagnetic Theory of Gravitation[END_REF] In 1957, Dicke further explored Wilson's model and discussed its possible relation to Mach's principle: the tentative and loosely defined notion that inertial forces are determined by the quantity and distribution of matter in the universe. [START_REF] Dicke | Gravitation without a principle of equivalence[END_REF] Here, we specify the conditions under which the generalized spacetime metric of a static BH shows consistency with the classical solar system tests. A method is described for computing the capture cross section for an incoming photon from infinity, which is applied to the Schwarzschild metric and the isotropic exponential spacetime metric.

2 Lagrangian approach

Lagrangian optics

Fermat's principle can be formulated as a principle of least action:

δ nd = n d dx dx = δ L γ dx = 0
where d is an infinitesimal displacement along the path and L γ is the optical Lagrangian:

L γ = n d dx = n 1 + dy dx 2
The Euler-Lagrange equation associated with the optical Lagrangian is:

d dx     ∂L γ ∂ dy dx     =

∂L γ ∂y

It follows that:

d 2 y dx 2 = 1 n ∂n ∂y 1 + dy dx 2 2 - 1 n dn dx dy dx 1 + dy dx 2
This equation will be useful in determining the gravitational deflection of light.

Conserved quantities

The Lagrangian can be recast in Hamiltonian form:

H = ∂L ∂ ṙ • ṙ -L = mc 2 f f - 1 c 2 (g ṙ2 + hr 2 φ2 ) (Hamiltonian)
The variable ϕ is absent from the Lagrangian:

d dt ∂L ∂ φ = ∂L ∂ϕ = 0
Hence, the canonical momentum conjugate to ϕ is conserved:

L z = ∂L ∂ φ = m φr 2 h f - 1 c 2 (g ṙ2 + hr 2 φ2 ) (Angular momentum)
which can be identified with the angular momentum along the axis perpendicular to the orbital plane.

Circular orbit

The orbital speed of a circular orbit with a radius r can be obtained by solving the Euler-Lagrange equation for ṙ = r = 0:

d dt ∂L ∂ ṙ = 0 ∂L ∂r = - mc 2 2 f - hr 2 φ2 c 2 df dr - 1 c 2 dh dr r 2 φ2 + 2hr φ2
which gives the following solution for the orbital speed of a circular orbit:

u circ = c    df dr 2h r + dh dr    1/2
Hence, the circular orbit Hamiltonian and circular orbit angular momentum are:

H circ = mc 2 f   f -h    df dr 2h r + dh dr       -1/2 L z,circ = mcrh    df dr 2h r + dh dr    1/2   f -    df dr 2h r + dh dr       -1/2

Effective radial potential

The equation of motion for the radius r in terms of t is:

ds dt 2 = -f c 2 + g dr dt 2 + hr 2 dϕ dt 2
where:

ds dt 2 = - mc 2 f H 2 c 2 dϕ dt 2 = L z c 2 f Hr 2 h 2
The equation of motion can be rewritten as:

m 1 2 dr dt 2 +m mc 2 H 2 c 2 2 f 2 g -m c 2 2 f g +m cL z H 2 c 2 2gh f r 2 = 0
which describes a particle moving in a one-dimensional effective radial potential:

U eff [r] = m mc 2 H 2 c 2 2 f 2 g -m c 2 2 f g +m cL z H 2 c 2 2gh f r 2
A stable circular orbit occurs at a local minimum of the effective radial potential and the innermost stable circular orbit (ISCO) radius can be obtained by minimizing the circular orbit angular momentum. The ISCO in the Schwarzschild metric is located at: r = 3r s .

3 Classical solar system tests

Gravitational redshift and gravitational deflection of light

The gravitational redshift can be accounted for if the following condition holds:

f 1 = -1 (Condition 1)
A commonly used method to approximate the deflection angle ∆ϕ is to integrate the curvature over an infinite straight path (dy/dx = 0):

∆ϕ ≈ ∞ -∞ d 2 y dx 2 y=r0 dx = ∞ -∞ 1 n ∂n ∂y y=r0 dx = r s r 0 1 -f 1 ) + O r s r 0 2
Since the observed angle of deflection is ∆ϕ ≈ 2r s /r 0 , we have:

g 1 -f 1 = 2 (Condition 2)
which, combined with Condition 1, yields:

g 1 = 1 (Condition 2*)

Gravitational time delay

A light signal that passes near a gravitational mass takes slightly longer to travel to a target than it would if the mass were not present. This time delay can be split into two different parts:

1. The Shapiro time delay, caused by a variation in the speed of a photon if it is submerged in a gravitational potential.

2. The geometric delay, caused by the increased length of the total light path from the source to the target, which is due to gravitational deflection.

The straight path approximation is correct in the first order, because the contribution from the geometric delay is of the second order. Hence, we use the straight path approximation to compute the gravitational time delay for an Earth-based light signal that passes the Sun, reaches a spacecraft and returns back to Earth along the same path:

∆t ≈ 2 c √ r 2 2 -r 2 0 - √ r 2 1 -r 2 0 h f + x 2 x 2 + r 2 0 g -h f -1 y=r0 dx = = (h 1 -g 1 ) r s c r 2 2 -r 2 0 r 2 + r 2 1 -r 2 0 r 1 ≈Constant +(g 1 -f 1 ) r s c ln r 2 2 -r 2 0 + r 2 -r 2 1 -r 2 0 + r 1 + O r 2 s
where r 0 is the distance of closest approach to the Sun, r 1 is the Earth-Sun distance and r 2 is the spacecraft-Sun distance. Although the anisotropy in the spacetime metric does enter into the prediction of the gravitational time delay, its contribution can be neglected due to its small time variation (small by today's standards, that is). Hence, the spacetime metric is consistent with the gravitational time delay test if Condition 2 holds: g 1 -f 1 = 2.

Perihelion precession of Mercury

The inverse square of the angular frequency ω ϕ of a circular orbit as a function of r is:

1 ω 2 ϕ = r u circ 2 = r c 2 r r s - 2 f 1 + 4f 2 -f 1 h 1 f 2 1 r s r + O r s r
If a perturbation is added to the circular orbit, then the square of the angular frequency ω r of the radial oscillation is given by:

ω 2 r = 1 m d 2 U eff dr 2 H=Hcirc Lz=Lz,circ = = mc 2 H circ 2 c 2 2 d 2 dr 2 f 2 g - c 2 2 d 2 dr 2 f g + cL z,circ H circ 2 c 2 2 d 2 dr 2 1 gh r 2 = = c r 2 r s r - f 1 2 + -f 2 1 + f 1 g 1 2 + 3f 1 h 1 4 r s r + O r s r 2
Hence, the advance of the perihelion in one revolution is:

∆ϕ = 2π 1 - ω r ω ϕ = 2π -f 1 + g 1 + h 1 2 + f 2 f 1 =3/2 r s r +O r s r 2
which closely matches the observed perihelion precession of Mercury if the following condition holds:

-f 1 + g 1 + h 1 2 + f 2 f 1 = 3 2 (Condition 3)
Combining these three conditions yields the following relation between h 1 and f 2 :

h 1 = 2f 2 (Condition 3*)
Indeed, in the Schwarzschild metric, we have: h 1 = f 2 = 0, while the isotropic spacetime metric requires: h 1 = 1 and f 2 = 1/2, as previously noted by Bragança and Lemos.[3] A well-known example of an isotropic spacetime metric that fits these prescribed criteria is the isotropic Schwarzschild metric:

ds 2 = 1 - r s 4r 2 1 + r s 4r 2 dt 2 -1 + r s 4r 4 (dr 2 + r 2 dϕ 2 )
which can be obtained by applying the following substitution to the Schwarzschild metric:

r → r 1 + r s 4r 2
Another example is the isotropic exponential spacetime metric:

ds 2 = exp - r s r c 2 dt 2 -exp r s r (dr 2 + r 2 dϕ 2 )
This spacetime metric contains no event horizon, but does feature a photon sphere of radius r γ = r s and an ISCO of radius r isco = r s (3 + √ 5)/2. Possible field equations corresponding to the isotropic exponential spacetime metric have been investigated by Ni [START_REF] Ni | Theoretical Frameworks for Testing Relativistic Gravity.IV. a Compendium of Metric Theories of Gravity and Their POST Newtonian Limits[END_REF] and Watt and Misner [START_REF] Watt | Relativistic scalar gravity: a laboratory for numerical relativity[END_REF].

Capture cross section 4.1 General method

The capture cross section for an incoming photon from infinity is the effective area within which the photon ends up captured by the BH. Let b be the impact parameter and β 0 the initial proper speed, then the Hamiltonian and angular momentum are given by:

H = mc 2 1 -β 2 0 L z = mcβ 0 b 1 -β 2 0
Hence, the effective radial potential can be written as:

U eff [r] = m 1 -β 2 0 c 2 2 f 2 g -m c 2 2 f g + mβ 2 0 b 2 c 2 2gh f r 2
In the limit β 0 → 1, the effective radial potential becomes:

lim β0→1 U eff [r] = -m c 2 2 f g + mb 2 c 2 2gh f r 2
If an incoming photon is to be captured by the BH, the height of the potential barrier has to be such that the particle doesn't bounce back and passes through the potential barrier. We solve the following equation for b:

lim β0→1 U eff [r γ ] = 0
Hence, the capture cross section for a photon is:

σ = πr 2 γ h[r γ ]/f [r γ ].

Schwarzschild metric and isotropic exponential spacetime metric

The capture cross section for a photon in the Schwarzschild metric is:

σ = πr 2 γ 1 - r s r γ -1 = π 27 4 r 2
s whereas the capture cross section for a photon in the isotropic exponential spacetime metric is:

σ = πr 2 γ exp 2 r s r γ = π exp[2]r 2 s
The fractional difference between the photon sphere radius in the Schwarzschild metric and the photon sphere radius in the isotropic exponential spacetime metric is:

r s -3r s /2 3r s /2 = - 1 3
while the fractional difference in the apparent photon sphere radius for a distant observer is of much smaller magnitude:

exp[1]r s - √ 27r s /2 √ 27r s /2 ≈ 0.046 5

Conclusion

The spacetime metric of a static BH can be generalized by introducing three scalar functions f [r], g[r] and h[r]:

ds 2 = f c 2 dt 2 -gdr 2 -hr 2 dϕ 2
where the scalar functions can be written as series expansions:

f = 1+ ∞ j=1 f j r s r j g = 1+ ∞ j=1 g j r s r j h = 1 + ∞ j=1 h j r s r j
We have determined the conditions under which the spacetime metric shows consistency with the four classical solar system tests:

An example of an isotropic spacetime metric that satisfies these conditions is the isotropic exponential spacetime metric. The main properties of the Schwarzschild metric and the isotropic exponential spacetime metric are summarized in the following

table :

 :

		Schwarzschild metric:	Isotropic exponential spacetime
							metric:
	Scalar function f :	1 -	r s r		exp -	r s r
	Scalar function g:		1 -	r s r	-1	exp	r s r
	Scalar function h:	1					exp	r s r
	Event horizon:	r s				N/A
	Photon sphere:	3 2	r s				r s
	ISCO:	3r s				r s (3 +	√	5)/2
	Capture cross section:	π	27 4	r 2 s		π exp[2]r 2 s