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

of transport of drugs and physiologically important compounds

Keywords: permeability, blood-brain barrier, quantum mechanics, molecular mechanism, quantitative models Running head: new model of blood-brain barrier permeability

Permeability of the blood brain barrier: molecular mechanism of transport of drugs and physiologically important compounds

Introduction

One of the biggest challenges of developing therapeutic agents for the treatment of neurodegenerative disorders is the blood-brain barrier (BBB). Effective drug delivery means that the BBB needs to be circumvented to achieve adequate drug doses in the brain. It has been noted that 98% of drugs fail in clinical trials due to inadequate BBB permeability [START_REF] Pardridge | Blood-brain barrier delivery[END_REF].

The BBB plays an important role in the homeostasis, or maintenance of the central nervous system CNS, by controlling the movement of nutrients and toxins to and from the CNS. Drugs that have non-CNS targets need to have characteristics that prevent transport across the BBB to avoid unwanted side effects.

The BBB is a highly selective permeable cellular phospholipid protein bilayer barrier that separates the circulating blood from the brain extracellular fluid in the CNS. The blood-brain barrier is composed of capillary endothelial cells, which are connected by tight junctions with an extremely high electrical resistivity of at least 0.1 Ω⋅m. The BBB also includes a thick basement membrane and astrocytic endfeet. The bloodbrain barrier allows the passage of water, some gases, and lipid soluble molecules by passive diffusion, as well as the selective transport of molecules such as glucose and amino acids that are crucial to neural function. The BBB also protects the brain from many common bacterial infections. Antibodies are too large to cross the blood-brain barrier, and only certain antibiotics are able to pass.

Compounds cross the BBB by a variety of mechanisms [START_REF] Mangas-Sanjuan | Drug penetration across the blood-brain barrier; an overview[END_REF][START_REF] Nau | Penetration of Drugs through the Blood-Cerebrospinal Fluid/Blood-Brain Barrier for Treatment of Central Nervous System Infections[END_REF][START_REF] Gabathuler | Approaches to transport therapeutic drugs across the blood-brain barrier to treat brain diseases[END_REF][START_REF] Banks | Characteristic of compounds that cross the blood-brain barrier[END_REF][START_REF] Pajouhesh | Medicinal chemical properties of successful central nervous system drugs[END_REF]:

1) Trans-membrane or trans-cellular passive non-saturable diffusion: usually molecules with high lipophilicity and low molecular size can passively diffuse across the BBB in the direction of the concentration gradient, without the input of energy. Paracellular diffusion is usually negligible because of the tight junctions between cells.

2) Active saturable transporters are integral membrane proteins (ATP dependent or ATP independent) which can transport drugs across the BBB against the concentration gradient. There are two types of transporters: (a) carrier mediated transporters, and (b) active efflux transporters (eg p-glycogen) which carry drugs and other compounds out of the brain. There are two classes of membrane transport proteins: carrier proteins, which carry specific molecules across, and channel proteins, which form a narrow pore through which ions can pass.

Channel proteins carry out passive transport, in which ions travel spontaneously down their gradients. Some carrier proteins mediate passive transport (also called facilitated diffusion), while others can be coupled to a source of energy to carry out active transport, in which a molecule is transported against its concentration gradient Facilitated diffusion is a process of spontaneous passive transport which does not require ATP energy, and differs from passive diffusion in relying on binding between the drug and carrier protein or membrane embedded channel, and it is a saturable process which is more reliant on temperature dependent binding processes than passive diffusion. The main role of the drug transporters is carrying the drugs and other xenobiotics into and out of the brain, and they are integral to other cell processes such as inflammation, differentiation of immune cells, cell detoxification, lipid trafficking, hormone secretion and development of stem cells.

3) Endocytosis and exocytosis whereby substances (proteins etc) are engulfed by the membrane and pass through the cell by vesicles and released on the other side.

4) Extracellular pathways.

Important molecular properties associated with BBB permeability: Pajouhesh [START_REF] Pajouhesh | Medicinal chemical properties of successful central nervous system drugs[END_REF] has reviewed various retrospective classification databases in the literature to determine the common attributes and their ranges that facilitate BBB permeability:

1) Experimental in vivo measures of permeability: log BB (which is a steady state equilibrium measure of the drug partitioning in the blood or brain) or log PS (obtained from in situ brain perfusion studies, usually using rats, is a kinetic rate measure of the volume cleared per unit time). An effective permeability >1x10 -6 cm/sec is considered a lower limit.

2) Lipophilicity has a positive correlation with ability to cross BBB: usually log P (o/w) for neutral compounds, with a minimal hydrophobicity (Clogp>5).

3) Hydrogen bonding of polarity has a negative correlation with ability to cross BBB: indicators include Abraham coefficients, or the number of acidic and basic atoms, or number of H-bond donor atoms <3, and number of H-bond acceptor atoms <7 4) Molecular weight <450, though there are exceptions [START_REF] Banks | Characteristic of compounds that cross the blood-brain barrier[END_REF] 5) Molecular topological polar surface area (TPSA): < 60-70 Å 2 6) Molecular shape: spherical shape preferred over rod shape, increased branching shows negative correlation with ability to cross BBB: McGowan characteristic volume for molecular size 7) Molecular flexibility has a positive correlation with ability to cross BBB, with the number of rotatable bonds being <8

8) The concentration of uncharged chemical species in water at the physiological pH level is critical, with the estimated pKa range for BBB permeability being 4-10 [START_REF] Fischer | Blood-brain barrier permeation: molecular parameters governing passive diffusion[END_REF] or 7.5-10.5 [START_REF] Pajouhesh | Medicinal chemical properties of successful central nervous system drugs[END_REF] The presence of a positive charge at pH 7-8, or compounds with a tertiary N atom, tend to enhance BBB permeability [START_REF] Goodwin | In silico prediction of blood-brain barrier penetration[END_REF]. Strong acids, including carboxylic acids, and bases are generally not easily transported across the BBB. 9) Metabolic stability with >80% remaining after 1 hour, since a high metabolic rate would remove the drug rapidly from the blood plasma. 10) Not being a high-affinity serum albumin ligand (K d < 10µM), since this would decrease the effective concentration of the drug in blood plasma.

Statistical (multiple linear regression) quantitative structure activity relationships (QSAR) models: The major descriptors [START_REF] Pajouhesh | Medicinal chemical properties of successful central nervous system drugs[END_REF][START_REF] Fischer | Blood-brain barrier permeation: molecular parameters governing passive diffusion[END_REF][START_REF] Jouyban | Blood Brain Barrier Permeation[END_REF][START_REF] Vilar | Prediction of passive blood-brain partitioning: straightforward and effective classification models based on in silico derived physicochemical descriptors[END_REF][START_REF] Goodwin | In silico prediction of blood-brain barrier penetration[END_REF][START_REF] Abraham | On the partition of ampholytes: Application to blood-brain distribution[END_REF][START_REF] Mehdipour | Brain drug targeting: a computational approach to overcoming blood-brain barrier[END_REF][START_REF] Kaznessis | A review of method for predicting blood-brain partitioning[END_REF][START_REF] Garg | In silico modelling of blood-brain barrier permeability predictions[END_REF] found to be important in QSAR models (which predominantly seek correlations with logBB) are:

1) Lipophilicity usually expressed as ClogP, has been found to be a critical factor relating to permeability. ClogP has a median value of 2.5 for successful CNS drugs. Alternatively, logD should be between 0-3 for smaller compounds.

2) It has been suggested that the molecular weight (MW) should be below 400-600 for successful CNS drugs (lower than the MW of drugs undergoing oral absorption. The mean MW for marketed CNS drugs is 310, whereas the median for orally active drugs is 377.

3) All the QSAR models include hydrogen bonding, either as polarity, polar surface area (PSA), hydrogen bond donor or acceptor coefficients (Abraham coefficients), or counting heteroatoms (O, N atoms) capable of hydrogen bonding. Generally CNS drugs tend to have lower PSA values than other drugs, usually falling within the range 60-90 Å 2 . There is also a trade-off relationship between polarity or PSA of a molecule and lipophilicity for larger organic compounds, where the polar component is counter balanced by the hydrophobic component of the molecule.

4) The consistent finding in QSAR modelling shows that lipophilicity is positively correlated, PSA is negatively correlated, hydrogen bonding is negatively correlated, and molecular size (or molecular weight) is negatively correlated to permeability. [START_REF] Pajouhesh | Medicinal chemical properties of successful central nervous system drugs[END_REF][START_REF] Fischer | Blood-brain barrier permeation: molecular parameters governing passive diffusion[END_REF][START_REF] Jouyban | Blood Brain Barrier Permeation[END_REF][START_REF] Vilar | Prediction of passive blood-brain partitioning: straightforward and effective classification models based on in silico derived physicochemical descriptors[END_REF][START_REF] Goodwin | In silico prediction of blood-brain barrier penetration[END_REF][START_REF] Abraham | On the partition of ampholytes: Application to blood-brain distribution[END_REF][START_REF] Mehdipour | Brain drug targeting: a computational approach to overcoming blood-brain barrier[END_REF][START_REF] Kaznessis | A review of method for predicting blood-brain partitioning[END_REF][START_REF] Garg | In silico modelling of blood-brain barrier permeability predictions[END_REF] There is some evidence that molecular volume might show a parabolic relationship to permeability, since a smaller volume positively increases diffusion, but a larger molecular volume might also increase lipophilicity, which is positively correlated with permeability. [START_REF] Garg | In silico modelling of blood-brain barrier permeability predictions[END_REF] 5) The order of permeability appears to be active uptake compounds > passive diffusion compounds > efflux compounds (by about one logPS unit in each case), and the effect of molecular charge for logPS passive diffusion was basic compounds > neutral compounds > acidic compounds. [START_REF] Liu | Development of a computational approach to predict blood-brain barrier permeability[END_REF] 6) Much of the effort in QSAR studies have focussed an finding and improving statistical relationships between logBB and variables such as lipophilicity, PSA, molecular size etc. However, given that the error in logBB (and logPS) experimental measurements is quite large, and a widely diverse range of compounds which have very different chemical structures, size, polarity etc have been examined, improvements in correlation coefficients may not necessarily be real (outliers may only be gross outliers). There are also significant errors in variables such as lipophilicity, polarity, size etc. Multiple regression analysis is particularly error prone where logBB is correlated with 3-5 variables. There is a significant error in the logPS values themselves, since experimental conditions can vary amongst different studies. It is suggested that only a molecular mechanistic approach which starts from a sound physical-chemical basis on a structurally similar range of compounds can be meaningfully correlated with experimental permeability measures such as logPS. In silico methods based on QM methods can help reduce errors in molecular properties, and errors from wet chemical methods used in logP lipophilicity measurements.

Huwyler et al [START_REF] Suenderhauf | Computational Prediction of Blood-Brain Barrier Permeability Using Decision Tree Induction[END_REF] recently used a decision tree analysis of 153 log PS data to find the dominant descriptors of permeability were lipophilicity (aLogP) and charge (PSA), with molecular geometry and connectivity also important factors. Their model also appeared to account for active transport as well as passive diffusion permeability. The property ranges used were: molecular weight 46-1201 Da, partition coefficient (aLogP) -4.3 to -2.4, polar surface area (PSA) 3.2-279 Å 2 , rotatable bonds count 0-18, hydrogen bond acceptor count 1-23. A broad distinction was found between between positive (CNSp+) and negative (CNSp-) molecules (compounds with logPS values ≥ -2 and ≤-3, respectively, with logPS values between -2.1 and -2.9 were exempt from consideration). Huwyler's model is generally consistent with the previous regression QSAR models, but includes a broader range of physiochemical properties, and the analysis is not constrained to parameters that are used in QSAR regression models.

A significant issue relating to the many QSAR studies of BBB permeability is the distinction between passive diffusion and active transport processes. It is not clear how many studies have made the distinction, since these two broad categories involve very different mechanisms. Recent work has shown that many of the higher molecular weight (volume) permeants utilize active transport mechanisms, such that true passive diffusion is not common. [START_REF] Pardridge | Blood-brain barrier delivery[END_REF][START_REF] Mangas-Sanjuan | Drug penetration across the blood-brain barrier; an overview[END_REF][START_REF] Nau | Penetration of Drugs through the Blood-Cerebrospinal Fluid/Blood-Brain Barrier for Treatment of Central Nervous System Infections[END_REF][START_REF] Gabathuler | Approaches to transport therapeutic drugs across the blood-brain barrier to treat brain diseases[END_REF][START_REF] Banks | Characteristic of compounds that cross the blood-brain barrier[END_REF][START_REF] Pajouhesh | Medicinal chemical properties of successful central nervous system drugs[END_REF][START_REF] Jouyban | Blood Brain Barrier Permeation[END_REF][START_REF] Pardridge | Drug transport across the blood-brain barrier[END_REF] The characteristics of the BBB itself clearly are dominant in any mechanistic considerations of permeability by drugs or other compounds such as amino acids, etc.

One important factor which affects the passage of highly polar and charged species is the dipole potential of the lipid bilayer membrane, [START_REF] Stowasser | The dipole potential of lipid membranesan overview[END_REF][START_REF] Peterson | Origin of membrane dipole potential: Contribution of the phospholipid fatty acid chains[END_REF][START_REF] Cattelotte | Changes in dipole potential at the mouse bloodbrain barrier enhance the transport of technetium sestamibi[END_REF][START_REF] Walter | Permeability of small nonelectrolytes through lipid bilayer membranes[END_REF][START_REF] Bezanilla | How membrane proteins sense voltage, Nature Revs[END_REF][START_REF] Heimburg | The capacitance and electromechanical coupling of lipid membranes close to transitions: the effect of electrostriction[END_REF][START_REF] Koerner | Electrodynamics of lipid membrane interactions in the presence of zwitterionic buffers[END_REF][START_REF] Cafiso | Influence of Charges and Dipoles on Macromolecular Adsorption and Permeability[END_REF] which has a phosphatidylcholine (PC) head attached to a long chain fatty acid bilayer. Charged molecules can modify the membrane dipole potential by electrostatically interacting with the BBB membrane by attraction or repulsion. Positively charged molecules interact with the membrane, causing the N + end of the head group to move towards the water phase, away from the lipid membrane surface. Conversely negative charged molecules cause the N + end of the head group to move towards to the lipid surface.

By changing the angle of the dipole with respect to the membrane surface, the membrane potential is altered. Alteration of the membrane potential affects the permeability of charged ions through the BBB. Exactly how charged and polar species interact with the BBB is unknown, but using model lipid bilayer membranes such as DPPC (dipalmitotylphosphatidylcholine) or DPHYPC (diphytanoylphosphatidylcholine) the membrane potential has been measured at 243 +/-4 and 228 +/-5 mV respectively. [START_REF] Cattelotte | Changes in dipole potential at the mouse bloodbrain barrier enhance the transport of technetium sestamibi[END_REF] However while formally charged lipophilic molecules such as the tetraphenylborate anion and tetraphenylphosponium cation interact with PC lipid membranes (possibly because high charge dispersal to the phenyl rings allows the dominant molecular lipophilicity to facilitate passive diffusion through the membrane), uncharged species such as phloretin also interact with the PC lipid membranes. The suggested interaction between phloretin (which is known to lower the membrane potential) and PC lipid bilayers, is a hydrogen bond between the phloretin and the P=O of the phospholipid, or the C=O of the lipid ester. [START_REF] Peterson | Origin of membrane dipole potential: Contribution of the phospholipid fatty acid chains[END_REF][START_REF] Cafiso | Influence of Charges and Dipoles on Macromolecular Adsorption and Permeability[END_REF] Charged species, including zwitterionic species such as amino acids such as glycine, alanine etc at the physiological pH 7.4 of blood serum can interact with the phospholipid membrane which can sense the charge on the interacting species [START_REF] Bezanilla | How membrane proteins sense voltage, Nature Revs[END_REF] or by affecting the capacitance of the membrane and causing electromechanical changes in the membrane [START_REF] Heimburg | The capacitance and electromechanical coupling of lipid membranes close to transitions: the effect of electrostriction[END_REF], or other similar electrodynamic processes [START_REF] Koerner | Electrodynamics of lipid membrane interactions in the presence of zwitterionic buffers[END_REF]. These electromechanical processes are related to the thermal fluctuation of defects, or small mobile free volumes in the hydrocarbon phase of membranes which might allow passage of small molecules through the membranes. [START_REF] Trauble | The movement of free molecules across lipid membranes: a molecular theory[END_REF] A molecular dynamics study [START_REF] Chew | Biggin PC Distribution and Dynamics of Adamantanes in a Lipid Bilayer[END_REF] of three adamantanes with a model PC lipid bilayer membrane (POPC, 1palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine) has shown that the protonated species interact with the PS lipid head group, creating a deformation of the membrane. The positively charged ammonium group faces the negatively charged phosphate moiety, and remain in this orientation until the adamantanes reach the centre of the lipid bilayer, then flips to face the PC headgroup of the other lipid leaflets. By computing the pK a as a function of lipid depth, it was concluded that deprotonation occurs, although it is unclear whether deprotonation occurred in the bulk solution of after initial adsoption into the interface region. This work will be shown to support a "preorganization" desolvation -dipole process prior to initial adsorption.

There are three separate potentials involved at the bloodmembrane interface: (1) the transmembrane potential, Δψ, is the potential difference between the aqueous solutions on either side of the membrane. It arises from concentration differences of ions;

(2) the surface potential, ψ S , is the potential difference between the membrane surface and the aqueous bulk. It arises from fixed charges at the membrane/water interface, affecting the negatively charged head groups of lipid molecules; (3) the dipole potential, ψ D , is the potential difference between the center of the bilayer and the membrane/water interface. It follows that any relationship between a charged (or highly polar) species and the BBB membrane potentially involves all three types of potentials which importantly includes water molecules as well as the drug or other physiologically important molecules. Cafiso [START_REF] Cafiso | Influence of Charges and Dipoles on Macromolecular Adsorption and Permeability[END_REF] has suggested that a significant proportion of changes to the dipole potential of lipid membranes may involve electrodynamic alignment of the dipoles of bound water molecules, as well as the dipoles of charged drugs.

The importance of the membrane dipole potential for charged, zwitterionic, or highly polar neutral molecules (phloretin) in permeating the BBB membrane indicates an important feature of CNS drugs may be the dipole moment in water. Also it has been recently shown that the free energy of solvation is a dominant factor in deciding the ability of statins to cross the BBB. [START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3-hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] The binding strengths of water molecules to the BBB membrane and the drug is expected to be crucial, if desolvation of charged or highly polar species are required before permeation can occur. Both passive diffusion and active transport processes through the membrane cells would be affected, particularly if the aqueous hydration strengths are energetically significant.

Active transport processes require carrier proteins, so desolvation would be required to facilitate protein-drug interaction.

Objects of this study: 1. To examine the molecular basis of BBB permeability focussing on the characteristics of both molecules that can potentially permeate, and models of the BBB membrane itself such as DPPC (dipalmitotylphosphatidylcholine), DPHYPC (diphytanoyl phosphatidylcholine), DLPC (dilauroyl phosphatidylcholine), DOPSE (1,2-dioleoyl phosphatidylserine), and POPC (1-palmitoyl-2-oleoyl-snglycero-3-phosphatidylcholine) etc.

2. The permeability characteristics of molecules will be examined by quantum mechanical (QM) methods are: the free energies of solvation, molecular volumes, atomic electrostatic charges, dipole moments, and measures of hydrogen bond donor / acceptor and cavity effects in water. Both passive and active transport processes will be examined.

3. An examination of the membrane potential of the model PC lipid bilayer membranes DPPC, DPHYPC, DLPC and POPC will be undertaken using QM methods to probe how the membrane dipole potential affect the dipole and desolvation of molecules, and hence permeability through the BBB.

4. LogPS experimental data will be used as the kinetic measure of BBB permeability. While there have been many QSAR investigations of logBB with variables such as lipophilicity, PSA, hydrogen bonding etc, logBB is an equilibrium measure which can be confounded with variables such as drug-blood protein interactions, metabolic disposal, etc. LogPS as a kinetic measure is less affected by these variables.

Experimental

All calculations were carried out using the Gaussian 09 package on optimised structures. Electrostatic potential at nuclei for solutions were calculated using the CHELPG method in Gaussian 09. The atomic charges produced by CHELPG are not strongly dependant on basis set selection. Using the B3LYP level of theory, calculated atomic charges were almost invariant amongst the basis sets 6-31G(d), 6.311(d,p), 6-311+(2d,2p), 6-311G++(3df,3dp) [START_REF] Martin | Charge Distribution in the Water Molecule -A Comparison of Methods[END_REF][START_REF] Kubelka | Population Analysis[END_REF]. Errors between calculated and experimental dipole moments were 3%. All solvent calculations were at the B3LYP/6-31G*(6d, 7f) level of theory, using optimised geometries, as this level has been shown to give accurate electrostatic atomic charges, and was used to optimize the IEFPCM/SMD solvent model. Where a solvent study was carried to compare different solvents, the same optimised solute geometry was used. 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] 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]. It has been found that the B3LYP / 6.31G+* combination gives reasonably accurate PCM and SMD solvation energies for some highly polar polyfunctional molecules, which are not further improved using higher level basis sets [START_REF] Forest | Accuracy of computational solvation free energies for neutral and ionic compounds: Dependence on level of theory and solvent model[END_REF]. Adding diffuse functions to the 6-31G* basis set (ie 6-31 + *) had no significant effect on the solvation energies with a difference of ca 1% observed, which is within the literature error range for the IEFPCM/SMD solvent model.

It should be noted that some very low correlation coefficients R 2 are shown in some of the regression equations. These low values are associated with very low slopes, ie the relationships with logPS are quite insensitive, which is a known statistical issue with regression correlations. The standard error of the estimate (logPS), SEE, is a better indicator of the precision of regression equations. In most cases examined, a lack of sufficient logPS observed data make the multiple regression equations less robust than desirable.

Compounds analysed by

Results and Discussion

It is clear from the many reviews of BBB permeability that drug and physiologically important compounds are primarily dependant on lipophilicity, polarity or charge, hydrogen bonding, and molecular size. In this study, these measures are calculated by QM solvent effects using the solvation free energy ΔG octanol for n-octanol as a measure of lipophilicity, the dipole moments or the calculated atomic charges (CHELPG charges) in water as measures of polarity or molecular charge, and the calculated molecular volumes. This approach has been previously applied to statins. [START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3-hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] Unfortunately, even though logP or logD in water-n-octanol (or other partitioning solvent combinations) is widely used to define drug lipophilicity, n-octanol contains 2.18M water in partitioning experiments at equilibrium, so most polar solutes would be solvated by this water, indicating that the log P, or log D values may be suspect.

Calculated ClogP values are based on experimental logP values. There are also significant errors in obtaining experimental logP values. It is also clear that n-octanol has significant hydrogen bonding capability, whereas n-octane has none. However, noctanol has been widely accepted as a membrane bilayer mimicking solvent, where the 2.18M equilibrium water concentration is consistent with the known water levels in cell membrane bilayer cores because of trans-bilayer transport. [START_REF] Balaz | Modelling kinetics of subcellular disposition of chemicals[END_REF] It has been previously shown [START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3-hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF] that desolvation effects (as measured by ΔG) can be dominant in BBB permeability of statins. There have been previous QSAR studies of the linear relationship between the free energy of solvation and logBB [START_REF] Lombardo | Curatolo WJ Computation of brain-blood partitioning of organic solutes via free energy calculations[END_REF][START_REF] Keseru | High-throughput prediction of blood-brain partitioning: a thermodynamic approach[END_REF] which have suggested that compounds with logBB > 0.3 cross the BBB, while those with logBB < -1.0 do not. Importantly, from a screening survey of 8700 CNS drugs, it was found that 96% of CNS active drugs had a ΔG higher than -12 kcal/mol. This study examines the desolvation of a wide range of drugs and their logPS permeability, but also concomitantly examines variables such as dipole moment or atomic charge, molecular volume, hydrogen bonding, and solvent cavity effects at the same time.

To gain mechanistic insight into the molecular basis of passive diffusion based permeability through the BBB, a closely related series of alcohols has been examined.

Unfortunately the QSAR approach of using a widely diverse range of compounds does not lend itself to mechanistic interpretation by linear free energy analysis (only statistical inference), as there is no control of molecular variability or transport processes. These results are shown in Table 1. It is clear that there are strong relationships between logPS and ΔG water (positive), a strong positive relationship with membrane lipophilicity as measured by ΔG octanol , a negative relationship with dipole moment D water , and a weaker negative relationship with molecular volume. It is clear that the value for water (and ethylene glycol to a lesser extent) is a clear outlier for the ΔG water and volume relationships, which is consistent with the known anomalous properties of this unique highly polar solvent. All molecules were geometry optimised to give the lowest energy conformations. There is a weaker negative relationship between logPS and CDS in water. The CDS term is included in the overall ΔG water term, and is a measure of hydrogen bonding (based on Abraham's coefficients) and cavity effects (creation of a "hole" in the solvent in which to place the solute, plus other cavity interaction effects).

The data is consistent with the following model for BBB permeability:

1. Desolvation, the reverse of solvation, of the permeating drug is a dominant negative factor, in view of the large ΔG values, and is probably the rate determining step (RDS). LogPS = -0.54 ΔG water -2.96 with R 2 0.79, SEE 0.98 where ΔG water is the water desolvation free energy (equation 1) 2. Lipophilicity as measured by ΔG octanol is significant and highly positively correlated with the permeation through the lipid bilayer. As n-octanol is a proxy for the membrane bilayer, this relationship implies that once desolvation of water has occurred, and the alcohols start to permeate the lipid bilayer, lipophilicity determines the rate of diffusion. LogPS = 1.28 ΔG octanol -2.83 with R 2 0.80, SEE

where ΔG octanol is the solvation free energy in n-octanol, or lipophilicity (equation 2)

3. The dipole moment D in water is negative correlated, probably due to the effect of the membrane dipole potential as the drug approaches the cell wall. This effect is much smaller than desolvation or lipophilicity. The model is consistent with previous QSAR models, but puts together the four factors for the first time, particularly using the free energy of desolvation and the dipole moment which have not been previously considered. Hydrogen bonding properties are captured in the solvation/desolvation terms. A dependency on lipophilicity and molecular size is consistent with previous QSAR models. This approach is unique in using a full in silico QM approach.

It should be noted that this model applies to BBB diffusion processes, where the molecular volumes are relatively small (12-77.4 cm 3 /mol), and all compounds are neutral species. If active transport processes are involved, [START_REF] Pardridge | Drug transport across the blood-brain barrier[END_REF] desolvation must still occur prior to any carrier protein-drug interaction, based on the magnitude of ΔG water .

It is unlikely on energy grounds that a large charged molecule could permeate a cell membrane in a manner that smaller ions (eg Na + ) can enter ATP driven ion pores.

Lipophilicity and molecular volume are also likely factors, based again on energy considerations and previous QSAR results. [START_REF] Suenderhauf | Computational Prediction of Blood-Brain Barrier Permeability Using Decision Tree Induction[END_REF][START_REF] Hutt | Drug chirality and its pharmacological consequences, Ch. 5, in Smith and Williams' Introduction to the Principles of Drug Design and Action[END_REF][START_REF] Pardridge | Drug transport across the blood-brain barrier[END_REF] The dipole moment of the drug is a vector measure of the molecular charge separation, and since protein-drug interaction is essentially an electrostatic interaction, it seems intuitively likely that the dipole of the drug should be an important factor in any drug-protein carrier interaction. Charged and zwitterionic molecules will be influenced by the membrane dipole potential which has a negatively charged PC head group with a positively charged lipid bilayer tail (see section 7 below). So the membrane dipole potential can exert a force on larger charged molecules that might facilitate desolvation processes prior to passive or active transport into the cell membrane, as shown previously shown. [START_REF] Chew | Biggin PC Distribution and Dynamics of Adamantanes in a Lipid Bilayer[END_REF] Active transport processes involve a protein-drug interaction where a neutral drug species can electrostatically interact with the carrier protein. Desolvation of water from the drug, which may be in a charged or zwitterionic state in blood plasma at pH 7.4 has to first occur, which is energy expending. Passive transcellular or paracellular diffusion are also favoured by lower charge which increases lipophilicity, so desolvation of charged or zwitterionic species to give a neutral species in a lipophilic environment favours passive permeation. By examining the effect of pH on the (active) permeability of D-glucose, L-and D-lactate, and nicotine (passive) it was found that it is the uncharged species that exhibits much higher permeation rates for both active transport and passive transport. [START_REF] Oldendorf | pH dependence of blood-brain permeability to lactate and nicotine[END_REF] To test this model, some comparisons have been made with closely related compounds or series of compounds which are known to show significant differences in permeability. These tests include: These equations 6-9, using only 5 logPS data points and consequently of low statistical robustness, are similar to those equations using the alcohol data (equations 1-4). The main difference is that the relationship with molecular volume (which range from 85 to 147 cm 3 /mol) shows an inverse relationship from that of the much smaller alcohols (which range from 12 to 77.4 cm 3 /mol). This is consistent with the observation that the relationship between permeability and molecular size has a parabolic relationship. [START_REF] Garg | In silico modelling of blood-brain barrier permeability predictions[END_REF] Despite the less than rigorous statistical basis (because of insufficient logPS data), the model performs reasonably well in closely related series of compounds being transported by facilitated and passive diffusion processes. Caffeine, theophylline and hypoxanthine are known to be transported by both active and passive modes.

[ [START_REF] Mccall | Blood-brain barrier transport of caffeine[END_REF][START_REF] Habgood | Permeability of the developing and mature blood-brain barriers to theophylline in rats[END_REF][START_REF] Spector | Hypoxanthine transport through the blood-brain barrier[END_REF] 2) Morphine and related derivatives: Morphine, heroin, codeine and morphine-6glucuronide (M6G) are all related: heroin has the two hydroxyls of morphine replaced by acetyl groups, but shows a 31-100 fold penetration rate of the BBB compared to morphine. M6G has a glucoronide group substituted for the 6 hydroxyl group of morphine and has a decreased BBB penetration rate of 57 times compared to morphine under identical conditions. [START_REF] Wu | Blood-Brain Barrier Permeability to Morphine-6-Glucuronide is Markedly Reduced Compared with Morphine[END_REF] M6G is the major active metabolite of morphine, and thus heroin, and is responsible for much of the analgesic effect. Codeine which has the phenol hydroxyl group of morphine replaced by a methoxy group penetrates the BBB 10 times faster than morphine.

Morphine is actively transported across the BBB by P-glycoprotein. [START_REF] Liu | Development of a computational approach to predict blood-brain barrier permeability[END_REF] Codeine crosses the BBB by passive paracellular diffusion. [START_REF] Jenkins | Pharmacokinetics of specific drugs[END_REF] The logPS values of morphine, heroin, codine and M6G are -2.7, -1.2, -1.7, -4.5 respectively. These compounds all exist predominantly as the cations or as a zwitterion (M6G) at physiological pH levels.

Examination of the ΔG water , ΔG octanol , D, and V values reveals that the zwitterionic M6G has larger desolvation ΔG water value (49.0 kcal/mol greater) and lower lipophilicity for the neutral species as measured by -ΔG octanol (18.7 kcal/mol) but larger D (by 2.36 times) and larger V (by 1.65 times) which appears to explain why it permeates 57 times as slow as the morphine ion under identical conditions. M6G also is also 187 times less lipid soluble than morphine as measured by octanol/water partitioning. [START_REF] Wu | Blood-Brain Barrier Permeability to Morphine-6-Glucuronide is Markedly Reduced Compared with Morphine[END_REF] Examination of the difference between morphine ion and heroin ion does not reveal such large differences as seen for M6G. Morphine is more soluble in water, and has a smaller desolvation energy 0.4 kcal/mol, a higher neutral lipophilicity by 0.6 kcal/mol, a lower D by 12.4% and a smaller V by 33% than the heroin ion.

All these factors suggest that morphine should permeate faster than heroin.

However heroin is more soluble than morphine in lipids because of the two acetyl groups. [START_REF] Jenkins | Pharmacokinetics of specific drugs[END_REF] These data suggest the 30.6 (to 100) times faster permeation rate for heroin [START_REF] Liu | Development of a computational approach to predict blood-brain barrier permeability[END_REF][START_REF] Oldendorf | Blood-brain barrier: penetration of morphine, codeine, heroin, and methadone after carotid injection[END_REF] is dominated by the heroin-lipophilic protein solubility.

Comparison of the morphine ion with the codeine ion shows that codeine has a smaller desolvation energy by 1.9 kcal/mol, a higher neutral lipophilicity by 0.7 kcal/mol, a higher D by 6.1% and a larger V by 2.5%, which is consistent with the observed difference in logPS.

In summary, morphine, heroin, codeine and M6G which are transported across the BBB by active and passive processes appear to be consistent with the developed transport model when solubility is taken into account.

3) Antipyrine and iodoantipyrine have experimental logPS values of -2.0 and -1.1 respectively, although the difference is only an I atom substituting for a H atom.

The ΔG water for antipyrine is lower by 0.9 kcal/mol (therefore requiring a lower desolvation energy, which is a positive factor for permeation), ΔG octanol is lower by 0.4 kcal/mol (therefore less lipophilic which is a negative factor for permeation), the dipole moment in water is lower by 0.3D (which is a positive factor for permeation), and the molecular volume is larger by 19% in n-octanol (which is a negative factor for permeation) compared to iodoantipyrine.

Apparently the greater molecular volume in octanol and increased lipophilicity override the desolvation and dipole effects to make iodopyrine permeate faster.. 4) Urea and thiourea have experimental logPS values of -3.8 and -3.4 respectively, although the difference is only an S atom substituting for a O atom. The ΔG water for urea is higher by 2.2 kcal/mol, ΔG octanol is lower by 0.3 kcal/mol, the dipole moment in water is lower by 0.2D, and the molecular volume is larger by 16% in n-octanol compared to thiourea. The large difference in ΔG water is due to a greater ΔG CDS for urea, probably due to a greater hydrogen bonding interaction. It appears that the larger desolvation for urea is the main cause of its lower permeation rate. 5) Nicotine (logPS -1.0) in its protonated form shows a greatly decreased brain uptake index (BUI) from 109 (pH 7.2) to 49 (pH 4.7). [START_REF] Oldendorf | pH dependence of blood-brain permeability to lactate and nicotine[END_REF] The difference in ΔG water for nicotine and the protonated species is 57.5 kcal/mol, ΔG octanol is 53.4 kcal/mol, the dipole moment in water differs by 7.7D, though the molecular volume of nicotine is larger by 12.8% in n-octanol. Similarly, the two N-methyl salts of nicotine (quaternized at either the pyridine or pyrrole N atoms) showed BUI values in rats of 3 compared to nicotine 120 at pH 7.4. The differences in ΔG water for nicotine and the N-methyl species are 59.9 or 45.8 kcal/mol, ΔG octanol are 52.2 or 43.6 kcal/mol, the dipole moments in water differs by 6.6D or 6.5D, and the molecular volumes of nicotine are smaller by 18.5% (pyrrole N-methyl) or larger by 6.6% (pyridine N-methyl) in n-octanol respectively. Overall these data are consistent with the higher passive diffusion uptake index for the neutral nicotine species, clearly demonstrating large desolvation, lipophilicity and dipole factors are operating. 6) Stereoselectivity of transport across the BBB: there are many examples of the chirality of drugs affecting their pharmacology, [START_REF] Hutt | Drug chirality and its pharmacological consequences, Ch. 5, in Smith and Williams' Introduction to the Principles of Drug Design and Action[END_REF] ranging from enantiomeric differences in binding to plasma proteins to transport across the BBB. As these enanatiomers have almost identical physical and chemical properties, they represent a good test of any theory being developed for BBB permeability. It is generally thought that the origins of stereoselectivity are electrostatic in origin, possibly during protein binding which involves conformational selectivity in the active transport process. [START_REF] Hutt | Drug chirality and its pharmacological consequences, Ch. 5, in Smith and Williams' Introduction to the Principles of Drug Design and Action[END_REF] Several examples of stereoselective permeation have been investigated: I. 4-Fluoro-L-phenylalanine has a logPS -1.7 compared to 4-fluoro-Dphenylalanine log PS -2.9: The ΔG water for 4-fluoro-L-phenylalanine (zwitterion at pH 7.4) is higher by 0.6 kcal/mol, ΔG octanol neutral species is lower by 1.9 kcal/mol, the dipole moment in water is almost the same, and the molecular volume is smaller by 6.4% in n-octanol than the 4-fluoro-Dphenylalanine zwitterion. These data do not appear consistent with the log PS data, unless the molecular volume term dominates. The ΔG octanol value has been used as a proxy for lipophilicity of the membrane bilayer in passive diffusion permeation. However for active transport (by the large amino acid transporter, LAT1 [START_REF] Pardridge | Drug transport across the blood-brain barrier[END_REF]) where a drug-carrier protein interaction is involved, it is unclear whether the carrier protein-drug interaction is hydrophobic driven, or hydrophilicly driven where hydrogen bonding dominates. The experimental result could be explicable if the positive desolvation and smaller molecular volume is supported by a postive contribution to the permeability rate by a lower ΔG octanol implying that the protein-drug interaction has a dominant hydrophilic rate determining effect.

II. D & L amino acids:

The transport of amino acids across the BBB are by active transport processes, [START_REF] Oldendorf | Amino acid assignment to one of three blood-brain barrier amino acid carrier[END_REF][32] Using the brain uptake index (BUI) as a rate measure of BBB permeability, it was found that the L-enantiomers of aspartic and glutamic showed higher BUI than the D-enantiomers. The ΔG water for L-aspartic acid (anion) is higher by 1.1 kcal/mol, ΔG octanol for the neutral species is lower by 2.4 kcal/mol, the dipole moment in water is identical, and the molecular volume is virtually the same in n-octanol than the D-aspartic acid anion. The ΔG water for L-glutamic acid (anion) is lower by 0.3 kcal/mol, ΔG octanol (neutral species) is the same, the dipole moment in water is higher by 1.3D, and the molecular volume is smaller by 18% in noctanol than the D-glutamic acid anion. As an organic anion transporter is the active transporter, these data appear inconsistent with the simple model, and show the same pattern as that for 4-fluoro-phenylalanine. It appears that active transport of neutral amino acids by LAT1 is clearly more complex than for passive diffusion transport. III. Lactic acid: the L-enantiomer of lactic acid showed higher BUI than the Denantiomers. [START_REF] Oldendorf | Amino acid assignment to one of three blood-brain barrier amino acid carrier[END_REF] Lactic acid is predominantly transported across the BBB by the monocarboxylic acid transporter type 1, MCT1. [START_REF] Pardridge | Drug transport across the blood-brain barrier[END_REF] The ΔG water for L-lactic acid (anion) is higher by 3.2 kcal/mol, ΔG octanol (neutral species) is higher by 1.0 kcal/mol, the dipole moment in water is lower by 2.8D, and the molecular volume is smaller by 3% in n-octanol than the D-lactic acid anion. These data are consistent with the model, and the smaller molecular size is consistent with facilitated diffusion transport. It has been previously

shown that the neutral species that penetrates the BBB fastest [START_REF] Oldendorf | pH dependence of blood-brain permeability to lactate and nicotine[END_REF] which is consistent with desolvation being an important "preorganization" factor for facilitated transport, before permeation initiates.

IV. Glucose: D-glucose is transported across the BBB by the GLUT-1 transporter about 100 times faster than its stereoisomer L-glucose (logPS 2.5 versus 5.0 respectively). The only differences between the stereoisomers are a larger molecular volume in water and octanol by 34.4 and 8.6% for the open chain forms of D-and L-glucose. This situation is reversed for the Dand L-pyranose form. This apparent anomaly must be due to a stereospecific glucose-GLUT-1 interaction as it is not explainable on the basis of any "preorganization" processes related to changes from the bulk solvent (blood serum) or prior interaction between glucose-BBB before permeation of the BBB start to occur (eg desolvation, lipophilic solubility, or dipole). Glucose transport at the BBB appears to be dependent on and regulated by a serial chain of membrane-bound and intracellular transporters and enzymes (permeases that change their conformations during the transport processes). [START_REF] Mcallister | Mechanisms of glucose transport at the blood-brain barrier[END_REF] V. Baclofen: R-Baclofen (a CNS muscle relaxant) was shown to have a rat BBB transport rate 4.3 times as fast as the S-isomer, probably by using the large neutral amino acid carrier, since it is a zwitterion at pH 7.4 and has low lipophilicity. [START_REF] Van Bree | Stereoselective transport of Baclofen across the blood-brain barrier in rats as determined by the unit impulse response method[END_REF] The ΔG water for R-baclofen (zwitterion) is higher by 0.7 kcal/mol, ΔG octanol neutral is higher by 2.5 kcal/mol, the dipole moment in water is higher by 3.5D, and the molecular volume is higher by 4% in noctanol than the S-baclofen zwitterion. As active transport (large neutral amino acid transporter) is involved, these data are difficult to interpret unambiguously using the passive transport model. VI. Mefloquine: Mefloquine is a chiral neurotoxic antimalarial agent showing stereoselective brain uptake in humans and rats. It is a substrate and an inhibitor of the efflux protein P-glycoprotein. (-)Mefloquine had a lower blood and brain apparent volume of distribution and a lower efflux clearance from the brain, resulting in a larger brain-blood ratio compared to (+)mefloquine. [START_REF] Barraud De | Cerebral uptake of mefloquine enantiomers with and without the P-gp inhibitor elacridar (GF1210918) in mice[END_REF] The ΔG water for (-) mefloquine (cation at pH 7.4) is higher by 2.2 kcal/mol, ΔG octanol neutral is higher by 1.0 kcal/mol, the dipole moment in water is lower by 0.5D, and the molecular volume is smaller by 13% in n-octanol than the (+)-mefloquine cation. The data is ambiguous as the higher desolvation energy and smaller volume favour the faster permeation by the (+) stereoisomer, but the ΔG octanol and D support a faster rate for the (-) stereoisomer. VII. Ritalin: Ritalin is widely prescribed for attention deficit hyperactivity disorder (ADHD). The d-threo isomer is the pharmacologically active species, but the l-threo isomer crosses the BBB faster by a factor of about 2:1 to 5:1. [START_REF] Ding | Brain kinetics of Methylphenidate (Ritalin) enantiomers after oral administration[END_REF] The ΔG water for l-threo isomer (cation at pH 7.4) is lower by 1.0 kcal/mol, ΔG octanol (neutral) is lower by 1.0 kcal/mol, the dipole moment in water is higher by 0.4D, and the molecular volume is larger by 15% in noctanol than the d-threo isomer cation. As the active monoamine transporter is involved [32] the data is ambiguous in terms of which stereoisomer should permeate faster. VIII. Quinine (an antipyretic, anti-inflammatory, antimalarial) logPS -2.6 and quinidine (an antirrhythmic) logPS -3 are stereoisomers. The ΔG water for quinine (cation at pH 7.4) is lower by 0.2 kcal/mol, ΔG octanol neutral is almost identical, the dipole moment in water is lower by 0.3D, and the molecular volume is larger by 3% in n-octanol than the quinidine cation.

Active transport is probably involved in view of the large molecular sizes. IX. It is apparent from the experimental data above (6.I to 6.VIII) that there are relatively small differences in the desolvation ΔG water , lipophilicity ΔG octanol , dipole moment in water, and molecular volume in n-octanol for all the stereoisomers examined. There is no consistent pattern since all these examples involve active transport processes, where there are differences in the trans-membrane proteins and their interactions with different drugs. In the case of glucose, it is clear that a specific stereoselective active transport process prevails, which is not dependent at all on any "pre-organisation" of the permeant prior to commencement of permeation into the BBB. The test data where facilitated diffusion is involved for smaller sized permeants (xanthines, ureas, antipyrines, lactic acids) appear to support the four factor passive diffusion model (equations 1-5).

7) The permeability model developed above assumes that n-octanol is a proxy for the lipid bilayer membrane of the BBB, and is a measure of lipophilicity of the BBB.

For charged or zwitterionic species, it has been assumed that it is the neutral species that passively permeates or is actively transported by a carrier protein, and so ΔG octanol is calculated for the neutral species. Also it has been shown above that the dipole moment of the drug is one of the four critical factors in the permeation model for the BBB. To test the assumption about n-octanol as a valid proxy for lipophilicity in the BBB, and to explore physical mechanisms that explain why the dipole moment should be a critical factor in permeation, the model lipid bilayer membranes, DPPC (dipalmitotylphosphatidylcholine), DPHYPC (diphytanoylPC), dilauroyl-sn-glycero-3-phosphocholine (DLPC), 1,2-dioleoyl phosphatidylserine (DOPSE) and 1-palmitoyl-2-oleoyl-sn-glycero-3phosphatidylcholine, (POPC) and their interactions with various compounds have been examined. III. It has been shown that organic ions such as tetraphenylborate TPB -and tetraphenylphosphonium TPP + can adsorb to and permeate lipid membranes owing to their hydrophobic nature and the strong delocalisation of charge to the phenyl rings. [START_REF] Cafiso | Influence of Charges and Dipoles on Macromolecular Adsorption and Permeability[END_REF] TPB -permeates about 10 6 times faster than TPP + and the difference in the free energy of binding to the membrane is about -5 kcal/mol. Hydrophobic anions generally bind more strongly than hydrophobic cations, and permeate through membranes faster. [START_REF] Stowasser | The dipole potential of lipid membranesan overview[END_REF][START_REF] Cattelotte | Changes in dipole potential at the mouse bloodbrain barrier enhance the transport of technetium sestamibi[END_REF][START_REF] Cafiso | Influence of Charges and Dipoles on Macromolecular Adsorption and Permeability[END_REF] The positive dipole potential of the mouse BBB restrict the permeability of cationic compounds by active transport mechanisms, but neutral compounds like phloretin or anionic species like TPB -can enhance the permeability of cationic species. [START_REF] Cattelotte | Changes in dipole potential at the mouse bloodbrain barrier enhance the transport of technetium sestamibi[END_REF] DPHYPC-TPB -(where the B weakly interacts with the P at 4.3Å) has been compared to DPHYPC-TPP + (where the P of TPP + weakly interacts with the O of the PO 4 group at 2.6Å): the dipole moments are 39.9D and 50.4D respectively, and ΔG water are -117.4 and -115.9 kcal/mol respectively. The DPHYPC-TPB -PC group has a charge of -4.67V on the zwitterionic N and PO 4 atoms, compared to DPHYPC-TPP + -5.1V. These data support the experimental greater permeation rate and of TPB - since the charge separation (as per the D values) is far greater for the DPHYPC-TPP + , as well as the lower salvation energy and N and PO 4 charges of the DPHYPC-TPB -. These data support the importance of a electrostatic relationship affecting the interaction between drugs and other permeants with the membrane potential, and consequently BBB permeability. Certain polar neutral compounds, like phloretin, and ionic compounds that have a high lipophilicity (and highly dispersed formal charge) can facilitate and permeate PC lipid membranes, so it is likely that similar relationships may apply to the BBB.

An important aspect of how the membrane potential can interact with permeants is the concept of voltage sensing. [START_REF] Bezanilla | How membrane proteins sense voltage, Nature Revs[END_REF] Charged molecules can reorient in the electric field of the membrane (particularly at the negatively charged PC head of the membrane), as has been found in voltage gated ion channels. Sch a process might facilitate desolvation of permeants, as previously discussed above. [START_REF] Fong | Statins in therapy: Understanding their hydrophilicity, lipophilicity, binding to 3-hydroxy-3-methylglutaryl-CoA reductase, ability to cross the blood brain barrier and metabolic stability based on electrostatic molecular orbital studies[END_REF][START_REF] Chew | Biggin PC Distribution and Dynamics of Adamantanes in a Lipid Bilayer[END_REF] This process is particularly important for active transport processes.

Carpenter et al [START_REF] Carpenter | A Method to Predict Blood-Brain Barrier Permeability of Drug-Like Compounds Using Molecular Dynamics Simulations[END_REF] have shown from molecular dynamics studies that an initial stabilizing interaction of up to 3 kcal/mol occurs as the drug moves from the bulk solvent to the PC headgroup in the model membrane DOPC (about 2.5-3.0 nm from the bilayer centre). The most stabilising region is about 1.0-1.5 nm from the bilayer centre, which reflects the hydrophobic stabilization as the drug penetrates the bilayer. Solvation energies were not explicitly investigated.

IV. Adamantidine has been used in the treatment of Parkinson's disease and influenza. It crosses the BBB logPS -3.1. Molecular dynamics modelling of the interaction of adamantidine with the model membrane POPC, 1palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine, has shown that adamanatidine firstly interacts with the negatively charged PC head group via the charged ammonium group, and is deprotonated as it penetrates the centre of the lipid bilayer. [START_REF] Chew | Biggin PC Distribution and Dynamics of Adamantanes in a Lipid Bilayer[END_REF] This work has focussed on the desolvation of charged or zwitterionic species in blood serum at pH 7.4 before permeants enter the cell membrane, so the neutral species is the permeating species. This is based on energy considerations since the desolvation of charged or zwitterionic species is highly energy intensive. The interaction of the dipole moment of permeants is also critical to the process of desolvation and hence permeation. A QM study of the adamantidine-POPC interaction shows: respectively in octanol, (or 20.7D and 24.9D no solvent). This large difference of 6.5D clearly illustrates the energy gradient as adamantidine permeates the lipid bilayer of POPC and then flips orientation and starts to interact with the charged head group of the other lipid leaflet that comprises the cell membrane. This finding is an accord with the previous MD study of POPC-adamantidine permeation. [START_REF] Chew | Biggin PC Distribution and Dynamics of Adamantanes in a Lipid Bilayer[END_REF] The magnitude of the change in dipole is large illustrating substantial energy changes to the lipid bilayer structure during permeation, presumably by electromechanical forces. [START_REF] Stowasser | The dipole potential of lipid membranesan overview[END_REF][START_REF] Peterson | Origin of membrane dipole potential: Contribution of the phospholipid fatty acid chains[END_REF][START_REF] Cattelotte | Changes in dipole potential at the mouse bloodbrain barrier enhance the transport of technetium sestamibi[END_REF][START_REF] Walter | Permeability of small nonelectrolytes through lipid bilayer membranes[END_REF][START_REF] Bezanilla | How membrane proteins sense voltage, Nature Revs[END_REF][START_REF] Heimburg | The capacitance and electromechanical coupling of lipid membranes close to transitions: the effect of electrostriction[END_REF][START_REF] Koerner | Electrodynamics of lipid membrane interactions in the presence of zwitterionic buffers[END_REF][START_REF] Cafiso | Influence of Charges and Dipoles on Macromolecular Adsorption and Permeability[END_REF][START_REF] Trauble | The movement of free molecules across lipid membranes: a molecular theory[END_REF][START_REF] Chew | Biggin PC Distribution and Dynamics of Adamantanes in a Lipid Bilayer[END_REF] (d) Adamantidine and other positively charged species can interact with either or both of the phosphate group of the PC head group or the carbonyl group of the lipid ester chain. It is also possible that interaction with the carbonyl could be via the neutral species, as the carbonyl group is directly attached to the lipid chains, so close to being in a hydrophobic environment.

The dipole moments for the neutral species are 27.2 and 27.3D in water and octanol, and for the charged species are 28.9 and 28.3D respectively. These small differences suggests that the neutral species preferentially with the carbonyl group. However the change in membrane potential by interaction with the carbonyl is clearly far smaller than the interaction with the phosphate group, so significant electromechanical changes in the membrane are probably not induced by permeant-carbonyl group interaction.

V. The diffusion of small nonelectrolytes through planar lipid bilayer membranes (egg phosphatidylcholine-decane) has been measured. [START_REF] Walter | Permeability of small nonelectrolytes through lipid bilayer membranes[END_REF] It was found by using an electrical (membrane voltage) technique to measure permeability that very small molecules (MW<50) diffused much faster than those with higher molecular weights, and the overall data was consistent with a solubility-diffusion model in the lipid bilayer (acting as a soft polymer), as the permeabilities were inversely related to molecular weight and strongly related to hydrophobicity of the solvent. In so far as the egg phosphatidylcholine-decane is a reasonable proxy for the BBB cell membranes, this experimental technique provide support to the notion that membrane potentials are important for BBB permeation.

VI. Given the importance of zwitterions (amino acids, some drugs etc) at physiological pH 7.4 levels, and the known effect of the membrane dipole potentials on permeation of cell membranes, it can be concluded that the dipole moment is an important characteristic of the ability of drugs (or other physiologically important molecules) to cross the BBB. To date there appears to be no consideration in the literature of this factor.

8) The passive diffusion model (equations 1-5) developed above was derived for alcohols and glycols from methanol to 1-butanol, as the logPS data for higher alcohols is not available. However there have been extensive studies of the effects of alcohols on cell membranes and the CNS of rats. [START_REF] Ingo´lfsson | Alcohol's Effects on Lipid Bilayer Properties[END_REF][START_REF] Lyon | A relationship between alcohol intoxication and the disordering of brain membranes by a series of short-chain alcohols[END_REF][START_REF] Mckarns | Correlation between hydrophobicity of short-chain aliphatic alcohols and their ability to alter plasma membrane integrity[END_REF][START_REF] Mccreery | Physico-chemical correlates of alcohol intoxication[END_REF][START_REF] Chiou | Interfacial dehydration by alcohols: hydrogen bonding of alcohols to phospholipids[END_REF][START_REF] Ho | Chronic ethanol intoxication induces adaptive changes at the membrane protein/lipid interface[END_REF] There is no dispute that alcohols modulate lipid bilayer properties. There is a chain length effect on alcohol-induced modulation of lipid bilayers, often referred to as a "cut-off" effect (where the increasing potency with increasing chain length effects eventually levels off, or decreases). These modulations can be temporary or permanent.

These "cut-off" effects are observed for many systems, from the formation of the photoactivated form of rhodopsin in 1-palmitoyl-2-oleyl-phosphatidylcholine lipid vesicles which occurs at chain length <6, the anaesthetic effects on tadpoles which reaches a maximum at C10, to the ataxia (intoxication) effects on rats which maximizes at C6-C7. [START_REF] Ingo´lfsson | Alcohol's Effects on Lipid Bilayer Properties[END_REF] To understand the molecular mechanisms which underpin equation 5, several studies using the PC lipid bilayer model membrane DPHYPC have been undertaken: I. A physiochemical modification of the membrane protein-lipid interface is known to occur which is based on a hydrogen bonding interaction between the alcohols and the phosphate moiety of the PC head of the membrane bilayer. [START_REF] Lyon | A relationship between alcohol intoxication and the disordering of brain membranes by a series of short-chain alcohols[END_REF][START_REF] Mckarns | Correlation between hydrophobicity of short-chain aliphatic alcohols and their ability to alter plasma membrane integrity[END_REF][START_REF] Chiou | Interfacial dehydration by alcohols: hydrogen bonding of alcohols to phospholipids[END_REF][START_REF] Ho | Chronic ethanol intoxication induces adaptive changes at the membrane protein/lipid interface[END_REF] A strong inverse relationship was found between the effective dose that produced ataxia and the membrane buffer partition coefficient (or logP) up to the cut-off point. [52.52] The "cut-off" effect could also be due to steric effects between the alcohol and membrane bilayer. It was found that short-chain alcohols (1-hexanol and shorter) cause volume increases when partitioning into dimyristoylphosphatidylcholine bilayers, whereas longer alcohols cause volume decreases. These cut-off effects tend to appear when the alcohol chain length is approximately equal to half the acyl chain length of the bilayer-forming lipids. [START_REF] Aagaard | Packing properties of 1-alkanols and alkanes in a phospholipid membrane[END_REF] II. The vesicle-forming lipid 1,2-dierucoyl-sn-glycero-3-phosphocholine (DCPC) was used with a fluorophore to determine the bilayer-modifying potency (D*) of a series of alcohols (where D* is the concentration at which the alcohol doubled the quenching rate). [START_REF] Mckarns | Correlation between hydrophobicity of short-chain aliphatic alcohols and their ability to alter plasma membrane integrity[END_REF] The data are shown in Table 2 along with the calculated ΔG octanol values for DPHYPC (DPHYPC is similar to DCPC but has a saturated C20 acyl chain instead of the C22 acyl chain with a double bond at C13-C14). A linear relationship exists between -ΔG octanol and D* up to the "cut-off" at 1-heptanol. Conversely a linear relationship exists between -ΔG octanol and logD* :

-ΔG octanol = -0. It is also clear that steric effects are active for the 3 alcohols which are not straight chain alcohols with a terminal hydroxyl group (2-propanol, 2butanol, and t-butyl alcohol), which are outliers to the 1-alcohol series.

These data are consistent with a steric effect being responsible for the "cutoff" behaviour.

III. In a study of the ability of alcohols to toxically break down or create reversible graded increases in cell membrane permeability in rat liver epithelial cells, it was found that a linear QSAR relationship existed between log(1/LDH50) and logP as the measure of hydrophobicity. LDH or lactate dehydrogenase release is correlated with the breakdown or change in membrane permeability. The LDH50 values are defined as the concentrations which elicited a 50% increase of LDH50 release relative to the untreated control. [START_REF] Lyon | A relationship between alcohol intoxication and the disordering of brain membranes by a series of short-chain alcohols[END_REF] These data are shown in Table 2 along with ΔG octanol values for DPHYPC. A linear relationship exists between -ΔG octanol and LDH50 up to the "cut-off" at 1-pentanol. Conversely, a linear relationship exists between -ΔG octanol and logLDH50:

-ΔG octanol = -0.32 logLDH50 -6.99 (R 2 0.938, SEE 0.09) (equation 12)

IV. The passive diffusion model (equations 1-5) was developed for alcohols up to C5 in length. It is clear from 8.II and 8.III above that the "cut-off" point for alcohols permeating, or structurally modulating, the model PC lipid bilayer membranes is about C5 -C6 in chain length. The logPS data are usually derived from rats, and the data discussed above, are all consistent with the experimental ataxia finding in rats which showed a "cut-off" at C6 -C7 in alcoholic chain length. These data are also consistent with the logPS model for the xanthines (see 1. above) which permeate the BBB by both passive and facilitated diffusion, and which showed an inverse relationship with molecular volume, (see equation 9) compared to that found for the smaller alcohols (see equation 4). This is consistent with the "cut-off" point seen for alcohols which is controlled by steric forces.

V. These "cut-off" change of mechanism effects for the alcohols-DPHYPC series could possibly be due to a dipole effect, rather than a steric effect which increases as the aliphatic chain length increases. To eliminate this possibility, the relationship between the atomic charges on the phosphate group of the PC moiety and the -C(O)O-of the acyl chain have been examined. The following linear relationships were observed:

The straight chain aliphatic alcohols varied from methanol by one carbon The sum (+/-) charges is proportional to the dipole moment of DPHYPC in the various alcohols.

Equation 13:

Length carbon chain = -203. These relationships clearly demonstrate that no dipolar interaction nor hydrogen bonding, nor cavity effects between the model membrane and the alcohols is responsible for the "cut-off" effect. The outliers (which all involve alcohols with known steric hindrance solvent effects) to equations 13-15 also reinforce the conclusion that steric effects from incrementally increasing the aliphatic chain length (by one carbon at a time) are clearly responsible for the "cut-off "effect in these alcohols.

Analysis of literature logPS QSAR relationships

Recent studies of BBB permeability relationships [START_REF] Liu | Use of a Physiologically Based Pharmacokinetic Model to Study the Time to Reach Brain Equilibrium: An Experimental Analysis of the Role of Blood-Brain Barrier Permeability, Plasma Protein Binding, and Brain Tissue Binding[END_REF]63] It should be noted that some of the literature logPS values are for compounds which exist as charged ions or zwitterions at physiological pH 7.4: these compounds were treated as the solvated neutral species in deriving the regression equations. This implies that it is the desolvation of the neutral species that controls the kinetics, since the very large ΔG water values for these charged species would effectively preclude any initiation of the permeation process. This situation is possible since there would be both solvated charged and neutral species at pH 7.4.

Conclusions

The permeability of the BBB is dependant on desolvation, lipophilicity, molecular volume and dipole moment. Previous models for BBB permeability have not considered desolvation and dipole moment as critical factors. The model applies to passive diffusion processes, and some facilitated diffusion processes. Passive diffusion transport processes for many common drugs appear to be less common than active transport processes, so BBB permeability models for passive transport may not apply to active transport processes, particularly where complex membrane protein binding processes (eg stereoselectivity) are involved. Model phosphatidylcholine (PC) lipid bilayer membranes have been used to evaluate how charged or polar neutral compounds can interact through their molecular dipoles with the cell membrane to induce electromechanical changes in the cell membrane which facilitate permeation.

The free energy of solvation in n-octanol has been shown to be a good measure of membrane lipophilicity by calculating the solvation free energy of a model PC lipid membrane in a series of closely related alcohols. The passive diffusion model for alcohols has been shown to correlate with previous studies of the modulation of membrane bilayers by alcohols which showed a "cut-off" point in potency, which is related to molecular size. The dominant species at physiological pH levels in blood serum is integrated into the model, and particularly affects desolvation energies for charged and zwitterionic species. For most drugs and related molecules, the neutral species are the permeating species.

Footnotes:

LogPS values from [START_REF] Goodwin | In silico prediction of blood-brain barrier penetration[END_REF][START_REF] Abraham | On the partition of ampholytes: Application to blood-brain distribution[END_REF][START_REF] Liu | Development of a computational approach to predict blood-brain barrier permeability[END_REF][START_REF] Suenderhauf | Computational Prediction of Blood-Brain Barrier Permeability Using Decision Tree Induction[END_REF] and reference therein. ΔG values in kcal/mol, molecular volume V values in cm 3 /mol, dipole moment values in D ΔG CDS in kcal/mol, include hydrogen bonding interactions, solute-solvent cavity interactions, and other non-electrostatic solute-solvent effects Log PS data for nicotine and derivatives from [START_REF] Oldendorf | pH dependence of blood-brain permeability to lactate and nicotine[END_REF][START_REF] Oldendorf | Blood-brain penetration abolished by Nmethyl quaternization of nicotine[END_REF].

Footnotes:

ΔG values in kcal/mol See equation 10 which defines the relationship between ΔG DPHYPC and ΔG Octanol Sum(+) Charges DPHYPC are the atomic charges in volts on the N and P atom of the PC head group. Sum(+/-) charges are the sum of atomic charges on the N, PO 4 , and the two oxygens of the two -OC(O)-groups of the acyl chains of DPHYPC. See equations 13 and 14. D* is the concentration at which the alcohol doubled the quenching rate of the fluorophore /1,2dierucoyl-sn-glycero-3-phosphocholine (DCPC) system used to determine the bilayer-modifying potency (D*) of a series of alcohols [START_REF] Mckarns | Correlation between hydrophobicity of short-chain aliphatic alcohols and their ability to alter plasma membrane integrity[END_REF]: see equation 11 LDH or lactate dehydrogenase release is correlated with the breakdown or change in membrane permeability. The LDH50 values are defined as the concentrations which elicited a 50% increase of LDH50 release relative to the untreated control [START_REF] Lyon | A relationship between alcohol intoxication and the disordering of brain membranes by a series of short-chain alcohols[END_REF]; see equation 12 ΔG CDS in kcal/mol, include hydrogen bonding interactions, solute-solvent cavity interactions, and other non-electrostatic solute-solvent effects: see equation 15.

  Liu 2004 [15]: Antipyrine, Caffeine, Threophylline, Threobromine, CP-141938*, Fluoxetine, Chloroambucil, Colchicine*, DPDPE*, Daunomycin*, Digoxin*, Dopamine, Glycine, Hypoxanthine, Xanthine, Levodopa*, Methotrexate, Morphine*, NFPS, Phenylalanine*, Phenytoin, SR141716, Quinidine*, Salicyclic Acid, Taurocholic Acid, Valproic Acid, Testosterone. Nine compounds* were identified as being actively transported across the BBB.

4 .

 4 , SEE 1.04 where D is the dipole moment in water (equation 3) Molecular volume V is negative correlated to permeability, probably due to how well the drug can physically (sterically) enter the lipid bilayer and diffuse through. LogPS = -0.82 V -41.96 with R 2 0.41, SEE 1.62 where V is the molecular volume in n-octanol (water excluded as an outlier) (equation 4). It should be noted that the alcohols studied here are relatively small in size, which might be expected to favour passive diffusion. The multiple regression model (equation 5) for BBB permeation is: LogPS ~ 0.20 ΔG water + 1.03 ΔG octanol -0.044 D + 0.006 V where R 2 0.855, SEE 1.04, F 4.43 This relationship is only indicative, since there are insufficient data points to be statistically robust. The linear relationships in 1. to 4. above are more statistically meaningful.

1 )

 1 Xanthines: Xanthine, caffeine, theophylline, theobromine are very closely related drugs structurally differing only in methyl groups substituting hot H atoms, and hypoxanthine is also closely related, but having only one carbonyl group instead of two as in the other xanthines. An analysis shows the flowing linear relationships exist: LogPS = -0.30 ΔG water + 2.08 with R 2 0.64, SEE 0.48 where ΔG water is the water desolvation free energy (equation 6) LogPS = 0.24 ΔG octanol + 0.24 with R 2 0.64, SEE 0.55 where ΔG octanol is the solvation free energy in n-octanol, or lipophilicity (equation 7) LogPS = -0.41 D -1.11 with R 2 0.30, SEE 0.67 where D is the dipole moment in water (the D of hypoxanthine is an outlier, but is still included in the analysis) (equation 8) LogPS = 0.03 V -6.22 with R 2 = 0.915, SEE 0.23 where V is the molecular volume in n-octanol (equation 9)

  I. DPHYPC was constrained to a structure whereby the two parmitoyl ester chains are as close to parallel as possible (after molecular mechanics optimisation) and pointing away from the phosphatidylcholine (PC) head group, to resemble as closely as possible the PC lipid bilayer of a cell membrane. The relationship between ΔG for the DPHYPC and n-octanol for the range of alcohol solvents used to develop the permeation model, plus other alcohols up to 1-decanol, shows that a strong linear relationship is observed: ΔG DPHYPC = 9.34 ΔG Octanol -6.98 (R 2 0.976, SEE 0.75) (equation 10)Water, and to a lesser extent, ethylene glycol, are outliers, presumably related to multiple hydrogen bonding effects. This result indicates that noctanol is a good proxy for a cell membrane bilayer.II. The electrical properties of the BBB have been examined by calculating the charge distribution of the model membranes in water, all of which have a zwitterionic phosphatidylcholine PC head group and long bilayer fatty acid tails. It is known that DOPSE which has a negatively charged PC head group moves against the direction of an applied electric field.[START_REF] Koerner | Electrodynamics of lipid membrane interactions in the presence of zwitterionic buffers[END_REF] Conversely, the model zwitterionic lipid bilayer membrane DLPC in the zwitterionic buffer 3-(N-morpholino)propanesulfonic acid (MOPS) at pH 7, becomes positively charged in an applied electric field. The addition of the ion Br -to DLPC vesicles resulted in a negative charge on DLPC. Using X-ray techniques, it was found that the buffer solutes (MOPS etc) enter the inter-lamellar space and modify (possibly by binding) interlayer interactions.[START_REF] Koerner | Electrodynamics of lipid membrane interactions in the presence of zwitterionic buffers[END_REF] These experimental results are consistent with electrostatic atomic charge models of DOPSE, DLPC and DLPC-MOPS which clearly show that the PC head group is overall negatively charged, and the bilayer lipid tail is slightly positively charged. For example, DOPSE has a overall -1.115V PC charge, and a 0.04V lipid tail. The DOPSE PC group has a charge of -3.69V on the zwitterionic N and PO 4 atoms. (Similarly, the negatively charged DLPC-Br where the Br -ion interacts with the P atom at a distance of 3.3Å has charge of -3.60V on the zwitterionic N and PO 4 atoms.) It is likely that charge in an electric field would be determined by this charge. DLPC-MOPS interacting through a weak electrostatic bond 2.6Å between the P atom of DLPC and the O of the SO 3 group of MOPS has an overall -1.38V PC charge, and 0.35V lipid tail. The DLPC-MOPS PC group has a charge of -1.89V on the zwitterionic N and PO 4 atoms. It appears that the difference of -1.8V between the DOPSE and DLPC-MOPS zwitterion charges is instrumental in deciding the overall movement in an electrical field. Movement in an electrical field would be a complex interplay of charge driven ionic factors for all cationic and anionic species present in solution. The dipole moments in water of DOPSE, DLPC-Br and DLPC-MOPS are 38.5D, 31.0D and 37.1D respectively. These data all suggest that the BBB membrane would electrically interact with charged species and hence affect trans-membrane transport.

  (a) the difference in desolvation energy between the protonated species and neutral species is 60 kcal/mol in water. (b) the zwitterionic species in water can interact with the phosphate group via an ionic NH 3 + ---OP(O)-interaction or with the carbonyl group of the ester fatty acid via a NH 3 + --O=C-hydrogen bonding interaction. The dipole moment of the {adamantidine-POPC} complex in water increases from 28.9D to 49.4D, (compared to the value for uncomplexed POPC of 27.3D). This dramatic change indicates that the dipole of POPC (and all the PC membrane models studied here) which is oriented from the negatively charged PC head group towards the positively charged lipid bilayer, has a greatly increased negatively charged PC head group upon complexing with the adamantidine zwitterion. Such a large energy change could electromechanically distort the membrane to facilitate permeation [21-25,43], as well as facilitate desolvation, since shedding hydrogen bonded water molecules is a means of lowering the energy and stabilising the complex. (c) permeation of the neutral adamantidine species into the lipid bilayer was studied by inserting the adamantidine molecule between the two lipid chains in two orientations, one with the -NH 2 moiety facing towards the PC head group, and the other with the -NH 2 group facing away from the PC head group. The dipole moments for the orientations are 23.9D and 30.4D

at a time up to 1 -

 1 decanol. Non-linear aliphatic chains such as 2-propanol, 2-butanol, tertiary-butanol and 2-methyl-1-propanol were clear outliers from the straight chain alcohol relationship. Sum(+) PC charges are the atomic charges in volts on the N and P atom of the PC head group. Sum(+/-) charges are the sum of atomic charges on the N, PO 4 , and the two oxygens of the two -OC(O)-groups of the acyl chains of DPHYPC.

40 logD* -8.16 (R 2 0.881, SEE 0.18) (equation 11)

29 Sum(+) PC charges + 267.69 (R 2 0.864, SEE 1.19) or

	Length carbon chain = 270.27 Sum(+/-) charges + 921.77 (R 2 0.991,
	SEE 0.28) if methanol is omitted as an outlier
	Equation 15:
	A linear relationship exists between the straight chain alcohols from
	ethanol to 1-decanol and the ΔG CDS value in kcal/mol (which include
	hydrogen bonding interactions between the alcohols and DPHYPC, the
	energy to create a cavity for the solute DPHYPC and other non-
	electrostatic solute-solvent interactions [59]).
	by omitting methanol as an outlier (probably due to
	its capacity to form multiple solvation interactions because of its smaller
	size)
	Length carbon chain = -305.25 Sum(+) PC charges + 398.61 (R 2
	0.942, SEE 0.33)
	Equation 14:
	Length carbon chain = -0.0015 Sum(+/-) charges + 6.0 (R 2 0.273, SEE
	2.74) or

Length carbon chain = 0.73 ΔG CDS + 7.59 (R 2 0.050, SEE 3.13)

	Or by omitting methanol as an outlier

Length carbon chain = 3.90 ΔG CDS + 17.96 (R 2 0.969, SEE 0.52)

  have been reanalysed using the four factors identified in this study, ΔG water , ΔG octanol , D, and V.[START_REF] Liu | Use of a Physiologically Based Pharmacokinetic Model to Study the Time to Reach Brain Equilibrium: An Experimental Analysis of the Role of Blood-Brain Barrier Permeability, Plasma Protein Binding, and Brain Tissue Binding[END_REF] Murikama, and Liu 2004 data, it can be seen that a similar pattern emerges. The linear regression equations are more accurate than the multiple regression equations which have less than the optimal data points to be robust. There is a negative dependence on the desolvation from water, a positive dependence on the lipophilicity, and smaller dependencies on molecular volume and dipole moment. The literature relationships are for a diverse and wide range of permeants that tend to cloud systematic structural

	logPS = -0.11 ΔG water -0.89 (R 2 0.683, SEE 0.52) logPS ~ 0.13 ΔG water + 0.27 ΔG octanol -0.11 D + 0.01 V -2.67 (R 2 0.564, SEE 0.63) logPS ~ -0.055 ΔG water + 0.038 ΔG octanol + 0.101 D + 0.006 V -2.77 (R 2 0.678, SEE
	logPS = 0.11 ΔG octanol -0.836 (R 2 0.434, SEE 0.69) By excluding all the 8 compounds that were identified as being actively transported, 0.58)
	logPS = -0.002 V -2.09 (R 2 0.032, SEE 0.91) the following relationships were found: The multiple regression relationship has improved significantly with the exclusion of
	logPS = -0.28 D -1.15 (R 2 0.588, SEE 0.59) logPS = -0.051 ΔG water -2.14 (R 2 0.518, SEE 0.67) the 9 actively transported compounds. (See experimental section regarding the low R 2
	logPS ~ 0.18 ΔG water + 0.36 ΔG octanol -0.37 D + 0.02 V -1.58 (R 2 = 0.754, SEE logPS = 0.060 ΔG octanol -2.03 (R 2 0.410, SEE 0.75) values)
	0.73) logPS = 0.001 V -3.50 (R 2 0.005, SEE 0.97) By comparing the regression equations derived for the alcohols (equations 1-5), the
	Gratton [15] examined the relationship between logPS and antipyrine, 2-propanol, logPS = 0.157 D -4.20 (R 2 0.197, 0.87) xanthines (equations 6-9) with those derived from using the Liu 2005, Gratton,
	95005, erythritol, mannitol, sucrose, thymine, ethanol, estradiol, thiourea, urea, and logPS ~ 0.15 ΔG water + 0.33 ΔG octanol + 0.13 D + 0.01 V -2.84 (R 2 0.895, SEE 0.37)
	ethylene glycol. The following relationships were found: The multiple regression relationship has improved significantly with the exclusion of
	logPS = -0.085 ΔG water -1.49 (R 2 0.592, SEE 0.94) the 8 actively transported compounds. (See experimental section regarding the low R 2
	logPS = 0.097 ΔG octanol -1.43 (R 2 0.434, SEE 1.10) values)
	logPS = 0.002 V -3.10 (R 2 0.012, SEE 1.47) Liu 2004 [15] examined the relationship between logPS and 28 compounds (see
	logPS = -0.08 D -2.53 (R 2 0.012, SEE 1.47) experimental).
	logPS ~ 0.14 ΔG water + 0.38 ΔG octanol -0.12 D + 0.02 V -1.34 (R 2 0.900, SEE 0.56) logPS = -0.036 ΔG water -2.09 (R 2 0.314, SEE 0.71) change (similar to those in linear free energy relationships) which is seen for the
	Murikami [15] examined the relationship between logPS and digoxin*, hypoxanthine, logPS = 0.037 ΔG octanol -2.10 (R 2 0.237, SEE 0.75) alcohols and xanthines series which are more closely structurally comparable. It is
	methotrexate, phenylalanine**, quinidine*, theophylline, valproic acid, mannitol, logPS = -0.002 V -2.42 (R 2 0.073, SEE 0.82 ) clear that including actively transported species in the regression relationships lowers
	sucrose, alanine**, cyclosporine A*, glibencamide, glucose**, iodoantipyrine, logPS = -0.06 D -2.49 (R 2 = 0.060, SEE 0.83) the correlations, indicating that these species are really outliers, or at best increase
	quinine, tolbutamide, vinblastine*, warfarin, cimetidine, vincristine*, and thiourea. logPS ~ -0.128 ΔG water -0.124 ΔG octanol -0.001 D -0.002 V -2.26 (R 2 0.374, SEE scatter. Based on the observations above for the actively transported stereoisomers
	The compounds* were considered to be transported by P-gp, and the compounds ** 0.72) (section 6 above) it is clear that where any significant binding interaction with
	were transported by uptake mechanisms. The following relationships were found, but By excluding the 9 compounds that were identified as being actively transported, the transport proteins in the BBB occurs, then no simple relationship with logPS is easily
	actively transported cyclosporine, digoxin, vinblastin were excluded as clear outliers. following relationships were found: distinguishable.
	examined how caffeine, theophylline, theobromine, fluoxetine, NFPS and CP-141938 logPS = -0.041 ΔG water -2.29 (R 2 0.346, SEE 0.69) logPS = -0.054 ΔG water -1.82 (R 2 0.399, SEE 0.72)
	permeated (logPS) the rat BBB and equilibrated in plasma. A physiologically based logPS = 0.043 ΔG octanol -2.39 (R 2 0.247, SEE 0.74) logPS = 0.046 ΔG octanol -1.96 (R 2 0.196, SEE 0.83)
	pharmacokinetic model was used to correlate an in vivo logPS with in situ Log PS (R 2 logPS = 0.001 V -3.27 (R 2 0.001, SEE 0.86) logPS = 0.001 V -2.42 (R 2 0.073, SEE 0.82 )
	0.83). The following relationships were found using Liu's 2005 data: logPS = 0.078 D -3.58 (R 2 0.053, 0.84) logPS = 0.075 D -2.94 (R 2 = 0.017, SEE 0.91)

Table 1 Selected logPS and input data used to derive equations 1-9 and to analyse closely related test compounds LogPS ΔG water -ΔG octanol V octanol D water ΔG CDS

 1

	Water	-2.1	8	-6.7	12	2.4	1.5
	Methanol	-1.6	3.6	-4.2	30.5	2.2	2.5
	Ethanol	-1.5	4.8	-5.1	53	2.2	2.4
	2-Propanol	-1.7	4.5	-5.5	56.9	2.2	2.7
	Ethylene Glycol	-3	10.1	-8.8	54.7	3	2.5
	Propylene Glycol	-2.5	8.8	-8.5	63.3	3.1	2.9
	1,4-Butanediol 1-Butanol	-3 -2.4	10.9 4.5	-10.7 -6.2	77.4 71.5	3.1 2.2	3 2.9
	Xanthine	-3.8	18.3	-14.6	85	4.8	2.4
	Caffeine	-2	15.2	-12	147.3	4.3	0
	Threophylline	-2.9	16.5	-13.2	112.7	4	0.8
	Threobromine	-3	15.1	-11.9	126.9	4.4	0.9
	Hypoxanthine	-3.5	19.1	-16.8	102	6.3	1.8
	Morphine	-2.7	17.7	-17.2	4.7	233.4	2.5
	Morphine Ion		70.1	-66.7	17.8	204	6.4
	Heroin	-1.2	8.2	-17.8	4.9	286.6	5.9
	Heroin Ion		69.7	-66.6	20	256.4	9.7
	Codeine	-1.7	16.8	-16.5	4.6	204.2	3.5
	Codeine Ion		68.2	-65	18.9	209.2	7.3
	Morphine-6-Glucuronide	-3	42.2	-35.9	4.8	292.3	5.8
	Morphine-6-Glucur Zwitt		119.1	-103.5	42.1	337.7	9.6
	Methadone	-1.3	6.1	-12.6	6.7	287.8	8.9
	Methadone Ion		52.4	-54.5	12	248.7	8.9
	Antipyrine	-2.0	11.4	-12.6	173.1	8.1	5.0
	Iodoantipyrine	-1.1	12.3	-12,2	140.1	8.4	4.6
	Urea
	Thiourea

Table 2 Comparison of DPHYPC (diphytanolphosphatidylcholine) and n- Octanol properties in a series of alcohols. ΔGs DPHYPC ΔGs

 2

			Octanol	D* Ref 52	LDH50 Ref 51	Sum(+) Charges	Sum(+/-) Charges	ΔG CDS DPHYPC
						DPHYPC	DPHYPC
	Gas	0	0			1.154
	Water	-71	-3.79					24.4
	Methanol	-93.9	-9.21	666	3.4
	Ethanol	-91.7	-9.04	147	1.7	1.298	-3.402	-4.1
	1-Propanol	-90.3	-8.89	35.5	0.47	1.295	-3.399	-3.9
	2-Propanol	-89.9	-8.94	93.3	0.16
	1-Butanol 2-Butanol t-Butanol 1-Pentanol 1-Hexanol 1-Heptanol 1-Octanol 1-Nonanol	-88.5 -88 -86.6 -87.4 -85.5 -84.1 -82.3 -80.1	-8.65 -8.72 -87.2 -85.5 -8.37 -8.20 -8.03 -7.81	11.9 23.7 47.3 4.7 1.2 0.4 0.4 0.2	0.16 0.36 0.19 0.047 0.012 0.0045 0.00089	1.293 1.291 1.287 1.284 1.280	-3.396 -3.392 -3.389 -3.386 -3.382	-3.3 -3.3 -3.2 -2.8 -2.7 -2.3
	1-Decanol	-77.8	-7.61	0.6		1.276	-3.377	-2.0
	1,2-Ethanediol	-76.1	-56.5			1.272	-3.372