**1. Introduction**

142 Toxicity and Drug Testing

Wolff, K. & Hay, A.W. Plasma methadone monitoring with methadone maintenance

Wolff, K., Rostami-Hodjegan, A., Hay, A.W.M., Raistrick, D. and Tucker, G. Population-

Wolff, K., Sanderson, M., Hay, A.W. and Raistrick, D. Methadone concentrations in plasma and their relationship to drug dosage. *Clinical Chemistry.* 1991. 37(2): 205-209. Verebely, K., Volavka, J., Mulé, S. & Resnick, R. Methadone in man: pharmacokinetic and

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treatment. *Drug Alcohol Depend.* 1994. 36(1): 69-71.

*Therapeutics.* 1975. 18(2): 180-190.

*Analytical Chemistry.* 2006. 78(11): 3571-3576.

potential clinical utility. *Addiction.* 2000. 95(12): 1771-1783.

444-444.

Application to Routine Drug Monitoring. *Pharmaceutical Research.* 1993. 10(3): 441-

based pharmacokinetic approach for methadone monitoring of opiate addicts:

excretion studies in acute and chronic treatment. *Clinical Pharmacology and* 

Levels of Acute Phase Proteins in the Plasma of Patients with Schizophrenia.

The term synergy is derived from the Greek *syn-ergos*, "working together". Synergies have been described in many settings and situations of life, including mechanics, technical systems, human social life, and many more. In all cases, synergy describes the fact that a system, i.e. the combination and interaction of two or more agents or forces is such that the combined effect is greater than the sum of their individual effects.

This definition implies that there are three possible ways of such an "interaction of agents or forces": these forces could simply add up, not affecting each other (no interaction), their combination could produce a greater than expected result (synergy), or the combination could lead to a result that is less than the sum of the individual effects. This "negative" summation is called antagonism.

Interactions of biologically active agents are an important aspect of pharmacology and biomedicine. In this context, interaction describes the biological activity that results from the presence of several drugs at the same time. Such situations occur in numerous clinical situations:


In all these cases, multiple drugs are administered, and will show some form of interaction, synergistic, antagonistic, or none. Methods to determine and quantify drug interactions are thus an essential tool in pharmacology. Historically, extracts from plants, animals, or even soils were the first classified pharmaceuticals. These were complex mixtures rather than single agents, and some ingredients may have interacted with others. Over the years of development of pharmacy, isolation, synthesis and marketing of single drugs became the accepted standard. Whether a complex mixture or a combination of drugs is used, the biological interaction of all active substances should be known. Synergy may be observed in simple systems – two drugs that only act on one target protein can show synergism. In such a case we can study the interaction of the drugs mechanistically and determine why and

Drug Synergy – Mechanisms and Methods of Analysis 145

In case of non-competitive inhibition, the inhibitor binds to a location on the enzyme different from the active site. Assuming that bound inhibitor converts the enzyme into an inactive (non product-forming) state, presence of a non-competitive inhibitor simply lowers the amount of active enzyme molecules (Fig. 1B). There are states, where both substrate and inhibitor are bound to the enzyme. The effect of several non-competitive inhibitors applied together raises the question if synergy can be observed in such a simple system. If two non-competitive inhibitors bind to the same site on their target enzyme, this inhibitory site can either be occupied by inhibitor A or B, but not by both inhibitors at the same time (Fig. 2). This would be a case of mutually exclusive binding of two inhibitors. If one inhibitor is present, and the second one added, one may observe indeed a greater extend of inhibition, but this would only be due to larger amounts of inhibitory molecules being present. At all times, we could predict the total amount of inhibition by summations. In the simplest molecular case, two inhibitors targeting the same site would

Fig. 2. Reaction scheme for two non-competitive inhibitors targeting the same site. Two noncompetitive inhibitors (squares) bind to the same site on the target protein. In this case their binding is mutually exclusive. Presence of the second inhibitor increases the total amount of

If, however, two different binding sites for non-competitive inhibitors exist on an enzyme, two inhibitors may bind simultaneously (Fig. 3). Inspection of the reaction schemes (Fig. 2, 3) shows that if two inhibitors have specific, independent sites on the enzyme, we will observe states where the enzyme indeed has two inhibitors bound (Fig. 3). These states cannot exist if both inhibitors bind to the same site (Fig. 2). Thus, if two inhibitors are able to bind simultaneously, we have a case of "mutually non-exclusive" binding. Here, presence of the second inhibitor will not only give an additive effect (increase of the number of inhibitory molecules), but will generate additional inhibited states of the enzyme. Therefore,

It should be noted that the considerations above are made following some basic assumptions, namely that binding of an inhibitor will convert the enzyme to an inactive state, binding of substrate and inhibitor is reversible, and binding of any compound is fully independent from all other compounds. Thus, the equilibrium binding constant for inhibitor A is the same whether A binds to the unliganded enzyme, or to the enzyme that has substrate and/or another inhibitor bound. Given these assumptions, the mechanisms for activation (Fig. 4A), and non-competitive inhibition (Fig. 4B,C) show the different states in which an enzyme exists in the presence of two non-competitive inhibitors that bind to the

inhibitor causing in increased inhibitory effect. This increase is only due to simple

on a molecular level we would expect a superadditive effect of two such inhibitors.

same site (Fig. 4B), or to different sites of the enzyme (Fig. 4B).

produce an additive effect only.

additivity, and not synergy.

how several drugs can reinforce each other (or why they do not). Synergy may also be observed in complex settings, such as patients receiving multiple medications. Usually, more than one biological target (protein, pathway, or even organ) are involved in such cases, and single mechanistic descriptions are not appropriate. Additional parameters to consider are drug absorption, tissue distribution, and clearance. It may be expected that many drugs interfere with metabolism of other drugs. Thus, a substance B that slows down clearance of an active drug A, say by blocking metabolizing enzymes or excretion, may lead to a higher effective concentration of A that remains in the body for a longer time. As a result, one would notice a greater effect of drug A when given together with B, although the two drugs have completely different modes of action. While certainly the combination of these two drugs would have a "combined effect is greater than the sum of their individual effects", their combination is synergistic in practical application, but not by the strict definition.

#### **2. Basic models and mechanisms – Synergy on a molecular level**

#### **2.1 A simple reaction scheme for enzyme inhibition**

Drug interaction and synergy has been intensively studied for more than 100 years, and some of the numerous concepts will be briefly introduced in this chapter. The simplest model cases will be presented, leading to a molecular definition of drug synergy.

Let us assume a simple enzyme following the laws of mass action and Michaelis-Menten kinetics. In the simplest case, this enzyme has an active site, where substrate is being converted into product, and possesses one or several specific binding sites for inhibitors (Fig. 1A). A competitive inhibitor by definition binds to the active site of the enzyme, displacing the substrate. Thus, a mixture of two purely competitive inhibitors will only ever target the active site. This is known as mutually exclusive binding. If only the simplest mechanistic case is considered, one would not expect a second competitive inhibitor to have any notable effect on the first one, other than raising the total amount of inhibitory molecules.

Fig. 1. Schematic representation of inhibition mechanisms (A) Competitive inhibition. Inhibitor (open circles) binds to the active site of the target protein. The agonist (solid circles) binds to the same site. By definition of competitive inhibition, all competitive inhibitors bind to the same (active) site. Thus, binding of two competitive inhibitors must be mutually exclusive, and they cannot act synergistically on the same target protein. (B) Noncompetitive inhibition. Inhibitor (open squares) binds to a site different from the active site of the target molecule. In pure non-competitive inhibition agonist binding is not affected by the inhibitor. Inhibition is due to conversion of the target protein into an inactive state.

how several drugs can reinforce each other (or why they do not). Synergy may also be observed in complex settings, such as patients receiving multiple medications. Usually, more than one biological target (protein, pathway, or even organ) are involved in such cases, and single mechanistic descriptions are not appropriate. Additional parameters to consider are drug absorption, tissue distribution, and clearance. It may be expected that many drugs interfere with metabolism of other drugs. Thus, a substance B that slows down clearance of an active drug A, say by blocking metabolizing enzymes or excretion, may lead to a higher effective concentration of A that remains in the body for a longer time. As a result, one would notice a greater effect of drug A when given together with B, although the two drugs have completely different modes of action. While certainly the combination of these two drugs would have a "combined effect is greater than the sum of their individual effects", their combination is synergistic in practical application, but not by the strict

**2. Basic models and mechanisms – Synergy on a molecular level** 

model cases will be presented, leading to a molecular definition of drug synergy.

Fig. 1. Schematic representation of inhibition mechanisms (A) Competitive inhibition. Inhibitor (open circles) binds to the active site of the target protein. The agonist (solid circles) binds to the same site. By definition of competitive inhibition, all competitive inhibitors bind to the same (active) site. Thus, binding of two competitive inhibitors must be mutually exclusive, and they cannot act synergistically on the same target protein. (B) Noncompetitive inhibition. Inhibitor (open squares) binds to a site different from the active site of the target molecule. In pure non-competitive inhibition agonist binding is not affected by the inhibitor. Inhibition is due to conversion of the target protein into an inactive state.

the first one, other than raising the total amount of inhibitory molecules.

Drug interaction and synergy has been intensively studied for more than 100 years, and some of the numerous concepts will be briefly introduced in this chapter. The simplest

Let us assume a simple enzyme following the laws of mass action and Michaelis-Menten kinetics. In the simplest case, this enzyme has an active site, where substrate is being converted into product, and possesses one or several specific binding sites for inhibitors (Fig. 1A). A competitive inhibitor by definition binds to the active site of the enzyme, displacing the substrate. Thus, a mixture of two purely competitive inhibitors will only ever target the active site. This is known as mutually exclusive binding. If only the simplest mechanistic case is considered, one would not expect a second competitive inhibitor to have any notable effect on

**2.1 A simple reaction scheme for enzyme inhibition** 

definition.

A

B

In case of non-competitive inhibition, the inhibitor binds to a location on the enzyme different from the active site. Assuming that bound inhibitor converts the enzyme into an inactive (non product-forming) state, presence of a non-competitive inhibitor simply lowers the amount of active enzyme molecules (Fig. 1B). There are states, where both substrate and inhibitor are bound to the enzyme. The effect of several non-competitive inhibitors applied together raises the question if synergy can be observed in such a simple system. If two non-competitive inhibitors bind to the same site on their target enzyme, this inhibitory site can either be occupied by inhibitor A or B, but not by both inhibitors at the same time (Fig. 2). This would be a case of mutually exclusive binding of two inhibitors. If one inhibitor is present, and the second one added, one may observe indeed a greater extend of inhibition, but this would only be due to larger amounts of inhibitory molecules being present. At all times, we could predict the total amount of inhibition by summations. In the simplest molecular case, two inhibitors targeting the same site would produce an additive effect only.

Fig. 2. Reaction scheme for two non-competitive inhibitors targeting the same site. Two noncompetitive inhibitors (squares) bind to the same site on the target protein. In this case their binding is mutually exclusive. Presence of the second inhibitor increases the total amount of inhibitor causing in increased inhibitory effect. This increase is only due to simple additivity, and not synergy.

If, however, two different binding sites for non-competitive inhibitors exist on an enzyme, two inhibitors may bind simultaneously (Fig. 3). Inspection of the reaction schemes (Fig. 2, 3) shows that if two inhibitors have specific, independent sites on the enzyme, we will observe states where the enzyme indeed has two inhibitors bound (Fig. 3). These states cannot exist if both inhibitors bind to the same site (Fig. 2). Thus, if two inhibitors are able to bind simultaneously, we have a case of "mutually non-exclusive" binding. Here, presence of the second inhibitor will not only give an additive effect (increase of the number of inhibitory molecules), but will generate additional inhibited states of the enzyme. Therefore, on a molecular level we would expect a superadditive effect of two such inhibitors.

It should be noted that the considerations above are made following some basic assumptions, namely that binding of an inhibitor will convert the enzyme to an inactive state, binding of substrate and inhibitor is reversible, and binding of any compound is fully independent from all other compounds. Thus, the equilibrium binding constant for inhibitor A is the same whether A binds to the unliganded enzyme, or to the enzyme that has substrate and/or another inhibitor bound. Given these assumptions, the mechanisms for activation (Fig. 4A), and non-competitive inhibition (Fig. 4B,C) show the different states in which an enzyme exists in the presence of two non-competitive inhibitors that bind to the same site (Fig. 4B), or to different sites of the enzyme (Fig. 4B).

Drug Synergy – Mechanisms and Methods of Analysis 147

bound inhibitor is completely inactive, ie it does not form product. Either inhibitor X or Y can bind at any given time. Presence of the second inhibitor can only exert an additive effect but is

In this section, a simple derivation of enzyme inhibition by one or two non-competitive inhibitors is given. To illustrate the consequences of mutually exclusive vs. non-exclusive

From the mechanism of a Michaelis-Menten-type enzyme (Fig. 4A) and the law of mass

*ES*

The enzyme E can only exist as free enzyme E, or enzyme-substrate-complex ES. The total enzyme concentration is [Etot], KM is the Michaelis constant. Only ES can form product, the

V0, the actual rate of product formation at a given concentration of substrate depends on the fraction of ES that is present in the equilibrium. Thus V0 can be expressed in terms of "ocupancy", or fES, the fraction of enzyme present in the enzyme-substrate complex ES.

> *ES ES <sup>f</sup> E ES*

*<sup>E</sup>* ,

1 1 *<sup>M</sup>*

 *ES X*, *ES <sup>f</sup> E ES EX EXS*

*S*

Note that this equation converts readily to the common form of the Michaelis-Menten

0 max

*V V <sup>K</sup>*

In the presence of a single non-competitive inhibitor, additional enzyme species are possible (EX, EXS in Fig. 4B). By definition of an inhibitor, these do not lead to any product

*ES*

*M E S*

*E E ES tot* (2)

*V kE* max 2 *tot* (3)

 0 max max *ES ES V V <sup>V</sup> <sup>f</sup> E ES*

*<sup>K</sup>* (1)

(4)

(6)

(7)

(5)

not synergistic. (C) Inhibition of a Michaelis-Menten enzyme by two non-competitive inhibitors X and Y, which are mutually non-exclusive, i.e. binding to different sites on the enzyme. Here, both inhibitors may bind simultaneously, giving rise to synergistic inhibition.

> *<sup>M</sup> E S*

*K*

*V k ES* 0 2

formation. Then fES becomes

*V V*

 0 max *tot ES*

equation, if the definition of KM (equation 1) is substituted into equation 5.

**2.2 Michaelis-Menten enzymes** 

action, we find:

binding, the simplest mechanisms are used.

maximum rate of product formation is Vmax.

Fig. 3. Reaction scheme for two mutually non-exclusive non-competitive inhibitors. Noncompetitive inhibition by two inhibitors (squares, triangles) binding to different sites on the target protein. Here, bindign of one inhibitor does not prevent binding of the other. Note that presence of the second inhibitor creates new inactive states of the target protein that are not possible if only one inhibitor is present. This applies even in the simplest theoretical case, where binding affinities of agonist and inhibitors are completely independent of each other. Thus, in the presence of two inhibitors of the same target protein that follow the rule of Bliss independence, i.e. mutually non-exclusive binding, synergy must be a necessary consequence.

Fig. 4. Mechanism of catalysis and non-competitive inhibition of a Michaelis-Menten enzyme. (A) Mechanism for the activation of an enzyme following Michaelis-Menten kinetics. E = enzyme, S = substrate, P = product, KM = Michaelis constant, k2 = rate of product formation from ES. A simplified MM kinetic scheme is used, assuming no backward reaction EP ES. (B) Inhibition of a Michaelis-Menten enzyme by two non-competitive inhibitors X and Y, which are mutually exclusive (e.g. binding to the same site). It is assumed that enzyme with

bound inhibitor is completely inactive, ie it does not form product. Either inhibitor X or Y can bind at any given time. Presence of the second inhibitor can only exert an additive effect but is not synergistic. (C) Inhibition of a Michaelis-Menten enzyme by two non-competitive inhibitors X and Y, which are mutually non-exclusive, i.e. binding to different sites on the enzyme. Here, both inhibitors may bind simultaneously, giving rise to synergistic inhibition.

#### **2.2 Michaelis-Menten enzymes**

146 Toxicity and Drug Testing

Fig. 3. Reaction scheme for two mutually non-exclusive non-competitive inhibitors. Noncompetitive inhibition by two inhibitors (squares, triangles) binding to different sites on the target protein. Here, bindign of one inhibitor does not prevent binding of the other. Note that presence of the second inhibitor creates new inactive states of the target protein that are not possible if only one inhibitor is present. This applies even in the simplest theoretical case, where binding affinities of agonist and inhibitors are completely independent of each other. Thus, in the presence of two inhibitors of the same target protein that follow the rule of Bliss independence, i.e. mutually non-exclusive binding, synergy must be a necessary

Fig. 4. Mechanism of catalysis and non-competitive inhibition of a Michaelis-Menten enzyme. (A) Mechanism for the activation of an enzyme following Michaelis-Menten kinetics. E = enzyme, S = substrate, P = product, KM = Michaelis constant, k2 = rate of product formation from ES. A simplified MM kinetic scheme is used, assuming no backward reaction EP ES. (B) Inhibition of a Michaelis-Menten enzyme by two non-competitive inhibitors X and Y, which are mutually exclusive (e.g. binding to the same site). It is assumed that enzyme with

consequence.

In this section, a simple derivation of enzyme inhibition by one or two non-competitive inhibitors is given. To illustrate the consequences of mutually exclusive vs. non-exclusive binding, the simplest mechanisms are used.

From the mechanism of a Michaelis-Menten-type enzyme (Fig. 4A) and the law of mass action, we find:

$$K\_M = \frac{[E][S]}{[ES]} \quad \rightarrow \quad [ES] = \frac{[E][S]}{K\_M} \tag{1}$$

The enzyme E can only exist as free enzyme E, or enzyme-substrate-complex ES. The total enzyme concentration is [Etot], KM is the Michaelis constant. Only ES can form product, the maximum rate of product formation is Vmax.

$$\begin{bmatrix} \begin{bmatrix} E\_{tot} \end{bmatrix} = \begin{bmatrix} E \end{bmatrix} + \begin{bmatrix} ES \end{bmatrix} \end{bmatrix} \tag{2}$$

$$V\_{\text{max}} = k\_2 \left[ E\_{tot} \right] \tag{3}$$

V0, the actual rate of product formation at a given concentration of substrate depends on the fraction of ES that is present in the equilibrium. Thus V0 can be expressed in terms of "ocupancy", or fES, the fraction of enzyme present in the enzyme-substrate complex ES.

$$f\_{ES} = \frac{\begin{bmatrix} ES \end{bmatrix}}{\begin{bmatrix} E \end{bmatrix} + \begin{bmatrix} ES \end{bmatrix}} \tag{4}$$

$$V\_0 = k\_2 \begin{bmatrix} ES \end{bmatrix} \ \Rightarrow \ V\_0 = V\_{\text{max}} \frac{\begin{bmatrix} ES \end{bmatrix}}{\begin{bmatrix} E\_{tot} \end{bmatrix}} \ \ \text{ \ } \ V\_0 = V\_{\text{max}} \frac{\begin{bmatrix} ES \end{bmatrix}}{\begin{bmatrix} E \end{bmatrix} + \begin{bmatrix} ES \end{bmatrix}} = V\_{\text{max}} f\_{ES} \tag{5}$$

Note that this equation converts readily to the common form of the Michaelis-Menten equation, if the definition of KM (equation 1) is substituted into equation 5.

$$V\_0 = V\_{\text{max}} \frac{1}{1 + \frac{K\_M}{S}} \tag{6}$$

In the presence of a single non-competitive inhibitor, additional enzyme species are possible (EX, EXS in Fig. 4B). By definition of an inhibitor, these do not lead to any product formation. Then fES becomes

$$f\_{ES,X} = \frac{\begin{bmatrix} ES \end{bmatrix}}{\begin{bmatrix} E \end{bmatrix} + \begin{bmatrix} ES \end{bmatrix} + \begin{bmatrix} EX \end{bmatrix} + \begin{bmatrix} EXS \end{bmatrix}} \tag{7}$$

Drug Synergy – Mechanisms and Methods of Analysis 149

1

*K S V S X Y S V K K <sup>V</sup> K X Y SK K*

*XY XY X Y*

The difference between mutually exclusive and non-exclusive inhibitors can directly be seen from an experiment where the concentration of inhibitor X is held constant, and only [Y] is

> 1 1 *XY X Y X S XY X S KK K*

Compared to the case of mutually exclusive inhibitors, the curve of S0/SX,Y. Y is shifted upwards by a constant concentration of X, and the slope of the curve also increases by a

The ratio method shown here applies to the simplest case of synergistic action of drugs, two substances binding to the same target. It requires some basic kinetic data to be collected and gives a simple linear graph that can be quickly inspected for a qualitative result whether two substances act on the same or on different sites on an enzyme, and thus whether these two substances can be synergistic on their target or not. It should be noted that by taking the ratios, the control signal (uninhibited case, i.e. the largest signal) is divided by a signal that becomes progressively smaller and thus carries a higher error. It is needed to detect whether two curves have the same slope (mutually exclusive binding, additive effect), or different slopes (mutually non-exclusive binding, synergy). This difference has to be clearly demonstrated from experiment and data analysis, requiring data of sufficient quality to make this distinction.

1. The value of KX, the inhibition constant, is unchanged if two inhibitors are only additive, and is decreased (~ higher inhibitory potency) in the presence of the second inhibitor. Therefore, we have a clear, mechanism-derived definition of synergy on the

2. Conversely, the method allows to determine whether two inhibitors bind to the same, or to different sites on an enzyme. This may be an important result for drug development, and is obtained without need of structural data. (Note: strictly speaking, the result only tells whether binding of two inhibitors is mutually exclusive or non-

The method has originally been presented for ligand-gated ion channels by Karpen and Hess (Karpen, Aoshima et al. 1982; Karpen and Hess 1986), and subsequently been used for the study of action of multiple inhibitors on ion channels (Karpen, Aoshima et al. 1982;

The basic mechanism presented here is by far not sufficient to describe multimeric enzymes, enzynmes requiring cofactors, and various modes of inhibition. Enzymes may form multimers, binding of one inhibitor may affect binding of other others, and binding sites my overlap. More complex mechanisms of inhibition of Michaelis-Menten enzymes have been discussed, including those of several inhibitors acting on a single enzyme (Palatini 1983). Action of several inhibitors as well as antagonistic interaction of enzyme inhibitors have

Karpen and Hess 1986; Breitinger, Geetha et al. 2001; Raafat, Breitinger et al. 2010).

*M*

*X Y*

1 1 <sup>1</sup>

(13)

(14)

1

1 11

max

*V*

*M*

0 ,

The technique provides two important pieces of information:

0 0

, 0, , max

varied. Equation 13 can be rearranged to:

factor of (1+X/KX).

molecular level.

exclusive)

The rate of product formation in presence of one non-competitive inhibitor is

$$V\_{0,X} = V\_{\text{max}} \frac{1}{\left(1 + \frac{K\_M}{S}\right) \left(1 + \frac{X}{K\_X}\right)}\tag{8}$$

For two mutually exclusive inhibitors X and Y (Fig. 4B), one obtains:

$$V\_{0,X,Y} = V\_{\text{max}} \frac{1}{\left(1 + \frac{K\_M}{S}\right) \left(1 + \frac{X}{K\_X} + \frac{Y}{K\_Y}\right)}\tag{9}$$

And for two mutually non-exclusive inhibitors (Fig. 4C), the rate equation is

$$V\_{0,X,Y} = V\_{\text{max}} \frac{1}{\left(1 + \frac{K\_M}{S}\right) \left(1 + \frac{X}{K\_X}\right) \left(1 + \frac{Y}{K\_Y}\right)}\tag{10}$$

There is a simple technique to determine the type of enzyme inhibition by two inhibitors, and whether their action on the enzyme is synergistic. To this end, the ratio of the initial rates in the absence (control, V0), and in the presence of inhibitor (V0,X) is measured. S0 is the control signal, SX is the signal obtained in the presence of inhibitor.

$$\frac{V\_{\text{max}}}{S\_{X}} = \frac{V\_{0}}{V\_{0,X}} = \frac{1}{V\_{\text{max}}\frac{1}{S} + \left(1 + \frac{K\_{M}}{S}\right)}\tag{11}$$

$$\frac{V\_{\text{max}}}{\left(1 + \frac{K\_{M}}{S}\right)\left(1 + \frac{X}{K\_{X}}\right)} = \left(1 + \frac{X}{K\_{X}}\right)\tag{12}$$

Thus, a straight-line curve is obtained when S0/SX is plotted against [X], the (varied) concentration of inhibitor X. The slope of this line (Fig. 6) gives the inhibition constant KX. This plot is linear over the entire range of inhibitor concentration.

In the case of two mutually exclusive inhibitors, the ratio becomes

$$\frac{V\_{\text{max}}}{S\_{X,Y}} = \frac{V\_0}{V\_{0,X,Y}} = \frac{\left(1 + \frac{K\_M}{S}\right)}{V\_{\text{max}}\frac{1}{\left(1 + \frac{K\_M}{S}\right)\left(1 + \frac{X}{K\_X} + \frac{Y}{K\_Y}\right)}} = \left(1 + \frac{X}{K\_X} + \frac{Y}{K\_Y}\right) \tag{12}$$

Presence of the second inhibitor only results in an additional term (Y/KY) that shifts the S0/SX,Y curve upwards. This term indicates additivity of the two inhibitors, but inhibitory potency (slope of the curve) is not altered.

For two mutually non-exclusive inhibitors, the ratio is

1

*K X S K*

1

1

*K X Y SK K*

<sup>1</sup> <sup>1</sup>

*X*

<sup>1</sup> <sup>1</sup>

*X Y*

1 11

1

 

*M*

*K X S K*

1

*<sup>V</sup> <sup>K</sup> S V S X S V <sup>K</sup> <sup>V</sup>*

1 1

Thus, a straight-line curve is obtained when S0/SX is plotted against [X], the (varied) concentration of inhibitor X. The slope of this line (Fig. 6) gives the inhibition constant KX.

1

 

*K S V S X Y S V K K <sup>V</sup>*

*K X Y S KK*

*M*

1

*XY XY X Y M*

Presence of the second inhibitor only results in an additional term (Y/KY) that shifts the S0/SX,Y curve upwards. This term indicates additivity of the two inhibitors, but inhibitory

*X X X M*

*M*

There is a simple technique to determine the type of enzyme inhibition by two inhibitors, and whether their action on the enzyme is synergistic. To this end, the ratio of the initial rates in the absence (control, V0), and in the presence of inhibitor (V0,X) is measured. S0 is the

max

max

*V*

1 1

*K X Y S KK*

*X*

*X Y*

*X Y*

(8)

(9)

(10)

(11)

(12)

1 1

1 1

*M*

*M*

The rate of product formation in presence of one non-competitive inhibitor is

0, max

*X*

For two mutually exclusive inhibitors X and Y (Fig. 4B), one obtains:

0, , max

And for two mutually non-exclusive inhibitors (Fig. 4C), the rate equation is

*X Y*

*V V*

0, , max

control signal, SX is the signal obtained in the presence of inhibitor.

0, max

This plot is linear over the entire range of inhibitor concentration. In the case of two mutually exclusive inhibitors, the ratio becomes

0 0

0 0

potency (slope of the curve) is not altered.

, 0, , max

For two mutually non-exclusive inhibitors, the ratio is

*X Y*

*V V*

*V V*

$$\frac{S\_0}{S\_{X,Y}} = \frac{V\_0}{V\_{0,X,Y}} = \frac{\left(1 + \frac{K\_M}{S}\right)}{V\_{\text{max}}\frac{1}{\left(1 + \frac{K\_M}{S}\right)\left(1 + \frac{X}{K\_X}\right)}} = \left(1 + \frac{X}{K\_X}\right)\left(1 + \frac{Y}{K\_Y}\right) \tag{13}$$

The difference between mutually exclusive and non-exclusive inhibitors can directly be seen from an experiment where the concentration of inhibitor X is held constant, and only [Y] is varied. Equation 13 can be rearranged to:

$$\frac{S\_0}{S\_{X,Y}} = 1 + \frac{X}{K\_X} + \frac{Y}{K\_Y} \left(1 + \frac{X}{K\_X}\right) \tag{14}$$

Compared to the case of mutually exclusive inhibitors, the curve of S0/SX,Y. Y is shifted upwards by a constant concentration of X, and the slope of the curve also increases by a factor of (1+X/KX).

The ratio method shown here applies to the simplest case of synergistic action of drugs, two substances binding to the same target. It requires some basic kinetic data to be collected and gives a simple linear graph that can be quickly inspected for a qualitative result whether two substances act on the same or on different sites on an enzyme, and thus whether these two substances can be synergistic on their target or not. It should be noted that by taking the ratios, the control signal (uninhibited case, i.e. the largest signal) is divided by a signal that becomes progressively smaller and thus carries a higher error. It is needed to detect whether two curves have the same slope (mutually exclusive binding, additive effect), or different slopes (mutually non-exclusive binding, synergy). This difference has to be clearly demonstrated from experiment and data analysis, requiring data of sufficient quality to make this distinction. The technique provides two important pieces of information:


The method has originally been presented for ligand-gated ion channels by Karpen and Hess (Karpen, Aoshima et al. 1982; Karpen and Hess 1986), and subsequently been used for the study of action of multiple inhibitors on ion channels (Karpen, Aoshima et al. 1982; Karpen and Hess 1986; Breitinger, Geetha et al. 2001; Raafat, Breitinger et al. 2010).

The basic mechanism presented here is by far not sufficient to describe multimeric enzymes, enzynmes requiring cofactors, and various modes of inhibition. Enzymes may form multimers, binding of one inhibitor may affect binding of other others, and binding sites my overlap. More complex mechanisms of inhibition of Michaelis-Menten enzymes have been discussed, including those of several inhibitors acting on a single enzyme (Palatini 1983). Action of several inhibitors as well as antagonistic interaction of enzyme inhibitors have

Drug Synergy – Mechanisms and Methods of Analysis 151

*D*

*K*

*open*

*SIF I*

we can then obtain

non-exclusive inhibitors.

 <sup>2</sup>

 2 <sup>2</sup> ( ) *RL RL open*

1

*D*

*K L*

1 1

0 max max 2

Fig. 5. Mechanisms of activation and non-competitive inhibition of ion channel receptors (A) Minimum mechanism for the activation of a ligand-gated ion channel. Note that the channel-opening reaction comprises two elementary steps, ligand binding (dissociation constant KD) and conformational change to the open state (open-close equilibrium ). R = receptor, L = activating ligand. In this example binding of two ligand molecules is needed prior to channel opening. (B) Inhibition of an ion channel receptor by two non-competitive inhibitors X and Y, which are mutually exclusive (e.g. binding to the same site). Either inhibitor X or Y can bind at any given time. Presence of the second inhibitor can only exert an additive effect but is not synergistic. (C) Inhibition of an ion channel receptor by two noncompetitive inhibitors X and Y, which are mutually non-exclusive, targeting different sites on the receptor. Synergism is then observed as a necessary consequence of two mutually

*RL L*

*RL* (19)

(21)

(20)

been studied (Asante-Appiah and Chan 1996; Schenker and Baici 2009), and a major development in drug interaction analysis was the detailed mathematical treatment of enzyme kinetics and inhibition by Chou and Talalay (Chou 1976; Chou and Talalay 1977; Chou and Talalay 1981; Chou 2006; Chou 2010), covering the mechanistic Michaelis-Menten approach as well as logistic approaches.

#### **2.3 Ligand-gated ion channel receptors**

Ligand-gated ion channels are principal mediators of rapid synaptic transmission between nerve cells and in the neuromuscular junction. Compared to Michaelis-Menten type enzymes, their mechanism of activation is more complex, requiring an additional transition (Hess 1993; Colquhoun 1998). First step of ion channel activation is binding of the activating ligand (a neurotransmitter), which is governed by the principle of mass action (Hess 1993; Colquhoun 1998). Usually, more than one ligand molecule is required; depending on receptor type, models with two or three ligands binding prior to efficient channel opening have been discussed. Ligand binding induces an conformational change, where the receptor protein converts from the closed to an open ion-conducting state (Fig. 5A) (Hess 1993;

Colquhoun 1998). Only the passing ions generate an electric signal and this signal can be recorded using patch-clamp techniques. Similar to the ES complex in enzymes, only the liganded receptor can undergo the opening transition. The mechanisms of non-competitive inhibition by two inhibitors binding to the same (Fig. 5B), or different (Fig. 5C) sites have been given. A similar derivation to the one for MM-enzymes can then be made.

The signal in this case is not a rate of product formation, but an ionic current, namely the rate of ion translocation through the open channel. Assuming a constant transmembrane voltage, and only one conducting state (ie only one channel size, in reality several conductance levels have been observed for each ion channel receptor).

The observed signal SL would then be:

$$S\_L = I\_L = \mathbf{n}\_{\text{Ch}} I\_{\text{ion}} \tag{15}$$

where IL is the observd current, nCh is the number of open channels, and Jion is the ion translocation rate. The maximum current signal would be observed if all ion channel were open at the same time. Fopen, the fraction of open channels, would then be equal to 1 (a theoretical value only).

$$S\_{\text{max}} = I\_{\text{max}} F\_{\text{open}} \tag{16}$$

Assuming that only receptors with two bound ligands can undergo the opening transition (Fig. 5A), we can define the fraction of open channels as

$$F\_{open} = \frac{\overline{\left[RL\_2(open)\right]}}{\left[R\right] + 2\left[RL\right] + \left[RL\_2\right] + \overline{\left[RL\_2(open)\right]}}\tag{17}$$

Using the law of mass action, we can define

$$K\_D = 2\frac{\left[R\right]\left[L\right]}{\left[RL\right]}\tag{18}$$

$$K\_D = \frac{\begin{bmatrix} RL \end{bmatrix} \begin{bmatrix} L \end{bmatrix}}{\begin{bmatrix} RL\_2 \end{bmatrix}} \tag{19}$$

$$\phi = \frac{\begin{bmatrix} RL\_2 \end{bmatrix}}{\begin{bmatrix} RL\_2 \text{(open)} \end{bmatrix}} \tag{20}$$

we can then obtain

150 Toxicity and Drug Testing

been studied (Asante-Appiah and Chan 1996; Schenker and Baici 2009), and a major development in drug interaction analysis was the detailed mathematical treatment of enzyme kinetics and inhibition by Chou and Talalay (Chou 1976; Chou and Talalay 1977; Chou and Talalay 1981; Chou 2006; Chou 2010), covering the mechanistic Michaelis-Menten

Ligand-gated ion channels are principal mediators of rapid synaptic transmission between nerve cells and in the neuromuscular junction. Compared to Michaelis-Menten type enzymes, their mechanism of activation is more complex, requiring an additional transition (Hess 1993; Colquhoun 1998). First step of ion channel activation is binding of the activating ligand (a neurotransmitter), which is governed by the principle of mass action (Hess 1993; Colquhoun 1998). Usually, more than one ligand molecule is required; depending on receptor type, models with two or three ligands binding prior to efficient channel opening have been discussed. Ligand binding induces an conformational change, where the receptor protein converts from the closed to an open ion-conducting state (Fig. 5A) (Hess 1993; Colquhoun 1998). Only the passing ions generate an electric signal and this signal can be recorded using patch-clamp techniques. Similar to the ES complex in enzymes, only the liganded receptor can undergo the opening transition. The mechanisms of non-competitive inhibition by two inhibitors binding to the same (Fig. 5B), or different (Fig. 5C) sites have

been given. A similar derivation to the one for MM-enzymes can then be made.

conductance levels have been observed for each ion channel receptor).

(Fig. 5A), we can define the fraction of open channels as

Using the law of mass action, we can define

The signal in this case is not a rate of product formation, but an ionic current, namely the rate of ion translocation through the open channel. Assuming a constant transmembrane voltage, and only one conducting state (ie only one channel size, in reality several

where IL is the observd current, nCh is the number of open channels, and Jion is the ion translocation rate. The maximum current signal would be observed if all ion channel were open at the same time. Fopen, the fraction of open channels, would then be equal to 1 (a

Assuming that only receptors with two bound ligands can undergo the opening transition

<sup>2</sup> ( ) *open*

*RL open <sup>F</sup>*

*K*

 2

> *<sup>D</sup>* <sup>2</sup> *R L*

( )

2 2

*S I nJ L L Ch ion* (15)

*S IF* max max *open* (16)

*RL* (18)

*R RL RL RL open* (17)

approach as well as logistic approaches.

**2.3 Ligand-gated ion channel receptors** 

The observed signal SL would then be:

theoretical value only).

$$S\_0 = I\_{\text{max}} F\_{\text{open}} = I\_{\text{max}} \frac{1}{\left(\frac{K\_D}{L} + 1\right)^2 \phi + 1} \tag{21}$$

$$\begin{array}{ccccc} \mathsf{A} & \mathsf{R} & \xleftarrow{\mathrm{K}\_{\mathsf{D}}} & \mathsf{RL} & \xleftarrow{\mathrm{K}\_{\mathsf{D}}} & \mathsf{RL}\_{2} & \xleftarrow{\Phi} & \overline{\mathrm{RL}\_{2}} \text{(open)} \end{array}$$

Fig. 5. Mechanisms of activation and non-competitive inhibition of ion channel receptors (A) Minimum mechanism for the activation of a ligand-gated ion channel. Note that the channel-opening reaction comprises two elementary steps, ligand binding (dissociation constant KD) and conformational change to the open state (open-close equilibrium ). R = receptor, L = activating ligand. In this example binding of two ligand molecules is needed prior to channel opening. (B) Inhibition of an ion channel receptor by two non-competitive inhibitors X and Y, which are mutually exclusive (e.g. binding to the same site). Either inhibitor X or Y can bind at any given time. Presence of the second inhibitor can only exert an additive effect but is not synergistic. (C) Inhibition of an ion channel receptor by two noncompetitive inhibitors X and Y, which are mutually non-exclusive, targeting different sites on the receptor. Synergism is then observed as a necessary consequence of two mutually non-exclusive inhibitors.

Drug Synergy – Mechanisms and Methods of Analysis 153

**mutually non-exclusive**

Fig. 6. Ratio method graph. Graph of signal ratio S0/SX,Y vs inhibitor concentration for the case of one inhibitor (black curve), two mutually exclusive inhibitors (gray curve), and two mutually non-exclusive inhibitors (light gray curve). In case of mutually non-exclusive binding the inhibitory potency of inhibitor Y is increased in the presence of inhibitor X, as indicated by the lower value of KY computed from the slope of the inhibition ratio curve. Note that the formalism described here becomes mechanism-independent and applies to Michaelis-Menten type enzymes as well as to more complex mechanisms of ion channel

So far, a simple description of the action of two inhibitors on a common target has been derived. The mechanisms were based upon (i) a common binding site for two inhibitors, leading to mutually exclusive binding (Fig. 2), or (ii) two independent binding sites, leading to mutually non-exclusive binding (Fig. 3), Indeed, these simple models underlie (i) the principle of Loewe additivity (Loewe 1953; Berenbaum 1989), also referred to "similar", or "homodynamic" action of drugs. Here, the expectation value for zero interaction is just additivity. Independent inhibitor sites (Fig. 3), in contrast, correspond to Bliss independence, "dissimilar", "heterodynamic", or "independent" action of drugs (Bliss 1939; Berenbaum 1989). The combined effect of two such drugs will be more than additive, fulfilling the basic criterion of synergy. It has been recognized that these are the two limiting mechanisms for drug interaction (Bliss 1939; Finney 1942; Plackett and Hewlett 1948), and indeed both models

are being used in the literature as zero interaction reference (Greco, Bravo et al. 1995).

of "both" drugs would be similar, and thus we have a perfect model of additivity.

It is intuitive, and favoured by this author to view the concept of Loewe additivity as the zero interaction reference, and noting the superadditive response from Bliss independence as synergism. This definition is widely accepted (Segel 1975; Chou and Talalay 1977; Berenbaum 1989). Furthermore, it allows for a very intuitive definition of zero interaction, proposed by Loewe: if drug A and B are the same, B being a dilution of A. Naturally, action

**X + Y**

**mutually exclusive**

**[Inhibitor Y]**

**Y alone**

**X + Y**

**SL SL,X,Y**

receptor inhibition.

In the presence of one non-competitive inhibitor X, we obtain the following equation for the signal SX:

$$S\_X = I\_{\max} \frac{1}{\left[ \left( \frac{K\_D}{L} + 1 \right)^2 \phi + 1 \right] \left( 1 + \frac{X}{K\_X} \right)} \tag{22}$$

where KX is the inhibition constant, L is the concentration of activating ligand, and X the concentration of inhibitor. One can now readily compute the ratios of control current signal to signal in presence of inhibitor:

$$\frac{I\_{\text{max}}}{I\_{S\_{X}}} = \frac{1}{\frac{\left[\left(\frac{K\_{D}}{L} + 1\right)^{2}\phi + 1\right]}{1}} = 1$$

$$\frac{I\_{\text{max}}}{I\_{\text{max}}} \overline{\left[\left(\frac{K\_{D}}{L} + 1\right)^{2}\phi + 1\right]} \left(1 + \frac{X}{K\_{X}}\right)} = 1 + \frac{X}{K\_{X}}\tag{23}$$

In case of two inhibitors binding to the same site (mutually exclusive), the ratio again becomes

$$\frac{S\_0}{S\_{X,Y}} = 1 + \frac{X}{K\_X} + \frac{Y}{K\_Y} \tag{24}$$

For two non-exclusive inhibitors, targeting different sites on the recpetor, this ratio then is

$$\frac{S\_0}{S\_{X,Y}} = 1 + \frac{X}{K\_X} + \frac{Y}{K\_Y} \left(1 + \frac{X}{K\_X}\right) \tag{25}$$

Equations 23 – 25 are identical to equations 11-14.

Similar to the treatment of Michaelis-Menten enzymes, we obtain again a system of linear equations that describes the action of one or two inhibitors of ion channel receptors. If the concentration of inhibitor X is held constant, and the concentration of the second inhibitor, Y, is varied, the ratio S0 / SX,Y is shifted up by a constant amount X/KX but the slope (1/KY) is unchanged. The slope of the ratio curve represents the inhibitory potency, and the constant upward shift is due to the additive effect of two mutually exclusive inhibitors.

In the presence of two mutually non-exclusive inhibitors, the slope (ie inhibitory potency) is increased by a factor of (1 + X/KX). Thus, if the mechanism underlying this analysis were followed, the "amount of synergy" could be calculated as 1 + X/KX. Often, quality of the data does not permit this quantitation, although the qualitative demonstration of synergy (increased inhibitory potency of drug A in the presence of drug B) is statistically safe. Thus, by taking the ratios of control and inhibited signals, we arrive at an equation that becomes mechanism-independent and corresponds to the principal equations used to describe drug interactions. The ratio method results in a simple graph that describes the type of joint action of two inhibitors on a common enzyme, neurotransmitter receptor, or general target protein (Fig. 6).

In the presence of one non-competitive inhibitor X, we obtain the following equation for the

1

1 11

*K X L K* 

where KX is the inhibition constant, L is the concentration of activating ligand, and X the concentration of inhibitor. One can now readily compute the ratios of control current signal

1

 

1 11

In case of two inhibitors binding to the same site (mutually exclusive), the ratio again

1 *XY X Y S X Y S KK*

For two non-exclusive inhibitors, targeting different sites on the recpetor, this ratio then is

1 1 *XY X Y X S XY X S KK K*

Similar to the treatment of Michaelis-Menten enzymes, we obtain again a system of linear equations that describes the action of one or two inhibitors of ion channel receptors. If the concentration of inhibitor X is held constant, and the concentration of the second inhibitor, Y, is varied, the ratio S0 / SX,Y is shifted up by a constant amount X/KX but the slope (1/KY) is unchanged. The slope of the ratio curve represents the inhibitory potency, and the constant upward shift is due to the additive effect of two mutually exclusive inhibitors. In the presence of two mutually non-exclusive inhibitors, the slope (ie inhibitory potency) is increased by a factor of (1 + X/KX). Thus, if the mechanism underlying this analysis were followed, the "amount of synergy" could be calculated as 1 + X/KX. Often, quality of the data does not permit this quantitation, although the qualitative demonstration of synergy (increased inhibitory potency of drug A in the presence of drug B) is statistically safe. Thus, by taking the ratios of control and inhibited signals, we arrive at an equation that becomes mechanism-independent and corresponds to the principal equations used to describe drug interactions. The ratio method results in a simple graph that describes the type of joint action of two inhibitors on a common enzyme, neurotransmitter receptor, or general target

*X X*

*S L X S K <sup>I</sup> K X L K*

1 1

<sup>1</sup> <sup>1</sup>

*X*

(24)

*X*

(22)

(23)

(25)

max 2

max 2

*K*

*D*

max 2

*I*

*D*

0 ,

0 , *D*

*X*

0

Equations 23 – 25 are identical to equations 11-14.

*S I*

signal SX:

becomes

protein (Fig. 6).

to signal in presence of inhibitor:

Fig. 6. Ratio method graph. Graph of signal ratio S0/SX,Y vs inhibitor concentration for the case of one inhibitor (black curve), two mutually exclusive inhibitors (gray curve), and two mutually non-exclusive inhibitors (light gray curve). In case of mutually non-exclusive binding the inhibitory potency of inhibitor Y is increased in the presence of inhibitor X, as indicated by the lower value of KY computed from the slope of the inhibition ratio curve. Note that the formalism described here becomes mechanism-independent and applies to Michaelis-Menten type enzymes as well as to more complex mechanisms of ion channel receptor inhibition.

So far, a simple description of the action of two inhibitors on a common target has been derived. The mechanisms were based upon (i) a common binding site for two inhibitors, leading to mutually exclusive binding (Fig. 2), or (ii) two independent binding sites, leading to mutually non-exclusive binding (Fig. 3), Indeed, these simple models underlie (i) the principle of Loewe additivity (Loewe 1953; Berenbaum 1989), also referred to "similar", or "homodynamic" action of drugs. Here, the expectation value for zero interaction is just additivity. Independent inhibitor sites (Fig. 3), in contrast, correspond to Bliss independence, "dissimilar", "heterodynamic", or "independent" action of drugs (Bliss 1939; Berenbaum 1989). The combined effect of two such drugs will be more than additive, fulfilling the basic criterion of synergy. It has been recognized that these are the two limiting mechanisms for drug interaction (Bliss 1939; Finney 1942; Plackett and Hewlett 1948), and indeed both models are being used in the literature as zero interaction reference (Greco, Bravo et al. 1995).

It is intuitive, and favoured by this author to view the concept of Loewe additivity as the zero interaction reference, and noting the superadditive response from Bliss independence as synergism. This definition is widely accepted (Segel 1975; Chou and Talalay 1977; Berenbaum 1989). Furthermore, it allows for a very intuitive definition of zero interaction, proposed by Loewe: if drug A and B are the same, B being a dilution of A. Naturally, action of "both" drugs would be similar, and thus we have a perfect model of additivity.

Drug Synergy – Mechanisms and Methods of Analysis 155

Here, E0 is the observed effect, Emax is the maximum signal, EC50 is the concentration of ligand L that produces 50 % of the maximum response, and n is a coefficient defining the steepness of the dose-response curve. The similarity to Michaelis-Menten type enzyme kinetics is obvious, yet the logistic formalism is not based on any mechanism. Indeed, complex clinical situations require use of mechanism-free models to analyze drug interactions (Chou 1976; Berenbaum 1978; Berenbaum 1980; Chou and Talalay 1981;

In the following section some principles and formalisms for the analysis of drug synergism are briefly reviewed. An exhaustive review of all concepts is outside the scope of this text, readers are directed to several excellent, comprehensive reviews (Berenbaum 1989; Greco, Bravo et al. 1995; Tallarida 2001; Chou 2002; Toews and Bylund 2005; Chou 2006; Tallarida

**3. Mechanisms and techniques of synergy testing in complex biomedical** 

it obviously assigns synergism to the effects of several drugs too readily.

would be just additive, and more than 20 would mean synergy.

varying doses of each drug alone, and in combination.

An example, modified from Berenbaum (Berenbaum 1989) is that of a woodcutter, able to cut 10 trees in a day. He is joined by a second woodcutter, also able to cut down 10 trees in a day. Together, they manage to cut 15 trees in one day. How do we describe this situation? One approach is that cutter A achieves 10 trees per day, our expectation value. Addition of cutter B results in 15 trees being cut, so there is synergy. Such an approach has been proposed e.g. by Gaddum, who only considered the effect of one agent and whether it was affected by another one being added (Gaddum 1940). This formalism is not used widely, as

Conversely, one would say that with two cutters, each able to cut 10 trees per day, the expectation value is 20 trees/day. If only 15 are achieved, they are antagonising each other. This is the application of additivity, and clearly, the combined effect is sub-additive, 20 trees

Mechanistically, one might argue that if cutter A works on a tree, then cutter B would not work on the same tree. Their action would be mutually exclusive, and the additive result would be expected. If, however, they are willing to work at the same tree together, they will be able to cut this tree in a much shorter time. In this case, they would be able to cut more than 20 trees in a day and their action would be mutually non-exclusive, leading to synergy. As stated above, pure mechanistic analysis is not sufficient (and not possible) for most clinical cases, so a general, mechanism-free analysis of drug interaction is needed. Berenbaum (Berenbaum 1989) has pointed out the similarity to non-parametric statistical tests that do not require information about the meaning of the values, or the distribution of populations from where the values originate. The equivalent in dose-response analysis is a logistic equation, that just describes a dose-response curve without any requirement of a mechanism. In such a setting, one would just define the desired outcome (enzyme inhibition, cell death, reduction of virus titer, ...), and then measure the effect achieved by

The mechanisms shown above illustrate just the simplest mechanistic model. In real life, the situation is more complicated, as mechanisms of enzyme or receptor acitivity are more complex. Furthermore medical intervention is not only directed at single proteins, but at entire pathways or controlling structures, such as transcription factors, that initiate or control biochemical processes. Some therapies, such as cancer chemotherapy even aim at

Berenbaum 1989; Tallarida 1992; Greco, Bravo et al. 1995).

2006; Bijnsdorp, Giovannetti et al. 2011).

**settings** 

However, what happens if we already know that drug A and B have completely different modes of action? Two drugs could be targeting different enzymes in a biochemical pathway. Of such a combination of drugs – having dissimilar action – we would expect superadditive behaviour. Can we call this synergy, or is it just expected from the mechanism and is now our zero reference? Arguments can be found for either view, and both models (and many more) are thus used and debated in the literature.

Once we move to more complicated systems, mechanism-based analysis is no longer feasible, and more general descriptions of drug interaction are needed. However, they all relate to the basic models of additivity and independence that were described above.

Equation 25 can be rearranged into the form

$$\frac{S\_0}{S\_{X,Y}} = 1 + \frac{X}{K\_X} + \frac{Y}{K\_Y} + \frac{X}{K\_X} \frac{Y}{K\_Y} \tag{26}$$

This equation is similar to a general equation that describes describing the joint action of two drugs on a specific target or biochemical process, presented by Greco et al. (Greco, Bravo et al. 1995).

$$1 = \frac{D\_1}{ID\_{X,1}} + \frac{D\_2}{ID\_{X,2}} + f\left(\frac{D\_1}{ID\_{X,1}}, \frac{D\_2}{ID\_{X,2}}, a, p\right) \tag{27}$$

Here, D1 and D2 are concentrations of drug 1 and 2 in a mixture; IDX,1 and IDX,2 are the concentrations that produce a certain effect (corresponding to EC50, or IC50 values); is the synergism/antagonism parameter and p represents additional parameter(s) describing the "interaction" (joint action) of the two drugs.

The models and derivations given above are indeed the simplest approach to synergism between drugs. At this time, we do not even have a complete description of the action of every drug. It has been pointed out that under physiological conditions, it is expected that indeed presence of a drug will always result in an altered state of metabolism and thereby affect other drugs (Gessner 1974). In many patients multiple drug regimes have to be given, and the metabolism of a critically ill person may differ from a healthy "control" volunteer. Taken together, medical reality is not sufficiently described by simplified models. However, as shown above, even from simple model cases we can understand mechanisms of synergy and can derive mechanism-independent formalisms to determine the type of joint action of drug combinations.

In biomedical modelling, an alternative approach is the use of a mechanism-free description of activity, such as enzyme activity, ion channel function, the throughput of an entire biochemical pathway, or even cell survival in toxicity assays. The most common approach is the use of logistic equations that simply connect concentration of an effector (agonist or inhibitor) to the measured effect (enzyme activity, product formation, cell survival). The most comonly used formalism is that of the Hill equation.

$$E\_0 = E\_{\text{max}} \frac{1}{1 + \left(\frac{EC\_{50}}{L}\right)^n} \tag{28}$$

However, what happens if we already know that drug A and B have completely different modes of action? Two drugs could be targeting different enzymes in a biochemical pathway. Of such a combination of drugs – having dissimilar action – we would expect superadditive behaviour. Can we call this synergy, or is it just expected from the mechanism and is now our zero reference? Arguments can be found for either view, and both models (and many

Once we move to more complicated systems, mechanism-based analysis is no longer feasible, and more general descriptions of drug interaction are needed. However, they all

> *XY X Y X Y S X Y XY S K K KK*

This equation is similar to a general equation that describes describing the joint action of two drugs on a specific target or biochemical process, presented by Greco et al. (Greco,

> 1 2 12 ,1 ,2 ,1 ,2 1 , , , *X X XX D D DD <sup>f</sup> <sup>p</sup> ID ID ID ID*

Here, D1 and D2 are concentrations of drug 1 and 2 in a mixture; IDX,1 and IDX,2 are the concentrations that produce a certain effect (corresponding to EC50, or IC50 values); is the synergism/antagonism parameter and p represents additional parameter(s) describing the

The models and derivations given above are indeed the simplest approach to synergism between drugs. At this time, we do not even have a complete description of the action of every drug. It has been pointed out that under physiological conditions, it is expected that indeed presence of a drug will always result in an altered state of metabolism and thereby affect other drugs (Gessner 1974). In many patients multiple drug regimes have to be given, and the metabolism of a critically ill person may differ from a healthy "control" volunteer. Taken together, medical reality is not sufficiently described by simplified models. However, as shown above, even from simple model cases we can understand mechanisms of synergy and can derive mechanism-independent formalisms to determine the type of joint action of

In biomedical modelling, an alternative approach is the use of a mechanism-free description of activity, such as enzyme activity, ion channel function, the throughput of an entire biochemical pathway, or even cell survival in toxicity assays. The most common approach is the use of logistic equations that simply connect concentration of an effector (agonist or inhibitor) to the measured effect (enzyme activity, product formation, cell survival). The

> 1 *<sup>n</sup> E E*

50

1

*EC L*

 

(26)

(27)

(28)

relate to the basic models of additivity and independence that were described above.

1

0 ,

more) are thus used and debated in the literature.

Equation 25 can be rearranged into the form

"interaction" (joint action) of the two drugs.

most comonly used formalism is that of the Hill equation.

0 max

Bravo et al. 1995).

drug combinations.

Here, E0 is the observed effect, Emax is the maximum signal, EC50 is the concentration of ligand L that produces 50 % of the maximum response, and n is a coefficient defining the steepness of the dose-response curve. The similarity to Michaelis-Menten type enzyme kinetics is obvious, yet the logistic formalism is not based on any mechanism. Indeed, complex clinical situations require use of mechanism-free models to analyze drug interactions (Chou 1976; Berenbaum 1978; Berenbaum 1980; Chou and Talalay 1981; Berenbaum 1989; Tallarida 1992; Greco, Bravo et al. 1995).

In the following section some principles and formalisms for the analysis of drug synergism are briefly reviewed. An exhaustive review of all concepts is outside the scope of this text, readers are directed to several excellent, comprehensive reviews (Berenbaum 1989; Greco, Bravo et al. 1995; Tallarida 2001; Chou 2002; Toews and Bylund 2005; Chou 2006; Tallarida 2006; Bijnsdorp, Giovannetti et al. 2011).
