activity relationships

195

Analyrrcu Chrmrca Acre, 235 (1990) 195-207 Elsevter Sctence Publishers B.V., Amsterdam - Pnnted m The Netherlands

Subcellular pharmacokinetics and quantitative structure/ time/ activity relationships STEFAN BALbj Depanment

**a, MICHAEL

of Medzcrcmnland Pharmaceutrcal

WIESE, HAN-LIN

CHI b and JOACHIM

Chemistry, Institute for Experrmental

K. SEYDEL

Blo(ogv and Medzcme, D-206/

Borstel (F R G)

(Recaved 7th July 1989)

ABSTRACT A reahsttc form of the cbsposttion function 1s denved to descnbe the kmetics of the mtraceIIular distrtbutton of extraneous compounds on the basis of their hydrophobtcity, actd-base properties, affimty to protems, and the rate parameters of ehmmation Comparative stmphctty of the expressions was achieved by usmg an expenmentally venfted time hierarchy for the processes that compounds undergo after their apphcation to btosystems. The function agrees well wrth published experimental data on the mmrobial degradatron of orgamc compounds. Apphcation of the approach to btosystems of varying complexity is discussed. The possibihty of usmg the resuIts for the construction of mechanistic quantitative structure/ time/ actrvlty relatronships (QSTAR) IS dlustrated by examples for non-tomzable compounds or compounds havmg identical acid-base properties (btoacttvtty/Iipopfuhctty profiles measured at a constant exposure time, and kmettcs of analgesic effects of fentanyl denvatives m rats).

Knowledge of relationships between the structure of extraneous compounds and their intracellular distribution is of great importance for various branches of biological science (analytical biochemistry, pharmacology, drug design, ecological toxicology, etc.). This problem has been treated frequently; diffusional transport [l-lo], accumulation in the membranes and other lipoid phases [l-12], protein binding [13] and also elimination in some cases [5,8,14,15] have been considered as the main processes determining the fate of lowmolecular-weight substances in biosystems. The main achievements of these attempts are six-fold: (1) formulation of the relation between the rate parameters of the transport in simple two-phase systems and the partition coefficient [3,6,16-181; (2) experimental proof of this relation

a Permanent address. Slovak Polytechmcs, Department of Btochemrcal Technology, CS-81237 Bratislava, Czechoslovakra. b Permanent address: Instrtute of Matena Medtca, Chmese Academy of Medical Science, Pekmg, Chma. 0003-2670/90/$03.50

0 1990 - Elsevter Science Publishers B.V.

not only for homologous and electroneutral molecules [19] but also for structurally diverse and/or ionizable compounds and ion-pairs [6]; (3) equations for the type of saturation [20], convex (parabolic [21], bilinear [4], composed of three or four linear parts [9]), and double-parabolic [5,22] dependences of the intracellular drug concentration at a certain moment after drug admmistration on the partition coefficient, which are valid for a series of compounds with zero, identical or hydrophobicity-dependent elimination rates; (4) modelbased relations for the equilibrium distribution of compounds differing in acid-base propertres [12]; (5) substitution of the static probability of reaching a predefined point in the cell withm an allotted time [21] by the dynamic disposition function, using as variables physicochemical properties of the compounds and the exposure time [23]; and (6) explicit expressions for the kinetics of the distribution in the water/ membrane/ water system [24], and for the unidirectional transport in an arbitrary biosystem [25], possibly combined with protein binding [13] and elimination [26].

196

Although much work has been done, the influence of the structure and physicochemical properties of compounds on their intracellular movements cannot be considered as completely understood. The models reported under (2) and (3) do not take into account the exposure time and possibly varying elimination rates within the series studied, in contrast to the models under (5) which, however, contain too many adjustable parameters to be evaluated easily with experimental data. The main aim of the present paper is to design and verify a kinetic description of the intracellular disposition of compounds which overcomes these deficiencies and to apply the results to the construction of quantitative structure/ time/ activity relationships (QSTAR).

METHODS

Time hierarchy The transport rate of low-molecular-weight compounds depends on their properties and on the complexity of the biosystem. For nonamphiphilic and neither extremely hydrophilic nor extremely hydrophobic molecules, the process is completed within an interval ranging from milliseconds to minutes in simple biosystems like liposomes [27,28], subcellular particles [29] and cells [30]. For amphiphilic solutes [31] and very hydrophilic or hydrophobic solutes [31,32] in simple biosystems, and for non-amphiphilic and optimally hydrophobic drugs in large biosystems such as organs or organisms [33], the period needed for the achievement of lipo-hydrophilic equilibrium

s BALAi

ET AL

may be as long as several mmutes or even hours after the drug administration. Nevertheless, the transport of the drug molecules must be faster than their elimination, otherwise they could not persist practically in the mtracellular space [34]. Provided that the rates of the two processes differ sufficiently, transport is able to maintain the actual drug concentrations in individual cellular phases permanently near the lipo-hydrophilic equilibrium. A corresponding morphologically compartmentalized biosystem is depicted in Fig. 1. System components In the aqueous phases (Fig. l), both ionized (D’) and unionized (D) molecules are present, only the latter being assumed to cross the lipid core of the membranes readily [35]. Both ionized (primed symbols) and unionized molecules can form non-covalent complexes with two classes of proteins, namely with inert proteins (DP and DP’) and with metabolizing enzymes (DM and DM’). Both ionized and unionized molecules can be eliminated via: (1) enzymatic reactions preceded by non-covalent protein binding (as DM or DM’); (2) spontaneous chemical reactions as D and D’ (e.g., hydrolysis, reactions with low-molecularweight cell constituents such as amino acids, glutathion, etc.); and (3) excretion (as D and D’). There is no need to differentiate between enzymatically catalyzed metabolism and covalent binding to proteins (e.g., alkylating agents) as both processes play, from the viewpoint of the extraneous molecules, the same role: they cause the irreversible loss of the compound.

Fug. 1. Outhne of the morphologtcally compartmentahzed btosystem. The drug molecules are present m each aqueous compartment m the free state (D and D’), or bound to mert protems (DP and DP’) or to metabohzmg enzymes (DM and DM’); the pnmed symbols mdrcate tonized spectes. Only umonized molecules enter the hprd core of the membrane. Two-srded arrows represent fast reverstble non-covalent processes, and one-sided arrows ttme-dependent elmunatlon steps.

SUBCELLULAR

197

PHARMACOKINETICS

Model assumptions The non-covalent binding to proteins and the ionization reactions are finished within a few seconds at a maxrmum [36]. Provided that transport 1s much faster than elimination, all the reversible processes (indicated by two-sided arrows in Fig. 1) can be considered as instantaneous; irreversible elimination (normal arrows) then remains the only time-dependent step. Under such condrtions, all the phases of the biological system become kinetically indistinguishable and the or&al N-compartment system loses its structure and collapses to a one-compartment open system. Only the free umonized species move freely in such a system, therefore only their concentratron is identical at each moment in all the aqueous phases and in all the lipid phases, the two concentrations obeying Nernst’s distribution law. For low apphed doses of the compounds, tt can be assumed that: (1) activities are equal to concentratrons; (2) the dependence of the rate of enzymatic reactions on the drug concentratton IS linear; and (3) the decrease in the concentrations of inert proteins, metabolizing enzymes, and protons caused by interacttons with the drug molecules is negligible.

Descrlptron of the model The pseudo-equtlibrmm drug distribution outlined m Fig. 1 can be characterized in the usual way by using the mass-action law. The loss of the drug amount, n, by elirnmation 1s the summation of excretion, reactions with low-molecular-weight cell constrtuents, and enzymatic metabolism for both the ionized and unionized molecules m all the aqueous compartments: -dn/dt=C.,{V;(k,CA+~,,[DMl,+k:[D’l,

+d,,[DM’l,))

(1)

As mentioned above, the symbols D’, DM, and DM’ stand for the free ionized drug, the unioruzed and ionized drug bound to a metabolizmg enzyme, respectively. The brackets denote concentrations, defined with regard to the volumes, V,, of the individual compartments. Throughout this paper, the prime refers to parameters and

symbols of ionized species and the subscript I to the I th compartment. The rate parameter for the enzymatic metabolism, urn,, 1s defined as the rate of the saturated reaction when the enzyme is present m umt concentration. This makes it possible to take account of possibly varying concentrations of metabolizing enzymes, e,, m individual compartments. When a species is mvolved in multiple elimination processes in the same compartment, the corresponding rate parameter can be considered as the sum of the rate parameters for indtvidual processes. All the concentratrons in Eqn. 1 except cA (the concentratton of the free unionized drug in the aqueous phase) can be substituted by the expressions resulting from the definition of the correspondmg equilibrium constants and containmg cA to yield

-dn/dt = cACaq{ K[k,+ u,,e,/fL

+q,(k,‘+ &,e,'/K&)l}

(2)

where the ratro of the ionized and unionized free species, q, = [D’],/[D],, is equal to K,/[H+], for acids with the dissociatton constant K, and to K,[H+], for bases with the association constant constant for bmdmg to KtV The association metabolizing enzymes present at concentration e, is approximated by the reciprocal value of the corresponding Michaelis constant, K,,. To solve Eqn. 2, the same variables are needed on both sides. The relation between n and cA 1s grven by the definition of the “btological” partition coefficient, P,:

=

[n

-

(3)

C,,(C,K)]/CCA~~)

Here cL is the drug concentration m the hptd phases having the total volume V,, nA is the drug amount in the aqueous phases, and c, 1s the total drug concentratton in the I th aqueous compartment. The mass-balance equation for the I th aqueous phase is c, = cA + [DP],

+ [DM],

+ [D’],

+ [DP’],

+ [DM’I, where

DP stands

(4) for the drug bound

to an inert

5 BALM

198

protein. Using the same procedure converting Eqn. 1 to Eqn. 2 gives

as above

for

C,=CA[l+K,~!+er/Kmr + 4, (1

+

K’P,’

Here K, 1s the inert proteins binding sites. Eqns. 3 and 5 CA =

+

e,‘/%,

)I

(5)

association constant for binding to and p, is the concentration of the Then c, can be expressed from as (6)

VI/,,,,

where Vcorr =P,V=+Ca,{l/;[l+K,p,+e,/Kmr + q, (I+

K,‘P,’ + e,‘/KA, >I )

can be considered as the apparent volume of the biosystem with respect to the free unionized molecules, corrected for their accumulation in the membranes, ionization, and protein binding. Division of Eqn. 2 by V,,,, gwes t = ke,cA

- dc,/d

with the global as

kl =

c

aq

{

(7)

elimination

constant,

k,,, defined

Y [k,+ %l,e,/Knu

+q,(k:+u~,e:/K~,)I}/~/,,,, Equation ‘A

=

7 is solved easily to yield

&‘%Vcorr)

exd

-kd)

=

%A(

P,, t)

(8)

The linear term represents the initial concentration of the free unionized compound in the aqueous phases as given by Eqn. 6, with the initial amount, c,,V, instead of the actual one, n. Obviously, ca is the applied concentration (dose) and V is the total volume of the organism or the sample in the case of a cell suspension or tissue in in zntro experiments. A( p,, t) is a general notation for the so-called disposition function, which uses as variables: (1) physicochemical properties, p,, of the drugs, (2) the location of the receptor compartment m the biosystem, and (3) the exposure time, t. This function relates the internal drug concentration to the applied dose [23]. In Eqns. 13 and 14 given later, the proportionality factor between cA and c0 represents a specific form of the

ET AL

disposrtion function, A( p,, t), valid for the condttions used for derivation of these equations. The above Eqn. 8 provides a complete description of the fate of extraneous compounds in btosystems under the pseudo-equilibrium conditions of the time hierarchy adopted. Although only the concentration of the free unionized drug in the aqueous phases, cA, is given explicitly, the concentration of any species in an arbitrary compartment can be obtained by multiplication of cA by the proper factor resulting from the definition of the respective eqtulibrium constant (i.e., q, for the free ionized molecules, K,p, and q,K,‘p: for the unionized and ionized molecules, respectively, bound to inert proteins, and e/K,,,, and q,e,‘/K&, for the unionized and ionized molecules, respectively, bound to metabolizing enzymes). The global elimination rate parameter, k,, (Eqn. 7), is not a pure summation of contributions of individual ehmination processes. The rate parameters in individual compartments are weighted by the respective volume, v, to scale their contribution to the total elimmation and the denominator, VcoTT,accounts for the actual concentrations of the compounds available for elimination. Other features Extrathermodynamlc assumptions. The physicochenncal properties of the compounds can be mtroduced easily into the model via their substitution instead of model parameters m Eqn. 8 by using plausible extrathermodynamic assumptrons [12,21]. The biological partition coefficient, P,, can be related [37] to the 1-octanol/water partition coefficient, P, by P, = CWPP

(9)

where LYand j3 are constants under proper condrtions. The non-covalent binding to proteins has also been shown [38] m many cases to be structure-independent and related to P: K, P, = Y,J”~

(10)

Another equation has been shown [39] to hold sometimes for non-covalent binding to metabolizing enzymes: er/Kmr = C,PC, with different

(11) constants

(shown

by Greek

letters).

SUBCELLULAR

199

PHARMACOKINETICS

Kmetlcs of biological actwrty. Biological activity is related to the drug concentration in the receptor surroundings provided that the drug effect is the immediate consequence of a one-step, fast, and reversible 1: 1 drug-receptor interaction and is proportional to the fraction of the receptors occupied [23]. Under these assumptions, Eqn. 8 can be reformulated [23] to express directly the time course of biological activity:

tion about the biosystem and the compounds studied or on a hypothesis to be tested. The possibilities are numerous and the procedure is comparatively straightforward, therefore the actual equations are not given. The concept will be illustrated for some simple cases. These case studies were also intended to check the validity of the model assumptions used.

l/c, = KNP,, t>(l - x)/x

Hydrophoblclty-dependent dtstnbutlon In the most frequently reported QSAR studies on complex biosystems, bioactivity (and consequently the distribution) at a specified exposure time is a more or less smooth non-linear function of hydrophobicity [40-421. It follows from Eqn. 8 that such relations can be obtained only if all the rate and equilibrium parameters involved are either zero, constant or hydrophobicity-dependent. As non-ionizable drugs or ionizable drugs with approximately the same acid-base properties were mostly used in these experiments, the term q, accounting for ionization in Eqns. 6 and 7 is either zero or constant. The exponents in Eqns. 10 and 11 may frequently have identical values in all the aqueous compartments. Then Eqn. 8 can be wntten as follows:

=WXorr)Kl - X)/Xl exd-k,t) (12) where K is the drug-receptor association constant isoeffective conand cx is the time-dependent centration eliciting the fraction X of the maximal effect. Flttmg equations to expertmental data. The values of the adjustable parameters were optimized by a combination of linear and non-linear regression methods m a specially written Pascal program (M.W.).

RESULTS

Combination of Eqns. 9-11 with Eqn. 8 results in the general (within the framework of the used time hierarchy) kinetic expression for the drug disposition as a function of hydrophobicity, acidbase properties and elimination rate parameters. The rate parameters may be replaced by either the rate constants of in-uitro reactions imitatmg the metabolic reactions of the extraneous substances, or proper extrathermodynamic relations. However, it should be kept in mind that for enzymatic metabolism very complicated relationships between the rate parameters and physicochemical and/or molecular properties are to be expected, especially when compounds of diverse structures are considered. The association constants for binding to inert protems and metabolizing enzymes can also be determined in in-vitro expenments, provided that Eqns. 10 and 11 do not hold for a particular case. To adapt Eqn. 8 to reality, a number of simplifications can be used, based either on informa-

c, = c,,[ A/(

B,PP + B,P” + B,P3 + l)]

xexp[-(CPl+D)t /‘( B,PB + B,P” + B,Pr + l)]

(13)

where A=

mJw+dl;

B, = AaVL/V; B, =Ac.,[%

+ q,y,‘)l/V:;

The unknown model parameters, assumed to be constant under given conditions, are here (and also will be in following equations) collected in the adjustable parameters A-D, which are to be fitted by regression analysis of expenmental data. Un-

F%BAL.kk

200

less specifically stated otherwise, the parameters A-D in the following equations are not mutually related. Equation 13 can be further simplified when some of the exponents p, 8, and { have identical values. If this is true for all the exponents, then

xexp[-(CP+D)t/(BP+l)]

04)

With regard to the limited number of compounds in the series exhibiting hydrophobicity-dependent distribution and bioactivity, Eqn. 14 with its five adjustable parameters (at maximum) seems to be a reasonable compromise between exactness of the description and accessible experimental information.

ET AL

At this point, some properties of the present model can be illustrated by graphical presentation (Fig. 2) of Eqn. 14 with the sum of the metabolic rate parameters, u,,,, instead of C (cf. Eqn. 13). The modified Eqn. 14 is valid for compounds with invariant acid-base properties, and hydrophobicity-dependent binding to membranes, inert proteins and metabolizing enzymes, according to Eqns. 9, 10 and 11, respectively, with the exponents equal to /3 = 1, metabolized solely by enzymatic reactions (i.e., D = 0 in Eqn. 14). In order to consider the concentration of the drug bound to inert proteins (which may well represent the receptors), Eqn. 14 must be multiplied by K,p, (i.e., by P in this case). The concentration/hydrophobicity profiles at a constant rate of enzymatic metabolism u,,,, will be

Rg 2 Relations between the concentration, c, of the drug bound to Inert protems, the partltlon coeffxlent, P, and the sum of the rate parameters of saturated enzymatx reactions, u,. Calculated from Fqn. 14, multlphed by P with p = 1, q, = 1, A = 1, E = 0 01, C = u,, D = 0,and t= lo-' (A), 1 (B), lo3 (C) and lo6 (D) time umts.

SUBCELLULAR

201

PHARMACOKINETICS

considered first. For low values of u,, there 1s no significant difference between the initial and actual concentrations and the relationships have the bilinear form characteristic of the lipo-hydrophilic equthbrium [11,12,14]. For higher rates of metabolism, the equilibnum dependences become distorted in the regton of htgh hydrophobmty, because of tighter binding of hydrophobic compounds to the metabolizing enzymes, and assume a tnlinear shape, winch is particularly visible as the upper contour of Fig. 2D. The dependences of the concentration of the compound-protein complex on the rate of metabolism (the cross-sections for constant hydrophobicities, P) have a simpler shape. The concentration, c, 1s independent of the rate of enzymatic metabolism, u,, till u, reachs a certain value, and thereafter drops suddenly. Thts limiting value of u, is constant for hydrophobic compounds having log P > I; for other compounds, it is inversely proportional to hydrophobicity. The series of plots A-D (Ftg. 2) reflects the influence of the exposure time on the drug distribution. The apparent concentrations of hydrophilic compounds bound to the hydrophobic regtons of the inert proteins remain practically mtact even after long mcubation periods. Tins comparatively slow elimination of hydrophilic solutes is caused by their low affinity to the metabolizing enzymes. Moreover, the aqueous compartment, contaming most of the hydrophilic molecules, can serve as a depot supplying them to the inert protems when their concentration decreases. This compensation mechanism works until the amount eliminated is no longer negligible in compartson with the total amount of drug; thus the concentration of hydrophtlic compounds bound hydrophobically to the inert protems is maintained practically constant for some time, whereas the concentration of hydrophobic solutes having identical elimination rate parameters is significantly decreased. Plots analogous to those gtven in Fig. 2 can be constructed for any particular example. Their main features can be described semiquantitatively as follows. At first, the concentration/ hydrophobmty profiles have equilibnum shapes (i.e., they are bihnear with slopes of 0 for low log P values and

- 1 for high values in the aqueous phase, and with slopes of 1 and 0 in the lipotd phase). At longer times, the concentrations of the compounds decrease, but not to an equal extent for all of them. Hydrophobic compounds are preferably ehminated tf hydrophobictty is a prerequisite for bmding to metabolizing enzymes, whereas mainly hydrophilic solutes disappear during elimmation of the free molecules, e.g., by excretion or spontaneous hydrolysis. Accumulation of the drugs in states inaccessible to elimination is responsible for such behaviour because it lirmts the concentration of a drug available for elimination and because it allows the origination of the drug depots. For hydrophobicity-dependent bioactivtty measured at a constant exposure time, Eqn. 12 can be rewritten by using Eqn. 14 instead of Eqn. 8, to describe the drug disposttion via the disposition function A(p,, t) log(I/c,)

= ,!? log P - log( BPp + 1) -(CP”+D)/(BPP+

log(l/c,)

= -log( /@Pa+

1) +A

(15)

BPB + 1) - (CPp + D) 1) +A

(16)

The adjustable parameters C and D now also contain the constant exposure time, 1. The simplifying assumptions used in the derivation of Eqn. 14 also apply for Eqns. 15 and 16. In addition, the drug-receptor association constant, K (Eqn. 12), is expected to be etther constant or hydrophobicity-dependent, since bioactivity is a smooth function of hydrophobictty. Equation 15 holds if the relation between K and P IS given by Eqn. 10 with the exponent p, and Eqn. 16 is valid if K IS constant within the series tested. The accumulation in the membranes, as well as binding to the receptors, inert protems and metabolizing enzymes, are all assumed to depend on P according to Eqns. 9-11 with the same value of the exponent (p in Eqns. 15 and 16). Equations 15 and 16 describe the situation where elimination mvolves more steps, provided that at least one of these steps is hydrophobicttydependent enzymatic metabolism (Eqn. 11) and the others have identical rate parameters within the tested series. Frequently, one type of elirmna-

3

202 TABLE

BALAi

ET AL

1

Fits of Eqn 15

with D = 0 to antIbacterIa data from the hterature (The sources of data and the statlstlcs are gven In Table 4) Eqn. a

Compound

Bloactlvlty b

BXlOoo

cx1000

P

A

2 614 *0039 2 867 P aerugrnosa ( C6HSCH,N+R(CH& 18 * 0.081 MIC 3.690 19 Cl welchrr, kOO46 MKC 3 758 Cl w&ha, 20 C,H,CH,N+R(CH& +0066 MIC P uulgans, 2 896 21 C6HSCH2N+RCH& MKC f 0.076 3 321 22 RCHBrCOOD pneumonrae, MKC, pH 6.5 +0205 2 710 23 RCHBrCOOD pneumonrae, *0226 MKC, pH 7 5 24 2 453 RCHBrCOOD pneumonrae, MKC, pH 8.5 +0218 25 RCHBrCOOS hemolytrcus, 2.931 MKC, pH 6 5 + 0.184 26 RCHBrCOOS hemolytrcus, 2 305 +0222 MKC, pH 7 5 S hemolykw, 27 RCHBrCOO2.002 MKC, pH 8 5 +0221 _ __s rtiese equation numbers are @ven for ldkntltkatlon purposes m ‘&t!l’e4. ‘. MKC, mmlmai Uing concentration; MIC, mmlmai mhlbltory concentratton 17

C,H,CH,N+R(CH,),

2 831 +0473 21.34 +7384 2.813 *o 734 3 456 + 1.299 5 958 +1730 0 129 *0.111 0.111 +O 106 0 114 f 0.106 0.125 +0095 0 076 *0071 0 075 *0070

P aerugrnosa )

MKC

tion predominates and then Eqns. 15 and 16 can be simplified by presetting either D or C to zero, according to whether the elirnmation rate IS hydrophobicity-dependent or constant within the tested series. The decision which term is to be

0 929 *0035 0 821 +O 108 0.999 +0048 0 922 fO068 0 697 f 0.067 0 839 *0099 0 840 kO.110 0 849 &O109 0 803 +0083 0 858 *o 104 0 859 *o 104

7 534 * 1.078 35.75 &-9846 4 393 k0 805 5.441 k 1416 17 13 +4619 0.581 f 0.473 0 476 +0429 0448 *O 389 0 637 rt 0.462 0 352 *O 312 0 349 f 0.308

neglected can be made from visual comparison of the experimental bioactivity/hydrophobicity profiles with the theoretical curves. A pragmatic rule can be formulated as follows: If the left-hand portion of the expenmental bioactlvity/hydro-

TABLE 2 Fits of Eqn 16 with C = 0 to antlbactenal data from the hterature (The references and statlstlcs are gwen m Table 4)

Eqn

Compound

Bioactlwty

28

6-Alkoxyqummes

B drphrenae

29

RCHBrCOO-

30 31

BXlOOO

P

D

A

5 045 *2231 4458

0 588 * 0.034 0 651

pH 8 5, MKC

+2535

*0349

RCHBrCOO-

B leprseptrcus,

113 8

RSO; Na+

pH 7 5, MKC S aureus

3 529 +o 134 3 140 *0399 3 133 kO204 5 341 +O 163

6 905 f 0.083 4 342 +o 133 5.032 *o 134 5 270 f 0.075

V cholerae,

+542 6 882 k3458

0 936

+0082 0.436 * 0.040

SUBCELLULAR

TABLE

PHARMACOKINETICS

20.1

3

Fits of Eqn

15 wth

(The references

D = 0 to the hterature

and statlstlcs

data on bloactwlty

are gwen m Table

agamst fung

Eqn

Compounds

Bloactwty

32

C,HsCH,N+R(CH,),

C albrcnns,

BXIOOO

RHN:

0.719

Mandehc

acid esters

pig Ileum

n-Alkanes

Mice,

3144 *0049

0 584

44.36

2.339

153

* 31.15

*O 213

0.013 1.042

* 0.043

-1000 +o

fO005

0 639

50214

phobicity profile has the strictly equilibrium shape, then D = 0; if this is true for the right-hand side, then C = 0. Data from the extensive compilations by Hansch and coworkers [40-421 as well as by Kubinyi [43] were fitted by using Eqn. 15 with the value of the adjustable parameter D preset to zero and to Eqn. 16 with C = 0. The results for anti-

2 014

*0055

0 482

A

$0.381

0 817

+0021

LD,,

TABLE

*o

0 045

spasmolysls 35

1 056

*1150

Gumea

c x 1000

*0040

16 05

L leprdeus, MIC

34

btosystems

P

+O 381

MKC 33

and hgher

4)

*0447

121

0.217 f 0.016

bacterial activities are summarized m Tables 1 and 2, and the results for more compartmentalized biosystems m Table 3. In all cases, a satisfactory fit was obtained, the statistical indices being sumlar to, and in some case even better then, those obtained with the commonly used parabolic or bilinear equation (Fig. 3, Table 4). The results can be interpreted withm the frame-

4

StatIstIcal comparison

of the present,

parabohc

and bilinear

models

Present

Parabohc

Blhnear

Ref

&n

”

F

r

SD

r

SD

r

SD

17

12

155

0.994

0 090

0 962

0 219

0 990

0 122

18

9

11

0.959

0 136

0 880

0.208

0 937

0 168

40,14

19

12

152

0.995

0 107

0 980

0 190

0 992

0 125

40,14

20

12

64

0.987

0 154

0 966

0.230

0 983

0 172

40,14

21

10

27

0 978

0 138

0 960

0 172

0 973

0 153

40.14

22

8

14

0 974

0 339

0.898

0 593

0 939

0 519

40.14

23

8

11

0 967

0 376

0.893

0.596

0.934

0 530

40.14

24

8

11

0 969

0.360

0.909

0.542

0.942

0.488

40,14

19

40,14

25

8

0 981

0 309

0.898

0.620

0.959

0.444

40,14

26

8

12

0.971

0 371

0.872

0.685

0 942

0 523

40,14

27

8

13

0.972

0 369

0.872

0.686

0.943

0 522

40.14

28

17

98

0.985

0 152

0.936

0300

0.972

0 206

41,14

29

6

8

0.985

0 154

0.959

0.206

0.980

0 180

40,14

30

6

18

0 993

0 139

0.959

0.277

0 991

0.158

40,14

31

10

328

0 998

0 105

0.981

0.310

0 994

0 180

41 b

32

11

146

0 995

0.095

0 961

0 243

0 984

0 10

40.14

33

6

3

0 960

0 224

0.903

0 281

0 938

0 276

42 =

34

11

76

0 990

0.154

0 958

0 298

0 990

0 160

43 b

35

13

94

0 990

0.065

0 865

0 217

0 973

0 104

14 b

a The first study contams parabohc calculated

and bilinear here.

the data and the parabohc

equations

were calculated

here

equation,

the second

’ Thus contains

study the blhnear

equation

the data and the parabolic

b This contains

equation;

the bllmear

a

the data, equation

the was

204

5 BALAi

ET AL

degradation of alkyl esters of p-aminobenzotc acid by Pseudomonas fluorescens [46] were used to test our hypotheses. Because only the total drug concentrations were monitored m this study, Eqn. 8 has to be transformed accordingly. The actual drug amount, n, is defined by Eqn. 6 as (c,V,,,,). For the actual drug concentration, c = n/V, Eqn. 8 can be rewritten as log( c/co)

= - k,, t/in

10

= -(CPP+D)t/(BPfl+l)

(36)

work of the proposed model as follows: (1) for the data described by Eqn. 15, binding to both receptors and metabolizing enzymes is hydrophobicitydependent; (2) for the data fitted to Eqn. 16, neither the receptor binding nor metabohsm depend on hydrophobbty; rather, both processes proceed with identical rate and equihbrmm parameters for all the drugs tested. Inspection of the statistical indices, summarized for the present, parabolic and bilinear models in Table 4, shows that the first model provides the best fit to the analyzed data, with regard to both correlation coefficients and standard deviations. An mdication of deviation from the parabolic and bilinear models can also be seen in the bioactivity/ hydrophobicity profile presented in Fig. 3.

The second equality in Eqn. 36 comes from the exponent in Eqn. 13. This equation may represent an adequate description in this case, because the tested compounds are electroneutral under physiological conditions. Regression analysis of the data obtained from Fig. 2 of the original work [46] (the 1-octanol/water partition coefficients from the study [47]) provided the following values of the adjustable parameters: B = (1.220 k 0.218) X lo-‘, C = (1.163 f 0.197) x 10e3 mu-‘. Because of the good agreement between the model and the experimental data (the values of the statistical indices were n = 40, r = 0.988, SD = 0.040 and F = 793) optirmzation of further parameters was not needed and their values could be preset to D = 0 and p = 1. From the results, it can be inferred that binding of the compounds to metabolizing enzymes IS hydrophobictty-dependent and that the biological partition coefficients of the compounds are linearly related to the l-octanol/water partition coefficients (p = 1 in Eqn. 36). Further examples of application of the present approach to the description of the microbial degradability of organic compounds are available elsewhere [48].

Mtcroblal degradation of orgamc chemicals Ecologically orientated studies on this SubJect represent a valuable source of kinetic data on the fate of compounds m microbial suspensions. The time course of the substrate concentration is usually mono-exponential [44] unless the initial concentrations are htgh enough to saturate the pertinent enzyme [45]. This is m good agreement with the present model (Eqn. 8). The kinetic data on

Kmetlcs of blologlcal actwty Under the conditions stated above, biological activity can be related to the presence of drugs m the receptor compartment. The pertinent Eqn. 12 was tested by using kmetic data on the analgesic effects of fentanyl derivatives m rats obtained m the tail withdrawal test [49]. The best fit to the data, converted to a molar basis, was obtained by

log P Fig 3. Inhlbltory actlwty of alkylsulphomc acids agamst S nureu.~ vs the 1-octanol/water partltlon coeffwent The hne corresponds to Eqn 16 wth C=O and the values of other adjustable parameters gwen m Table 2 (Eqn. 31)

SUBCELLULAR

205

PHARMACOKINETICS

usmg the following combination 14 (with C = 0)

of Eqns. 12 and

log(l/ED,,) = -log(BPfl+

1) -Dt/(BPP+

1) +A

(37)

where EDso is the effective dose in mol/kg causing the 50% increase in the reaction time of the animals. The optimized values of the adjustable parameters were B = (1.873 k 0.737) x 10e3, j3 = 1.148 + 0.094, D = (5.966 & 0.322) x 10-l h-‘, and A = 9.154 k 0.205. The values of the statistical indices (n = 48, r = 0.932, SD = 0.340, and F = 70.8) indicate the satisfying quality of the fit. The data on analgesic potency between 0.25 and 6 h were used; five derivatives with the common substructures R, = CH,, R, = H or COOCH,, numbenng according to the original study [49], were excluded. The values of the 1-octanol/ water partitron coefficients were calculated by using the measured value for fentanyl (log P = 2.35) and the 7r-values of the substituents [50]. The results indicate that the hydrophobicity of the compounds in the lipoid phases of the tested biosystern is related to that in 1-octanol by Eqns. 9 and 10 with exponent /I = 1.15, and that the compounds, except for those excluded, have a common mechanism of elimination.

DISCUSSION

This study aims at developing a simple but reahstic kinetic description of the intracellular distribution of extraneous substances. The presented form of the disposition function (Eqn. 8) is of use in any field where the intracellular movements of small molecules are of interest. The results are especially important for drug design as they allow the construction of mechanistic QSTAR expressions based on extrathermodynarmc principles and on common assumptions about transformation of the receptor modification into biological response. The basic assumption in the derivation of the final Eqn. 8 is the experimentally verified time sequence of the processes determining the fate of drugs in biosystems [27-331. Advantage of the fact

that transport is much faster than elimination, at least m small biosystems, has been taken prevlously by Martin and Hackbarth m their pioneering work [12] on model-based equilibrium models for the bioactivity of iomzable compounds. In fact, the present paper can be considered as a direct extension of their studies. The same time hierarchy of drug distribution has been used by PliSka [34] for a phenomenological description of the inuiuo kinetics of drug action. In order to verify the assumptions used m derivation of the model, Eqn. 8 was modified to describe easily accessible data from the hterature, namely the bioactivity/hydrophobicrty profiles measured at a constant exposure time (Eqns. 15 and 16) the kinetics of microbial degradation of organic compounds (Eqn. 36) and the kinetics of biological activity (Eqns. 12 and 37). Equations 15 and 16 comprise all the three regularly observed types of bioactivity/hydrophobicrty profiles, i.e., saturation-type [l], parabolic [21] and bilinear [4] relations (Fig. 2A) and fit the experimental data well (Tables 1-4, Fig. 3). However, their good agreement with biologrcal reahty does not mean that they should replace the well-established and simpler parabolic and bilinear equations for the description of smooth relationships between concentration or biological activity and hydrophobicity unless these deviate clearly from the convex relations and have a trilinear shape (Fig. 3). A number of mechanistic explanations of the convex bioactivity/ hydrophobicity profiles has been summanzed by Hansch and Clayton [40]. Mathematical treatment has shown that two of these hypotheses, which seem to have the most general validity, namely the transport [4-10,1322,24-261 and the equilibrium model [11,12,20], are able to produce curves of the required shape. This study presents a further possibility: that a hydrophobicity-governed accumulation of drugs in the lipoid phases representing both membranes and hydrophobic regions of proteins, influences significantly the actual rate of elimination (Fig. 2). This well known phenomenon [51] has not been considered in this context previously. The expenmental bioactivity/hydrophobicity profiles are the result of an interplay of various factors. All the mentioned approaches, including the present one,

206 try to explain this complex phenomenon in a rather simplistic way. Therefore, it is difficult to decide which model is better under which circumstances. Nevertheless, in the case of simple biosystems like cellular suspensions, the models assuming fast attainment of the lipo-hydrophilic equilibnum seem to be more realistic, because convex bioactlvity/ hydrophoblcity profiles are frequently observed several hours after drug administration [40-421 whereas the results of uptake experiments indicate that transport 1s finished within a few seconds or minutes [27-311. The suitability of Eqn. 8 for a particular series of compounds and a particular biosystem can be tested by pharmacokinetic experiments. The concentration of the extraneous substance 1s easily monitored in the external medium of a simple biosystem in m uitro conditions (e.g., cell suspension or tissue) or in the blood or urine of organisms. The time course of the drug concentration is, in principle, biphasic: the first part, called the distribution phase or a-phase [52], corresponds to the drug transport and the second, mostly mono-exponential part (disposition phase or P-phase [50]), represents elimination. In the uptake experiments with simple biosystems, the distribution phase is often too short to be monitored without special equipment [27-311. The decisive criterion for applicability of Eqn. 8 is the duration of the first, distribution phase. If tlus phase represents only a small portion (say, l/5 or less) of the total period during which kmetics of uptake is observed, Eqn. 8 describes the distribution sufficiently precisely during the disposition phase when the process exhibits monoexponential kinetics. The presented results indicate a good agreement between the proposed model and experimental data. If further studies confirm the applicability of the proposed model, it may contribute to a better understanding of chemicobiological interactions.

S.B. thanks the Humboldt Foundation and the Heads of the Faculty of Chemical Technology, Slovak Polytechnic, Bratlslava, for support.

$ BALAi

ET AL

REFERENCES

1 J T. Mol. 2 J W 3 SH 210.

Penmston, L Beckett, 0 L. Bentley and C Hansch, Pharmacol , 5 (1969) 333 McFarland, J. Med Chem., 13 (1970) 1192 Yalkowsky and G L. Flynn, J Pharm Scl., 62 (1973)

4 H Kubmyi, J Med Chem, 20 (1977) 625 5 J C Dearden and MS Townend, m R Franke and P. Oehme (Eds ), Quantltatwe Structure-Actlwty Analysis, Akadenue-Verlag, Berhn, 1978, p 387 6 H van de Waterbeemd, P. van Bake1 and A Jansen, J. Pharm. Scl , 70 (1981) 1081 7 E.R. Cooper, B Bemer and R C Bruce, J Pharm SCI , 70 (1981) 57 8 L Aarons, D. Bell, R Walgh and Q Ye, J Pharm. Pharmacol , 34 (1982) 746. 9 3. Balti, M. Hrmovg, M Breza and T LlptaJ, Eur J Med Chem , 19 (1984) 167 10 M Txh$ and Z. Roth, m J K Seydel (Ed ), QSAR and Strateges m the Design of Bloactwe Compounds, Verlag Chenue, Wemham, 1985, p 190. 11 T. Hlguchl and S.S. Daws, J Pharm Sci., 59 (1970) 1376 12 Y C Martin and J J. Hackbarth, J. Med. Chem., 19 (1976) 1033 13 s Bal& E Sturdik and J Augustin, Biophys Chem., 24 (1986) 135 14 H Kubmyl, m E Jucker (Ed.), Progress m Drug Research, Vol 23, Bnkhauser, Basel, 1979, p 97 15 8 Balfi, E &urdik and J Augustin, Gen. Physlol Biophys , 6 (1987) 65. 16 B.J. Zwohnslu, H Eynng and C E Reese, J. Phys CoIlold Chem , 53 (1949) 1436 17 A.J L De Meere and E Tomhnson, Int J Pharm, 22 (1984) 177 18 F H N. De Haan, AC A Jansen, Int J Pharm., 29 (1986) 177 19 H Kubmyi, J Pharm Scl., 67 (1978) 262. 20 R.M Hyde, J. Med Chem , 18 (1975) 231. J. Am. Chem. Sot., 86 (1964) 21 C Hansch and T Fqlta, 1616 22 s. BalSlZ and E Sturdik, m M. Tichq (Ed.), QSAR m Toxicology and Xenobmchenustry, Elsewer, Amsterdam, 1985, p 257. 23 5. BaIti, E. Sturdik and M. Tich$, Quant Struct.-Act Relat , 4 (1985) 77. 24 s. Balti and E Sturdik, Gen. Physlol Blophys., 4 (1985) 105 25 5 Balti, E Sturdik and B Sklrka, Co11 Czechoslovak Chem Commun ,49 (1984) 1382 26 s Balti, E. Sturdik and J Augustin, Bull Math Blol, 50 (1988) 367. 27 R Welti, L.J. Mulhkm, T Yoshlmura and G.M Helmkamp, Jr., Biochenustry, 23 (1984) 6086. 28 s Balti, A. KuchBr, J DfevoJhnek, J Adamcova and A Vrbanovg, J. Blochem. Biophys Meth , 16 (1988) 75

SUBCELLULAR

PHARMACOKINETICS

29 s Bal%, E Sturdik, E i)ur~ov& M. Antalik, and P. Sulo, Blochlm. Blophys Acta, 851 (1986) 93 30 J Brahm, J Gen Physlol, 81 (1983) 283. 31 P H Hmderlmg, J Pharm Scl , 73 (1984) 1042 xi z_z Chnwhan,.T Ynt.wLyana@ami_W I_ HJgU&l_. I_ PhiUxn. SCI., 62 (1973) 221 21 H_ I_~llJman~~,. P % M_W M_ Tunm~~mans,. GM_ U!akat_ and A Ziegler, J Med Chem., 23 (1980) 560 34 V PhSka, Drug. Res., 16 (1966) 886 ZT e A. Sbhnrp,. e e Bxodx and CA. M. H@&. I Phaxm. Exptl. Therap, 119 (1957) 361 36 C AM van Gum&en_ IJI .I M. van Rnssum (Ed),. Kmetux of Drug Actlon, Spnnger, Berlm, 1977, p 357 17 R.. CnlJan~~. Acta Chem Scxnd,. I (L'Xl-).774. 14 C &JJxh_ and W .I nunn. rrr,. I Ph,%xn Six ,_fil_ (l_WZj. I 39 C Hansch, m F J. DI Carlo (Ed ), Drug Metabohsm Reviews, Vol 1, Dekker, New York, 1973, p 1 40 C Hansch and J M Clayton, J Pharm SCI., 62 (1973) 1. 4I E-I rxn,. C HxJ.%Ih. ami S-M_ Andf?x%lq. I_ M&i CbPJn. __ 11 (1968) 430 42 C Hansch and E J Lien, J Med Chem., 14 (1971) 653

207 43 H Kubmyl, m M Kuchaf (Ed ), QSAR m Design of Bloactlve Compounds, Prous, Barcelona, 1984, p 321 44 Y Urushlgawa and Y Yonezawa, Chemosphere, 8 (1979) 317

46 J.R Parsons, A Opperhulzen and 0 Hutzmger, Chemosphere, 16 (1987) 1361 4.7 D Mxkq.,. A_ &&es. and E Ptierson,. J. Phaxn. %I... 74 (1985) 1236 4X $ BaJ& M WUX. M &my,. Haa-Lm. ChL and J-YSeydel, Chemosphere, 19 (1989) 1677 49 W F M van. &ver,. CJ E N~s,. K H L. Sr&e&kens,. and P A_L J_ans%q. 0x1u.g I&s_,. 26. (1976). 15A8.. 50 C Hansch and A Leo, Substltuent Constants for Correlation Analysis m Chemistry and Biology, Wiley, New York, 1979

52 F H. Dost, Der Blutsplegel,

Thleme,

Lelpug,

1953.