2022 年 22 巻 p. 13-20
Diffusion is a spontaneous process and one of the physicochemical phenomena responsible for molecular transport, the rate of which is governed mainly by the diffusion coefficient; however, few coefficients are available for small molecules. We have constructed a simple and convenient experimental system with agar-gel to measure the diffusion coefficients of sugars.
A theoretical method has also been developed to estimate diffusion coefficients by a combination of molecular modeling and the Stokes–Einstein equation, by which the coefficients of several sugars, amino acids, and drug molecules have been obtained. This time we have applied both experimental and theoretical approaches to estimate the diffusion coefficients of an additional 10 amino acids. The measured and calculated values are consistent with small deviations, i.e., the diffusion coefficients estimated by molecular modeling correspond well to the experimental data, which suggests that the potential use of the diffusion coefficient as an additional molecular property in drug screening has been enhanced.
Diffusion is one of the molecular transport mechanisms by which a substance is distributed, and it is a spontaneous physicochemical phenomenon in itself. Diffusion also plays an important role in the life sciences [1,2]. For example, following the administration of a pharmaceutical, drug molecules are transported via the bloodstream and distributed to organs by active and passive transport. Diffusion is the driving force behind passive transport [2,3]; therefore, physicochemical information on drug diffusion is useful for the analysis of drug delivery systems and pharmacokinetics, as well as investigation of the distributions or diffusion velocities of central nervous system drugs in the brain after they penetrate the blood-brain barrier [4–6]. One of our goals is the quick and simple estimation of diffusion coefficients that can be used in drug screening as an additional physicochemical molecular descriptor [7].
With regard to the seeking of drug seeds from bacteria, which is one of the initial steps in drug discovery, a multidimensional diffusion-based gradient culture system for bacteria has been devised [8]. This circular culture apparatus with agar-gel represents one application of diffusion in the life sciences. In relation to this culture system, we have recently developed a method to estimate the translational diffusion coefficients (hereinafter referred to as diffusion coefficients) of small molecules using diffusion experiments in agar-gel [9]. Although not a few experimental values of diffusion coefficients related to dialysis have been reported [10,11], the number of published diffusion coefficients of small molecules are not many. One possible reason for this is that diffusion coefficients are measured based on complicated procedures such as the use of a synchronous pump under a stationary state [12,13]. The developed experimental agar-gel method is characterized by its simplicity and takes advantage of the vast inclusion of water by agar-gel.
We have more recently developed a theoretical method to estimate diffusion coefficients by the combination of molecular modeling and the Stokes–Einstein equation, by which we have obtained consistent results for several sugars and amino acids (alanine, proline, threonine, leucine, aspartic acid, and arginine) compared with those obtained by the agar-gel experiments [14]. In the present study, we applied both experimental and theoretical approaches to estimate the diffusion coefficients of an additional 10 amino acids and these values are reported here.
For the agar-gel measurements, amino acid samples were diffused for 3 days using a polycarbonate tube with a diameter of 1.0 cm that consists of a 13 mm long cylinder of agar-gel containing the sample at a concentration of 4–10 mg/mL in contact with a 127 mm long cylinder of agar-gel alone that acts as a diffusion layer, as shown in Figure 1. The agar concentration of both layers was around 0.8 w/v%. The diffusion coefficients Dc, were estimated from the relationship between the diffusion distances at 2 mm intervals and the relative concentrations of the samples there based on the reported method [9].
Figure 1. Schematic overview of the agar-gel diffusion experimental apparatus
For molecular modeling, the Molecular Operating Environment (MOE) software system developed by the Chemical Computing Group was used [15]. The carboxy group and the amino group were treated as the ionized forms. The imidazole group of histidine was treated as the nonprotonated form. The stable conformations of molecules were calculated using the Low Mode MD module with the force field MMFF94x [16]. Approximated spherical molecules were built based on the van der Waals volume vdwV, for the stable conformations with ΔE < 3 kcal/mol, as shown in Figure 2. The simple approximated radius rs, was then derived by averaging the radii ri of the approximated molecules by taking the Boltzmann distribution based on ΔE into account [14]. Ds in water was finally calculated for rs using the Stokes–Einstein equation:
| （1） |
where kB is the Boltzmann constant, T is the absolute temperature, andη0 = 0.8902 mN·s/m2 is the water solvent viscosity [17].
Figure 2. Derivation of the simple approximated radius rs of a molecule
Approximated spherical molecules were built based on the van der Waals volume vdwV for the stable conformers with ΔE < 3 kcal/mol. The simple approximated radius rs is then derived as the weighted average of the approximated spherical molecules. The weight (wi) is calculated by the Boltzmann distribution on the basis of ΔEi.
Diffusion coefficients were estimated for the 10 amino acids listed in Table 1. These results show the number of conformers with ΔE < 3 kcal/mol, the simple approximated radius rs, and the diffusion coefficient Ds obtained via molecular modeling, in addition to the experimental diffusion coefficients Dc. Due to the extent of the flexibility of lysine and methionine, the numbers of their conformers with ΔE < 3 kcal/mol are relatively greater. The number of isoleucine conformers with ΔE < 3 kcal/mol is the same as that of leucine obtained in our previous study [14]. According to the Stokes–Einstein equation, the diffusion coefficient of a molecule is expected to increase in inverse proportion to its approximate radius. Therefore, a larger molecular weight is related to a larger approximate radius, so that the diffusion coefficient is generally smaller. Although this broad trend is readily discernible in the data presented here, a close perusal of Table 1 reveals some disorder, probably related to the molecular conformations.
Table 1. Approximated radii and diffusion coefficients of amino acids and drugs
| Molecule 1 | MW 2 | NoC 3 | rs | Diffusion coefficient (×106 cm2/s) | ||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Ds | Dc 4 | drvD0 5 | Ds − drvD0 | drv2D0 6 | Ds – drv2D0 | |||||
| serine | 105 | 1 | 2.72 | 8.82 | 8.89 | 9.54 | -0.72 | |||
| valine | 117 | 3 | 3.05 | 8.03 | 7.70 | 8.35 | -0.32 | 8.14 | -0.11 | |
| isoleucine | 131 | 7 | 3.20 | 7.66 | 7.45 | 8.10 | -0.44 | 7.89 | -0.23 | |
| asparagine | 132 | 3 | 3.00 | 8.20 | 7.50 | 8.15 | 0.05 | 7.94 | 0.26 | |
| lysine | 146 | 8 | 3.28 | 7.47 | 6.97 | 7.62 | -0.15 | 7.41 | 0.08 | |
| glutamic acid | 147 | 3 | 3.08 | 7.96 | 7.45 | 8.10 | -0.14 | 7.89 | 0.06 | |
| methionine | 149 | 6 | 3.21 | 7.64 | 7.41 | 8.06 | -0.42 | 7.85 | -0.21 | |
| histidine | 155 | 2 | 3.21 | 7.62 | 7.21 | 7.86 | -0.24 | 7.65 | -0.03 | |
| phenylalanine | 165 | 2 | 3.40 | 7.21 | 6.79 | 7.44 | -0.23 | 7.23 | 0.02 | |
| tryptophan | 204 | 4 | 3.60 | 6.81 | 6.27 | 6.92 | -0.11 | 6.71 | 0.10 | |
| alanine* | 89 | 1 | 2.72 | 9.01 | 9.21 | 9.86 | -0.85 | |||
| proline* | 115 | 2 | 2.97 | 8.25 | 7.74 | 8.39 | -0.14 | 8.18 | 0.09 | |
| threonine* | 119 | 2 | 2.95 | 8.31 | 7.99 | 8.64 | -0.33 | 8.43 | -0.12 | |
| leucine* | 131 | 7 | 3.20 | 7.66 | 7.00 | 7.65 | 0.01 | 7.44 | 0.22 | |
| aspartic acid* | 133 | 2 | 2.92 | 8.39 | 7.90 | 8.55 | -0.16 | 8.34 | 0.05 | |
| arginine* | 174 | 2 | 3.41 | 7.19 | 6.80 | 7.45 | -0.26 | 7.24 | -0.05 | |
| Aspirin* | 179 | 4 | 3.37 | 7.27 | 6.98 | 7.42 | -0.15 | |||
| Salbutamol* | 239 | 6 | 3.91 | 6.27 | 6.01 | 6.45 | -0.18 | |||
| Loxoprofen* | 246 | 8 | 3.89 | 6.30 | 5.91 | 6.35 | -0.05 | |||
| Fast Green | 763 | 30 | 5.41 | 4.54 | 3.65 | 4.09 | 0.45 | |||
1 Data for the molecules with an asterisk were obtained in our previous work [14]. 2 Molecular weight. The carboxy and amino groups were treated as free and protonated forms, respectively. 3 Number of conformers with ΔE < 3 kcal/mol. 4 Diffusion coefficients estimated from measurements with agar-gel. The concentration of amino acids was 10 mg/mL except for glutamic acid at 6 mg/mL and that of the drug was around 10 mg/mL. 5 Values extrapolated to the infinitely dilute solution are derived as Dc + 0.65 based on previous work [14]. 6 Values extrapolated to the infinitely dilute solution are derived as Dc + 0.44 in this study.
D0 is the diffusion coefficient extrapolated to the infinitely dilute solution based on the diffusion coefficients obtained from solutions with different concentrations, Dc. Dc for finite concentration solutions is generally observed to be smaller than D0 with no inter-solute interactions. Because no intermolecular interactions are taken into account in the modeling, D0 corresponds to Ds derived by the theoretical method. D0 is only reported in the literature for sugars [18,19], so D0 for other molecules was reasonably approximated by adding the shift value of 0.65×10-6 cm2/s to Dc in our previous study [14]. Therefore, D0 derived in the same way are listed as drvD0 in Table 1, together with Ds − drvD0. In most cases, the absolute values of Ds − drvD0 are small, so that Ds and Dc are considered to be consistent. The difference Ds − drvD0 for serine, which has the smallest molecular weight of 105 among the 10 amino acids, as well as rs < 3 Å, was larger than that of the other molecules. The diffusion coefficients of molecules with a small radius of less than 3 Å may be corrected as suggested by Edward [20]. In the case of serine, the multiplication of Ds (8.82) by 1.1, as with alanine in our previous study, provided better agreement with drvD0 (9.54). Excluding serine, a rather small average deviation of 0.23×10-6 cm2/s was obtained, similar to that of 0.24×10-6 cm2/s in our previous work [14].
Table 1 also lists data previously obtained for 6 amino acids and 3 drugs (aspirin, salbutamol, and loxoprofen). Figure 3 also shows the correlation between Dc and Ds for all the 14 amino acids in this and the previous study, except for alanine and serine, for which separate corrections were necessary. The blue line in Figure 3 is the linear regression for the data points that represent each molecule, with the squared correlation coefficient, R2 = 0.91. The approximate line is almost parallel to the dashed line of Dc = Ds. Therefore, if the shift value of 0.44×10-6 cm2/s that was calculated to minimize the average error between Ds and D0 for the 14 molecules is used, Figure 4A is obtained, which shows that Ds well reproduces drv2D0 for all the compounds within the 95% prediction interval shown by the red dashed lines. Here, drv2D0 was derived by using the shift value of 0.44×10-6 cm2/s and drv2D0 as well as Ds – drv2D0 are added to Table 1.
From a different point of view, Dc can be well predicted by subtracting 0.44×10-6 cm2/s from Ds for amino acids with a concentration of around 10 mg/mL. Amino acids have a variety of functional groups, just as those that constitute drug molecules; therefore, this approach is also reasonable for drug molecules. The current shift value was then applied to the 4 drug molecules (Chart 1) listed in Table 1, and the results are shown in Figure 4B. All the molecules were within the 95% prediction interval. Fast Green was included and was nominally classified as a drug in the previous study because its molecular weight corresponds the upper limit of the drug candidates [21,22]. Although the deviation of Fast Green was larger than those of the other 3 drugs, it is still within the 95% prediction interval, which suggests that this approach can be reasonably applied to most drug candidates.
The previous shift value of 0.65×10-6 cm2/s was derived for sugars and the experimental concentrations of sugars (50 mg/mL) were around five times higher than those of the amino acids and drugs; the actual shift value for amino acids is considered to be smaller than 0.65×10-6 cm2/s if the concentration dependency of the diffusion coefficient is similar between sugars and other molecules. Therefore, the smaller shift value of 0.44×10-6 cm2/s applied to obtain Figure 4 does not contradict the above reasoning. If this shift value is used, the average deviation between Ds and drv2D0 is 0.14×10-6 cm2/s.
The following regression equation between molecular weight MW and reported diffusion coefficient D is presented by Takezawa et. al [10]:
| (2) |
The average deviation between D derived by the above equation and Dc is 3.57×10-6 cm2/s. If the same approach is applied for the compounds in Table 1 excluding serine, the similar regression equation shown below is obtained:
| (3) |
In this case, the average deviation becomes 0.18×10-6 cm2/s, which is as small as 0.14×10-6 cm2/s derived above. The slight difference of deviations might arise from the molecular conformations considered in our novel approach.
Figure 3. Correlations between Dc and Ds for the 14 amino acids
Figure 4. Correlation between (A) Ds and drv2D0 with a shift value of 0.44×10-6 cm2/s for the 14 amino acids, and (B) between Ds and drv2D0 for the 4 drug molecules In both cases, the red dashed lines indicate the 95% prediction interval.
Chart 1. Chemical structures of the four drugs examined in our previous study
It was thus reconfirmed that the diffusion coefficients estimated by molecular modeling correspond reasonably well with the experimental data, i.e., both experimental and theoretical approaches can provide consistent diffusion coefficients. The experimental method makes full use of the nature of agar-gel. Due to the inclusion of water of more than 99 w/v%, diffusion in the agar-gel is observed as with that in liquid water, even though it appears to be a semi-solid. For the molecular modeling, a series of calculations can be performed rather quickly. As a result, the number of diffusion coefficients elucidated have been doubled with respect to amino acids. In drug screening, it is advisable to have sufficient physicochemical molecular descriptors, such as pka and clogP [7,23]. We envisage that these approaches to estimate diffusion coefficients will be used for that purpose. Molecular dynamics simulations of small molecules in bulk water are currently in progress, and it is expected that refined stable conformations will be obtained and firmly hydrated water molecules that may be moved with solutes can be examined.
The diffusion coefficients of 10 amino acids were determined by an experimental method with agar-gel and by a theoretical approach with molecular modeling. The results show that the diffusion coefficients estimated by molecular modeling correspond well to the experimental data when the shift value of 0.44×10-6 cm2/s is used to derive D0, which implies that the potential use of the diffusion coefficient as an additional molecular property in drug screening has been enhanced.
