Candice Lemaire

Candice LEMAIRE University of Burgundy, France Madrid, Universidad Complutense ("Staging American Sounds" international conference, 9-10 May 2013) "Song of the wave": Frostian poetics of resurgence As New England poet Robert Frost writes in the preface to one of his volumes, "a poem is best read in the light of all the other poems ever written. We read A the better to read B (we have to start somewhere; we may get very little out of A). We read B the better to read C, C the better to read D, D the better to go back and get something more out of A. Progress is not the aim, but circulation",1 putting into relief the dense generic intertextuality of his own poetic works which he purposefully chose to establish at the crossroads of the old poetic tradition and his own poetic modernity, constantly borrowing from both, and taking a slight formal and thematic distance from these influences. Such statement thus sums up his own writing technique of repetition and variation, of constant circulation between forms and models and poetic appropriation, making for challenging and puzzling reading. But the poet's statement also highlights a more complex intratextual game at stake in all the volumes: the issue of double, circular reading of the poems because the collections form a vast chamber of echoes, a hide-and-seek game on multiple scales and levels-from the stage of the volume, down to the metric foot. This paper wishes to consider the Frostian play on intratextual echoes as a space of resurgence, in which the poems become the frame for a return or a revival of poetic occurrences, motifs and sounds, but re-used, re-appearing, circulating again with a difference and some typographic variations. We would first like to consider this poetics of resurgence through the technique of intratextual echoes in the original edition of Frost's fourth collection, New Hampshire (1923), in which the staging of echoes is made possible through the interweaving of notes and grace notes. Such staging of Frostian sounds and of poetic inspiration (through the attention paid to the journey of the poetic breath in the volumes) also requires a system of microscopic, embedded reading in which the reader is guided from one line to another and expected to guess and trace the echoes of lost sounds.

"The nearest boundary to escape across": 2 interweaving notes

The hide-and-seek technique of Frostian resurgence bases itself on a dense network of intratextual echoes-understood here both in its first sense of repetition of a sound, and the return of an implicit reference. Reading one collection by Frost is actually accepting to adopt a circular approach and read each poem as closely interwoven to the next, as closely understandable through the next-reaching the meaning is indeed "escaping across the nearest boundary" of each poem to adopt an overall view of the intratextual and self-referential quality of the work. We would like to focus more particularly on Frost's original edition of his fourth Pulitzer Prize-winning 1923 collection New Hampshire, as the most elaborate example of such technique of purposefully interweaving references and echoes. 3 When it first came out, New Hampshire was originally structured as a triptych with the lengthened title New Hampshire: A Poem with Notes and Grace Notes. Such body of critical notes combined with some extra ornemental notes (the grace notes, as in a music score) altogether inscribed the volume into a playful mode thanks to the openly parodic and mockingly scholarly aspect of its structure and title.

The title page of New Hampshire carries the mock-scholarly subtitle 'A Poem with Notes and Grace Notes', indicating its tripartite division: the title poem, a section of 'Notes' (including thirteen poems, mostly dramatic monologues and dialogues), and a final section of 'Grace Notes' (featuring thirty shorter lyrics). Many items in the title poem are footnoted, and the reader is led to other poems in the 'Notes' section that 'gloss' these items ... The section 'Grace Notes', a musical term that means notes in a composition that are purely ornamental and do not further the exposition of a theme, features a range of poems that do not bear directly on the contents of New Hampshire but fill out the volume with a diversity of forms and themes. (Tyler B. Hoffman, in Lewis 230-31) The parodic intertext is in this case a reference to T.S. Eliot's long poem The Waste Land which had been published a few months before, in December 1922, and whose impressive body of critical notes was equally elaborate and obscure, paradoxically multiplying the number of possible interpretations of one item when the first role of a critical footnote is to frame it. Frost wrote his notes and grace notes the way Eliot did: for one given line, each note started by "Cf.", followed by an intratextual paginated reference to another poem of the same collection for Frost (and to another intertextual work in Eliot).

"Said some of the best things we ever said: 'Cf. page 162, 'A Star in a Stoneboat'; and page 201, 'I Will Sing You One-O' ". (Poirier 168) As Tyler B. Hoffman explains about the eponymous threshold-poem, the opening poem of the collection, "New Hampshire" (which is given the largest amount of notes and grace notes), the volume so acquires its own internal logic, and functions as a closed circuit, being its own critical echo and an open joke on the part of Frost against Eliot's opaque critical gloss in The Waste Land.

For instance, when the title poem mentions 'a man who failing as a farmer / Burned down his farmhouse for the fire insurance', we are directed to a footnote to 'The Star-Splitter' in 'Notes', a poem that tells the story of that farmer in greater detail. Unlike Eliot's footnotes, which refer readers to a list of scholarly citations meant to elucidate symbols within the poem but which often obscure them further. Frost's footnotes lead the reader to other poems in the same book, as if to insist that outside knowledge is not necessary to interpret and appreciate the figures that he imagines. Frost thereby announces his text as a self-enclosed intellectual system and invites a wider range of readers than Eliot does. (Tyler B. Hoffman, in Lewis 230) Such system of resurgence implying interwoven intratextual notes is only one side of the intricate Frostian chamber of echoes. The elaborate game of hide-and-seek initiated at the level of the volume is also taken up on a smaller scale, reaching the frame of the poem, the line, and down to the syllable itself. The strategy of resurgence thus becomes a system of microscopic, embedded reading: the Frostian reader is required to read into, guess words within words, and trace echoes of lost sounds from one metric foot to another. Some of the poet's uncollected poems actually confront the reader with a dizzying practice of minute sound variations.

"The aim was song": 4 tracing echoes of lost sounds

The 1957 untitled sestet "Her husband gave her a ring"-belonging to the series of Uncollected Poems-illustrates the poet's exploration of the repetition / variation diptych, thanks to an intricate play on recurring rhythmic cells. Sounds are staged to repeat themselves and to be pronounced hidden within other words of the poem, and no longer independently: thus aggregating the phonemes, the sestet establishes a network of deformed echoes between the signifiers without ever resorting to the signified.

Her husband gave her a ring

To keep her a virtuous thing. But the fellow to whom I'm referring He gave her an earring for erring. He also gave her a necklace For being so sinfully reckless.

(Poirier 559) The dizzying sound tricks of the sestet revolve around the reflection on the jewel symbol: crossed isotopies of luxury and lust are tackled here on a clearly ironical and playful mode.

Modern Diderot-like "indiscreet jewels"5 -the ring, the earring and the necklace-are the lavish and extravagant instruments of an adultery they condemn through their very combination of phonemes, and their strategic position in the stanza. These three lexical items superimpose onto the voice of the persona ("I"), acting as moral judge of such scene: the phrase "I'm referring" (line 3) can thus be heard both as a desire to name the seducer-lover ("to refer to someone", to make allusion to someone), and as "referee", the umpire of the game who guarantees the respect of the law and of good moral standards. The resurgence technique builds up a system of portmanteau words used as regular lexical items-the boundaries between signifier and signified are blurred in scattering clues of the adulterous fault, under the form of sound and visual echoes within the words themselves. Indeed, none of the characters of this love triangle is identified in this sestet in which the issue of naming is paradoxically solved when observing the clues given by the words carefully chosen by the poet. For instance, the adulteress's identity-while being referred to as "her" (/h3:r/) five times in the stanza-will then be revealed by the repetition of the same phonemic combination /3:r/, in the terms "referring" and "erring". The verb "referring" therefore enables to unveil the male seducer because it gathers together the sound of the adulterous misconduct ("erring", used here as verbal ending), the jewel which led the woman to temptation (the "earring", because of the doubled "r"), and the first letter of his identity (the "f" in "fellow", line 3): "referring".

The following line (line 4) elliptically sums up such powerful phonemic resurgence phenomenon: "an earring for erring", combining the guilty object ("earring"), sharing the doubled consonant "r" with the sin it has provoked ("erring"), and textually embracing the culprit of such crime (the "fellow") present through the letter "f" in the preposition "for".

When pronounced quickly, this very expression in line 4, "an earring for erring", can also be heard as "an earring for a ring", thus balancing the guilty jewel of adultery (the earring) and the lawful jewel of marital love (the wedding ring). Presented in strong anaphoric position through the personal pronoun "he" in lines 4 and 5 ("He gave her", "He also gave her"), the seducer is always the one poisoning the poem with the adulterous fault on a thematic level (because of the two jewels he offers the lady-the necklace and the earring), but also on a visual level. In offering her the necklace, he brings the vowel "a" ("necklace") back into the stanza when this very scarlet letter "a" was already cankering the first jewel (the "earring"), and the verb itself ("gave"). 6 The last line of the sestet uses the resurgence phenomenon to picture the adulterous woman's tragic and irreversible fall into sin: "For being so sinfully reckless". The gradation of the double adverb, "so sinfully", placed in central position in this tetrameter, foreshadows the dysphoric suffix "-less" in the adjective "reckless". The ternary rhythm of the sibilants ("so / sinfully / reckless") both connotes a seducing serpent hidden under the jewel gifts, and the sarcastic mockery of the persona commenting on the scene. Within a condensed, framed, narrow stage of six lines, this poem thus paradoxically offers plenty of room for the Frostian poetics of resurgence to be displayed through the variation of repeated vowels or consonants, and the dissemination of homophonous phonemic combinations.

Both a riddler and a craftsman of the poetical line, endowing each poem with challenging, almost invisible sound effects, Frost enlarges his poetics of resurgence and takes it to a more figurative level with the introduction of two motifs-the wind and the wave-as possible emblems of his staging of the poetic breath and its flow of inspiration. The wind and the wave elements indeed compete to superimpose the resurgence phenomenon onto a metaphorical wandering along the lines, mimicking in their own twists and turns both the reader's mental meanders to escape the Frostian acoustic maze, and the musical journey of the poetic pneuma.

"The breath of parted lips": 7 staging poetic inspiration

The Frostian staging of sounds is most perceptible in poems mimicking a music score, as Frost's vibrant interest for music, and for the gradual precedence of song over speech, of sound over sense, is accounted for in many titles throughout the collections: "Our Singing Strength" (Poirier 220), "Five Nocturnes" (Poirier 346), "I Will Sing You One-O" (Poirier 201), "Pan With Us" (Poirier 32), "The Valley's Singing Day" (Poirier 217). Inside this Frostian music box, some poems are specifically dedicated to a reflection on the birth and journey of the poetical breath, the poetical inspiration: the resurgence technique notably appeals to air and water imagery, incorporating the music-wind of "The Aim Was Song" or the orchestrated breaking of a wave in "Song of the Wave".

In "The Aim Was Song", the course and meanders of the wind follow a four-four time beat, developing the wind's modulations and vagaries over four quatrains of tetrameters.

Before man came to blow it right

The wind once blew itself untaught, And did its loudest day and night

In any rough place where it caught.

Man came to tell it what was wrong: I hadn't found the place to blow; It blew too hard-the aim was song.

And listen-how it ought to go! He took a little in his mouth, And held it long enough for north To be converted into south, And then by measure blew it forth.

By measure. It was word and note, The wind the wind had meant to be-A little through the lips and throat.

The aim was song-the wind could see.

(Poirier 207) In this agon between the wind and man wishing to bend it, to control and discipline it, the multiplication of the verb "to blow" in the first two quatrains, and its gradual disappearance thanks to the softening of plosives into sibilants, both pave the way to a progressive metamorphosis of the wind first into pneuma -poetic breath of creation-then into "song". The plosive consonants in "before, blow, blew, loudest, day, place" slowly acquire a sibilant quality, "through, lips, throat, song, see". The rough, untamed beat of the gust of wind mutates into musical discourse ("word and note") as soon as the wind learns to follow the right pace-the plurivocal phrase "by measure" repeated twice in a row (lines 12-13) both suggests such apt tempo and a suitable moderation in the musical performance. The new pneuma which circulates in the poem thus found the right "place to blow" -from the original breath born from the cheeks in the north, down to the vibratory, controlled quality of the song being produced in the south, through the screen of vocal cords and lips ("through the lips and throat"). While learning the difference between sudden squall and song in the geography of the throat, the anthropomorphic wind imposes its binary evolution in lines 7 and 16, perfectly balancing and embracing its former mistake and its successful correction around the caesura: "It blew too hard-the aim was song" / "The aim was song-the wind could see".

Frost, Robert. "The Prerequisites", Preface to Aforesaid, 1954 (Poirier 815).

Quotation from the poem "New Hampshire", in the New Hampshire collection (Poirier 158).

Such auctorial decision is made obvious in the fact that the later editions of New Hampshire no longer presented such technique, Frost having decided to erase all the critical body of notes, and shortened the volume's title.

Title of a New Hampshire poem (Poirier 207).

The Indiscreet Jewels (Les Bijoux Indiscrets) is the first novel by Denis Diderot, published anonymously in 1748. It is an allegory that portrays Louis XV as the sultan Mangogul of the Congo who owns a magic ring that makes women's genitals ("jewels") talk, and reveal all the adulterous and scandalous secrets of the Court.

Our underlining.

The Breath of Parted Lips: Voices from the Robert Frost Place is a collection of 130 poems contributed by Frost Place residents, many of whom have risen to prominence. Some contributors' poems were written during their tenures on the farm of Franconia, New Hampshire (known as "the Frost Place").

Such progressive discovery and mastering of the poetical breath and inspiration is taken up again through the use of water imagery to depict the Frostian resurgence techniqueits hide-and-seek playful quality is beautifully enhanced by a three-step backwash in the "Song of the Wave". "Rolling, rolling, o'er the deep, Sunken treasures neath me sleep As I shoreward slowly sweep.

Onward peacefully I roll, Ever thoughtless of the goal, Sea-bells round me chime and toll.

There is peace above, below, Far beneath me sea-weeds grow, Tiny fish glide to and fro, Now in sunlight, now in shade, Lost within some ocean glade By the restless waters made.

Pushing onward as before, Now descry the distant shore, Hear the breakers sullen roar; In this youth poem in tercets Frost wrote when he was sixteen, the resurgence of the wave strictly follows the pattern of a three-step waltz, in a constant ternary movement of birth, breaking and death of this wave on the breakwater. The strong punctuation ending and framing each tercet initiates this impression of perpetual repetition, of endless return of the same flow / short pause / ebb pattern between the stanzas, as stanza 1 exemplifies: the formation of the roller ("Rolling, rolling, o'er the deep") is briefly suspended by the insertion of a comment of the persona-wave ("Sunken treasures neath me sleep"), and finally breaks towards the shore in an imitative harmony of sibilants ("As I shoreward slowly sweep"). The twists and turns of the resurgence phenomenon are in accordance with the melody in "r" this "Song" performs, and the transformations undergone by the verb "to roll" throughout the poem: from the spreading of the "roll"(er), in the haste and eagerness to "reach" the shore, to its violent, fatal crash on the mole ("roam no more", "roar"). Such linguistic verbal metamorphosis parallels the thematic metamorphosis of the wave itself, and finds its climax in the final tercet in which a coda is added-forming a quatrain, and the held final chord of the waltz. Such lengthened chord is formed of one distich pronounced by the persona-wave ("I must now my proud crest lower / And the wild sea roam no more") omnisciently commented on in the following distich, depicting the dreary sound of the dying wave after closing the quotation marks ("Hark !"). The technique of Frostian resurgence with a slight variation is complete with the circular return of the elided preposition "o'er" in the last line of the "Song", in which its original euphoric spatial meaning of powerful rise over the surface of the ocean is degraded to simply metaphorically symbolize the fallen, dying roller crashing into a myriad of water drops ("Then the wave's short life is o'er").

The Frostian game of repetition and variation typified by his poetics of resurgence in the volumes is thus played on different scales, whether his staging of poetic sounds addresses a whole volume through a system of intratextual notes and somewhat opaque references (as in New Hampshire), or the zooming scale of the poem, the line, and down to the metric foot.

Having to accept such hide-and-seek rule in the Frostian game, his reader soon becomes the poet's "partner in rhyme"-forced to guess and read words into words, tracking back the original sounds now lost in variation. As he becomes Frost's poetic co-player in developing his auditory and visual memory, the reader endorses and treasures the poet's saying: "It takes all sorts of in and outdoor schooling / To get adapted to my kind of fooling" (Poirier 478).

Works cited


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

Improved Edaravone delivery to the brain and crossing the blood brain barrier: using quantum mechanics

Keywords: Edaravone, Edaravone tautomers, drug delivery, cucurbit[7]uril, oxidative stress, free radicals, quantum mechanics

Edv or Edv-keto Edaravone, Edv-enol Enol tautomer of Edaravone, Edv-amine Amine tautomer of Edaravone , CB[7] Cucurbit[7]uril, CB[8] Cucurbit[8]uril , {Edv-keto@CB[7]} Complex of Edv and CB[7] , {Edv-enol@CB[7]} Complex of Edv-enol and CB[7], and {Edvamine@CB[7]} Complex of Edv-amine and CB[7], BBB Blood brain barrier, OS oxidative stress, ROS reactive oxygen species, ET electron transfer, ΔG desolv,CDS free energy of water desolvation, ΔGlipo,CDS lipophilicity free energy, CDS cavity dispersion solvent structure of the first solvation shell, HOMO highest occupied molecular orbital, LUMO lowest unoccupied molecular orbital, LD50 drug toxicity, TS transition state.

Introduction

Edaravone (Edv) is a free radical scavenger FDA approved in 2017 for the treatment of amyotrophic lateral sclerosis (ALS). It has been previously investigated for the treatment of ischemic stroke, reperfusion injury, and myocardial infarction as it possesses antioxidant and anti-apoptotic properties. It was approved in Japan in 2001 as a drug to treat acute phase cerebral infarction, and in 2015 it was approved for ALS. It can cross the blood-brain barrier BBB to exert nootropic and neuroprotective effects. [START_REF] Watanabe | How is edaravone effective against acute ischemic stroke and amyotrophic lateral sclerosis?[END_REF] The drug is an antioxidant, and oxidative stress is considered to be part of the process that kills neurons in people with ALS. [START_REF] Petrov | ALS Clinical Trials Review: 20 Years of Failure. Are We Any Closer to Registering a New Treatment[END_REF] However, like most therapeutic drugs, drug delivery to the target site is a major impediment to good therapeutic outcomes. The in vitro binding rates of Edv to human serum protein and albumin are 92% and 89-91%, respectively, with no concentration-dependence. About 0.7-0.9% of the dose is excreted as unchanged drug and 71.0-79.9% of the dose is excreted as metabolites through renal elimination.

[3] So it is clear that efficient Edv delivery to molecular targets in the brain is a major impediment, and the improving the ability to cross the BBB could improve therapeutic outcomes.

Lipid soluble drugs such as nicotine, caffeine, barbiturate drugs can passively permeate the BBB with ease, but more hydrophilic or charged drugs (ions) do not. The tight junction between endothelial cells of the BBB allows only small molecules, lipid-soluble molecules, and some gases to pass freely through the capillary wall and into brain tissue, oxygen and carbon dioxide are not lipid-soluble but are actively transported across the BBB to support normal cellular function of the brain. A rule of thumb says if the drug has a molecular weight <400 Da and forms <8 hydrogen bonds (ie has a lower water desolvation energy) then permeation of the BBB is easier. The BBB becomes more permeable during inflammation, allowing antibiotics and phagocytes to move across the BBB. We have previously developed a model which can successfully predict the ability of drugs to cross the BBB by diffusion or facilitated diffusion. [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF] Various drug delivery systems have been used to protect the drug from unwanted chemical degradation, where the biological effect is pharmacokinetic. [START_REF] Norouzy | Synthetic Macrocyclic Receptors as Tools in Drug Delivery and Drug Discovery[END_REF][START_REF] Nmb Smeets | Designing Responsive Microgels for Drug Delivery Applications[END_REF][START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF][8][START_REF] Jin | Edaravone-Encapsulated Agonistic Micelles Rescue Ischemic Brain Tissue by Tuning Blood-Brain Barrier Permeability[END_REF][START_REF] Saleh | Cucurbiturils in Drug Delivery And For Biomedical Applications[END_REF][START_REF] Nj Wheate | Cucurbit[n]urils as excipients in pharmaceutical dosage forms[END_REF] Drug delivery to targets in the brain can also be enhanced by synthetic and naturally occurring drug delivery systems. Polymeric nanoparticles are potential carriers for CNS drug delivery by encapsulating drugs, hence protecting them from excretion and metabolism, as well as delivering drugs across the BBB. [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] Lipid nanoparticles can facilitate crossing the BBB to enter the brain by endocytosis. [8] Edv encapsulated micelles have been shown to target the BBB. [START_REF] Jin | Edaravone-Encapsulated Agonistic Micelles Rescue Ischemic Brain Tissue by Tuning Blood-Brain Barrier Permeability[END_REF] The use of cucurbiturils CB in drug delivery and as excipients has been widely examined. [START_REF] Saleh | Cucurbiturils in Drug Delivery And For Biomedical Applications[END_REF][START_REF] Nj Wheate | Cucurbit[n]urils as excipients in pharmaceutical dosage forms[END_REF][START_REF] Vd Uzunova | Toxicity of cucurbit[7]uril and cucurbit[END_REF][START_REF] Fong | Cisplatin cyclodextrin complexes as potential free radical chemoradiosensitizers for enhanced cisplatin treatment of cancers: a quantum mechanical study[END_REF][START_REF] Fong | Cucurbiturils as potential free radical chemoradiosensitizers for enhanced cisplatin treatment of cancers: a quantum mechanical study[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] Appel | Enhanced stability and activity of temozolomide in primary glioblastoma multiforme cells with cucurbit[n]uril[END_REF] The solubility of cucurbiturils is increased by Na + ions, which is applicable in many solution-based pharmaceutical dosage forms which contain NaCl as the excipient to achieve isotonicity with blood serum. [START_REF] Nj Wheate | Cucurbit[n]urils as excipients in pharmaceutical dosage forms[END_REF] The solubility of cucurbituril inclusion complexes tends to increase in water compared to the free cucurbituril, particularly if the guest molecule is positively charged. [START_REF] Saleh | Cucurbiturils in Drug Delivery And For Biomedical Applications[END_REF][START_REF] Nj Wheate | Cucurbit[n]urils as excipients in pharmaceutical dosage forms[END_REF] In vivo studies have shown broad short term safety with high maximum tolerated doses when administered via intravenous injection (250 mg kg -1 ) for CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] and orally (600 mg kg -1 ) for a mixture of CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] and CB [8]. [START_REF] Vd Uzunova | Toxicity of cucurbit[7]uril and cucurbit[END_REF] A salient feature of macrocyclic, polymeric or nanoparticle drug delivery is whether the {encapsulated drug-carrier} complex can cross the BBB, or whether the complex merely acts to protect the drug from unwanted chemical degradation, and the biological effect is pharmacokinetic. The probability of {encapsulated drug-carrier} complexes crossing the normally impervious BBB can be aided by observations that in both HIV-1 encephalitis (HIVE) and Alzheimer's disease (AD), blood-borne activated monocyte/macrophages and lymphocytes are thought to migrate through a disrupted blood-brain barrier. [START_REF] Fiala | Cyclooxygenase-2-positive macrophages infiltrate the Alzheimer's disease brain and damage the blood-brain barrier[END_REF] Drugs encapsulated by CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] have been shown to be efficiently internalized by macrophages indicating their potential for the intracellular delivery of drugs. Also CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]-bound drug molecules can be released from the container to find their intracellular targets, human embryonic kidney 293 cells (HEK 293) and murine macrophage cell line (RAW 264.7). [START_REF] Hettiarachchi | Toxicology and Drug Delivery by Cucurbit[n]uril Type Molecular Containers[END_REF] CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] and CB [8] complexes with acridine orange and pyronine Y complexes can cross the cell membrane of 3T3 cells. [START_REF] Montes-Navajas | Cucurbituril complexes cross the cell membrane[END_REF]

Results and Discussion

Edv has two tautomers, as shown in Figure 1. These are named as Edv-keto (Edaravone), Edvenol (the enol tautomer of Edv) and Edv-amine (the amine tautomer of Edv). In aqueous solution, at physiological pH 7.0-7.4, Edv can exist as the neutral and anionic forms as shown in Figure 2. Their relative concentrations at pH 7.4 are 28.5% and 71.5% respectively in water. The anionic form can lose an electron to form the highly reactive free radical which can undergo a number of side reactions. The oxidation potential decreases with the pH ie as the proportion of anions increases. [START_REF] Watanabe | How is edaravone effective against acute ischemic stroke and amyotrophic lateral sclerosis?[END_REF] In terms of Edv being able to exert its maximum therapeutic efficacy in the brain, it is clear that the anionic form of Edv is primarily responsible for chemical degradation of Edv by various side reactions. The anionic form is calculated to be 8.6 times as reactive as the neutral form, and Edv is an excellent HO• scavenger. [START_REF] Pérez-González | OH Radical Scavenging Activity of Edaravone: Mechanism and Kinetics[END_REF] The in vitro binding rates of Edv to human serum protein and albumin are 92% and 89-91% respectively. It is not clear which of the tautomers of Edv are most reactive with the various biological species in the human body. It is not known which of the neutral Edv tautomers or the anionic form are the causes of the known side effects of Edv after intravenous injection.

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 | 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 tirapazmine and the effect on the extravascular penetration of tirapazamine into tumours. [START_REF] Fong | Screening anti-colorectal cancer drugs: free radical chemotherapy[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 | 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] 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. Equation 2.

Oxidative Stress or Oxidative Properties = ΔG desolv,CDS + ΔG lipo,CDS + Dipole Moment + Electron Affinity

Previous studies clearly indicate that it is the enol tautomer which at physiological pH 7.0-7.4 can form the anionic species of Edv that is responsible for much of the chemical degradation after intravenous infusion. [START_REF] Watanabe | How is edaravone effective against acute ischemic stroke and amyotrophic lateral sclerosis?[END_REF] The antioxidant properties of Edv which are responsible for its chemotherapeutic efficacy against ALS, ischemic stroke and possible its nootropic and neuroprotective properties can be assessed by adiabatic electron affinity of the tautomers and the use of eq 2. [START_REF] Montes-Navajas | Cucurbituril complexes cross the cell membrane[END_REF][START_REF] Pérez-González | OH Radical Scavenging Activity of Edaravone: Mechanism and Kinetics[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] The chemical reactivity of the tautomers 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 | 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] Table 1 list the key molecular properties of Edv and its tautomers in water. It can be seen that the three neutral tautomers have very similar properties suggesting that all three would have similar abilities to permeate the BBB, assuming eq 1 holds for these species. [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF] However the Edv-enol tautomer is inherently less reactive than the Edv-keto and Edv-amine tautomers based on its HOMO -LUMO gap, and has a lower reducing potential, but these factors are offset at physiological pH levels where the enol tautomer easily forms the anionic species which is more chemically reactive than the neutral tautomers. The properties of the anionic form and the free radical Edv• are also shown in Table 1.

An attractive means by which the clinical efficacy of Edv could be enhanced by increasing its delivery efficiency to the brain is by encapsulating Edv in cucurbiturils, namely cucurbit [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]uril [CB7] and cucurbit [8]uril [CB8] which have known abilities to protect drugs from unwanted side reactions which reduce the availability of the drugs to its intended targets. The three inclusion complexes of Edv with CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] have been examined: {Edv-keto@CB[7]}, {Edv-enol@CB[7]}, and {Edv-amine@CB[7]}. Also {Edv-keto@CB [8]} was cursorily examined for comparison with the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] complex.

Figures 3, 4 and5 shows {Edv-keto@CB[7]}, {Edv-enol@CB[7]}, and {Edv-amine@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complexes in water. It can be seen that the complexes are quite similar in overall structure. There are many possible configurations for {Edv-keto@CB[7]}, {Edv-enol@CB[7]}, and {Edvamine@CB[7]} and it is not possible to test for all possibilities. The configurations shown in Figures 3,4 and 5 are similar, with the phenyl ring protruding out of the top portal of CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] and the rest of the molecules buried within the cavity. These configurations then allow a direct comparison of how well the Edv-keto, Edv-enol and Edv-amine tautomers can interact with the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] container. Energy minimizations to the transition states for insertion of the tautomers into the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] can then be compared using the same dispersion corrected DFT method. It is noted that chosen DFT methods may not be the optimum methods but should allow examination of comparative differences between the stability and reactivity of the three complexes in similar configurations.

Table 2 list the ΔG absolute stabilities of the complexes, the ΔG activation energies, and the HOMO-LUMO energy gaps for the complexes, relative to the {Edv-keto@CB[7]} complex. Since there are many possible configurations it is more informative to compare activation energies than various ground state configurations as an indicator of the relative ability to the tautomers to bind with the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] and CB [8]. The configurations shown in Figures 3,4 and5 are quite similar with respect to the orientations of the Edv tautomers to the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]. The dipole moments of the complexes are also shown.

Thermodynamic properties were calculated using two methods: (a) DFT/B3LYP/6-31+G(d) high level // DFT/B3LYP/3-21G low level with the empirical dispersion = GD3BJ correction in water, and (b) DFT/B3LYP/6-31+G(d) high level // UFF/Qeq/electronic embedding low level with the empirical dispersion = GD3BJ correction in water. The Qeq charge equilibration methods evaluates the electrostatic potential by assigning point charges to each atom. Method (a) allows molecular orbital interaction between the Edv tautomers and the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] whereas (b) allows only electrostatic interaction between the Edv tautomers and the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] with appropriate GD3BJ dispersion corrections.

As seen in Table 1, the thermochemical absolute free energy stabilities ΔG in water of the {Edvketo@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} and {Edv-enol@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complexes are comparatively very similar using the full QM method (a), whereas the {Edv-amine@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]}is far less stable, due to hydrogen bonding between the N-H and a O=C of CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]: Edv-amine (N-H---O=C) CB7, 1.55Å. Egress of Edvamine occurs when {Edv-amine@CB7}is energy minimized with no Edv-amine (N-H---O=C) CB7 H bonding with method (b) which uses the electrostatic Qeq/UFF force field to describe the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]. It is noted that no egress of Edv-keto or Edv-enol from the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] occurs during energy minimization when using method (b). It is clear that the {Edv-amine@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complex is less stable than the {Edv-keto@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} and {Edv-enol@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complexes based on its lower absolute stability, the lower HOMO-LUMO energy gaps, and higher dipole moment (more polarized).

The higher TS energy for the {Edv-amine@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complex compared to the {Edvketo@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} and {Edv-enol@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complexes is a result of the H bond Edv-amine (N-H---O=C) CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]. All TSs had negative ΔG values. A non-transition state configuration of {Edvamine@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} without imaginary frequencies had a binding energy in water of -15.7 kcal/mol, with a Edv-amine (N-H---O=C) CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] H bond length of 1.60Å (compared to a value of 1.55Å in the TS).

It can also been seen in Table 1, that {Edv-keto@CB [8]} complex is very similar in properties as the {Edv-keto@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} and {Edv-enol@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complexes, but noting that the configuration examined in Figure 6 has a side orientation with the Edv-keto placed lengthwise. A more vertical orientation of the Edv-keto would be expected to show a lower TS energy and greater HOMO-LUMO energy gap as the CB [8] has a larger portal than the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF].

Conclusions

It has shown that the three neutral tautomers of Edv have very similar molecular properties suggesting that all three would have similar abilities to permeate the BBB. [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF] The Edv-enol tautomer is inherently less reactive than the Edv-keto and Edv-amine tautomers based on its HOMO -LUMO gap, and has a lower reducing potential, but these factors are offset at physiological pH levels in the body where the enol tautomer easily forms the anionic species which is more chemically reactive than the neutral tautomers. There is no evidence in the literature whether the Edv-keto or the Edv-amine tautomer is the more active species therapeutically. [START_REF] Watanabe | How is edaravone effective against acute ischemic stroke and amyotrophic lateral sclerosis?[END_REF][START_REF] Petrov | ALS Clinical Trials Review: 20 Years of Failure. Are We Any Closer to Registering a New Treatment[END_REF][3] This study cannot differentiate between the radical scavenging reactivity or ability to permeate the BBB by the Edv-keto or the Edv-amine tautomers.

In has been shown that the tautomers of Edv can form inclusion complexes with CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] and CB [8]. The {Edv-keto@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} and {Edv-enol@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complexes have similar stabilities, but the {Edv-amine@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]}is significantly less stable than the {Edv-keto@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} and {Edvenol@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complexes. The {Edv-amine@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complex forms a hydrogen bond between the Edv-amine N-H and a O=C of CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF].

These results demonstrate that improved drug delivery of Edv therapy could be achieved by encapsulating Edv within CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] such that the Edv-enol form would be protected against the anionic forming side reaction at physiological pH levels in the body. The Edv-keto and Edv-enol tautomers have similar propensities to form inclusion CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] complexes in water.

The Edv-amine tautomer forms a less stable CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] complex, and the formation of a H bond between the Edv-amine N-H and a O=C of CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] could be disrupted at slightly acidic pH by protonating the Edv-amine N-H moiety within the CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] cavity. This route offers a potential pathway into investigating the different therapeutic efficacy of the Edv-amine tautomer compared to that of the Edv-keto and Edv-enol forms.

There is potential for altering the relative concentrations of the Edv tautomers in the inclusion complexes by using polar aprotic solvents to form the {Edv-keto@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complex preferentially over the formation of the {Edv-enol@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} which is expected to be favoured in water. This selectivity is not available using micelles delivery systems which are water based systems. [START_REF] Jin | Edaravone-Encapsulated Agonistic Micelles Rescue Ischemic Brain Tissue by Tuning Blood-Brain Barrier Permeability[END_REF] There is some literature evidence that supports the potential for {Edv@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complexes to be transported across the BBB possibly by macrophages [START_REF] Fiala | Cyclooxygenase-2-positive macrophages infiltrate the Alzheimer's disease brain and damage the blood-brain barrier[END_REF][START_REF] Hettiarachchi | Toxicology and Drug Delivery by Cucurbit[n]uril Type Molecular Containers[END_REF].

Finally, this study has demonstrated that it may be possible to differentiate which of the Edv tautomers has greater therapeutic efficacy and drug delivery in the brain.

Experimental Methods

All calculations were carried out using the Gaussian 09 package. Energy optimizations were at the DFT/B3LYP/6-31G(d,p) (6d, 7f) or DFT/B3LYP/3-21G (for larger molecules) level of theory for all atoms. 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).

Edaravone and its tautomers encapsulated inside cucurbit [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]uril [CB7] and cucurbit [8]uril [CB8] were optimized using the ONIOM methods: (1) DFT/B3LYP/6-31+G(d) high level // DFT/B3LYP/3-21G low level with the empirical dispersion = GD3BJ correction in water and (2) at the DFT/B3LYP/6-31+G(d) high level // UFF/Qeq/electronic embedding low level with the empirical dispersion = GD3BJ correction in water, and Similar comparative results were obtained with both methods, with the exception of the {Edv-enol@CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF]} complex where Edv N-H ---O=C CB [START_REF] Tosi | Polymeric nanoparticles for the drug delivery to the central nervous system[END_REF] hydrogen bonding occurs in the complex. Energy calculations of the CB complexes were conducted at the DFT/B3LYP/6-31+G(d) high level // DFT/B3LYP/6-31G(d) low level with the empirical dispersion = GD3BJ correction in water. Both SMD and CPCM solvation models were used with similar results.

Electron affinities (EA) in eV in water were calculated by the SCF difference between the optimized/relaxed neutral and optimized radical species method as previously described. [START_REF] Montes-Navajas | Cucurbituril complexes cross the cell membrane[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] It has been shown that the B3LYP functional gives accurate electron affinities when tested against a large range of molecules, atoms, ions and radicals with an absolute maximum error of 0.2 eV. [START_REF] Jc Rienstra-Kiracofe | Atomic and molecular electron affinities: Photoelectron experiments and theoretical computations[END_REF][38][39] 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 tautomeric species is the aim of the study.

Figure 1 .Figure 2 .

 12 Figure 1. The three tautomers of Edaravone

Figure 4 .Figure 5 .Figure 6 .

 456 Figure 4. {Edv-enol@CB[7]} Left image top view, right image side view

Table 1 Edv-keto Edv-enol Edv-amine Edv-anion Edv•

 1

		Figure 1	Figure 1	Figure 1	Figure 2	Figure 2
	ΔG desolv,CDS kcal/mol	4.59	4.54	5.36	4.50	4.56
	Desolvation energy
	ΔG lipo,CDS kcal/mol	-4.86	-4.57	-4.86	-4.72	-4.82
	Lipophilicity
	Dipole Moment D	5.14	3.95	9.00	7.21	3.37
	Molecular Volume	135	132	162	120	144
	cm 3 /mol
	Adiabatic Electron	1.57	1.18	1.55
	Affinity AEA eV
	HOMO -LUMO Gap	5.09	6.96	5.10	4.69	5.46
	eV

Table 2 {Edv-keto@CB[7]}

 2

			{Edv-enol@CB[7]}	{Edv-amine@CB[7]}	{Edv-keto@CB[8]}
		Figure 3	Figure 4	Figure 5	Figure 6
	Absolute stability ΔG	0 method 1	-8.3 less stable	-24.9 H bond
	kcal/mol	0 method 2	31.0 more stable	Edv-amine egress occurs
	Transition state ΔG	0 method 1	3.5 higher energy	35.8 H bond *	3.4
	kcal/mol	0 method 2	-9.1 lower energy	-1.4
	HOMO-LUMO Gap eV	0 method 1	0.13 more stable	-0.6 less stable	-0.2
		0 method 2	0	-0.6
	Dipole moment D	5.0 method 1	5.0	10.45	6.2
		2.3 method 2	3.7	10.6
	Footnotes: Method 1: DFT/B3LYP/6-31+G(d) high level // DFT/B3LYP/3-21G low level with the empirical dispersion = GD3BJ correction in
	water. Method 2: DFT/B3LYP/6-31+G(d) high level // UFF/Qeq/electronic embedding low level with the empirical dispersion = GD3BJ
	correction in water. H bonding: Edv-amine (N-H---O=C) CB7, 1.55Å. Egress of Edv-amine occurs when {Edv-amine@CB7}is energy
	minimized with no Edv-amine (N-H---O=C) CB7 H bond. * A non-transition state configuration of {Edv-amine@CB[7]} without imaginary
	frequencies had a binding energy in water of -15.7 kcal/mol, with a Edv-amine (N-H---O=C) CB[7] H bond length of 1.60Å.


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

Predicting hydrophobic alkyl-alkyl and perfluoroalkyl-perfluoroalkyl van der Waals dispersion interactions and their potential involvement in protein folding and the binding of drugs in hydrophobic pockets

Keywords: Protein folding, alkyl-alkyl dispersion interactions, perfluoroalkyl-perfluoroalkyl interactions, foldamers, solvophobicity, hydrophobicity, CDS, electrostatic potential surfaces, polarizability, solvent-foldamer dispersion, cavitation, repulsion, quantum mechanics

Predicting hydrophobic alkyl-alkyl and perfluoroalkyl-perfluoroalkyl van der Waals dispersion interactions and their potential involvement in protein folding and the binding of drugs in hydrophobic pockets

Clifford W Fong

Introduction

The hydrophobic effect is important in understanding protein folding since minimizing the number of hydrophobic side chains exposed to water is the principal driving force behind the folding process, although formation of hydrogen bonds within the protein also stabilizes protein structure. Water-soluble proteins have a hydrophobic core in which side chains are buried away from water, and charged and polar side chains are situated on the solvent-exposed surface where they interact with surrounding water molecules.

Molecular dynamics modeling of protein folding and foldamers is well advanced, but one area which is still problematical is the role of solvation in water which is highly intensive using explicit water models. Implicit salvation models based on sound physical properties can reduce the calculational intensity. The incorporation of hydrophobicity into force fields would also advantageous.

Solute-solvent interactions can be divided into (i) the bulk electrostatic contribution associated with the polarization free energy of the solute-solvent system, and (ii) the non-bulk-electrostatic contribution, called the CDS which includes cavitation, dispersion, and solvent-structure effects where the solvent structure contributions include effects such as hydrogen bonding solvent structure breaking, exchange repulsion, solute-solvent charge transfer, dielectric saturation, and electrostriction effects. The lumped term non-electrostatic CDS describes the first solvent shell and this shell rather than the bulk solvent properties would be expected to be the dominant solvatochromic environment influencing the interior regions of a folded protein, or for a drug accessing a hydrophobic pocket of a biological protein or enzyme. The widely used SMD solvent model allows the calculation of the CDS for many solvents.

The solvophobic effect applies to interactions between polar solvents (particularly water in biological systems) and non-polar solutes. Pure solvents have strong cohesive forces between the solvent molecules due to hydrogen bonding or other polar interactions, so non-polar solutes tend to be insoluble in polar solvents because these solvent-solvent binding interactions must be overcome first. The classical hydrophobic effect involved in protein folding and drug protein binding in hydrophobic pockets includes solvophobic effects.

A recent comprehensive study of the hydrophobic alkyl-alkyl interaction using a molecular model (see Figure 1) which allows the folding and unfolding of two alkyl chains of the molecule in a wide variety of non-aqueous solvents has shown that the intramolecular hydrophobic van der Waals alkyl-alkyl interaction is smaller than the solute-solvent dispersion interaction. The crystal structure of the model compound has shown that the folded conformation is the most stable state. The analogous perfluoroalkyl-perfluoroalkyl interaction has been compared to the alkyl-alkyl interaction using the same molecular scaffold has also been studied in solvents, and it has been concluded that solvophobic effects were most important in aqueous and the highly polar organic solvents, but differences in van der Waals dispersion forces were important in apolar organic and fluorous solvents. Dispersion interactions in apolar solvents were found to be strongly attenuated in solution compared to the gas phase, but still play a major role in fluorous and organic solvents. [START_REF] Yang | How much do van der Waals forces contribute to molecular recognition in solution[END_REF][START_REF] Adam | Partitioning Solvophobic and Dispersion Forces in Alkyl and Perfluoroalkyl Cohesion[END_REF] Mecinovic [START_REF] Mecinović | Fluoroalkyl and Alkyl Chains Have Similar Hydrophobicities in Binding to the "Hydrophobic Wall" of Carbonic Anhydrase[END_REF] found that alkyl and perfluoroalkyl chains had similar hydrophobicities in carbonic anhydrase binding.

The importance of hydrophobic alkyl-alkyl interactions when drugs bind to proteins (proteinligand interactions) is well known. [4][START_REF] Hj Dyson | The role of hydrophobic interactions in initiation and propagation of protein folding[END_REF][6][START_REF] Daze | Molecular interaction and recognition, Encyclopedia of Physical Organic Chemistry[END_REF] The hydrophobic effect related to a single methyl group of a drug initially exposed to water that gets buried within a hydrophobic binding pocket of a protein has a predicted stabilizing free energy gain of ca 21-25 cal/mole per Å 2 which translates into an affinity enhancement of ca 50 fold. [START_REF] Milo | Cell Biology by the Numbers[END_REF] Fluorinated drugs are known to confer major pharmacological effects compared to their alkyl analogues particularly influencing potency, selectivity, absorption and metabolism. The impact of fluorine substitution on lipophilicity (log P and log D), cell membrane permeability and pKa can be large but variable. These effects can be attributed to the size of the fluorine atom (van der Waals radius of 1.47Å), compared to hydrogen atom (van der Waals radius of 1.20Å), the high electron withdrawing ability of fluorine, the greater stability of the C-F bond compared to the C-H bond, and the greater lipophilicity of fluorine compared to hydrogen. [START_REF] Shah | The role of fluorine in medicinal chemistry[END_REF] It is noted that lipophilicity and hydrophobicity are often used interchangeably in the pharmaceutical industry, but they are not exactly the same, and one definition used in the drug industry is that lipophilicity = hydrophobicity -polarity. [START_REF] Testa | Lipophilicity and Related Molecular Properties as Determinants of Pharmacokinetic Behaviour[END_REF] Hence molecular polarizability of the drug in solution might be expected to influence solvophobicity.

Non-covalent dispersion interactions (commonly called van der Waals or London interactions) can arise from induced dipoles between neutral molecules that bear no permanent dipolar attraction or from an intramolecular mutual recognition folding processes in proteins for example. Dispersive interactions originate from quantum mechanical correlated motions of electrons in approaching molecules or from intramolecular moieties in close proximity. These distance related forces are electrostatic in nature. [START_REF] Daze | Molecular interaction and recognition, Encyclopedia of Physical Organic Chemistry[END_REF] Study Objectives:

To determine if a relationship can be found between the free energy of alkyl-alkyl interaction and free energy of solvation as it applies to the first solvent shell of the model molecule. To determine whether a solvent mediated model can predict the folding alkyl-alkyl and perfluoroalkyl-perfluoroalkyl interaction in an aqueous environment as would be applicable in biological processes such as protein folding or drug binding in hydrophobic pockets of enzymes or related biological entities.

Results

Solvation studies of alkyl-alkyl interaction

Table 1 shows the calculated free energies of solvation for the model compound (Figure 1) in the folded state with the experimentally determined free energies of alkyl-alkyl interaction (ΔG DMC and ΔG Exp ) from reference 1 in 15 solvents. ΔG DMC and ΔG Exp are two different methods used by Yang et al to experimentally determine the free energies of alkyl-alkyl interaction [START_REF] Yang | How much do van der Waals forces contribute to molecular recognition in solution[END_REF][START_REF] Adam | Partitioning Solvophobic and Dispersion Forces in Alkyl and Perfluoroalkyl Cohesion[END_REF]. Correlations were sought between the experimental free energy alkyl-alkyl interactions (ΔG DMC and ΔG Expt ) and the total free energy of solvation (ΔG Solv ) , the CDS free energy (ΔG CDS ) the using the SMD solvent model, the dipole moment and molecular volume of the model compound. The only significant relationships found are shown in equations 1 and 2 which are very similar, with eq 1 being the most precise and significant. Polarizability is the ability to induce instant electric dipole moments in a molecule, which can apply to solvents and solutes. Polarizability plays an important role in determining the hydrophobic force acting between weakly polar surfaces. [START_REF] Bresme | On the influence of solute polarizability on the hydrophobic interaction[END_REF][START_REF] Mh Newt | Molecular Dynamics Calculation of the Effect of Solvent Polarizability on the Hydrophobic Interaction[END_REF] Linear correlations were found with ΔG DMC and ΔG Expt with the isotropic molecular polarizability of the model compound in the various solvents. Equations 3 and 4 illustrate the relative contributions of ΔG CDS and the polarizability (raw values have been scaled to allow direct comparison of the relative magnitudes of the CDS salvation and the polarizability). It can be seen that the solvation effect is about 4 to 10 times as great as the polarization effect. The alkyl-alkyl and perfluoroalkyl-perfluoroalkyl interactions of the folded model compound (Figure 1) in water were determined by (1) optimizing the model compounds in water or other solvent, (2) then rotating around the aryl-aryl bond as shown in Figure 1(b) to unfold the model molecule as shown in Figure 2, (3) then creating fragments (X=Y=H) for both the folded and unfolded states, (4) then calculating the gas phase ΔG values for all entities using the dispersion corrected wB97XD functional with the ONIOM lower level 6-31G(d) for the fragment as per (3) and 6-311++G(d,p) for the high level alkyl-alkyl or perfluoroalkyl-perfluoroalkyl fragments. Finally (5) using thermodynamic cycles, the solvation energy (ΔG CDS ) was added to derive the ΔG Interact, sol .

The following ΔG Interact, sol values were obtained:

Alkyl-alkyl interaction: ΔG Interact, water -1.11 kcal/mol, ΔG Interact, hexane 0.27 kcal/mol The alkyl-alkyl calculated values for water and n-hexane can be compared to the extrapolated experimental value for water of -1.01 kcal/mol and experimental value 0.03-0.14 kcal/mol for nhexane [START_REF] Yang | How much do van der Waals forces contribute to molecular recognition in solution[END_REF][START_REF] Adam | Partitioning Solvophobic and Dispersion Forces in Alkyl and Perfluoroalkyl Cohesion[END_REF] The extrapolated value for water from equations 1 and 2 are 0.14-0.29 kcal/mol Figures 2345show the calculated electrostatic potential surfaces (EPS) for the alkyl-alkyl model in the unfolded and folded states in gas, water and n-hexane solvents. EPS were calculated in two modes, firstly from the total SCF electron density, and secondly from electron densities from CHELPG partial atomic charges. The CHELPG EPS's show a region of electron density around the end of the alkyl chains, in the unfolded state, and with greater density in the folded alkylalkyl chains in the gas, water (Figures 3 and4) and particularly showing increased density for the n-hexane solvent (Figure 5). The electrostatic potential surface EPS from the total SCF electron density for example is shown in Figure 2 as -7.425e -2 to +7.245e-2 in water, or alternatively by showing the EPS at different electron densities from CHELPG partial atomic charges, (green -1.662e -2 to +1.662e -2 . The wide variability of the ranges of electron density in the EPS shown in Figures 234567891011illustrates the extent of differentiation that can be resolved by using EPS.

Fluoroalkyl-fluoroalkyl interaction: ΔG Interact, water not applicable, ΔG Interact, hexane -2.71 kcal/mol

The experimental value for the perfluoroalkyl model in n-hexane was -0.01 [START_REF] Yang | How much do van der Waals forces contribute to molecular recognition in solution[END_REF][START_REF] Adam | Partitioning Solvophobic and Dispersion Forces in Alkyl and Perfluoroalkyl Cohesion[END_REF]. It was found that the non-folded perfluoroalkyl foldamer was more stable in the gas phase (and water) than the folded conformation by a range of 11.3 -26.29 kcal/mol depending on whether the foldamer is fully relaxed or transitioning from the less stable folded state. This result is undoubtedly due to the higher polarity of the perfluoroalkyl compound compared to the alkyl model compound. Inspection of Figures 6 to11 which show the unfolded and folded perfluoroalkyl foldamers which clearly illustrates the higher electron density on the perfluoroalkyl moieties compared to that shown for the alkyl chains in Figures 2345. Figure 7 for the perfluoroalkyl foldamer in water shows the influence of solvated water on the EPS compared to Figures 6 or 8 in the gas phase or n-hexane solvent.

Solvent-solute dispersion, solvent-solute cavity formation, solvent-solvent repulsion

It was previously found by Cockroft [START_REF] Yang | How much do van der Waals forces contribute to molecular recognition in solution[END_REF][START_REF] Adam | Partitioning Solvophobic and Dispersion Forces in Alkyl and Perfluoroalkyl Cohesion[END_REF] that solvophobic effects were most important in aqueous and the highly polar organic solvents, but differences in van der Waals dispersion forces were important in apolar organic solvents. Dispersion interactions in apolar solvents were found to be strongly attenuated in solution compared to the gas phase, but still play a major role in organic solvents. In light of the strong correlations found in eqs 1 and 2 between the ΔG CDS and the experimental dispersion interactions in solution, it is instructive to examine water-foldamer dispersion interaction separately from the SMD ΔG CDS correlation. The solvent PCM model (which is the basis for the SMD solvent model) allows the calculation of the solvent-foldamer interaction in water and n-hexane using the method of Floris and Pierotti (see experimental) which separates the solute-solvent interaction into the ΔG Dis dispersion, ΔG Cav cavity and ΔG Rep repulsion free energies.

G Dis values for alkyl-alkyl foldamers folded in water were -50.20, unfolded in water -59.21, and folded in hexane -43.18 and unfolded in hexane -50.72 kcal/mol. ΔG Dis for the unfolded and folded perfluoroalkyl-perfluoroalkyl foldamers in water were -56.94 and -57.14 kal/mol. It is clear that the difference in ΔG Dis |-9.01| kcal/mol between the unfolded and folded alkyl-alkyl foldamers in water (offset by the +ΔG Cav and +ΔG Rep ) is a significant factor in stabilizing the folded conformation in water, compared to the negligible difference in ΔG Dis |~ 0| kcal/mol between the unfolded and folded perfluoroalkyl-perfluoroalkyl foldamers in water. This result is consistent with the finding above in calculating ΔG Interact, water that the non-folded perfluoroalkyl foldamer was more stable in the gas phase and water than the folded conformation. These findings confirm and clarify earlier findings [START_REF] Yang | How much do van der Waals forces contribute to molecular recognition in solution[END_REF][START_REF] Adam | Partitioning Solvophobic and Dispersion Forces in Alkyl and Perfluoroalkyl Cohesion[END_REF] that solvent-foldamer dispersion interactions override intramolecular alkyl-alkyl and especially fluoroalkyl-fluoroalkyl interactions. The difference in ΔG Dis |-7.54| kcal/mol between the unfolded and folded alkyl-alkyl foldamers in the more solvophilic hexane compared to the value of |-9.01| in solvophobic water may be indicative of the interactions experienced in a more hydrophilic interior of a protein compared to the more hydrophobic water-exposed exterior.

Discussion

We have previously used the ΔG CDS value to determine the free energy of desolvation in water (defined as ΔG desolv , CDS ) and the lipophilicity of drugs (ΔG CDS in octane or octanol, defined as ΔG lipo,CDS ) in a number of quantitative structure activity studies of various biological activities involving drug-cell receptor binding and drug-cell membrane transport activities. [START_REF] Fong | Free radical anticancer drugs and oxidative stress: ORAC and CellROXcolorectal cancer cells by quantum chemical determinations[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 | 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 free radicals in PARP inhibition of cervical carcinogenesis: PARylation of eLa cells, and inhibition of hPARP-1 enzyme[END_REF][START_REF] Fong | Platinum anti-cancer drugs: free radical mechanism of Pt-DNA adduct formation and anti-neoplastic effect[END_REF][START_REF] Fong | Platinum based radiochemotherapies: Free radical mechanisms and radiotherapy sensitizers[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 | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[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] Lipophilicity as determined by ΔG lipo,CDS is linearly related to logP (or logD) and other experimental measures of lipophilicity [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF]. ΔG CDS is a lumped term description of the nonelectrostatic first solvent shell which incorporates the dielectric constant, refraction index, hydrogen bonding, surface tension, the carbon aromaticity, and fraction of non-hydrogenic solvent atoms that are F, Cl, or Br. [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 Surface Tensions[END_REF] A major difficulty exist in differentiating the intramolecular dispersion interaction between folded alkyl groups in solvents, and the solvent-solute dispersion of foldamers. The highly parameterized ΔG CDS appears to be a good measure of most factors which can influence solventsolute interactions, particularly in the critical first solvent shell. The relationships between the experimental alkyl-alkyl dispersion interactions in solution and ΔG CDS shown in eq 1 and 2 indicate that ΔG CDS is a good measure of solvent-solute interactions. Eq 3 and 4 show that the molecular polarizability of the foldamers is also influenced by the polarizability of the solvents used to determine the experimental ΔG DMC and ΔG Expt as expected. Cockcroft showed that ΔG DMC and ΔG Expt were linearly related to the cohesive energy density and polarizability of the solvents, both of which are bulk solvent factors, rather than first solvation shell factors. [START_REF] Yang | How much do van der Waals forces contribute to molecular recognition in solution[END_REF][START_REF] Adam | Partitioning Solvophobic and Dispersion Forces in Alkyl and Perfluoroalkyl Cohesion[END_REF] The calculated ΔG Interact, water and ΔG Interact, hexane values derived from the unfolded and folded foldamers are in good agreement with the experimental ΔG DMC values derived from Wilcox molecular balances double-mutant cycle approach, and the alternative ΔG Expt approach. [START_REF] Yang | How much do van der Waals forces contribute to molecular recognition in solution[END_REF][START_REF] Adam | Partitioning Solvophobic and Dispersion Forces in Alkyl and Perfluoroalkyl Cohesion[END_REF] This congruence gives added confidence that all approaches are fundamentally sound.

The independent derivation of the solvent-foldamer ΔG Dis interactions in water and hexane clearly shows that solvent-foldamer dispersion interactions outweigh intramolecular alkyl-alkyl van der Waals dispersion interactions, particularly in water the medium most applicable to biological systems. This aspect could only be inferred from the ΔG DMC and ΔG Expt studies. In so far that a hexane solvent environment may approximate a hydrophobic pocket deep inside a protein, and the dominance of solvent-solute dispersion interactions over that for an intramolecular alkyl-alkyl dispersion interaction, then it follows that the removal of water from a hydrophobic binding pocket is required before any hydrophobic binding can occur.

Conclusion

It has been shown that the hydrophobic van der Waals dispersion alkyl-alkyl folding free energy interaction in a wide range of polar and non-polar solvents is strongly correlated with the cavitation, dispersion, solvent-structure (CDS) solvation free energy and to a lesser extent the molecular polarizability of the foldamer.

The predicted folding alkyl-alkyl free energy interaction in water calculated from the unfolded and folded foldamers by dispersion corrected quantum mechanical modeling is in agreement with the calculated values derived from solution experiments. Electrostatic potential surfaces in aqueous and hexane solution are very useful in differentiating amongst foldamers.

The solute-solvent dispersion interaction overrides the solvent free intramolecular alkyl-alkyl dispersion interaction, but the latter may still be a significant effect deep within the hydrophobic interior of a folded protein.

The analogous perfluoroalkyl-perfluoroalkyl foldamer interaction shows different behavior in water compared to n-hexane where the unfolded state is more stable in water but the folded state is more stable in n-hexane.

The use of the continuum SMD solvent model using water and non-polar solvents to examine protein folding processes may have advantages over more intensive explicit solvation studies. Quantum mechanical ONIOM methods using the SMD solvent model can be used to examine fine details of dispersion interactions at a high level of theory, while describing less critical molecular interactions at lower levels of theory such as using molecular mechanics.

Experimental

All calculations were carried out using the Gaussian 09 package. Energy optimizations were at the DFT/wB97XD/6-31G(d) (6d, 7f) levels of theory for all atoms in all solvents. Selected optimizations at the DFT/wB97XD/6-31 + 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 negative frequencies. Single point solvent and gas calculations were at the DFT/wB97XD/6-31G(d) (6d, 7f) or DFT/wB97XD/6-311++G(d,p) (6d, 7f) levels of theory.

Frequency ONIOM calculations of the folded and unfolded conformations as shown in Figure 1(a) and (b) were conducted at the low level at DFT/B3LYP/6-31G(d,p) (6d, 7f) and at the high level of theory at DFT/wB97XD/6-311G++(d,p) (6d, 7f) with optimized geometries in solvents, using the SMD solvent model. ONIOM calculations at the low level at DFT/wB97XD/6-31G(d,p) (6d, 7f) and at the high level of theory at DFT/wB97XD/6-311G++(d,p) (6d, 7f) gave very similar results. The high level ONIOM calculations were applied to the n-heptyl, n-hexyl and perfluorohexyl side chains of the folded and unfolded conformations as shown in Figure 1(a) and (b). Partial atomic charges were calculated using the CHELPG method with energy calculations where atomic charges are fitted to reproduce the molecular electrostatic potential at a number of grid points around the molecule.

With the 6-31G(d) 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 Surface Tensions[END_REF] The 6-31G(d,p) 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(d) basis set (ie 6-31 + G(d,p)) 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.

Solvents in the SMD solvent model are defined using seven physical factors that determine the non-electrostatic CDS term: the dielectric constant, the square of the refraction index, hydrogen bond acidity and basicity, the macroscopic surface tension at a liquid-air interface, the carbon aromaticity or fraction of non-hydrogenic solvent atoms that are aromatic carbon atoms, and the electronegative halogenicity or fraction of non-hydrogenic solvent atoms that are F, Cl, or Br. PCM solvent ΔG Cav cavitation, ΔG Dis dispersion and ΔG Rep repulsive energies were calculated according to the methods of Floris [START_REF] Floris | Dispersion and repulsion contributions to the solvation energy: Refinements to a simple computational model in the continuum approximation[END_REF] and Pierotti [START_REF] Ra Pierotti | A scaled particle theory of aqueous and nonaqueous solutions[END_REF] For alkyl-alkyl foldamers (kcal/mol) ΔG Cav values folded in water were 93.34, unfolded in water 101.39, folded in hexane 65.28, unfolded in hexane 69.67. ΔG Dis values for alkyl-alkyl foldamers folded in water were -50.20, unfolded in water -59.21, folded in hexane -43.18, unfolded in hexane -50.72. ΔG Rep values for alkyl-alkyl foldamers folded in water were 2.77, unfolded in water 3.43, folded in hexane 2.21, unfolded in hexane 2.70.

For perfluoroalkyl-perfluoroalkyl foldamers (kcal/mol) ΔG Cav , ΔG Dis and ΔG Rep values in water were 100.01, -57.14 and 3.42 for the folded conformation, and 108.66, -56.94 and 4.09 for the unfolded conformation respectively. Dipole isotropic polarizabilities (W= 0.000000) were calculated in Bohr 3 in water.

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 related species is the aim of the study. The literature values for ΔG DMC and ΔG Exp used to develop the multiple regression LFER equations have much higher experimental uncertainties than the calculated molecular properties. The statistical analyses include the multiple correlation coefficient R 2 , the F test of significance, standards errors for the estimates (SEE) and each of the variables ΔG CDS SE(ΔG CDS ), molecular polarizability Pol SE(ΔG Pol ) as calculated from "t" distribution statistics. Residual analysis was used to identify outliers.

Equation 1 Equation 2

 12 Relationship between the alkyl-alkyl interaction ΔG DMC and solvent ΔG CDS of the model compound for 15 solvents ΔG DMC = -1.43ΔG CDS -19.17 Where R 2 = 0.764, SEE = 0.168, SE(ΔG CDS ) = 0.009, F=42.15, Significance=0.0000 Relationship between the alkyl-alkyl interaction ΔG Exp and solvent ΔG CDS of the model compound for 15 solvents ΔG Exp = -1.47ΔG CDS -22.64 Where R 2 = 0.668, SEE = 0.220, SE(ΔG CDS ) = 0.012, F=26.16, Significance=0.0002

Equation 3 Equation 4

 34 Relationship between the alkyl-alkyl interaction ΔG DMC and isotropic polarizability Pol of the model compound in 15 solvents ΔG DMC = -1.56ΔG CDS + 0.16Pol -29.71 Where R 2 =0.771, SEE= 4.13, SE(ΔG CDS )=0.31, SE(Pol)= 0.26, F=20.19, Significance=0.00014 Relationship between the alkyl-alkyl interaction ΔG Expt and isotropic polarizability Pol of the model compound in 15 solvents ΔG Expt = -1.88ΔG CDS + 0.51Pol -56.81 Where R 2 =0.725, SEE= 4.98, SE(ΔG CDS )=0.37, SE(Pol)= 0.32, F=15.83, Significance=0.00043 Dispersion studies of alkyl-alkyl and analogous perfluoroalkyl-perfluoroalkyl interactions Solvent effect on alkyl-alkyl and perfluoroalkyl-perfluoroalkyl interactions (ΔG Interact, sol )

Figure 1 (Figure 1 Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 . 2 Figure 9 .Figure 10 .Figure 11 .

 112345678291011 Figure 1(a) Model compound used to evaluate the folded n-heptyl-n-hexyl interaction (left) and representation of folding process showing the unfolded and folded interchange with solvent mediation (right) adapted from reference 1

Table 1 .

 1 SolventΔG DMC ΔG Exp ΔG Solv ΔG CDS Footnotes: ΔG DMC and ΔG DMC from reference 1, scaled to 100 times kcal/mol. ΔG Solv and ΔG CDS in kcal/mol. Molecular dipole in D, molecular volume in cm 3 /mol, isotropic molecular polarizability scaled to 0.1 times Bohr 3 . Values in red for water are calculated values from equations 1 and 2.

			Molec
	Molec	Molec	Polariz
	Dipole	Volume	-ability