1)RUG P O T E N C Y .\SD HYUROPHOHICITY
given concentration will enter a lipid phase is k while the rate constant for the reverse process is 1. The partition coefficient is, of course, k / l . It is assumed here that the nature of the intervening aqueous phases are essentially the same a s aqo, and correspondingly the nature of the intervening lipid phases are essentially the same as lipl. Thus, regardless u here the drug is in the maze of inter- and intracellular fluids and membranes, k and 1 remain r,sentially unaltered. If thr rate :It which the drug-receptor complex forms is slow with respect to the speed that an equi1ibr:um condition is approached in these intervening phases, then the various lipid barriers will have little influence upon the potency of a drug because the concentration of drug in all aqueous phases will be essentially the same. Thus, under equilibrium conditions, the potency of a drug should be independent of its partition coefficient insofar as getting the drug to the receptor site is the only role the partition weficient p l a j x Where the partition coefficient is a direct reflection of a drug's hydrophobic bonding capacity in the drug-receptor complex, then it will once again have a n important influence on the drug's potency. AIore often than not, however, the drug does not have sufficient time to establish a n equilibrium condition in a biological system. Besides getting to its intended target, it also wanders to other corners of the system where it is metabolized or excreted. Only a small fraction of the total drug administered actually reaches the receptor site, and it must do so within a limited time or else the reservoir of drug in aq, will be depleted through other losses to the point where it will no longer be able to maintain a n adequate concentration a t the more distant receptor site aq,. Under the conditions of determining the potency of a drug, just enough compound is administered to effect a submaximal response. I:rom the foregoing discussion one mould expect and frequently observes that the response appears soon after the administration of the drug, i.e., when the drug in the reservoir phase is a t its maximum concentration, and then the response fades as the concentration is reduced through various losses. I n this situation the standard biological response should occur before a significant amount of drug has left aqo. Therefore, it can be assumed that the concentration of drug molecules a t the receptor site is quite small with respect to the concentration in the reservoir phase, and it can be calculated by multiplying the concentration in aqo times the probability of a drug molecule reaching the receptor site. The coiicentration in aqo is, of course, that which gives the standard response and forms the basis of the potency index, log l/C. I t remains for us to estimate the probability factor. Naturally one should not expect to do this in an absolute sense, but there should be some means to estimate the relative probabilities of the various members of a drug series from the model given in IYgure 1. The following is offered as such a method. The probabilitj- of a molecule reaching aq, from aqo is given by eq 4. Eq 4 assumes that there is only one
path to the receptor and that there is a n exact (although unknown) number of aclueous-lipid interfaces to crass. IYhile it is highl). improbable that there is only one
Journal of Jledicinal Chewiistrg, 1970, F'ol. 13, S o . 6 1193
pathway available to a drug, it is convenient to develop this equation a t this point. Branched and alternate pathways will be discussed later on. As a simplifying assumption we xi11 consider that the probability of a neutral molecule moving from a n aqueous phase into a lipid phase is the same thrcughcut the system, such that eq 5 is true. P0.l
= P2.3 = P n - - s , n - l
Similarly, the reverse situation is given by eq 6. By P1.2
=
P3.4
(6)
= Pn-l,?l
combining eq 5 and 6 with eq 4, eq 7 is obtained. It
Po,, = (po,l)n'2. (Pl,2)"'2 (7) will be assumed here that the probability of a drug transferring from aqo to lip1 is determined at the interface of the phases. S o t all drug molecules reaching the interface will succeed in penetrating it; the preference of entering lip, or being reflected to aqo is determined by the relative values of k and 1. When k is larger than 1 most drug molecules will readily enter lip, and a lesser fraction will return to aqo; u hen 1 is greater than k the reverse situation obtains. Thus, within a particular time span the number of molecules entering lipl will be proportional to 12, n hile the total number of molecules presented the opportunity to enter lip, mill be proportional to the sum of k and 1. Hence, Po.lis given by eq 8. Division of the upper and lower parts of
k
P0,l
= __
k+l
the fraction in eq 8 by 1 gives eq 9. I n the reverse (9)
situation, Pl,ais the probability of a molecule in lip, getting into aqo, and it is given by eq 10. However, P1,O
=
1 - Po1
P1,2
=
1-
(10) according t o the model given in Figure 1, Pl,o is not different from PI,?.hence: (11)
P0,l
Substituting this new expression into eq 7 we have: P o , n = (Po,1)n'2(1 -
Po n
=
($;-l)n'2(1
Po,,=
Po,l)n'2
-
k/l
(k//)n
(kll
n'2
i,il+l)
(2 )"'*('1--) k!l + 1 k'l + 1 P o n =
(12)
+ 1F
ni2
(13)
(14) (15)
Thus, according t o eq 15, the probability of a drug reaching a receptor site is a function of its partition coefficient and the number of intervening aqueouslipid interfaces between aqo and aq,. P s eq 15 stands it is difficult to envision even qualitatively how the probability is changing as k / l and n vary. H o ~ e v e r s?me , insight is gained by examining Table I mhich gives solutions to eq 15 for selected values of k/Z and n. I n situations where the drug must Dass through
-
-
:it least one lipid phase, then increasirig difficulty is met
( i ) the partition coefficient becomes very large or very small, m d (ii) the number of lipid barriers increases. Where / I > 1* a maximum probability is found when k l l is eyual to unity (1.00). Where ) i = 2 or more, the probability of 21 drug reaching the receptor region drops to nil when the partition coefficient is infinite 01' is zero. This mathematical conclusion is intuitively satisfying, since if ii drug has an infinite partition coefficient, we can hon- the drug would he readily taken up by lipl, but it would never be able to leave that ph:ise; on the other hatid, ti drug which as :I parition coefficient of zero CHI^ iievcr get into lip, in thci first place. m d hence is bloclied from reaching ays. When the probabilities from Table I :ire plotted against log li ,I the family of c u r ~ win 1'igiir.c. 2 is oh:IS
I t is of interest to see if' the probability nrgument o f the present work gives conclusions simihr to thoso of the Hansch school of thought, L i s ii first step on(' c:iii convert the probabilitiey of Table I illto their Iogartlimic form :is has been clorie in Table 11. X plot of thcs(1 tiumhers ngainst the corresponding vdues f ~ i I, ( J1%. ~1 rrsults i n thc family of r ~ i r v shown (~ in Iigure 3. ( ) I I
I
P
I
n-2
-I
.o
I
-I ,54
-2.0-
I
-2.5-
-3.0-
0.05\
0.00
qJA\ -3.0
-2.0
0.0
-1.0 log
1.0
2.0
3.0
212
Figiire "--The probability of a drug reaching a rerptor wgiorr fiinrtioti ( J f the log partit,ioIi coeffirierit (log k : l ) and of I ! , the iirimher of interfaces separating the site of driig appliraiioti f ri)m the rewpt 01'region. ( ~ 1 , a~s)I
tnined. Each member of this family has its maximum value at log k l l = 0, :ind i.: remiriiwwt of a normal dist rihution curve.
iiiipection o11(' IS tempted to believe that the curve.: :ire p:iraljolus. Hut claw anal> reveal- that thi.; i h not the caw. I'or an? particular bet of data i n Table I1 n here u > 1, a parabola can be fitted to the set of point. log Po n , log 1, / Regardless of the value of chosen. the correlatioii coefficient of the regression curve 1: ~ I w a > s) = 0.985, I e . , 97.0% of the variance is : counted for hj. regression. The "unexplained" 3.0 of the variance t i too large to be result of round-off' error. However. \\hen one examines the differences betneeti the input values of log Po and the value3 calculated from the regression equation it becomes obvious that the differences themselves are a function of log k I and ) / (see Figure 4). Because these differmceh :ire not r:iti(Iorn it must be concluded th:it thc
TABLE I1 LOGARITHM OF PROBABILITY OF DRUG CROSSING n INTERFACES: LOGPo,, k/1
n = 2
n =I
n = O
n = 3
0.000 - Inf - Inf -1nf 0.000 -3.00 -3.00 -4.50 -2.52 -2..52 -3.79 0.000 -3.01 -2.00 -2.00 0.000 -1.54 -1.A5 -2.32 0.000 0,000 -1.04 -1.08 -1.62 -0.64 -0.75 -1.13 0.000 1 -0.30 -0.60 -0.90 0 . 000 -1.09 3 - 0 , 13 -0.73 0 . 000 0,000 -0.04 -1.08 10 -1.62 30 -0.014 -1.51 -2.26 0.000 -0.0044 -2.01 -3.01 100 0,000 0.000 -0.0017 -2.48 -3.72 300 0.000 -0,0004 -3.00 -4.50 1000 I nf 0.000 0.000 -1nf - Inf a Values of Table I converted into logarithmic form.
0 0 0 0 0
000 001 003 01 03 0 1 0 3
1og Po, n = 4
n = 5
n = 6
n = 6
-1nf -6.00 -5.05 -4.00 -3.10 -2.17 -1.51 -1.20 -1.4.5 -2.17 -3.00 -4.00 -4.95 -6.00 - Inf
- Inf -7.50 -6.31 -6.02 -3.87 -2.70 -1,89 -1.51 -1.82 -2.70 -3.76 -3.02 -6.20 -7.50 - Inf
- Inf -9.00 -7.57 -6.03 -4.65 -3.2;i -2.22 -1.81 -2.10 -3.25 -4.52 -6.03 -7.46 -9.00 - Inf
-1nf -12.00 -10.10 -8.03 -6.19 -4.34 -3.00 -2.41 -2.91 -4.34 -6.02 -8.03 -9.92 -12.00 -Tnf
1.0-
0.9-
0 80.70 60 .s-
0.40.3-
0.2-
-
0.0-0.1-0.2-
-0.3-0.4-
-0.5-0.6-
-0.7-0.8I
-3.0
n = 10
log h / l
- Inf
- Inf - 3 00 -2 ,i2 - 2 00 - 1 32 - 1 00 - 0 52 0 00 0 48 1 00 1 48 2 00 2 48 3 00 Inf
-15.00 -12.63 -10.04 -7.74 -5.41 -3.7,5 -3.01 -3.63 -5.41 -7.63 -10.04 -12.40 - 1.5.00 - Illf
similar to a normal distribution curve with its maximum at log k/Z = 0.00. The sum of such curves is still another curve of the same form,4and hence the presence of branched and alternate routes to the receptor site does not alter the basic relationship of Po,nto log k l l . The normal distribution curve therefore approximates this relationship for the probability considering all routes as well as an individual route. I n real situations there may be several independent routes involving the samenumber of interfaces, 1 1 . These routes of equal probability can be considered degenerate in analogy to quantum mechanics’ degenerate states, ie., states of equal probability. Thus, for molecules traveling in a set of degenerate routes, the probability of reaching a receptor is independent of the path taken. More important, however, routes involving the fewest interfaces are the routes of highest probability. Hence, these minimum pathways will bear the greatest traffic as far as drug molecules reaching the receptors are concerned. Obviously, the minimum pathway could be comprised of several degenerate routes, and this set of routes would be the primary access a drug molecule would have t o the receptors. Only a negligible number of molecules will reach the receptors by routes having
A !Og P
0.1
7
-2.0
- 1.0
0.0
log
1.0
2.0
3.0
&Id
Figiire 4.-The difference between values of log P in Table I1 and those calculated from a regression parabola used to corre1at)e the data.
relationship between log PO,, and log k / l is not, strictly speaking, parabolic, however, for all practical situations it may as well be. At this point the question of branched and alternate pathways can be taken up again. When more th$n one route of travel is open to a drug molecule then the probability of a drug molecule reaching its target will be the sum of the probabilities of the various routes available (see eq 16). I n eq 16 the left hand superscript of each
+
+
Po = (‘PO *Po,, 3P0,n. . . ’Po,,) (16) probability term within the parenthesis designates one identifiable route. However, each jP0,, term in eq 16 is a function of log k / l as illustrated by the family of curves in Figure 2 . Each member of this family is
(4) It would be difficult t o prove this assertion mathematically. but it can be readilydemonstrated t o be true for the case a t hand. .4 typical regression equation is t h a t for n = 2 (see eq i). log Po,? =
n
=
13
- 10.261
i 0.013) (log k / l ) 2
r2 = 0.970
-
(0.002 =iz 0.0221) (loo. k / l ) - 0.804 ( i ) s = 0,149 Fi.io = 191.6 I’ < 0.0008
Let us now consider t h e probabilities for three routes consisting of different values of n, s a y n = 2, n = 4, a n d n = 6. H y surnrnina these proha!iilities for different values of log k / l , a n d taking t h e logarithms thereof, rye can eventually arrive a t t h e regression eqiiation eq ii. log (Pa,?f PO,^
+ Po,a) =
-(0.271 i 0.016) (log k I ) ?
-
(0.00‘2 zt 0.026) (log k ? ) - 0.739
n = 13
r2 = 0.960
?,
= 0.177
F?.N = 146.6
I’
ls due to the greater round-off error inherent in eq ii. hlost interesting are t h e differenres butween the “observed” a n d calculated values given b) 130th equations. T h e differences associated with eq ii show t h e same regiilar foiirth-order dependence on log k / l a s those associated with eil i . These differences are observed for all values of n (see Figure 4). T h e addition of higher x.alued terms in n (e.g., Pa.*)t o eq ii will have very little effect on the outcome since such terms make a negligible contribution t o the overall probability relative to t h e lower ordered terms.
