Laurent Nottale
email: laurent.nottale@obspm.fr

Theoretical prediction of neutrino and electron masses in the theory of scale relativity

, which writes to lowest order 8 3 α ln(m P /m e ) = 1 (where m P is the Planck mass). This relation is improved by taking account of threshold effects on the running charge and mass at the electron Compton scale, leading to an agreement with the experimental values at the 10 -5 level. Then we suggest the existence of a similar relation valid for neutrinos, which writes 2α ln(m P /m ν ) = 1. From this relation, we theoretically predict a lightest neutrino mass m ν = m P exp(-α -1 /2) = 0.0214 eV. The masses of the two heavier neutrinos, 0.0231 eV and 0.0552 eV, can then be obtained from experimental results of neutrino oscillations.

Introduction

The discovery of neutrino oscillations (see Ref. [START_REF] Zyla | [END_REF] and references therein) proved that neutrinos are not massless. However, only differences of square masses are known through these experiments and observations, not the masses themselves.

The current view about the origin of elementary particle masses attributes them to their Yukawa coupling to the Higgs field. This is based on the fact that the coupling constants and the masses are equal. However, one may remark that the acquisition of masses by the W ± and Z 0 bosons through the spontaneous breaking of the SU (2) × U (1) electroweak (EW) symmetry leads to an explicit prediction of these masses, while this is not the case for the fermion masses. One can therefore interpret these facts in the reverse way, namely, that the fermion masses are determined by a different mechanism and that they subsequently define the Yukawa couplings with the Higgs field.

Among the reasons which lead one to suspect that the HIggs / spontaneous symmetry breaking mechanism is not sufficient to understand fermion masses, there is the huge difference between the e, µ, τ masses and that of their associated neutrinos, despite their similar electroweak quantum numbers. Actually, the neutrinos should be precisely massless with the Standard Model [START_REF] Zyla | [END_REF]. On the other hand, it is remarkable that the masses of ν e , e and u, d quarks are in increasing order while they respectively are subjected to the (weak), (weak and electromagnetic), then (weak, electromagnetic and strong) forces. This suggests that the origin of fermion masses could be linked to the gauge fields themselves.

In the theory of scale relativity [3,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF], a geometric interpretation of the nature of gauge transformations, of gauge fields and their associated charges has been proposed [START_REF] Nottale | Relativity in General[END_REF][START_REF] Nottale | [END_REF], [START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]Chapt. 7]. In this framework, one naturally obtains an universal relation between couplings (i.e. square of charges) and mass scales ratios with the Planck scale (or equivalently Compton-length-scale ratios).

Such a relation between the electron to Planck mass ratio and the fine structure constant has been long ago proposed [START_REF] Nottale | Relativity in General[END_REF]. In the present paper, we suggest the existence of a similar relation for neutrinos, which allows us to obtain a theoretical prediction for the lightest neutrino mass, then to derive the other masses from experimental results of neutrino oscillations.

Mass-charge relation for the electron 2.1 Gauge theories in scale relativity

In the theory of scale-relativity and fractal space-time [3,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF], the geometry of spacetime is generalized to a non-differentiable continuum. This implies its fractality, i.e. its explicit dependance on the resolution scale (going as far as divergence when the scale interval tends to 0). In this framework, we add the space-time resolutions (which may become realized through the measurement resolutions) to the fundamental variables that characterize in a relative way the state of the coordinate system (in addition to position, orientation and motion) [7,3,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF]. Then particles are identified with the geodesics of this fractal space-time: more specifically, the velocity field V of these geodesics provides a geometric interpretation of the nature of the wave function (e.g. mV = -i ∇ ln ψ in the simplest case where ψ is solution of a Schrödinger equation) [START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | [END_REF]8,[START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF].

This has led us to a new interpretation of gauge transformations and gauge fields [START_REF] Nottale | Relativity in General[END_REF][START_REF] Nottale | [END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF] in which the variable conjugate to the charge is just the resolution scale. In other words, charges find their origin, according to Noether's theorem, in the symmetries of the space of resolution-scales (internal to the geodesics which are identified to the particle).

As a consequence the wave function of a free electron can be written as:

ψ = ψ 0 × e i[(px-Et+σφ)/ +2π α(r) ln(λe/r)] , (1)
where α(r) is a running coupling, r is a running resolution length-scale and λ e = /m e c is the Compton scale of the electron. We know that angles φ vary between 0 and 2π, so that spin differences δσ are universally quantized in units of . In the same way, in the theory of special scale-relativity (SSR) where the Planck length-scale λ P is re-interpreted as a minimal, undepassable scale of resolution, invariant under dilations and contractions [3], the scale variable is limited as ln(λ e /r) ≤ ln(λ e /λ P ), so that the charge is quantized according to the relation:

α ln λ e λ P = 1. ( 2
)
We have suggested, since the running occurs up to the Planck scale, that the coupling appearing in this relation should be the effective electromagnetic coupling of the electroweak theory, α = 8α/3, where α is the fine structure constant. This mass-coupling relation can be understood as yielding the mass of the electron from its charge.

Bare charge 1/2π

The electron charge itself can be derived through its running from its infinite energy value (i.e. Planck length-scale value in SSR) to the electron energy 0.511 MeV. In the Standard Model (SM), the value of the charge at infinite energy is divergent. This is no longer the case in SSR, where an infinite energy corresponds to the finite Planck length-scale (in similarity with special motion relativity, where an infinite value of energy corresponds to a finite value v = c of velocity), according to the log-Lorentz relation [3,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF]:

ln m m e = ln(λ e /λ)
1 -ln 2 (λ e /λ)/ ln 2 (λ e /λ P ) ,

where the referernce scale has be taken here to be the electron scale. It is easy to check in this formula that m → ∞ when λ → λ P and that the Planck length λ P = G/c 3 becomes, in the SSR framework, a limit scale, unreachable and invariant under dialtions.

We have argued [9,[START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF] that the infinite energy value (which can be identified with its "bare" value) of the inverse electromagnetic coupling (including the electroweak factor 3/8) is expected to have the value 4π 2 . Let us briefly recall the argument.

The force between two charges can be computed as:

F = δp δt , ( 4
)
where δp is the momentum exchanged by the intermediate boson during the interaction time δt. Since the boson is assumed to be of zero mass and moving at light velocity, the time interval is δt = r/c, where r is the distance between the two charges. This force can be decomposed into a radial and an anguler part, F = F r × F θφ . As concerns the radial part of the force, it can be established by describing the exchanged boson in terms of an harmonic oscillator, according to the second quantization theory. A general method making use of the concept of information entropy has been devised by Finkel [10] for constructing any exact Heisenberg relation between any couple of variables and to determine the kind of wave function which achieves the limit of the inequality.

In the situation considered here, δx = r is an interval of distance. In this case, the Finkel method [10] allows one to show that the Heisenberg relation writes r δp ≥ /π, and that the limit /π is just reached by the harmonic oscillator distribution. Therefore we find that the radial part of the momentum variation in the exchange of null mass bosons over a distance r is given by

δp = πr . (5)
As regards the angular part of the force, it is naturally given by spherical harmonics |Y m l (θ, φ)| 2 , i.e. in the isotropic case considered here, by

|Y 0 0 (θ, φ)| 2 = 1 4π . ( 6
)
The force created by the exchange of null mass bosons writes

F = c πr 2 × 1 4π = c 4π 2 r 2 . (7)
By comparing Eq. ( 7) with the standard expression of the force written in terms of a coupling constant, F = α c/r 2 , one finally obtains a "natural" value for the coupling constant [9]:

α -1 ∞ = 4π 2 = 39.478... (8)
We have shown [START_REF] Nottale | Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity[END_REF]9,[START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF] that the running of the effective electromagnetic coupling in the electroweak theory, α 0 (r) -1 = 8α(r) -1 /3 = 5 3 α 1 (r) -1 + α 2 (r) -1 fairly supports such an expectation: we obtained α -1 0∞ = 39.489 ± 0.013 in [START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]Sec. 11.1.3.4], which differs from 4π 2 by less than 1σ.

The running of this coupling from this value (4π 2 ) at infinite energy scales (i.e. Planck length-scale in SSR) to low energy scales determines the low energy value of the electric charge, i.e. of the fine structure constant (this running depends on the whole content of charged elementary particles). Then its intersection with the running mass fixes the mass of the electron to its low energy value (see Fig. 2).

Note that this universal value 4π 2 of an inverse coupling at infinite energy can also be combined with the mass-coupling relation Eq. ( 2) to yield a fundamental mass-scale [9,[START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF] given by ln

m P m W Z = 4π 2 . ( 9
)
This mass-scale is 1.397 × 10 17 smaller than the Planck mass, i.e. m W Z = 87.4 GeV [9], which is typical of the electroweak scale since it is intermediate between the W (80.379

GeV) and Z boson (91.1876 GeV) masses. Such a relation may therefore help solving the hierarchy problem (i.e. the large ratio between the GUT scale, which is the Planck mass-scale in SSR, and the electroweak scale). Another application of this mass-coupling relation concerns the strong inverse coupling, which can be shown, in the SSR framework, to reach the value 4π 2 just at the scale where it crosses the gravitational inverse coupling [START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF].

Lowest order mass-coupling relation

The ratio of Compton lengths between electron and Planck scales can be replaced by the mass ratio:

ln

λ e λ P = ln m P m e . (10)
Therefore, the mass-charge relation we theoretically obtained for the electron [START_REF] Nottale | Relativity in General[END_REF] reads:

8 3 α ln m P m e = 1. (11)
The present recommended values of the fine structure constant and of the electron mass [11,[START_REF] Zyla | [END_REF] are α -1 = 137.035999084(21), m e = 0.51099895000(15) × 10 -3 GeV, (12) while the Planck mass determination has been now improved to [11,[START_REF] Zyla | [END_REF] m P = 1.220910(29) × 10 19 GeV.

These values yield numerically

8 3 α ln m P m e = 1.0027, (14)
which differs by less than 0.3 % from the predicted value 1.

It should be remarked that a similar relation was proposed long ago by Rosen [12] in the framework of a quantum theory of gravitation, with a slightly different and arbitrary constant 0.362 ('to be computed from the details of the theory'), instead of the universal value 3/8 = 0.375 (justified by the electroweak theory), used here. Moreover, the Planck scale enters in Rosen's formula through a cut-off linked to quantum gravity, while in the SSR framework the Planck length-scale is a limit, impassable scale invariant under dilations which fully justifies the quantization of charge.

Account of threshold effects on the running charge

However, this zeroth order calculation does not account for the threshold effects that occur at the electron scale. The fine structure constant is a large length-scale value measured at the atom scales. Actually, the electron mass and the electron charge both increase below the Compton length of the electron λ e toward small length-scales (i.e., beyond the electron mass m e toward large mass-scales) because of radiative corrections. Therefore, there is a smooth transition around the electron Compton-length between the constant large scale value and the variable small scale running coupling, which is known since Uehling and Serber first calculation in 1935 to diverge asymptotically only logarithmically (see Fig. 1). Consequently, the scale at which the asymptotic running coupling reaches the value of the fine structure constant α differs from λ e .

The mass-charge relation is actually an equation relating the running electron coupling and electron mass in function of the scale r

8 3 α(r) ln m P m(r) = 1, ( 15
)
whose solution is expected to be the electron scale. Therefore, the quantities entering in this relation are the asymptotic functions, which involve offsets, given by numerical constants, with respect to the precise electron Compton scale. The asymptotic running QED coupling is given by [START_REF] Landau | Relativistic quantum theory[END_REF][START_REF] Itzykson | Quantum Field Theory[END_REF] α as (r

) = α 1 + 2α 3π ln λ e r -γ + 5 6 , (16)
where γ = 0.5772156... is Euler's constant. This expression means that the asymptotic running coupling reaches the value of the fine structure constant at a mass scale exp(γ + 5/6) = 4.098 ≈ 4 times the electron mass. This result is quite in agreement with the fact that the scale dependance of the coupling comes from the e + e -pairs of mass 2m e , which therefore define the center of the transition between the two regimes (logarithmically scale-dependant at high energies vs constant at low energies). Therefore

α as (r = λ e ) = α 1 - 2α 3π γ + 5 6 . (17)
Using this value in the mass-charge relation yields:

8 3 α 1 - 2α 3π γ + 5 6 ln m P m e = 1.00052, (18)
an improvement by a factor ≈ 5 from 2.7 × 10 -3 to 5.2 × 10 -4 of the agreement between the theoretical prediction and the experimental value. including the threshold effect (drawn here in terms of length-scale instead of mass-scale). Red curve: running electromagnetic inverse coupling during the electron-scale transition toward the inverse fine structure constant α -1 = 137.036. Green line: running scale factor (8/3) ln(r/λ P , which reaches the value of the inverse electromagnetic coupling at the electron scale. Brown line: asymptotic inverse running coupling α -1 as (r) = α -1 + 2 3π ln(r/λ e ) + γ + 5 6 ]; it crosses 8 3 ln m P me = 137.408 (green horizontal line) at the scale where the running electron mass is assumed to reach its low energy constant value, m(r) = m e (red point, ln(r/λ e ) = -1/3).

Account of threshold effects on the running mass

There is also a numerical constant in the expression for the logarithmic divergence of the running mass [START_REF] Weisskopf | [END_REF][START_REF] Itzykson | Quantum Field Theory[END_REF],

m r (m) = m e 1 + 3α 2π ln m m e + 1 4 . ( 19
)
The value of α as should therefore be taken, no longer at exactly the electron Compton length λ e = /m e c itself, but at the scale where m = m e , i.e. m = m e ×e -1/4 = 0.7788 m e , corresponding to a length-scale ln λ = ln λ e + 1/4. One obtains a corrected relation:

8 3 α 1 - 2α 3π γ + 5 6 + 1 4 ln m P m e = 1.00013, ( 20
)
still improved by a factor ≈ 4 and now reaching the 10 -4 precision. However, with the now improved value of the Planck mass, which is the main uncertainty in this calculation (coming in its turn from the badly known Newtonian constant of gravitation), this result remains far from error bars. One gets δ ln(m P /m e ) = δm P /m P = 2.4 × 10 -5 , so that with 8α/3 ≈ 1/50, the experimental current uncertainty is 4.6 × 10 -7 , a value which can be expected to be still improved in the near future.

Possible improvement of the running mass contribution

An improvement of this relation can be obtained by realizing that the effective running mass intervening in the full Lagrangian has a slightly different constant [START_REF] Itzykson | Quantum Field Theory[END_REF] from the 1/4 present in the mere mass renormalization:

m r (m) = m e 1 + 3α 2π ln m m e + 3 8 . ( 21
)
By taking again the value of α as at the scale where m r (m) = m e , one obtains a still improved relation:

8 3 α 1 - 2α 3π γ + 5 6 + 3 8 ln m P m e = 0.999939, ( 22
)
which differs by only 6×10 -5 from the expected value 1, thus achieving a new improvement by a factor ≈ 2 of the agreement with the theoretical prediction.

From this relation one can derive a theoretical expectation for the mass of the electron from the fine structure constant:

m e (pred) = m P exp - 3 8 α -1 1 + 2α 3π γ + 29 24 = 0.5096 MeV, (23)
which lies within 0.3% of the experimental value m e = 0.5110 MeV.

Next order

A new improvement could be expected by going to the next order in α expansion. By taking the running EM coupling to two loops [16] and assuming that the correct constant determining the scale at which m = m e is 1/3 (intermediate between the two previous possible determinations 1/4 and 3/8), we would obtain (see Fig. 1):

8 3 α 1 - 2α 3π + α 2 2π 2 γ + 7 6 ln m P m e -1 = -7 × 10 -7 , (24)
which is of the order of size of the main uncertainty coming from the Planck mass ( 8 3 α δm P /m P ≈ 5 × 10 -7 ). This opens the hope that we actually deal with an exact relation and that we just need to get a correct and precise description of the electron Compton transition between constant mass and charge at large length-scales to their scale dependance toward small scales.

Generalization to neutrino masses

As recalled hereabove, the mass-coupling relation obtained for the electron reads to lowest order (i.e. disregarding small corrections due to threshold effects):

8 3 α ln m P m e = 1, (25)
where α = α e is the low energy fine structure constant. The factor 8/3 comes from the electroweak theory. It results from the fact that the photon γ, which carries the electromagnetic interaction, is a low energy residual of the four high energy electroweak bosons from the SU (2) L field (W 1 , W 2 , W 3 ) and U (1) Y field (B). At W Z scale (≈ 90 GeV), these bosons are combined through the Higgs mechanism into three massive bosons (W ± and Z 0 ), this spontaneous symmetry breaking leaving only the U (1) EM field, carried by the massless photon γ.

As a consequence the electromagnetic coupling, which can be defined as a linear combination of the U (1) (α 1 ) and SU (2) (α 2 ) couplings,

α -1 0 = 5 8 α -1 1 + 3 8 α -1 2 , (26)
is abruptly decreased by a factor 3/8 at the W Z scale, leaving the running fine structure "constant"

α -1 = 8 3 α -1 0 = 5 3 α -1 1 + α -1 2 . ( 27
)
The running electron mass-charge relation Eq. ( 25) is therefore just expressed in terms of this high energy coupling α 0 (r), i.e. it can be written as

ln m P m(r) = 5 8 α 1 (r) -1 + 3 8 α 2 (r) -1 , (28)
where r is a running length-scale. Extrapolated to low energies, it is solved in terms of the electron mass and the electron charge [START_REF] Nottale | Relativity in General[END_REF][START_REF] Nottale | Scale Relativity and Fractal Space-Time: a New Approach to Unifying Relativity and Quantum Mechanics[END_REF]. Under its form Eq. ( 25), this kind of mass-charge relation seems not to be applicable to the neutrino, which is devoid of electric charge. However, it owns SU (2) and U (1) Y charges similar to that of the electron (and constitutes a doublet with the left-handed electron) so that the EW form Eq. (28) can also be meaningful for the neutrino. Namely, t 3 = -1/2 and y = -1 for the left-handed electron, so that Q e = t 3 + y/2 = -1 while t 3 = 1/2 and y = -1 for the neutrino, so that Q ν = 0.

The electron, having a weak and electric charge, interacts with the 4 bosons. The neutrino, on the contrary, being electrically neutral, interacts with only the 3 weak bosons, W ± and Z 0 , but not the γ. We therefore suggest that a multiplicative factor 3/4 should to the Universe length-scale (defined by the cosmological constant [START_REF] Nottale | Solution to the cosmological constant problem from dark energy of the quark vacuum[END_REF]). In the SSR theory, the Planck length-scale is a minimal scale, invariant under dilatations. The natural value of an inverse coupling at this scale is 4π 2 (see text, horizontal black line). The electromagnetic inverse coupling starts from this value (39.478), runs toward the W Z electroweak scale where it jumps by a factor 8/3 and reaches the value ≈ 128 (because of the Higgs mechanism leading to the emergence of the W + , W -and Z 0 masses), and then increases due to the effects of the various elementary particle pairs, up to the electron scale where it acquires its large scale value 137.036 (black dashed thick curve). Four scale-coupling relations, which can be written as ln(r/λ P ) = α be applied to the coupling in the above relation, i.e a factor 4/3 to the inverse coupling, in order to apply it to the lowest mass neutrino:

ln m P m(r) = 5 6 α 1 (r) -1 + 1 2 α 2 (r) -1 . (29)
Extrapolating to low energies, this expression can be written as 5 6 α 1 (r) -1 + 1 2 α 2 (r) -1 = 1 2 α(r) -1 , which becomes constant beyond the electron Compton scale where we recover the standard fine structure constant α. This finally results in the following conjectured mass-coupling relation for the neutrino:

2 α ln m P m ν = 1. ( 30
)
This relation allows us to derive a lowest neutrino mass from the fine structure constant and the Planck mass:

m ν1 = m P exp - 1 2α = 1.7499 × 10 -30 m P = 0.0214 eV. (31)
The uncertainty on this theoretical prediction is difficult to estimate. From the lowest order relation for the electron, one derives a mass m e = m P exp(-3α -1 /8) = 0.587 MeV, which differs by 15 % from the experimental electron mass m e = 0.511 MeV. Assuming a similar uncertainty for the neutrino lowest order relation, one obtains for the smallest neutrino mass m ν = (0.0214 ± 0.0032) eV.

The two other neutrino masses can now be derived from the experimental results on neutrino oscillations. From ∆m 2 21 = (7.39 ± 0.20) × 10 -5 eV 2 and ∆m 2 32 = (2.51 ± 0.03) × 10 -3 eV 2 [START_REF] Zyla | [END_REF], one obtains m ν2 = 0.0231 eV, m ν3 = 0.0552 eV, (

i.e. a mass ordering m ν1 ≈ m ν2 < m ν3 and a total mass of 0.0997 ≈ 0.1 eV. Accounting for the uncertainties, one finds m ν2 = (m ν1 + 0.0017) ± 0.0030 eV (always larger than m ν1 ), m ν3 = (0.0552 ± 0.0013) eV and a total mass (0.100 ± 0.008) eV. These values are quite compatible with our current knowledge about neutrinos (in particular, getting for one of the neutrino masses ≈ 0.05 eV occurs in several scenarios) and in good agreement with a Majorana origin of the mass which requires 0.016 < Σ j m νj < (0.061 -0.165) eV [START_REF] Zyla | [END_REF].

Conclusion

In this paper, we have recalled the mass-coupling relation we derived for the electron in the scale-relativity framework [START_REF] Nottale | Relativity in General[END_REF], α ln(m P /m e ) = 3/8 to lowest order, and attempted to improve it by a more complete description of the transition occuring around the electron Compton scale.

Then we have suggested the existence of a similar relation for the lightest neutrino, reading α ln(m P /m ν ) = 1/2. From this relation, we obtain a theoretical prediction for the lightest neutrino mass m ν = m P exp(-1 2 α -1 ) = 21.4 meV. This corresponds to an inverse ratio 2.39 × 10 7 with the electron mass, i.e. to a Compton length λ ν = /m ν c = 9.22 µm, of the order of size of a living cell.

The two other neutrino masses can be derived from this mass and from experimental results of neutrino oscillations. We find m ν2 = 23.1 meV and m ν3 = 55.2 meV, yielding a total mass of 0.1 eV, compatible with the current experimental limits.

It will be possible to put this expectation to the test by future neutrino experiments allowing direct mass measurements. The present upper limit by KATRIN, m ν < 1.1 eV [START_REF] Aker | [END_REF], is still ≈ 50 times the mass predicted in the present work. If no neutrino mass signal is found, the KATRIN sensitivity after 3 years of measurements will be m ν < 0.2 eV (90% CL.), which is closer to our theoretical prediction but still larger.

Figure 1 :

 1 Figure 1: Behavior of the electron scale-coupling relation around the electron Compton scale,

Figure 2 :

 2 Figure 2: Scale-coupling relations in a global diagram ranging from the Planck length-scale

- 1 ,

 1 are considered in this diagram.[START_REF] Nottale | Relativity in General[END_REF] Taking α -1 = 4π 2 yields the W Z scale (marked Z in the diagram). (2) With α -1 = α -1 , one obtains a cosmological scale (U in the diagram). (3) With α -1 = (3/8)α -1, where α is the low energy fine structure constant,one gets the electron scale. (4) Finally, taking α -1 = (1/2)α -1 , one obtains a new length-scale that can be identified with the lightest neutrino Compton-length (nu in the diagram), yielding a predicted mass of 0.0214 eV.


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

Caco-

-2 cells in apical to basolateral direction P app(AB), Permeability across Caco-2 cells in basolateral to apical direction P app(BA) , Efflux Ratio ER

Keywords: Predicting drug permeability Predicting drug permeability and relationship with COVID-19 infected Caco-2 cells Caco-2, drug cellular permeability, passive transcellular and paracellular diffusion, active transport, COVID-2019 or SARS-CoV-2, ACE2 receptor binding, spike serine proteases, S-RBD, TMPRSS2, M pro , IC 50 , host cell membrane fusion or endocytosis, translatome and proteome proteomics, linear free energy relationships, HOMO-LUMO, quantum mechanics

Caco-2: Predicting drug permeability and relationship

with COVID-19 infected Caco-2 cells

Introduction

Caco-2 is a cell line of human colorectal adenocarcinoma cells usually used as a cell culture monolayer. The Caco-2 monolayer is widely used as an in vitro model of the human small intestinal mucosa to predict the absorption of orally administered drugs. There is a well established correlation between the in vitro apparent permeability (P app ) across Caco-2 monolayers and the in vivo oral fraction of drugs absorbed. Caco-2 results can predict of the human absorption for drugs which display active uptake or efflux or passively pass through the cell membrane by transcellular or the paracellular route. Transport in both directions (apical to basolateral (AB) and basolateral to apical (BA) across the cell monolayer allows an efflux ratio to be determined which provides an indicator as to whether a compound undergoes active efflux. The Caco-2 cell monolayer has most of the known intestinal transporters in a pattern similar to that for the small intestine, and can give information on the mechanism of intestinal drug absorption and drug-drug interactions. In absence of transporter and enzyme involvement in the Caco-2 monolayer, transport in the A-B and B-A direction is symmetrical: P app (AB) = P app (BA). The efflux ratio ER = {P app (BA) / P app (AB)} > 2 normally indicates significant involvement of apical efflux transporters, although the ER is dependent on passive permeability as well. ER values between 1-2 are less determinate. Figure 1 schematically describes the various transport mechanisms. [START_REF] Sun | The Caco-2 cell monolayer: usefulness and limitations[END_REF]

Figure 1. Transport mechanisms in Caco-2 cells and ACE2 and TMPRSS2 receptors

The overall permeation rate P app for various drugs is the sum of all involved transport mechanisms: (a) passive diffusion across the monolayer, either directly through the cells (transcellular route) or through the water-filled pores in tight junctions interconnecting the cells (paracellular route), (b) facilitated diffusion or be actively transport against the concentration gradient either in the apical to basolateral (absorptive) or in the basolateral to apical (secretive) direction, (c) receptor mediated endocytosis or transcytosis, more common to peptides and larger molecules, and is of less importance for small molecules.

The characterization of passive permeability of drugs has usually been ascribed to their physicochemical properties in various QSAR studies. However, owing to the intrinsic heterogeneity of the parental cell line, and multiple clonal cell lines, culture-related conditions, as well as the different Caco-2 cell lines utilized in different laboratories, often makes it extremely difficult to compare various QSAR results in the literature. Inter-laboratory variations in P app for various drugs are known to be large. One study showed that the standard deviation of published measurements for the same compound can reach almost 0.6 log units. [START_REF] Pham-The | In silico assessment of ADME properties: Advances in Caco-2 Cell Permeability Modeling[END_REF][START_REF] Lanevskij | Physicochemical QSAR Analysis of Passive Permeability Across Caco-2 Monolayers[END_REF][START_REF] Jb Lee | Quantitative analysis of lab-to-lab variability in caco-2 permeability assays[END_REF] Since efforts to identify accurate predictive structure activity relationships for the permeation of drugs across the Caco-2 cell membrane is known to be restricted by the large experimental errors that exist amongst different testing laboratories, [START_REF] Pham-The | In silico assessment of ADME properties: Advances in Caco-2 Cell Permeability Modeling[END_REF][START_REF] O'hagan | The apparent permeabilities of Caco-2 cells to marketed drugs: magnitude, and independence from both biophysical properties and endogenite similarities[END_REF] several studies have used the extensive experimental data of 52 widely disparate drugs from Yazdanian [START_REF] Yazdanian | Correlating partitioning and Caco-2 cell permeability of structurally diverse small molecular weight compounds[END_REF] to determine such predictive relationships. [START_REF] Ponce | A new topological descriptors based model for predicting intestinal epithelial transport of drugs in caco-2 cell culture[END_REF][START_REF] Milanetti1 | Prediction of the permeability of neutral drugs inferred from their solvation properties[END_REF][START_REF] Oa Santos-Filho | Combined 4D-Fingerprint and Clustering Based Membrane-Interaction QSAR Analyses for Constructing Consensus Caco-2 Cell Permeation Virtual Screens[END_REF] Lanevskij [START_REF] Lanevskij | Physicochemical QSAR Analysis of Passive Permeability Across Caco-2 Monolayers[END_REF] analyzed by multistep non-linear multiple regression a literature database of 1366 Caco-2 permeability coefficients for 768 diverse drugs and drug-like compounds from varying experimental conditions (pH from 4.0 to 8.0) to develop a permeability model comprised of a minimal set of physicochemical descriptors: octanol-water log D, pKa, hydrogen bonding potential, and molecular size. Kim [START_REF] Kim | A correlation between the permeability of a series of peptides using an in vitro cell culture model Caco-2 and those using an in situ perfused rat ileum culture model of the intestinal mucosa[END_REF] found that the permeability coefficient for a series of model peptides in Caco-2 cells was dependent on octanol-water lipophilicity and hydrogen bonding potential. Milanetti [START_REF] Milanetti1 | Prediction of the permeability of neutral drugs inferred from their solvation properties[END_REF] used the Yazdanian data set [START_REF] Yazdanian | Correlating partitioning and Caco-2 cell permeability of structurally diverse small molecular weight compounds[END_REF] and found that an analysis of the dynamic distribution of water molecules around 23 neutral drugs at pH 7.4 by molecular dynamics simulations, correlated well with the permeability measured using the Caco-2 cells. Ponce [START_REF] Ponce | A new topological descriptors based model for predicting intestinal epithelial transport of drugs in caco-2 cell culture[END_REF] used the Yazdanian data set [START_REF] Yazdanian | Correlating partitioning and Caco-2 cell permeability of structurally diverse small molecular weight compounds[END_REF] and found that the permeability coefficients in Caco-2 cells for 18 structurally diverse drugs (plus a training set of 33 drugs) were described by the quadratic indices of the molecular pseudograph's atom adjacency matrix as molecular descriptors. Santos-Filho [START_REF] Oa Santos-Filho | Combined 4D-Fingerprint and Clustering Based Membrane-Interaction QSAR Analyses for Constructing Consensus Caco-2 Cell Permeation Virtual Screens[END_REF] have used a membrane interaction QSAR approach based on the Yazdanian data set using 11 drugs (plus a training set of 30 drugs) quantum mechanically interacting with dimyristoylphosphatidylcholine (DMPC) molecule as the membrane monolayer surrogate model. It was concluded that Caco-2 cell permeation is governed by the spatial distribution of hydrogen bonding and nonpolar groups over the molecular shape of the DMPC monolayer.

In a wide and comprehensive review of 696 separate measurements from 15 collections of P app (AB) of FDA approved drugs, O'Hagan [START_REF] O'hagan | The apparent permeabilities of Caco-2 cells to marketed drugs: magnitude, and independence from both biophysical properties and endogenite similarities[END_REF] concluded that while inter-laboratory measurement variation varied by factors of 2-5, the P app of marketed approved drugs is poorly correlated with either simple biophysical properties, the extent of molecular similarity to endogenous metabolites (endogenites), or any specific substructural properties. The octanol:water partition coefficient, logP, shows negligible correlation with Caco-2 permeability. It was concluded that the data are best explained on the basis that most drugs enter (and exit) Caco-2 cells via a multiplicity of transporters of comparatively weak specificity. [START_REF] O'hagan | The apparent permeabilities of Caco-2 cells to marketed drugs: magnitude, and independence from both biophysical properties and endogenite similarities[END_REF] Epithelial tight junctions are known to contain size-and charge-selective pores that control the paracellular movement of charged and non-charged solutes. Van Itallie [START_REF] Van Itallie | The density of small tight junction pores varies among cell types and is increased by expression of claudin-2[END_REF] showed that the paracellular (Papp) of a continuous series of non-charged polyethylene glycols (PEGs) across the monolayers of five different epithelial cell lines (including Caco-2), and porcine ileum, indicated two distinct pathways: (a) high-capacity small pores and (b) a size-independent pathway for larger solutes. All cell lines and ileum shared a pore aperture radius of 4 Å. Caco-2 cells have the largest numbers of pores as well as the greatest permeation through the size independent pathway of all epithelial cell lines. Camenisch [START_REF] Camenisch | Estimation of permeability by passive diffusion through Caco-2 cell monolayers using the drugs' lipophilicity and molecular weight[END_REF] found that the permeability of a diverse range of structurally different drugs pass through Caco-2 cells by passive diffusion which can be predicted from the drugs' distribution coefficient in 1-octanol/water (log D(oct)) and its molecular weight (MW).

The lipophilicity of cell membranes is usually taken to be well represented by logD (to provide for charged species or logP for neutral species) in octanol at pH 7.4. Octanol is thought to be a suitable surrogate for phospholipid membranes as it mimics the hydrogen bonding ability of cell membranes. [START_REF] Camenisch | Estimation of permeability by passive diffusion through Caco-2 cell monolayers using the drugs' lipophilicity and molecular weight[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 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 | 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] We have previously used ΔG lipo,CDS the free energy in n-octane as a measure of lipophilicity of drugs interacting with various targets, but for cell membrane targets, ΔG lipo,CDS in octanol is a better measure. Octanol has been widely accepted as a membrane bilayer mimicking solvent, where the 2.18M equilibrium water concentration of n-octanol is consistent with the known water levels in cell membrane bilayer cores cause by some trans-bilayer water transport. [START_REF] Balaz | Modelling kinetics of subcellular disposition of chemicals[END_REF] Octane has no hydrogen bonding capability. The lipophilicity of the cell membrane also has an effect on drugs trying to enter the membrane, so some degree of desolvation of water surrounding the drugs is required to facilitate entry into membrane. ΔG desolv,CDS the free energy is a measure of water desolvation, the drug-water short-range interactions which includes contributions from hydrogen bonding capacity, as well as dispersion, exchange repulsion etc. An important aspect that can have a significant effect on the Caco-2 passive permeability of various drugs is the potential ionization at physiological pH 7.4, particularly since many approved drugs are weak bases rather than acidic drugs. The pH partition theory suggests that while many drugs will be partly ionized at pH 7.4, it is mainly the neutral species that will be passively transported by 10 2 to 10 4 times as fast as their cationic or anionic counterparts. [START_REF] Palm | Effect of molecular charge on intestinal epithelial drug transport: pH-dependent transport of cationic drugs[END_REF][START_REF] Neuhoff | pH-dependent bidirectional transport of weakly basic drugs across caco-2 monolayers: implications for drug-drug interactions[END_REF][START_REF] Neuhoff | pH-Dependent passive and active transport of acidic drugs across caco-2 cell monolayers[END_REF] Lee et al [START_REF] Kj Lee | Effect of experimental pH on the in vitro permeability in intact rabbit intestines and Caco-2 monolayer[END_REF] similarly found that the neutral species of 14 acidic and basic drugs were preferentially transported across Caco-2 cells at pH 7.4 (and 6.5) rather than the ionized forms. Less is well known about active transport across Caco-2 cells. For example, digoxin, quinidine, verapamil, acebutolol, talinolol, Erythromycin, Ranitidine, cimetidine, Ritonavir, saquinavir, indinavir, nelfinavir, Taxol, doxorubicin, Zolmitriptan, Pimozide, reserpine, Chloroquine, domperidone, terfenadine, etc are actively intestinally transported by P-gp (MDR1, ABCB1) in Caco-2 cells, whereas the charged Estrone-3-Sulfate and Carnitine are transported by OATP2B1 and OCTN2 respectively. [START_REF] Sun | The Caco-2 cell monolayer: usefulness and limitations[END_REF] Atenolol is known to be actively transported by OCT1 in MDCK cells, whereas metoprolol is thought to be transported by facilitated diffusion. [START_REF] Mimura | Functional identification of organic cation transporter 1 as an atenolol transporter sensitive to flavonoids[END_REF][START_REF] Dixit | Standardization of an ex vivo method for determination of intestinal permeability of drugs using everted rat intestine apparatus[END_REF] Li [START_REF] Li | A fast screening model for drug permeability assessment based on native small intestinal extracellular matrix[END_REF] in their study of a new fast Caco-2 permeability test, identified that fexofenadine, cephalexin, digoxin, sulfasalazine, estradiol were known to be actively transported, and examined three transporters (PEPT1, OATP1A2, and P-gp) as well as their mediated influx or efflux and the increased gene expression of two excretive transporters (BCRP, MRP2).

A critical aspect of any quantitative structural activity study of the permeability of various drugs into Caco-2 cells is that a number of parallel mechanisms can occur: passive transcellular and paracellular diffusion, facilitated diffusion, and active transport processes. Comparing various data from different studies is confounded by possible variations in the actual Caco-2 cell line used, which may have inherent differences in membrane characteristics (affecting transcellular and paracellular diffusion) and active transporters. It is likely that multiple transport mechanisms may occur in transport studies of drugs in Caco-2 cell lines, for example both transcellular and paracellular diffusion for smaller neutral and less polarized drugs along with active transport for larger charged or more polarized drugs.

Fredlund [START_REF] Fredlund | In Vitro Intrinsic Permeability: A Transporter-Independent Measure of Caco-2 Cell Permeability in Drug Design and Development[END_REF] determined the apparent intrinsic permeability P app (AB) across Caco-2 cell monolayers in the presence of an optimized cocktail of chemical inhibitors toward the three major efflux transporters ABCB1, ABCC2, and ABCG2. The intrinsic P app values is thought to allow an estimate of passive permeability, which is independent of transporter expression levels and not limited by solubility or cell toxicity. Interestingly the drugs that showed passive permeability exhibited a similar (parallel) linear regression line for the intrinsic P app (AB) versus P app (AB) as did the drugs that were efflux inhibited.

Another topical use for Caco-2 cell assays is to identify repurposed therapeutic drugs that can be screened for efficacy against the SARS-CoV-2 virus. Ellinger [START_REF] Ellinger | Identification of inhibitors of SARS-CoV-2 in-vitro cellular toxicity in human (Caco-2) cells using a large scale drug repurposing collection[END_REF] has screened a large number of approved drugs against SARS-CoV-2 infected Caco-2 cells, as well as against infected Vero-6, Calu-3 (human lung) or BHK-21 (hamster kidney fibroblast) cells. It is known that the ACE2 receptor is the entry port for the SARS-Cov-2 virus, and that mechanism is facilitated by the TMPRSS2 membrane protease. Both ACE2 and TMPRSS2 receptors are present on the apical surface of Caco-2 cells. Infected Caco-2 cells may be more sensitive than Vero-6 cells to the TMPRSS2 inhibitors, camostat and nefamostat, and may be better test bed for testing efficacy against the SARS-CoV-2 virus. Caco-2 cells have also been effectively tested against the SARS-CoV and MERS coronaviruses. Figure 2 describes how these receptors are present in Caco-2 cells.

Figure 2. Membrane fusion and endocytosis mechanisms of viral cell entry showing potential receptor inhibitory sites: S-RBD, TMPRSS2, ACE2, M Pro and translation

Bojkova [START_REF] Bojkova | Proteomics of SARS-CoV-2-infected host cells reveals therapy targets[END_REF] used quantitative translatome and proteome proteomics to obtain the cellular response to SARS-CoV-2 infection in human Caco-2 cells. Ribavirin (IC 50 0.07 mM), NMS-873 (0.025 μM), 2-deoxy-D-glucose (9.09 mM), pladienolide B (0.007 μM), cycloheximide (0.17 μM), emetine (0.47 μM), inhibited viral infection by interfering with translation, proteostasis, glycolysis, splicing and nucleotide synthesis pathways at concentrations that were not toxic to the Caco-2 cells. However the inhibitory efficacy of these drugs is dependent upon how well the drugs pass through the infected Caco-2 cell membrane.

Klann [START_REF] Klann | Growth Factor Receptor Signaling Inhibition Prevents SARS-CoV-2 Replication[END_REF]studied SARS-CoV-2 infected Caco-2 cells and identified viral protein phosphorylation and phosphorylation-driven host cell signaling changes upon infection. Growth factor receptor signaling and downstream pathways are activated. Drug-protein network analyses revealed growth factor receptor signaling as a key target for viral inhibition by some approved anti-cancer drugs.

Study objectives:

(a) Develop a previously indentified LFER methodology based on quantum mechanical derived molecular descriptors to characterize the transport of a wide range of drugs into Caco-2 cells (b) Identify how the ACE2 and TMPRSS2 surface receptors on the Caco-2 cells might influence drug transport into SARS-CoV-2 infected Caco-2 cells. (c) Identify how various repurposed drugs can inhibit replication of M pro , translation, proteonomics and growth factor signaling of SARS-CoV-2 infected Caco-2 cells.

Results

We have previously shown that the general equation 1 can successfully describe passive or facilitated diffusion transport processes of drugs across the cell membrane. [START_REF] Fong | Permeability of the Blood-Brain Barrier: Molecular Mechanism of Transport of Drugs and Physiologically Important Compounds[END_REF][START_REF] Fong | The extravascular penetration of tirapazamine into tumours: a predictive model of the transport and efficacy of hypoxia specific cytotoxic analogues and the potential use of cucurbiturils to facilitate delivery[END_REF][START_REF] Fong | 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] Eq 1 Drug Transport = ΔG desolv,CDS + ΔG lipo,CDS + Dipole Moment + Molecular Volume

The four independent molecular specifiers in eq 1 can be useful in screening how drugs might be transported across the host cellular membrane or block the virus-host membrane interaction that is critical for progressing COVID-2019 infection.

Eq 2 can be applicable to the inhibition of the various proteases involved in the SARS-CoV-2, SARS-CoV and MERS-CoV replication process. We have previously shown [START_REF] Klann | Growth Factor Receptor Signaling Inhibition Prevents SARS-CoV-2 Replication[END_REF][START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF] that the molecular specifiers used in eq 1 and eq 2 can be useful screening tools to evaluate the potential efficacy of therapeutic drugs that may be active against the SARS-CoV-2 virus.

Eq 2 Inhibition COVID-19 = ΔG desolv,CDS + ΔG lipo,CDS + Dipole Moment + Molecular Volume + HOMO-LUMO where ΔG desolv,CDS is the free energy of water desolvation, ΔG lipo,CDS is the lipophilicity free energy in octane, DM is the dipole moment in water, Mol Vol is the molecular volume in water, and HOMO-LUMO is the energy gap in water.

(a) Permeability of drugs into Caco-2 cells

Analysis of P app (AB) for Caco-2 cells from Yazdanian's study [START_REF] Yazdanian | Correlating partitioning and Caco-2 cell permeability of structurally diverse small molecular weight compounds[END_REF] for 52 widely structurally diverse drugs (see Table 1) is shown below in eq 3(a). Given the results of O'Hagan [START_REF] O'hagan | The apparent permeabilities of Caco-2 cells to marketed drugs: magnitude, and independence from both biophysical properties and endogenite similarities[END_REF] who showed that active transporters may be the dominant form of transport across caco-2 cell membranes, the general form of eq 1 can be modified to include possible drug-carrier transporter interaction as identified by the HOMO-LUMO energy gap of the drugs, which reflect the HOMO drug à LUMO transporter molecular interaction, or the HOMO transporter à LUMO drug . It is clear that there is little significance in ΔG desolv,CDS , ΔG lipo,CDS, or Mol Vol as judged by the large SE values and their corresponding P-values (0.50, 0.79, and 0.49 respectively), whereas the DM and HOMO-LUMO independent variables have P-values of 0.017 each. It should be noted that all the 52 diverse drugs have been included in the regression analysis, with no statistical optimization methods such as training sets and test sets or machine/artificial learning techniques etc which artificially optimize test data such as used in QSAR methods which seek to find relationships with any independent variables that might fit the data. [START_REF] Ponce | A new topological descriptors based model for predicting intestinal epithelial transport of drugs in caco-2 cell culture[END_REF][START_REF] Oa Santos-Filho | Combined 4D-Fingerprint and Clustering Based Membrane-Interaction QSAR Analyses for Constructing Consensus Caco-2 Cell Permeation Virtual Screens[END_REF] [Ponce, Santos-Filho] The correlation coefficient R 2 is a measure of the variability or scatter around the best fit regression line, whereas a low P-value for any independent variable indicates that the coefficient for that variable is statistically not zero (ie reject the null hypothesis that the coefficients are zero). The low P-values of 0.017 indicate that DM and the HOMO-LUMO in eq 3(a) are highly significant compared to the values for ΔG desolv,CDS , ΔG lipo,CDS (for both n-octane and n-octanol) or Mol Vol at 0.50, 0.79 and 0.18, and 0.49 respectively. This LFER approach always uses the same 5 independent variables as molecular specifiers in seeking structure activity relationships.

No outliers were excluded in eq 3(a). The 52 drugs tested in eq 3(b) were the neutral forms only, but where charged forms were possible for 23 drugs, inclusion of these forms in addition to their neutral counterparts gave a very similar equation to eq 3(b) for the 75 neutral and charged drugs.

Eq 3(b) can be derived from eq 3(a) by eliminating the non-significant variables to give eq 3(b), which show that active transportation is the dominant mechanism based on the strong dependency on the dipole moment and HOMO-LUMO gap:

Eq 3(b) P app (AB) = -0.70DM - It is noted that 7 drugs have efflux ratio ER = {P app (BA) / P app (AB)} > 2, with another 8 having ER between 1-2, and two with ER <1 (caffeine and progesterone). Acebutolol 20.98 and chlorothiazide 13.89 have very large ERs. The ER data indicates significant involvement of apical efflux transporters, although the ER can also be dependent on passive permeability as well. [START_REF] Sun | The Caco-2 cell monolayer: usefulness and limitations[END_REF][START_REF] Pham-The | In silico assessment of ADME properties: Advances in Caco-2 Cell Permeability Modeling[END_REF] These ER ratios strongly support the HOMO-LUMO gap as being an indicator of active transport processes in Caco-2 cells, and the similarity between eqs 4 and 3(b) again indicate that active transport processes predominate in P app (AB) and P app (BA).

Analysis of Fredlund's data [START_REF] Fredlund | In Vitro Intrinsic Permeability: A Transporter-Independent Measure of Caco-2 Cell Permeability in Drug Design and Development[END_REF] (Table 2) for the permeability of 58 diverse neutral drugs in Caco-2 cells gives eq 5(a Fredlund also measured the apparent intrinsic permeability P app (AB) across Caco-2 cells of the same 58 drugs (Table 2) with a cocktail of chemical inhibitors of the three major efflux transporters ABCB1, ABCC2, and ABCG2. [START_REF] Fredlund | In Vitro Intrinsic Permeability: A Transporter-Independent Measure of Caco-2 Cell Permeability in Drug Design and Development[END_REF] The intrinsic P app values were thought by Fredlund's data set also includes the intrinsic permeability of 24 drugs (see Table 2) that were known to be actively transported by ABCB1, ABCC2, and ABCG2. These drugs are analyzed in eq 5(c), but are included in the 58 drugs tested in eq 5(b), so the remaining 34 drugs analyzed in eq 5(b) may or may not be transported by ABCB1, ABCC2, and ABCG2.

Eq 5(c) 24 drugs known to be transported by ABCB1, ABCC2, and ABCG2 Intrinsic P app (AB) = -0.98ΔG Since transportation by the ABCB1, ABCC2, and ABCG2 transporters is known to occur for the 24 drugs in eqs 5(c) and 5(d), these results show that non-dependency on the HOMO-LUMO gap is an indicator of inhibition of active transport processes by the ABCB1, ABCC2, and ABCG2 transporters. And when that active transport ceases, the remaining transport process is dependent only on the molecular volume of the drugs, ie paracellular diffusion still occurs. It would be expected that transcellular diffusion would show some dependency on the lipophilicity ΔG lipo,CDS as interaction must occur between the drug and the lipophilic Caco-2 cell membrane.

Li [START_REF] Li | A fast screening model for drug permeability assessment based on native small intestinal extracellular matrix[END_REF] has tested the "presumed" passive permeability of 19 drugs (see Table 3) in Caco-2 cells as analyzed in eq 6. It was found that P app (AB) was only dependent on molecular volume when evaluated against P-values for all other independent variables ΔG desolv,CDS , ΔG lipo,CDS (both octane and octanol), DM, and HOMO-LUMO gap. Eq 6 indicates that paracellular diffusion is the dominant transport mechanism. Camenisch [START_REF] Camenisch | Estimation of permeability by passive diffusion through Caco-2 cell monolayers using the drugs' lipophilicity and molecular weight[END_REF] found that the permeability of a diverse range of structurally different drugs (see give 47 drugs tested gave a similar equation to eq 7, but it is assumed that the neutral species permeate faster than their charged equivalents, so eq 7 is the preferred model. It is noted that the P-values of the coefficients of the ΔG desolvCDS, ΔG lipoCDS (octane) and Molecular Volume molecular specifiers in eq 8(b) are significant at 0.015, 0.018 and 0.003 levels. The low regression coefficient is partly due to the low sensitivity of the coefficients to IC 50 , since the regression coefficient is dependent on the slope of the regression line. Eq 8(b) and 8(a) also includes 9 positively charged drugs and their neutral counterparts, since these drugs can be charged at neutral pH conditions in vitro. Eq 8 shows dependency on ΔG lipoCDS (octane) but not with ΔG lipoCDS (octanol) and no dependency on the HOMO-LUMO gap.

Analysis of the limited data of Bojkova [START_REF] Bojkova | Proteomics of SARS-CoV-2-infected host cells reveals therapy targets[END_REF] for the inhibition of SARS-CoV-2 infection in human Caco-2 cells by interfering with translation, proteostasis, glycolysis, splicing and nucleotide synthesis pathways showed that IC 50 was highly correlated with the HOMO-LUMO gap or LUMO (but not HOMO) of the six inhibitors as per eq 9(a) or 9(b). No significant dependencies were found with the other independent variables. Both eq 9(a We have previously shown Fong [START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF][START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] that using the same LFER methodology, stepwise analysis of 23 inhibitors from Vatansever [START_REF] Vatansever | Targeting the SARS-CoV-2 Main Protease to Repurpose Drugs for COVID-19[END_REF] of the M pro of SARS-CoV-2 yields eq 11. It is noted that the p-values of the coefficient of the HOMO-LUMO molecular specifier is modestly significant at the 0.077 level.

)
Eq 11 IC 50 = 64.71(HOMO-LUMO) +60.40

Where R 2 = 0.14, SEE = 189.96, SE(HOMO-LUMO) = 34.84, F=3.49, Significance=0.077

Molecular docking is currently the mainstay of predictive computational methods to evaluate new and repurposed ant-virals for coronaviruses, and there have been some reports [START_REF] Vatansever | Targeting the SARS-CoV-2 Main Protease to Repurpose Drugs for COVID-19[END_REF][START_REF] Bobrowski | Computational Models Identify Several FDA Approved or Experimental Drugs as Putative Agents Against SARS-CoV-2[END_REF] that inhibitory structure activity relationships (SARs) do not always agree with docking results for the M pro . Analysis of the docking binding energy of 22 inhibitors to M pro (PDB 6LU7) from Vatansever [START_REF] Vatansever | Targeting the SARS-CoV-2 Main Protease to Repurpose Drugs for COVID-19[END_REF] was previously shown by Fong [START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF][START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] to give eq 12. It is noted that the corresponding correlation with HOMO is much poorer than that for LUMO in eq 12, and the dependencies on the other independent variables were poor.

Eq 12

Docking Binding Energy = 0.58LUMO -8.72

Where R 2 = 0.31, SEE = 0.75, SE(LUMO) = 0.19, F=8.65, Significance=0.008

It can be seen that in vitro IC 50 inhibitory assay studies in eq 11 are related to the theoretical docking binding energies in eq 12, but not directly related for this series of drugs, since eq 11 shows a dependency on the HOMO-LUMO gap whereas eq 12 is more strongly correlated to the LUMO only. Importantly, the experimental in vitro inhibition IC 50 of the these drugs is directly related to the calculated docking binding energy through the HOMO-LUMO gap in eq 11 or the closely related LUMO in eq 12.

Fischer et al [START_REF] Fischer | Potential Inhibitors for Novel Coronavirus Protease Identified by Virtual Screening of 606 Million Compounds[END_REF] computationally screened a library of over 606 million compounds for binding to the main protease (M pro ) of SARS-CoV-2 (PDB 6LU7) and identified 15 repurposed potential inhibitors of the M pro . We have previously shown [START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF][START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] that eq 13 is the best correlation. It can be seen that the docking binding energies to M pro are related to the molecular specifiers of the inhibitors used in eqs 11 and 12, but the HOMO-LUMO or LUMO are the dominant factors influencing the binding energies.

We have previously shown [START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF][START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] Fong] that docking studies of the angiotensin-converting enzyme 2 (ACE2) receptor can also be used to derive structural activity LFER relationships.

Terali has investigated the binding of 9 repurposed drugs [START_REF] Terali | Prioritizing potential ACE2 inhibitors in the COVID-19 pandemic: Insights from a molecular mechanics-assisted structure-based virtual screening experiment[END_REF] to the ACE2 receptor eq 14:

Eq 14 Docking BE (ACE2) = -0.81HOMO -0. Comparison of eqs 14 or 15 for the ACE2 binding energy and 16(b) for the S-RBD binding energy shows a reversal of sign for the three independent variables with far greater opposite dependence on the HOMO in eqs 14 or 15 compared to eq 16(b). These eqs indicate a substantial difference between the inhibitor binding interactions of the viral S-RBD and the ACE2.

Analysis of Smith's data [START_REF] Choudhary | Identification of SARS-CoV-2 Cell Entry Inhibitors by Drug Repurposing Using in silico Structure-Based Virtual Screening Approach[END_REF] which is a computational model of the RBD of the spike protein (Sprotein) of SARS-CoV-2 interacting with the human ACE2 receptor, for 14 repurposed inhibitors interacting with the S-protein of SARS-CoV-2 and the ACE2 receptor gives eq 17. [START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF][START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] This result is unusual in this study as there is no dependency on the HOMO-LUMO gap, when the inhibitors interact with the ACE2 and S-RBD receptor.

Eq 17 Docking BE (ACE2+S-RBD) = -0.40ΔG lipo,CDS + 0.017Dipole Moment -7.50

Where R 2 = 0.462, SEE = 0.107, SE(ΔG lipoCDS ) = 0.016, SE(DM) = 0.007, F=4.71, Significance=0.033

The transmembrane serine 2 protease (TMPRSS2) on the cell surface is involved in S protein priming in lung cells, particularly involving the ACE2 receptor which is the main entry point into host cells for the SARS-Cov-2 virus. [START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF][START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] Analysis of Roomi's docking of 24 repurposed drugs [START_REF] Ms Roomi | Potential Compounds for the Inhibition of TMPRSS2[END_REF] to the TMPRSS2 receptor gives eqs 18 We have previously shown [START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF][START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] that analysis of the binding energy of 10 inhibitors of TMPRSS2 from Idris [START_REF] Mo Idris | Computer-aided screening for potential TMPRSS2 inhibitors: a combination of pharmacophore modeling, molecular docking and molecular dynamics simulation approaches[END_REF] gives eqs 19

Discussion

This study has shown that our LFER methodology based on quantum mechanical derived molecular descriptors can successfully describe: (a) the transport of a wide range of drugs into Caco-2 cells (as described by eqs 3-7) (b) the inhibition of SARS-CoV-2 infected Caco-2 cells (as described by eqs 8-10) and (c) in vitro inhibition and docking binding energies of SARS-CoV-2 of M pro , ACE2, S-RBD, TMPRSS2 (as described by eqs 11-18). These equations were derived from applying the general form equation 2 to a large variety of inhibitors to find which of the five molecular specifiers can describe the transport of drugs into Caco-2 cells, or the inhibition of SARS-CoV-2 infected Caco-2 cells, or the in vitro inhibition or docking binding of inhibitors to the SARS-CoV-2 virus to host cells. It is noted that we have found that ΔG lipo,CDS (octanol) better describes the lipophilicity of the Caco-2 cell membrane rather the ΔG lipo,CDS (octane) molecular descriptor as previously identified. [START_REF] Camenisch | Estimation of permeability by passive diffusion through Caco-2 cell monolayers using the drugs' lipophilicity and molecular weight[END_REF][START_REF] Balaz | Modelling kinetics of subcellular disposition of chemicals[END_REF][START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF][START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] However we have used ΔG lipo,CDS (octane) to describe the lipophilicity involved in drug-receptor binding as usual as in eqs 11-18.

(a) Permeability of drugs into Caco-2 cells

It has been shown that apical to basolateral (AB) permeability, P app (AB) for Caco-2 cells for 52 widely structurally diverse drugs (from the Yazdanian data [START_REF] Yazdanian | Correlating partitioning and Caco-2 cell permeability of structurally diverse small molecular weight compounds[END_REF]) and the basolateral to apical (BA) permeability P app (BA) for 17 drugs are mainly dependent on the HOMO-LUMO gap and to a lesser extent, the dipole moments of the drugs, as shown in eq 3(a), 3(b) and 4. It is also noteworthy that eq 3(b) for the P app (BA) data is very similar to that for P app (AB), and apical efflux transport is well established in Caco-2 cells (see Figure 1). [START_REF] Sun | The Caco-2 cell monolayer: usefulness and limitations[END_REF] Of the 17 drugs used to derive P app (BA) in eq 4, 7 drugs have efflux ratio ER = {P app (BA) / P app (AB)} > 2, with another 8 having ER between 1-2, and two with ER <1 (caffeine and progesterone). Acebutolol 20.98 and chlorothiazide 13.89 have very large ERs. These ER data indicates significant involvement of apical efflux transporters, although the ER is also dependent on passive permeability as well.

Eqs 5(a) and (b) from Fredlund's data [START_REF] Fredlund | In Vitro Intrinsic Permeability: A Transporter-Independent Measure of Caco-2 Cell Permeability in Drug Design and Development[END_REF] describes the permeability P app (AB) and the apparent intrinsic permeability P app (AB) of the same 58 drugs with a cocktail of chemical inhibitors of the three major efflux transporters ABCB1, ABCC2, and ABCG2. It is seen that both equations show strong parallel dependencies on ΔG desolv,CDS , ΔG lipo,CDS , Mol Vol and the HOMO-LUMO gap.

Eq 5(d) for the intrinsic permeability of the 24 drugs known to be transported by ABCB1, ABCC2, and ABCG2 shows only a dependency on the molecular volume. The non-dependency on the HOMO-LUMO gap is an indicator of inhibition of active transport processes by the ABCB1, ABCC2, and ABCG2 transporters, and that when active transport ceases, the remaining transport process is dependent on the molecular volume of the drugs, ie paracellular diffusion then occurs. It would be expected that transcellular diffusion would show some dependency on the lipophilicity ΔG lipo,CDS as interaction must occur between the drug and the lipophilic Caco-2 cell membrane.

It appears that eqs 5(a) and (b) describe (i) transcellular passive diffusion (dependency on ΔG desolv,CDS and ΔG lipo,CDS ), (ii) paracellular diffusion (dependency on molecular volume) and (iii) active transport (dependency on the HOMO-LUMO energy gap. These three processes can operate concurrently.

The dependency on the HOMO-LUMO gap directly indicates electronic interaction between the HOMO and LUMO of the drugs and some membrane transporter, ie HOMO drug à LUMO transporter and HOMO transporter à LUMO drug . A passive transport process would not show such large associations with the HOMO-LUMO gap. This finding is broadly aligned with the previous finding by O'Hagan that most drugs enter (and exit) Caco-2 cells via a multiplicity of transporters of comparatively weak specificity. [START_REF] O'hagan | The apparent permeabilities of Caco-2 cells to marketed drugs: magnitude, and independence from both biophysical properties and endogenite similarities[END_REF] Eq 6 describes the "presumed" passive permeability of 19 drugs from [START_REF] Li | A fast screening model for drug permeability assessment based on native small intestinal extracellular matrix[END_REF] in Caco-2 cells where P app (AB) was only dependent on molecular volume. This result indicates that paracellular diffusion predominates in this cell line.

Camenisch [START_REF] Camenisch | Estimation of permeability by passive diffusion through Caco-2 cell monolayers using the drugs' lipophilicity and molecular weight[END_REF] found that the permeability of a diverse range of structurally different drugs pass through Caco-2 cells by passive diffusion based on dependency on log D octanol and molecular weight (MW) of the 36 tested drugs. Eq 7 shows that permeability was only statistically dependent on the ΔG desolv,CDS , ΔG lipo,CDS (octanol only), and molecular volume. This result is in broad agreement with the finding of Camenisch. It is noted that no dependency was found with the HOMO-LUMO gap, nor ΔG lipo,CDS (octane). This result is consistent with both transcellular and paracellular diffusion occurring together.

In summary from eqs 3-7, dependency on the HOMO-LUMO gap indicates that permeability is an indication of active transportation, but in some studies only passive transcellular or paracellular permeation is found. This appears to suggest that different cell lines of CACo-2 may involve different or reduced transporters being present in the various drug permeation studies. Octanol (with better hydrogen bonding capability) is clearly a better proxy to describe the lipophilicity of the cell membrane than octane (no hydrogen bonding ability). P-values can be used to predict which molecular specifiers are statistically dominant in describing cell membrane transportation processes, however where multiple processes such as transcellular, paracellular, or active transportation may be occurring concurrently for a given series of drugs, then the correlation coefficient will reflect significant dispersion about the best fit relationships found in equations 3-7. This LFER methodology can quantitatively differentiate between passive transcellular, paracellular and active transport of drugs into Caco-2 cells.

(b) SARS-CoV-2 infected Caco-2 cells: inhibition and drug transport into cells

In this study, three cases of inhibition of SARS-CoV-2 infected Caco-2 cells are reported in eq 8(a), 8(b), 9(a), 9(b) and 10. Eqs 8(a) and (b) are derived from data that includes inhibitors that affect cellular entry of the virus (ACE2, TMPRSS2 and possibly S-RBD) or possibly intracellular processes such as inhibition of viral replication via M pro . [START_REF] Ellinger | Identification of inhibitors of SARS-CoV-2 in-vitro cellular toxicity in human (Caco-2) cells using a large scale drug repurposing collection[END_REF] Eq 9(a) and (b) and 10 are derived from inhibition of intracellular translation and growth factor receptor signaling processes in virus infected Caco-2 cells. [START_REF] Bojkova | Proteomics of SARS-CoV-2-infected host cells reveals therapy targets[END_REF][START_REF] Klann | Growth Factor Receptor Signaling Inhibition Prevents SARS-CoV-2 Replication[END_REF] Eqs 9(a), 9(b), and 10 show strong IC 50 linear dependencies on HOMO-LUMO or LUMO only, and we have shown that drugs are actively transported across the cell membrane have a strong dominant dependency on the HOMO-LUMO gap. It follows that the HOMO-LUMO gap or LUMO are strong predictors of how certain drugs can enter Caco-2 cells and inhibit virus infected internal cell replicative processes. It then is possible that Ellinger's results analyzed in eq 8(b) which show dependencies on ΔG desolvCDS, ΔG lipoCDS (octane) and molecular volume and negligible dependency on HOMO-LUMO appear to be a result of mainly inhibition of Caco-2 cell entry processes involving ACE2, TMPRSS2 or S-RBD, rather than intracellular inhibitory processes. This notion is further discussed below and in eq 17 where Smith's docking study [START_REF] Smith | Repurposing Therapeutics for COVID-19: Supercomputer-Based Docking to the SARS-CoV-2 Viral Spike Protein and Viral Spike Protein-Human ACE2 Interface[END_REF] of inhibitors to the combined S-RBD and ACE2 binding pocket showed no dependence on the HOMO-LUMO gap, only dependency on the lipophilicity and dipole moment of the inhibitors.

(c) In vitro and docking binding energies of SARS-CoV-2 inhibition: M pro , ACE2, S-RBD, TMPRSS2

It is clear that the application of the LFER model to the in vitro inhibition of (a) M pro of SARS-CoV-2, (b) SARS-CoV-2 spike protein S-RBD, (c) ACE2 receptor, and the (d) TMPRSS2 receptor is well described over a wide range of repurposed drugs, as shown in eqs 11-19. [START_REF] Fong | Covid-19: Predicting inhibition of SARS-CoV-2 in Caco-2 cells: viral cellular entry[END_REF][START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] These results cover experimental IC 50 studies and computational docking binding energy/affinity studies. The overwhelming conclusion is that the HOMO, or LUMO or the HOMO-LUMO energy gap of the various inhibitors is the principal determinant of inhibition of M pro as well as the S-RBD of the virus or the binding interaction between the inhibitors and the host cell targets, ACE2 and TMPRSS2.

Eqs 17 derived from Smith's docking analysis [START_REF] Smith | Repurposing Therapeutics for COVID-19: Supercomputer-Based Docking to the SARS-CoV-2 Viral Spike Protein and Viral Spike Protein-Human ACE2 Interface[END_REF] of the inhibition of the S-protein of SARS-CoV-2 interacting with the human {ACE2-S-RBD} combined receptor is an exception to the rule that HOMO, or LUMO or HOMO-LUMO is the principal determinant(s) of the binding energy. It is unclear whether this result is due to the interface between the S-protein and the ACE2 being a "shallow" binding pocket as opposed to the usual "buried" binding pocket, or possibly due to counterposing contributions from inhibitor HOMO à receptor LUMO or receptor HOMO à inhibitor LUMO dominated interactions for both S-protein and ACE2. The latter postulate is likely correct since we have shown that the binding of inhibitors to the ACE2 receptor (eqs 14 or 15) show a negative dependency on the HOMO, whereas binding of inhibitors to the S-RBD (eq 16) shows a positive dependency on the HOMO.

Conclusions

Detailed studies of the permeability of a very wide range of drugs into Caco-2 cells shows that active transport, passive transcellular and passive paracellular transport can be separately identified using the LFER method involving ΔG desolv,CDS , ΔG lipo,CDS (both octane and octanol), dipole moment, molecular volume, and HOMO-LUMO gap. Active transport is dependent on the HOMO-LUMO energy gap. Transcellular passive permeability shows dependency on the cell membrane lipophilicity as measured by ΔG lipo,CDS in n-octanol and the free energy of water desolvation ΔG desolv,CDS . Paracellular passive permeability shows dependency on the molecular volume in water. There is evidence that different Caco-2 cell lines in various literature permeability studies can result in varying contributions of active and passive transport modes for a given range of drugs.

Inhibition of SARS-CoV-2 infected Caco-2 cells can be studied using the LFER method to separate Caco-2 cell entry processes involving ACE2, TMPRSS2 or S-RBD from intracellular inhibitory processes. The extensive study by Ellinger [START_REF] Ellinger | Identification of inhibitors of SARS-CoV-2 in-vitro cellular toxicity in human (Caco-2) cells using a large scale drug repurposing collection[END_REF] likely predominantly involves the inhibition of Caco-2 cell entry processes involving ACE2, TMPRSS2 or S-RBD. It is known that Caco-2 cells were the only human cell type of 13 tested refractory cell lines that supported efficient SARS-CoV replication and expression of the SARS-CoV receptor, ACE2, [START_REF] Ec Mossel | Exogenous ACE2 Expression Allows Refractory Cell Lines To Support Severe Acute Respiratory Syndrome Coronavirus Replication[END_REF] and Chu [START_REF] Chu | Comparative tropism, replication kinetics, and cell damage profiling of SARS-CoV-2 and SARS-CoV and implications for clinical manifestations, transmissibility, and laboratory studies of COVID-19: an observational study[END_REF] found both human Calu-3 cells and Caco-2 cells were best suited for studying SARS-CoV-2 replication. Ellinger's data for Caco-2 cells is a valuable source for evaluating the efficacy of SARS-CoV-2 therapeutics.

The in vitro IC 50 and docking binding energies of SARS-CoV-2 inhibition of M pro , ACE2, S-RBD, TMPRSS2 show that the overwhelming conclusion is that the HOMO, or LUMO or the HOMO-LUMO energy gap is the principal determinant of inhibition of M pro , S-RBD, ACE2 and TMPRSS2. The exception to this rule is the inhibitory binding to the human {ACE2-S-RBD} combined receptor which is likely to be a result of counterposing contributions from inhibitor HOMO à receptor LUMO or receptor HOMO à inhibitor LUMO dominated interactions for both S-protein and ACE2.

Materials and methods

All calculations were carried out using the Gaussian 09 package. Energy optimizations were at the DFT/B3LYP/6-31G(d) (6d, 7f) level of theory for all atoms in water. Selected optimizations at the DFT/B3LYP/6-311+G(d,p) (6d, 7f) level of theory gave very similar results to those at the lower level. Optimized structures were checked to ensure energy minima were located, with no imaginary frequencies. Energy calculations were conducted at the DFT/B3LYP/6-31G(d,p) (6d, 7f) for neutral and cationic compounds 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] Chu | Comparative tropism, replication kinetics, and cell damage profiling of SARS-CoV-2 and SARS-CoV and implications for clinical manifestations, transmissibility, and laboratory studies of COVID-19: an observational study[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] 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][START_REF] Rayne | 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+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). It is noted that high computational accuracy for each species in different environments is not the focus of this study, but comparative differences between various species is the aim of the study. Experimental errors in inhibitory and docking binding studies are substantially higher than those for calculated molecular specifiers.

Additional data related to eq 7: The relationship ΔG lipo,CDS (octanol) = -0. Footnote: Drugs which are known to be passively transported or actively effluxed are shown with attached parentheses with their efflux ratios (AB Intrinsic Papp/AB Papp) as per ref [START_REF] Fredlund | In Vitro Intrinsic Permeability: A Transporter-Independent Measure of Caco-2 Cell Permeability in Drug Design and Development[END_REF]. Drugs labeled 'efflux' were used in deriving eq 5(d).

  Where R 2 = 0.222, SEE = 10.59, SE(ΔG desolvCDS ) = 0.51, SE(ΔG lipoCDS ) = 0.87, SE(DM)= 0.29, SE(Mol Vol) = 0.034, SE(HOMO-LUMO) = 1.58, F=2.62, Significance= 0.036

Eq 6 P

 6 app (AB) = -0.042Mol Vol + 21.12 Where R 2 = 0.422, SEE = 4.50, SE(Mol Vol) = 0.011, F=12.40, Significance= 0.0026=P-Value

39 Where R 2

 392 = 0.516, SEE = 0.789, SE(ΔG desolvCDS ) = 0.083, SE(ΔG lipoCDS ) = 0.064, SE(Mol Vol) = 0.084, F=11.39, Significance= 0.00003(b) SARS-CoV-2 infected Caco-2 cells: inhibition and drug transport into cellsEllinger et al[START_REF] Ellinger | Identification of inhibitors of SARS-CoV-2 in-vitro cellular toxicity in human (Caco-2) cells using a large scale drug repurposing collection[END_REF] investigated a collection of 5632 compounds including 3488 compounds which have undergone clinical investigations (marketed drugs, phases 1 -3, and withdrawn) for their inhibition of viral induced cytotoxicity using the human epithelial colorectal adenocarcinoma cell line Caco-2 and a SARS-CoV-2 isolate obtained from an individual originally exposed to the virus in the Wuhan region of China. A total of 64 compounds with IC 50 <20 μM were identified in the primary screen, including 19 compounds with IC 50 < 1 μM. ACE2 is expressed in Caco-2, Calu-3 and Vero-6 cells on the apical membrane domains. Primary screening was performed at 10μM compound concentration, at a virus multiplicity of infection (MOI) of 0.01 and a virus incubation period of 48 h, to ensure multiple viral replication cycles. It is thought that the assay procedure is able to confirm the activity of clinically relevant compounds against both viral entry and replication mechanisms. Remdesivir was found to show the same inhibition of SARS-CoV-2 in Caco-2 and Vero-6 cells (IC 50 0.77 μM), and interestingly, higher potency IC 50 0.07 μM in SARS-CoV and MERS.Analysis of Ellinger's IC 50 data for SARS-CoV-2 in Caco-2 cells [Fong 28,[START_REF] Fong | Predicting inhibition of repurposed drugs for SARS-CoV-2 viral activity and cellular entry[END_REF] for 56 widely structurally diverse drugs is shown below in eq 8(a). No correlation was found with the dipole moment. The molecular volumes have been scaled by 35 times to allow a direct comparison of the relative magnitudes of the four molecular specifiers. Eq 8(a) IC 50 = 0.76ΔG desolv,CDS + 0.82ΔG lipo,CDS + 1.71Mol Vol -0.09(HOMO-LUMO) + 4.34 Where R 2 = 0.174, SEE = 5.55, SE(ΔG desolvCDS ) = 0.30, SE(ΔG lipoCDS ) = 0.34, SE(Mol Vol) = 0.54, SE(HOMO-LUMO) = 0.73, F=2.69, Significance= 0.041 It is clear that eq 8(a) shows a small dependency on the HOMO-LUMO gap, which can be eliminated to give eq 8(b) which is more significant by virtue of the elimination of the HOMO-LUMO variable: Eq 8(b) IC 50 = 0.76ΔG desolv,CDS + 0.83ΔG lipo,CDS + 1.71Mol Vol + 4.34 Where R 2 = 0.174, SEE = 5.55, SE(ΔG desolvCDS ) = 0.30, SE(ΔG lipoCDS ) = 0.34, SE(Mol Vol) = 0.54, F=3.64, Significance= 0.018

  or 9(b) indicate intracellular active processes are occurring. Eq 9(a) IC 50 = 2494.7(HOMO-LUMO) -13187 Where R 2 = 0.897, SEE = 1329.4, SE(HOMO-LUMO) = 422.5, F=34.84, Significance= 0.004=P-value Eq 9(b) IC 50 = 2880.9(LUMO) + 2898.8 Where R 2 = 0.850, SEE = 1607.4, SE(LUMO) = 606.4, F=22.6, Significance= 0.009=P-valueAnalysis of the limited data (5) of Klann[START_REF] Klann | Growth Factor Receptor Signaling Inhibition Prevents SARS-CoV-2 Replication[END_REF] for the inhibition of SARS-CoV-2 infected Caco-2 cells by inhibition of growth factor receptor signaling gives eq 10. No dependencies were found with HOMO, or HOMO-LUMO, or the other independent variables ΔG desolv,CDS , ΔG lipo,CDS, Dipole Moment, or Mol Vol. Eq 10 indicates intracellular active processes are occurring. Eq 10 IC 50 = 5.47(LUMO) + 10.94 Where R 2 = 0.946, SEE = 0.54, SE(LUMO) = 0.64, F=71.2, Significance= 0.0035=P-value Eqs 9(b) and 10 are very similar inhibitory studies of mechanism of virus replication processes in infected Caco-2 cells. (c) In vitro and docking binding energies of SARS-CoV-2 inhibition: M pro , ACE2, S-RBD, TMPRSS2

  561, SEE = 7.47, SE(ΔG desolvCDS ) = 1.02, SE(ΔG lipoCDS ) = 1.09, SE(HOMO-LUMO) = 5.11, F=4.69, Significance=0.024

  0.658, SEE = 1.07, SE(LUMO) = 0.57, SE(Molec Vol) = 0.16, F=20.20, Significance=0.00001 Eq 18(b) eliminating the weaker molecular volume specifier gives: Docking BE (TMPRSS2) = -2.97LUMO -13.87 Where R 2 = 0.547, SEE = 1.20, SE(LUMO) = 0.58, F=26.55, Significance=0.00003

  (a) or (b): Eq 19(a) Docking BE (TMPRSS2) = 1.09HOMO-LUMO -0.49 Molecular Volume -10.64 Where R 2 = 0.715, SEE = 0.85, SE(HOMO-LUMO) = 0.62, SE(Molec Vol) = 0.38, F=8.75, Significance=0.012 Eq 19(b) Docking BE (TMPRSS2) = 1.67HOMO-LUMO -15.00 Where R 2 = 0.649, SEE = 0.88, SE(HOMO-LUMO) = 0.62, F=14.77, Significance=0.005

  Where R2 = 0.208, SEE = 10.35, SE(DM)= 0.28, SE(HOMO-LUMO) = 1.35, F=6.42, Significance= 0.0033 Analysis of the P app (BA) for 17 drugs (see Table 1(a)) gives eq 4 with similar results as those

3.64(HOMO-LUMO) + 38.75 shown in eq 3(b) with negligible dependencies on ΔG desolv,CDS , ΔG lipo,CDS (both n-octane and noctanol) or Mol Vol: Eq 4 P app (BA) = -1.76DM -3.84(HOMO-LUMO) + 49.68 Where R 2 = 0.463, SEE = 10.83, SE(DM)= 0.54, SE(HOMO-LUMO) = 2.46, F=6.04, Significance= 0.012

  Fredlund to allow an estimate of passive permeability. The intrinsic P app values are analyzed in eq 5(b). P-values for ΔG desolv,CDS , ΔG lipo,CDS (octanol), Mol Vol and HOMO-LUMO were 0.0006, 0.063, 0.00011, 0.0055 were highly significant whereas the values for ΔG lipo,CDS (octane) 0.31 and DM 0.90 were not significant.

	Eq 5(b)
	Intrinsic P app (AB) = -6.83ΔG desolv,CDS -3.91ΔG lipo,CDS -7.14Mol Vol -16.27(HOMO-
	LUMO) + 119.58
	Where R 2 = 0.301, SEE = 25.45, SE(ΔG desolvCDS ) = 1.88, SE(ΔG lipoCDS ) = 2.06, SE(Mol Vol) = 1.71, SE(HOMO-LUMO) = 5.62,
	F=5.72, Significance= 0.00067

Table 4 )

 4 pass through Caco-2 cells by passive diffusion based on dependency on log D octanol and molecular weight (MW) of the 36 tested drugs. LFER analysis again found that permeability of 36 neutral drugs was only statistically dependent on the ΔG desolv,CDS , ΔG lipo,CDS (octanol only), and molecular volume as shown in eq 7, as evaluated against P-values for all other independent variables ΔG desolv,CDS , ΔG lipo,CDS, DM, and HOMO-LUMO gap. Interestingly no correlation was found with ΔG lipo,CDS (octane) with a P-value of 0.68. P-values for ΔG desolv,CDS , ΔG lipo,CDS (octanol), Mol Vol and HOMO-LUMO were 0.0020, 0.00001, and 0.00011 were highly significant. P-values for DM 0.31, HOMO-LUMO gap 0.42 and ΔG lipo,CDS (octane) 0.58 were not significant. Eq 7 indicates that both transcellular (dependency on ΔG desolv,CDS and ΔG lipo,CDS ) and paracellular transport (dependency on molecular volume) processes are occurring concurrently. The molecular volume coefficient was multiplied by 35 to allow scaling comparisons with the ΔG desolvCDS and ΔG lipoCDS (octanol) coefficients. A strong correlation was found between the calculated ΔG lipoCDS (octanol) and the experimental log D octanol for both 36 neutral and 11 charged species (see Materials and methods). Inclusion of 11 charged species to

18 Molecular Volume -8.16

	Where R 2 = 0.52, SEE = 0.45, SE(HOMO) = 0.56, SE(Molec Vol) = 0.09, F=3.27, Significance=0.109
	Using Choudhary's molecular docking binding energy [34] of 10 drugs to the ACE2 receptor
	gives eq 15;
	Eq 15
	Binding Energy (ACE2) = -1.14ΔG lipo,CDS -0.58Mol Vol -8.35HOMO -64.72
	Where R 2 = 0.73, SEE = 1.42, SE(ΔG lipoCDS ) = 0.33, SE(Mol Vol) = 0.30, SE(HOMO)= 3.08 , F=5.29, Significance= 0.040
	Eq 16(a)
	Binding Energy (S-RBD) = 2.585HOMO + 6.83
	Where R 2 = 0.958, SEE = 0.22, SE(HOMO)= 0.27 , F=90.98, Significance= 0.00067
	Eq 16(b) including ΔG lipo,CDS and molecular volume gives:
	Binding Energy (S-RBD) = 0.11ΔG lipo,CDS + 0.15Mol Vol + 2.845HOMO + 8.66
	Where R 2 = 0.981, SEE = 0.21, SE(ΔG lipoCDS ) = 0.075, SE(Mol Vol) = 0.01, SE(HOMO)= 0.32 , F=33.68,
	Significance= 0.029

Similarly using Choudhary's molecular docking binding energy

[START_REF] Choudhary | Identification of SARS-CoV-2 Cell Entry Inhibitors by Drug Repurposing Using in silico Structure-Based Virtual Screening Approach[END_REF]

of 6 drugs to the SARS-CoV-2 receptor binding domain of the virus spike protein (S-RBD) gives eq 16(a) or 16(b);

Table 1 . P app (AB) for Caco-2 cells [6] for 52 widely structurally diverse drugs: eq 3(b)

 1 63 logD + 1.15 where SE=1.95, F=14.54, significance 0.0005 was found for the experimental log D for 36 neutral species from ref 12, or by including 11 charged species in addition to the 36 neutral species ΔG lipo,CDS (octanol) = -0.705 logD + 1.23 where SE=2.05, F=23.20, significance 0.00005. The relationship ΔG lipo,CDS (octanol) = ΔG lipo,CDS (octane) + 2.345 where SE=2.25, F=2.28, significance 0.14 for 36 neutral species was far weaker.

		P(AB)	ΔG desolv,CDS	ΔG lipo,CDS	Dipole	Molec	HOMO-
		10^-6	kcal/mol	octanol	Moment	Volume	LUMO
		cm/s		kcal/mol	D	cm 3 /mol	gap eV
	Aminopyrine	36.5	-5.4	-0.12	6.93	173	5.07
	Caffeine	30.8	-5.29	0.74	5.7	126	5.04
	Nevirapine	30.1	-4.04	-1.07	3.67	216	4.33
	Phenytoin	26.7	-7.39	1.08	3.75	175	5.72
	Testosterone	24.9	-6.27	-1.9	4.98	212	5.29
	Phencyclidine	24.7	-3.64	-4.29	1.26	218	5.48
	Desipramine	24.4	-3.74	-3.66	2.35	209	5.05
	Progesterone	23.7	7.51	-1.48	7.61	280	5.23
	Clonidine	21.8	-0.44	-3.69	2.97	151	5.11
	Propranolol	21.8	-4.64	-1.75	4.14	206	4.74
	Chlorpromazine	19.9	-2.87	-4.28	2.96	271	4.44
	Meloxicam	19.5	-7.24	2.67	17.55	257	3.57
	Nicotine	19.4	-0.99	-1.71	3.19	133	5.23
	Pindolol	16.7	-4.79	-0.78	4.31	200	5.22
	Telmisartan	15.1	-7.91	-3.26	3.2	335	4.10

Table 1 (a). P app (BA) for Caco-2 cells [6] for 17 drugs: eq 4

 1

	P(BA)	ΔG desolv,CDS	ΔG lipo,CDS	Dipole	Molec	HOMO-
	10^-6	kcal/mol	octanol	Moment	Volume	LUMO
	cm/s		kcal/mol	D	cm 3 /mol	gap eV

Table 2 . P app (AB) and intrinsic P app (AB) for Caco-2 cells [24] for 58 drugs

 2

	P(AB)	P (AB)	ΔG desolv,CDS	ΔG lipo,CDS	Dipole	Molec	HOMO-
	10^-6		Intrinsic	kcal/mol	octanol	Moment	Volume	LUMO
	cm/s		10^-6		kcal/mol	D	cm 3 /mol	gap eV
			cm/s
	Acetaminophen (2.0	30	58	-5.25	1.76	5.74	111	5.51
	efflux)
	Acyclovir (0.88 passive)	0.19	0.16	-2.81	-0.39	18.55	187	5.51
	Amiloride (0.42)	1.5	0.64	-2.33	0.14	10.35	148	4.11
	Amitryptyline (1.0	12	13	-3.41	-4.12	1.73	212	5.27
	passive)
	Antipyrine (1.7)	36	60	-5.33	0.27	8.38	147	5.07
	Atenolol (0.83 efflux)	0.33	0.28	-5.88	0.64	3.44	247	5.80
	Atorvastin (2.0 efflux)	13	25	-13.73	0.29	6.52	394	4.75
	Atropine (0.93 passive)	4.3	4	-3.99	-2.27	3.78	251	5.54
	Betaxolol (1.6)	12	19	-6.03	-2.40	4.07	214	5.83
	Bumetanide (1.0)	1.6	1.7	-8.79	2.88	9.39	203	4.34
	Buspirone (1.2 passive)	11	13	-3.52	-4.89	4.78	300	4.75
	Bisoprolol (1.5)	6.8	10	-7.11	-1.39	4.6	293	5.87
	Caffeine (1.5 passive)	59	90	-5.29	0.74	5.7	126	5.04
	Chloramphenicol (1.1)	11	12	-8.63	2.58	8.31	153	4.02
	Chlorothiazide (0.91	0.39	0.35	-3.72	3.56	15.94	166	5.11
	passive)
	Cimetidine (1.1 passive)	0.6	0.68	-4.6	-2.89	4.05	184	6.26
	Clozapine (1.0 efflux)	15	15	-1.17	-4.48	7.4	231	4.52

: eqs 5(a), 5(b) and 5(d)