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ANESTHETIC PHARMACOLOGY

Additivity Versus Synergy: A Theoretical Analysis of Implications for Anesthetic Mechanisms

Steven L. Shafer, MD^*, Jan F. A. Hendrickx, MD, PhD^¶, Pamela Flood, MD^*, James Sonner, MD, and Edmond I. Eger, II, MD

From the *Department of Anesthesiology, Columbia University, New York, New York; Department of Anesthesia, Stanford University, Stanford, California; Departments of Biopharmaceutical Sciences and Anesthesia and Perioperative Care, University of California at San Francisco, San Francisco, California; and ¶Department of Anesthesiology and Critical Care Medicine, OLV Hospital, Aalst, Belgium.

Address correspondence and reprint requests to Steven L. Shafer, MD, Department of Anesthesiology, Columbia University Medical Center, 622 W. 168th St, PH 5-505, New York, NY 10032-3725. Address e-mail to sshafer{at}columbia.edu.

	Abstract

Top Abstract Introduction Theoretical Development DISCUSSION APPENDIX A: TWO DRUGS... APPENDIX B: RECEPTORS ARRANGED... APPENDIX C: TWO DRUGS... Appendix D: Derivations REFERENCES

 
BACKGROUND: Inhaled anesthetics have been postulated to act at multiple receptors, with modest action at each site summing to produce immobility to noxious stimulation. Recent experimental results affirm prior findings that inhaled anesthetics interact additively. Synergy implies multiple sites of action by definition. In this essay, we explore the converse: does additivity imply a single site of action?

METHODS: The interaction of one versus two ligands competing for the same binding site at a receptor was explored using the law of mass action. Circuits were then constructed to investigate how the potency of drugs and the steepness of the concentration versus response relationship is amplified by the arrangement of suppressors into serial circuits, and enhancers into parallel circuits. Assemblies of suppressor and enhancer circuits into signal processing units were then explored to investigate the constraints signal processing units impose on additive interactions. Lastly, the relationship between synergy, additivity, and fractional receptor occupancy was explored to understand the constraints imposed by additivity.

RESULTS: Drugs that compete for a single receptor, and that similarly affect the receptor, must be additive in their effects. Receptors that bind suppressors in serial circuits, or enhancers in parallel circuits, increase the apparent potency of the drugs and the steepness of the concentration versus response relationship. When assemblies of suppressor and enhancer circuits are arranged into signal processing units, the interactions may be additive or synergistic. The primary determinant is the relationship between the concentration of drug associated with the effect of interest and the concentration associated with 50% receptor occupancy, k_d. Effects mediated by very low concentrations are more likely to be additive. Similarly, inhaled anesthetics that act at separate sites are unlikely to exhibit additive interactions if anesthetic drug effect occurs at concentrations at or above 50% receptor occupancy. However, if anesthetic drug effect occurs at very low levels of receptor occupancy, then additivity is expected even among anesthetics acting on different receptors.

CONCLUSIONS: Additivity among drugs acting on different receptors is only likely if the concentrations responsible for the drug effect of interest are well below the concentration associated with 50% receptor occupancy.

	Introduction

Top Abstract Introduction Theoretical Development DISCUSSION APPENDIX A: TWO DRUGS... APPENDIX B: RECEPTORS ARRANGED... APPENDIX C: TWO DRUGS... Appendix D: Derivations REFERENCES

 
Several investigations studying anesthetic pairs find that inhaled anesthetics interact with other inhaled anesthetics in an additive manner.1–4 Using the minimum alveolar concentration (MAC) that suppresses movement to noxious stimulation as the measure of anesthesia drug effect, a companion manuscript in this issue of Anesthesia & Analgesia describes the additivity of 16 inhaled anesthetic pairs with divergent effects on known ion channels, and with a rigorous definition of additivity.5 The authors suggest that the lack of synergy is consistent with a unitary mechanism of inhaled anesthetic action, rather than a mechanism requiring diverse actions synergistically summing to generate immobility in response to noxious stimulation. A second companion paper in this issue of Anesthesia & Analgesia explores the interactions among many drug classes of IV and inhaled drugs used in anesthetic practice.6 The general rule, albeit with exceptions, is that drugs with different sites of action interact synergistically.

Eckenhoff and Johansson7 proposed that inhaled anesthetics produce immobility by summation of modest effects on multiple receptor systems. Given the synergy observed in human and animal trials with many drug combinations that act on different receptors,6 if Eckenhoff and Johansson are correct we would expect at least some inhaled anesthetic pairs that have divergent effects on specific receptors would act synergistically. However, this is unambiguously not the case.5 The interactions are additive.

This essay explores the hypothesis that synergy implies separate sites of action, and the corollary hypothesis that additivity implies a unitary mechanism of action.

	Theoretical Development

Top Abstract Introduction Theoretical Development DISCUSSION APPENDIX A: TWO DRUGS... APPENDIX B: RECEPTORS ARRANGED... APPENDIX C: TWO DRUGS... Appendix D: Derivations REFERENCES

 
Definitions
Drug interactions are classically defined using isobolographic analysis. Assessment of the interaction requires that we precisely define drug effect. It is experimentally and mathematically convenient to select half of the maximum possible effect because concentration versus effect curves are typically steepest at half-maximal effect, permitting the most precise measurement of the concentration associated with the stated effect, while requiring the fewest animals to assess the concentration associated with a 50% probability of an animal generating a binary (yes/no) response (e.g., movement to noxious stimulation). For inhaled anesthetics, the effect is usually MAC, the alveolar concentration at steady-state associated with a 50% probability of moving in response to noxious stimulation.

As shown in Figure 1, the next step is to determine the concentration of each drug that achieves the effect of interest. Concentration may be an instantaneous plasma concentration, a steady-state plasma concentration, or an effect-site concentration. For inhaled anesthetics, the "concentration" is often the steady-state end-tidal concentration of the anesthetic. For IV anesthetic drugs, "concentration" is typically the calculated effect-site concentration, which is, by definition, the plasma concentration at steady-state. In some studies a bolus dose or infusion rate may substitute for concentration.

View larger version (20K): [in this window] [in a new window]   Figure 1. The interaction between two drugs, A and B, can be defined in terms of the "isobole" for a given drug effect when the drugs are given together. If we normalize drug A to the concentration of A alone that produces the stated effect, and drug B to the concentration of B alone that produces the stated effect, then an "additive" relationship exists if the concentrations of A and B together that produce the same effect satisfy the equation A + B = 1. The line of equipotent dosing is the line where equal doses of A and B are given. The arbitrary definition of synergy in this manuscript is that when equal concentrations of A and B given concurrently produce stated effect, then synergy exists if A + B < 0.9, and infra-additivity exists if A + B > 1.1.

In Figure 1, the X and Y axes units are fractions of the concentration of drugs A and B, respectively, that produce the defined drug effect. For inhaled anesthetics, the "concentration" in Figure 1 corresponds to the MAC fraction.

"Isoboles" are lines that define the concentrations (in the present case, the concentrations normalized to the concentration that produces the target effect) of two drugs that, given together, produce exactly the same effect. For an additive relationship, a proportional increase in the concentration of drug B exactly compensates for a proportional reduction in the concentration of drug A, resulting in a straight (additive) isobole. For example, the point marked X on Figure 1 implies that half of the required concentration of drug A (e.g., a 0.5 fraction of the concentration of drug A that produces the target effect) given with half of the required concentration of drug B (e.g., the same 0.5 fraction for drug B), produces the drug effect produced by the required concentration of drug A alone or the required concentration of drug B alone. This defines additivity.

Experimentally, additivity is considered to result if test findings lie within a defined area surrounding the additivity isobole. Evidence for synergy results if the combined (normalized) concentrations of both drugs required to produce the stated drug effect are significantly less than predicted by additivity (Fig. 1). Evidence for "infra-additivity" results if the combined concentrations of both drugs required to produce the stated drug effect significantly exceed that predicted by additivity.

Infra-additivity differs from classical pharmacological antagonism. For example, naloxone is a classical pharmacological antagonist of opioid drug effect. No concentration of naloxone causes analgesia, which precludes constructing an isobole that relates naloxone analgesia and opioid analgesia. What we will call "infra-additivity" in this analysis is often called "antagonism" in drug interaction studies. We have no quibble with this definition, but in this essay we use the term "infra-additivity."

Many investigators define synergy or infra-additivity as occurring when a combination differs significantly from the line of additivity. We will focus our analysis on three points of synergy in Figure 2. Point X demonstrates statistically significant synergy, as implied by the confidence bars for the respective concentrations of drugs A and B. The synergy demonstrated by point X is probably clinically significant. Point Y also deviates significantly from the line of additivity, but this statistically significant synergy is so slight that clinically it cannot be distinguished from additivity. Point Z is the most problematic. Despite a distance from point Z to the line of additivity that suggests a clinically important synergy, statistical significance is not achieved. Many investigators would conclude that, since point Z does not differ statistically significantly from the line of additivity, point Z represents additivity. However, point Z may demonstrate true synergy, as discussed below.

View larger version (9K): [in this window] [in a new window]   Figure 2. Three different results from a hypothetical interaction experiment, to distinguish synergy (point X), additivity (point Y), and additivity by default (point Z).

For this essay, and the companion papers, we define "synergy" as resulting when the sum of normalized fractional concentrations of two drugs that combined produce a target effect (e.g., MAC) is significantly <0.9 (i.e., a 10% or more decrease from additivity that reaches statistical significance). The 10% definition is arbitrary, but is consistent with the companion manuscripts.5,6 Similarly, there is "additivity" when the fractional concentrations of the two drugs, in combination, sum to between 0.9 and 1.1. Finally, there is "infra-additivity" when the fractions of the concentration of each drug individually that produces the same effect, sum to significantly exceed 1.1 (i.e., a 10% increase from additivity that reaches statistical significance). If the fractional sum of the concentrations is <0.9, or exceeds 1.1, but this difference is not statistically significant, we define this as "additivity by default" meaning that the data do not allow exclusion of synergy (if <0.9) or infra-additivity (if more than 1.1).

The most accurate assessment of synergy, additivity, and infra-additivity is achieved by giving approximately equipotent concentrations, indicated by the line of equipotent dosing in Figure 1. Detection of synergy becomes difficult or impossible if the concentration of either drug approaches the concentration that given alone produces the defined effect. By our definition, synergy cannot be detected if one gives 91% of the concentration of drug A that produces the desired effect because the concentration fraction to produce that effect is already 0.91 (i.e., within 10% of 1.0). The sum of the concentration fractions, even if the concentration of drug B is 0, still exceeds 0.9, eliminating synergy.

Referring again to Figure 2, X represents "synergy" by our definition because the fractional concentrations sum to <0.9, and the 95% confidence bounds exclude the line of additivity. Although the 95% confidence bounds for Y exclude the line of additivity, Y represents "additivity" because the fractional sum lies between 0.9 and 1.1, Z represents "additivity by default." Z strongly suggests synergy. The relationship suggested by point Z may be synergistic, but the study has insufficient statistical power to assert this at P < 0.05. However, at the same time it is not correct to conclude that Z represents additivity. Indeed, because point Z is the location with the highest probability of defining the relationship, the relationship is probably synergistic.

By definition, additivity is how a drug interacts with itself. The combination of a normalized 0.5 concentration of drug A with a normalized 0.5 concentration of drug A, obviously produces exactly the same effect as a 1.0 normalized concentration of drug A, because the subject received a 1.0 normalized concentration of drug A. By tautology, a drug's interaction with itself must be additive.

Note that "synergy," "additivity," and "infra-additivity" always refer to adding drug concentrations (or doses). "Synergy," "additivity" and "infra-additivity" are inappropriate for discussing drug effects. Most concentration versus response relationships are nonlinear (e.g., doubling the concentration does not double the effect). The effect of adding two concentrations, either of one drug or multiple drugs, can deviate from the sum of the individual effects, depending on the position of each concentration in the concentration versus response relationship. Reverting briefly to the dose domain, 100 mg of vecuronium, added to 100 mg of rocuronium, produces a less than additive effect. Each dose produces 100% blockade, but the combination can only produce 100% blockade, not 200% blockade. Thus, the effects are not additive. Indeed, most drugs show a ceiling effect. Combining two drugs in the same class at doses that individually produce the ceiling effect would still give the same ceiling, which is clearly less than additivity. So "additivity" and "synergy" strictly refer to adding drug concentrations (or doses), never adding drug effects.

Returning to Figure 1, the lines separating additivity from synergy and infra-additivity indicate 10% deviations from the additivity isobole. The region of additivity is just 13% of the total area of the graph. Indeed, the only meaningful deviation is that which occurs along the line of equipotent dosing, and that is just 10% of the total possible deviation from additivity. This is the first suggestion of the punch line of the manuscript: additivity is unusual, except for a drug acting on itself, which is additive by definition.

Mass Action and Synergy: Individual Receptors to Simple Circuits to Single Processing Units
Individual Receptors
Appendix A analyzes drug interactions at the level of individual receptors. Many pharmacological texts present this analysis. Additivity can result when two drugs compete for binding to the same receptor. Fentanyl and alfentanil provide such an example. Each binds to the same site on the µ opioid receptor, and they compete for access to this site. Fentanyl and alfentanil are both full µ agonists. The strict additivity in the concentration versus receptor occupancy relationship is readily demonstrated in an isobologram. However, naloxone and fentanyl also compete for the same binding site. The fractional receptor occupancy for fentanyl and naloxone will follow the law of mass action and demonstrate strict additivity. However, since naloxone is not a µ agonist, the clinical effect will not be additivity, but pharmacological antagonism. Additionally, because naloxone exerts no analgesic effect, the C₅₀ for naloxone analgesia does not exist and one cannot construct an isobologram for naloxone and fentanyl.

The point relative to our examination of anesthetic mechanisms is that competition of full agonists for the same site of action will yield a strictly additive interaction. We will now explore the converse: when can two drugs display additivity of normalized concentrations yet act at separate sites?

Receptors Grouped into Circuits
Appendix B analyzes the behavior of individual receptors grouped into circuits. Circuits amplify the potency and the steepness of the concentration versus response relationship for drugs. This finding supports Eckenhoff and Johansson's contention that the binding potency of inhaled anesthetics to their sites of action may be weaker than their apparent in vitro potency or their clinical potency.7 However, it also shows that the law of mass action, linked to the joint probability of multiple ligands independently interacting with multiple receptors to affect signal propagation, limits the steepness of the concentration versus response relationship to approximately 1.5 (the Hill coefficient, represented as in the equations that follow) within an assembly of receptors into a simple neuronal circuit. It is notable that this steepness is far less than the Hill coefficient of four or more typically found for studies of MAC,8 showing a very steep concentration versus probability of no response.

Circuits Grouped into Signal Processing Units
Appendix C develops the mathematics of more complex circuits, which are referred to as signal processing units. The signal processing units involve two drugs, A and B, acting at "Site A" and "Site B," respectively. These sites of action are not individual receptors. Rather, Site A and Site B are each circuits of enhancers and suppressors of signal propagation, as described in Appendix B (Figs. A2 and A3), now assembled together into a signal processing unit, as shown in Figure 3. Site A and Site B can be arranged either in series (top diagram) or parallel (lower diagram). The sites interact to determine the probability of signal propagation in the signal processing unit.

View larger version (11K): [in this window] [in a new window]   Figure A2 The potency and magnitude of drug effect is amplified when suppressors (drugs that suppress the probability of signal propagation) are arranged in a serial circuit, as shown here. The relationship can be approximated by a conventional sigmoid e-max model (dotted line).

View larger version (14K): [in this window] [in a new window]   Figure 3. Two arrangements for signal processing units. The sites of action refer to a circuit of enhancers or suppressors, as described in Appendix B. Two arrangements for the signal processing unit are considered: a series arrangement (top), and a parallel arrangement (bottom).

Several parameters affect the performance of these signal processing units, including the concentration of drug associated with the effect of interest, the probability of a signal propagating past an unbound receptor, the number of receptors at Site A and Site B, and the efficacy of a drug in producing the maximal possible response (either reducing the probability of signal propagation to 0 for a suppressor, or increasing the probability of signal propagation to one for an enhancer). Appendix C demonstrates that the primary factor that determines whether the circuit demonstrates additivity or synergy is the concentration of suppressor or enhancer associated with the drug effect of interest, normalized to k_d, the dissociation constant (i.e., the concentration associated with 50% receptor occupancy). If the normalized concentration is <1%, then additivity is expected. If the normalized concentration is in the range of (or exceeds) k_d, the concentration associated with 50% receptor occupancy, then synergy is expected. The probability of signal propagation past an unbound receptor, the number of receptors in series or parallel, the intrinsic efficacy, and the geometry of the circuit are of considerably less importance than the relationship between the concentration associated with the drug effect of interest relative to the concentration associated with 50% receptor occupancy.

Receptor Occupancy Versus Concentration–Effect Relationships
Single Drug Interaction
By definition, a drug interacting with itself is additive. What does this additivity mean for receptor occupancy versus the clinical effect produced by this occupancy? Figure 4 shows two sigmoidal graphs. The curve to the left shows fractional receptor occupancy by a ligand as a function of the normalized ligand concentration. Based on the law of mass action, and assuming no cooperative binding, the relationship is defined as

View larger version (13K): [in this window] [in a new window]   Figure 4. Superimposed graphs of fractional receptor occupancy (left axis) and the probability of no response (right axis) when MAC >> kd.

, where f is the fraction of receptors occupied (see Y axis on the left), [L] is the concentration of the ligand, and k_d is the dissociation constant. The axis marked "concentration" shows the concentration of the ligand, normalized to the concentration producing 50% receptor occupancy, i.e., "concentration" = [L]/k_d. As a result, the concentration at "X" is 1.0, which is the concentration associated with 50% occupancy. The steeper curve on the right is a hypothetical curve for inhaled anesthetic effect defined by the relationship

where P_{no response} is the probability of no movement (see Y axis on the right), MAC is 10 (i.e., 10 fold higher than the concentration producing 50% receptor binding), and , the Hill coefficient (a measure of the steepness of the concentration versus response relationship), is four. The point marked "MAC" identifies the anesthetic concentration associated with 50% probability of no response. The curve for the probability of no response is known to be steeper (Hill coefficient approximately 4) than the curve for receptor binding associated with 50% receptor occupancy (Hill coefficient = 1). Placing the curve for inhaled anesthetic drug effect well to the right of the curve for fractional receptor occupancy permits exploration of the implications of the setting where MAC >> k_d.

Now consider two concentrations of the same inhaled anesthetic, designated by the two vertical arrows. A concentration of five ( MAC) results in 83% receptor occupancy but a negligible 6% probability of no response. The effect of a second concentration of five ( MAC) to the first is indicated by the shorter vertical arrow "SYMBOL". These two concentrations together equal 1 MAC. Thus, the second concentration increased the probability of no response from 6% to 50%, a huge effect. However, it accomplished this by increasing receptor occupancy by just 7% (to 90%). This is additivity by tautology: a concentration of five, plus an additional concentration of five, gives the same result as a concentration of 10, because they are the same thing. However, it indicates a huge gain programmed into the system: a slight change in receptor occupancy enormously increases drug effect.

View larger version (4K): [in this window] [in a new window]   Symbol. No caption available.

Figure 5 shows the same two curves, except that MAC is now 0.1, one-tenth of the concentration associated with 50% receptor occupancy. Again consider two concentrations of a single drug. The first concentration (0.05; first upward arrow) produces a 5% receptor occupancy and, as before, a negligible 6% probability of no response. The second concentration (0.1), "SYMBOL", is double the first and increases receptor occupancy to 9%, again with a huge increase in drug effect to 50% probability of no response.

View larger version (13K): [in this window] [in a new window]   Figure 5. Superimposed graphs of fractional receptor occupancy (left axis) and the probability of no response (right axis) when MAC << kd.

View larger version (4K): [in this window] [in a new window]   Symbol. No caption available.

As previously shown, neuronal circuits amplify drug potency and increase the steepness of concentration versus response relationships, compatible with the predictions of Figure 5. For this reason, MAC probably is far below k_d (as illustrated in Fig. 5), rather than greatly exceeding k_d (as in Fig. 4.) Additionally, a MAC occurring at concentrations far exceeding 50% receptor occupancy (Fig. 4) would require a circuit exquisitely tuned to abruptly amplify the signal with almost no change in fractional receptor occupancy. Further diminishing the likelihood of this scenario, the many organisms that show similar sensitivity to inhaled anesthetics would need to conserve such circuitry, to no apparent purpose.

Two Drug Interaction
Now consider two possibilities for interactions of two drugs when MAC is 10 fold greater than the concentrations associated with 50% receptor occupancy (Fig. 6). The upper left example assumes a single target of drug action, as in Figure 4. A concentration of MAC of drug A produces 83% receptor occupancy, and almost no effect. However, because it binds to the same receptor, the addition of a MAC concentration of drug B only slightly increases receptor occupancy (upper right example) but enormously increases drug effect. The added concentration of drug B is noted as "SYMBOL" and indicates the increase in receptor occupancy and effect expected from the concentration of B added to the MAC concentration of drug A. This is additivity, exactly paralleling what happens when two concentrations of MAC of the same drug are given (Fig. 4).

View larger version (15K): [in this window] [in a new window]   Figure 6. Two possibilities for interactions of two drugs when MAC is 10 fold greater than the concentrations associated with 50% receptor occupancy. If the two drugs act on the same target (top graphs) then the second drug doubles the effective concentration, and the gain of the circuit increases the drug effect from modest to 50% probability of no response despite the modest change in receptor occupancy. However, if two drugs act on different targets (bottom graphs), then the addition of the second drug produces a huge change in occupancy of its receptor that inexplicably causes the same effect as a very modest change in receptor occupancy for drug A alone.

View larger version (4K): [in this window] [in a new window]   Symbol. No caption available.

The lower example in Figure 6 considers two targets of drug action, one for drug A, and one for drug B. As before, a MAC concentration of drug A produces 83% receptor occupancy, with a minimal effect on MAC. Drug A has no effect on the receptor site for drug B, and thus that second site is unoccupied and cannot influence MAC. Because a MAC concentration of drug B acts on a separate target, it also produces an 83% receptor occupancy for its target. It is easy to envision that the effect with two targets would greatly exceed the effect with a single target. With drug B present at a MAC concentration, an 83% occupancy of an additional receptor (receptor B) would occur at the second site. If occupancy of site B produces an identical change in response as with a single site (i.e., there is additivity, even though there are two targets of drug action), then somehow the increase from 0 to 83% receptor occupancy of the second receptor system must produce the same physiological response as a change from 83% to 90% receptor occupancy when either drug is used by itself. This requirement of additivity between inhaled anesthetics hugely constrains the neuronal circuitry. An 83% increase in receptor occupancy at a second site must produce exactly the same response as a 7% increase at the single site of action of either inhaled anesthetic. Although we cannot exclude the possibility that a neural circuit could produce the above results, we cannot envision the construction of such a circuit. It also seems improbable that such a complex circuit would be conserved across the many organisms with nearly identical responses to inhaled anesthetics. Thus, if inhaled anesthetics show additivity, and if MAC greatly exceeds k_d, then there is almost certainly a single site of inhaled anesthetic action.

Figure 7 shows two possibilities for drug interactions when MAC is 10-fold less than the concentrations associated with 50% receptor occupancy. In the upper example, there is a single target of drug action, just as in Figure 5. A concentration of MAC of drug A produces 5% receptor occupancy, and almost no effect, as shown by the upper left graph. However, because it binds to the same receptor, adding a MAC concentration of drug B increases receptor occupancy to 9% (upper right figure), and enormously increases drug effect (to MAC). The concentration of drug B that is added is noted as "SYMBOL" in the figure to the upper right to indicate the changes that would be expected from the added MAC concentration of drug B in the presence of a MAC concentration of drug A. This is additivity, because it is exactly what happens when two drugs are combined in MAC concentrations or when two MAC concentrations of the same drug are combined. The key observation is that drug A produced 5% receptor occupancy, and drug B approximately doubled it to 9%.

View larger version (15K): [in this window] [in a new window]   Figure 7. Two possibilities for interactions of two drugs when MAC is 10 fold less than the concentrations associated with 50% receptor occupancy. If the two drugs act on the same target (top graphs) then the second drug doubles the effective concentration and very nearly doubles receptor occupancy as well. As in Figure 6, the gain of the circuit increases the drug effect from modest to 50% probability of no response despite the modest change in receptor occupancy. If two drugs act on different targets (bottom graphs), then the addition of the second drug produces nearly the same change in overall receptor occupancy as if the drugs acted on the same target, and thus it is not surprising that the circuit might produce the same change in drug effect as seen if both drugs act on the same target.

View larger version (4K): [in this window] [in a new window]   Symbol. No caption available.

The lower example considers two targets of drug action, one for drug A, and one for drug B. As before, MAC of drug A produces 5% receptor occupancy, with minimal effect. Because the MAC of drug B acts on a separate target, it also produces 5% receptor occupancy of its target.

It is easy to envision that two anesthetics that each produce 5% receptor occupancy at separate targets could have the same effect as twice the concentration of either drug alone acting on a single target to produce 9% receptor occupancy. Either way, the system will require considerable gain to translate the receptor occupancy (9% at a single target, versus 5% at 2 separate targets) to the huge increase in drug effect. However, the complexity of the circuitry is entirely in increasing the gain, which is an expected property of neural circuitry: additivity when MAC << k_d does not require the complex constraints that exist if MAC vastly exceeds k_d.

This analysis suggests that if MAC exceeds k_d, then additive interactions among inhaled anesthetics strongly imply a single site of anesthetic action. However, if MAC is well below k_d, then additive interactions among inhaled anesthetics do not necessarily imply multiple sites of action.

	DISCUSSION

Top Abstract Introduction Theoretical Development DISCUSSION APPENDIX A: TWO DRUGS... APPENDIX B: RECEPTORS ARRANGED... APPENDIX C: TWO DRUGS... Appendix D: Derivations REFERENCES

 
Kaminoh et al. examined pathways of receptors arranged in combined serial and parallel circuits and performed an analysis similar to ours.9 They considered the effect of a single antagonist. Similar to our results, receptors in series increased the potency of antagonists. They also found that receptors in parallel decreased the potency of antagonists, an analysis we did not perform. Most critically, they found that it required a combination of parallel and serial circuits of antagonists to generate the steep concentration response curve associated with inhaled anesthetics. For example, a circuit consisting of 16 parallel pathways, each pathway containing 16 elements in series, would have a Hill coefficient of 4.5, similar to what is observed in MAC determinations for inhaled anesthetics. However, they and we predicted that circuits consisting only of elements in series, or elements in parallel, cannot generate a Hill coefficient more than 1.5. This is not surprising, because we used the same mathematical analysis for the probability of signal propagation, both over units of the circuit, and for the arrangement of individual circuits in serial and parallel assemblies.

The Kaminoh et al. analysis, our analysis, and the analysis by Eckenhoff and Johansson7all suggest that the steepness of the concentration versus response relationship for anesthetics is partly derived from multiple receptors acting in circuits. Figures A2 and A3 show that the increasing steepness is associated with increasing potency. Thus, steep concentration versus response relationships for anesthetic drugs imply that a half-maximal effect (i.e., the clinical effect) occurs at concentrations much less than k_d.

We developed simple circuit examples to explore interactions among drugs acting at separate sites. Such circuits provide theoretical examples to understand basic principles. We have no illusions that they describe real neurological circuits. However, our choice of simple parallel or serial circuits was not arbitrary. Each neuron receives multiple dendritic connections. Presuming that more than one of these is activated, multiple dendritic connections are parallel assemblies. Groups of neurons running in parallel also may receive input simultaneously, again representing parallel transmission. However, presynaptic receptors arranged along a single axon are arranged in series.

Additionally, nerves connect with each other in series to propagate a signal. Since movement of an animal or human in response to noxious stimulation may occur at a distance from the stimulation (e.g., pinch the tail and the upper extremity moves), the signal must pass through serial assemblies of neurons, just as we have postulated serial assemblies.

Although circuits are most readily envisioned as neural structures, the analysis need not relate to nerves at all. Signals can propagate within a single cell through cascading enzymatic reactions or ion fluxes. Signals may even propagate within a single protein through multiple sites of receptor attachment.

Our analysis demonstrates that additivity is expected when the drug effect of interest occurs at concentrations associated with low fractional receptor occupancy. This conclusion was reached from two different lines of reasoning: analysis of the probability of signal propagation as a function of the concentration of suppressors or enhancers (Section 2, and Appendices A–D), and an analysis of the relationship between fractional receptor occupancy and clinical concentration versus response relationships (Section 3).

The next step in this line of reasoning is to verify these predictions against known values of k_d for different anesthetic drugs, and see whether our models accurately predict synergistic or additive reactions. Unfortunately, the value of k_d is known for very few anesthetics. Indeed, the few reported values are suspect because of artificial experimental conditions.⁷For this reason, verification of these findings against experimental data is not included herein. Instead, it is a topic for further research.

As have others,7 we conclude that the clinical potency of inhaled anesthetics is likely amplified by neuronal circuitry from the binding potency at the level of individual receptors. We further conclude that additivity is an expected result from competition of drug for a common binding site, or when the drug effect occurs at concentrations that are much less than the concentration associated with 50% receptor occupancy, k_d. If the drug effect is only found at concentrations close to or exceeding k_d, then the interactions are necessarily synergistic. Thus, inhaled anesthetics either share a single site of action, or the effect is a result of binding to multiple sites of action at concentrations that leave the vast majority of sites unbound.

	ACKNOWLEDGMENTS

 
The authors express their appreciation to the reviewers of this article for their considerable contributions. Insights and suggestions from the reviewers identified critical deficiencies in the first submission and resulted in virtually a complete reanalysis and rewriting of the article. Further efforts by the reviewers helped anchor the theoretical arguments and vet the presentation to make this complex analysis as clear as possible.

	APPENDIX A: TWO DRUGS ACTING AT THE SAME SITE

Top Abstract Introduction Theoretical Development DISCUSSION APPENDIX A: TWO DRUGS... APPENDIX B: RECEPTORS ARRANGED... APPENDIX C: TWO DRUGS... Appendix D: Derivations REFERENCES

 
When a single drug binds to a receptor, the binding follows the law of mass action:

where [D] is the concentration of the drug, [RU] is the concentration of unbound receptor, and [DR] is the concentration of bound receptor (actually, these are activities rather than concentrations, but for this essay we use the latter term). The rate of formation of [DR] is

. At steady-state, which is nearly instantaneous, the net rate of formation is 0, and thus [D][RU]k_on=[DR]k_off. If we define k_d, the dissociation constant, as k_off/k_on, and f, fractional receptor occupancy, as

, then

This is an asymptotic relationship that reaches 50% receptor occupancy when [D] = k_d.

This can be expanded to two drugs competing for the same receptor. With simple competition, drug A and drug B independently follow the law of mass action:

, and

. The steady-state relationships are therefore [D_A][RU]k_on,A=[D_AR]k_off,A and [D_B][RU]k_on,B=[D_BR]k_off,B. If we define k_d,A as k_off,A/k_on,A, k_d,B as k_off,B/k_on,B, and f, fractional receptor occupancy, as

, then

If we select 50% occupancy as the drug effect of interest, then we can solve for the concentration of drug B associated with 50% receptor occupancy as a function of drug A:

. If we normalize the concentration of drug A to k_d,A,

, and if we normalize the concentration of drug B to k_d,B,

, then we can simplify the relationship between the concentration of drugs A and B associated with 50% receptor occupancy to

This relationship is shown in Figure A1, and obviously defines strict additivity. Thus, the law of mass action dictates that if 1) two full agonists independently compete for the same receptor, and 2) fraction receptor occupancy determines drug effect, then the interaction between the two drugs follows strict additivity.

View larger version (6K): [in this window] [in a new window]   Figure A1 By definition, the additivity isobole is defined by the relationship B = 1 - A.

The derivations of equations 1, 2, and 3 are given in Appendix D, which is available as a web supplement at www.anesthesia-analgesia.org.

View larger version (14K): [in this window] [in a new window]   Figure A3 The potency and magnitude of drug effect is amplified when enhancers (drugs that enhance the probability of signal propagation) arranged in a parallel circuit, as shown here. The relationship can be approximated by a conventional sigmoid e-max model (dotted line).

	APPENDIX B: RECEPTORS ARRANGED IN CIRCUITS

Top Abstract Introduction Theoretical Development DISCUSSION APPENDIX A: TWO DRUGS... APPENDIX B: RECEPTORS ARRANGED... APPENDIX C: TWO DRUGS... Appendix D: Derivations REFERENCES

We will define P as the probability of signal propagation. We will define "suppressors" as drugs that decrease the likelihood of signal propagation, and "enhancers" as drugs that increase the likelihood of signal propagation. We define P_un as the probability that a signal will pass an unbound receptor. If a suppressor is applied, the probability (P) will decrease from P_un towards 0. If an enhancer is applied, the probability (P) will increase from P_un towards 1. We define E as the efficacy of the drug action. E defines how far the drug is capable of altering the probability of signal propagation towards the limits of P = 1 (for enhancers) or P = 0 (for suppressors). E ranges from 0 (no efficacy) to 1 (full efficacy).

For notational convenience, we define the concentration of a suppressor, Su, in units of the concentration that produces 50% receptor binding:

when the ligand, [L] suppresses signal propagation. We similarly define the concentration of an enhancer, En, in units of the concentration that produces 50% receptor occupancy:

when the ligand, [L], enhances signal propagation. Note that [L] and k_d must have the same units. Although this would typically be moles per liter, the actual units are not relevant since they cancel out in the definitions of Su and En. Although Su and En will be referred to as concentrations, they could also be considered fractions or multiples Kd, depending on whether they are less or greater than one.

As mentioned, the probability that a signal will pass an unbound receptor is P_un. P_un is most readily envisioned as one for a suppressor, and 0 for an enhancer. However, it is included in the equations to provide greater generality. We will assume that if a particular receptor is bound to a suppressor or enhancer, then the suppressor or enhancer is 100% likely to exert its effect. In other words, the probability of a signal traversing a receptor bound to a suppressor is P_un – E · P_un, and the probability of a signal traversing a receptor bound to an enhancer is P_un + E · (1 – P_un).

Since the probability that a single receptor is bound at a certain drug concentration is exactly the same as the fraction of receptors bound in a population at that concentration, the law of mass action states that the probability of binding for a suppressor is

, and the probability of binding for an enhancer is

. These equations can be derived by rearranging Equations 4 and 5 and substituting into Equation 1 (see Appendix D in the online supplementary material).

Given the above definitions, the probability that a particular signal passes a receptor in the presence of a suppressor is

, where Su is the concentration of the suppressor, as defined above, and E is the efficacy of the suppressor at reducing the probability of propagation to 0. The probability that a particular signal passes a receptor in the presence of an enhancer is

, where En is the concentration of the enhancer, as defined above, and E is the efficacy of the enhancer at increasing the probability of propagation to one.

The upper graph in Figure A2 shows multiple receptors arranged in series, each containing a binding site for a suppressor. The probability that a signal passes multiple receptors occupied by a suppressor is the product of the individual probabilities. If N is the number of receptors in series, then the probability that the signal propagates past N receptors is

To explore this relationship we arbitrarily define P_un as 1 and E as 0.2. The choice of 0.2 permits demonstration of amplification of drug effect with physiologically reasonable numbers of receptors. We can graph this relationship as a function of Su, the concentration of suppressor, and N, the number of receptors in series. The bottom graph in Figure A2 shows this relationship. Although the effect of binding a single receptor is modest, with each bound receptor only reducing the probability of propagation by 20%, the effect of binding multiple receptors is amplified over the course of transmission. With four suppressors in series, the probability of propagation decreases by 50%. With 16 suppressors in series the probability of propagation decreases to nearly 0. Figure A2 also shows that the increase in effect is accompanied by an increase in the apparent potency of the suppressor. For N = 1, half-maximal effect occurs at 1 (by definition). For N = 16, half maximal effect occurs at a concentration of approximately 0.25. The increasing effect and potency is accompanied by a modest increase in the steepness of the concentration versus response relationship.

The relationship for N = 16 can be modeled with a standard Hill equation,

, where Su₅₀ is the suppressor concentration associated with 50% maximum drug effect, Max is the maximum drug effect, and is the "Hill coefficient" that determines the steepness of the concentration versus response relationship. This is the dotted line (obtained using the Solver function of Excel), and it nearly exactly matches the shape of the adjacent curve

, similar to the analysis by Eger et al.8 However, Su₅₀ is 0.25, a roughly four fold increase in potency, and is 1.13. asymptotically approaches 1.5 for very large N (tested to N = 10,000). This limit is independent of E. Thus, aligning suppressors in series increases the apparent potency and steepness of the concentration versus response relationship.

We next turn to an analysis of enhancers. We cannot simply invert our model of suppressor in series to a model of enhancers in series. In an inverted model, P_un would be 0, and the model would be

, where 0.2 is the incremental increase in probability of propagation when the enhancer is bound into the receptor, and En is the concentration of the enhancer, normalized to the concentration associated with 50% receptor binding. In the inverted model, the probability of signal propagation is only 0.2 when N = 1. The probability of signal propagation further decreases for N > 1. Even if one assumes a very high prior probability of signal propagation past the unbound receptor, e.g., P =

, if N = 50, and En = 1 (50% receptor occupancy), the probability of signal propagation is <1%. One can overcome this by permitting the enhancer to nearly saturate the receptor (e.g., En >> 1), but as explained in the manuscript, if En >> 1 then the additivity necessarily implies a single site of action.

This is why the circuit for enhancers, shown in Figure A3, is arranged as a parallel circuit, rather than in a series circuit. Enhancers acting in parallel can be analyzed with similar assumptions to suppressors acting in series, except the overall probability of propagation in a series circuit of N receptors is P^N, while the overall probability in parallel circuit of N receptors is 1 – (1 – P)^N.

Given N receptors in parallel, the probability that the signal will propagate in a parallel circuit of N receptors is

. We will postulate a weaker effect for an enhancer, as the parallel circuit permits more enhancers to participate in the overall propagation of a single signal. If we define P_un as 0 (the signal cannot propagate without the enhancer) and E as 0.1, we can graph this relationship as a function of En, the concentration of enhancer normalized to the concentration associated with 50% receptor binding, and N, the number of receptors in parallel.

This relationship is shown in the bottom graph in Figure A3. Although the effect of each bound enhancer is weak, the effect is amplified by the parallel circuit. With 16 enhancers in parallel, the maximum probability of propagation increases to approximately 80%. With 256 enhancers in parallel, the probability of propagation reaches nearly 100%. This is accompanied by an increase in the apparent potency of the enhancer, as well as an increasingly steep concentration versus response relationship. Increasing the number of enhancers 256 fold increases the apparent potency by 40 fold, as evidenced in the leftward shift of the dose versus response relationship in Figure A3. This is accompanied by an increase in the steepness of the concentration versus response curve.

The relationship for N = 256 can be modeled with a standard Hill equation,

where En₅₀ is the suppressor concentration associated with 50% maximum drug effect, Max is the maximum drug effect, and is the "Hill coefficient," using the Solver function in Excel. This is the dotted line that nearly matches the shape of the adjacent curve

. However, En₅₀ is 0.025, an increase in potency of approximately 40 fold, and is 1.41. asymptotically approaches 1.5 for very large N (tested to N = 10,000, see the Excel spreadsheet "circuits.xls" available as a Web supplement on the Anesthesia & Analgesia website [www.anesthesia-analgesia.org]). This limit is independent of E. Thus, aligning enhancers in parallel increases the apparent potency and modestly increases the steepness of the concentration versus response relationship.

	APPENDIX C: TWO DRUGS IN SIMPLE SIGNAL PROCESSING UNITS

Top Abstract Introduction Theoretical Development DISCUSSION APPENDIX A: TWO DRUGS... APPENDIX B: RECEPTORS ARRANGED... APPENDIX C: TWO DRUGS... Appendix D: Derivations REFERENCES

 
Figure 3 shows two simple signal processing units, each with a site of action for drug A, and a site of action for drug B. These "sites of action" are either serial circuits comprised of many receptors for drugs that suppress signal propagation, "suppressors" as shown in Figure A2, or parallel circuits comprised of many receptors that enhance signal propagation, "enhancers" as shown in Figure A3.

We will examine the behavior of four signal processing units, each comprised of two drugs: 1) Two suppressor drugs, whose sites of action are in a series circuit relative to each other, 2) Two suppressor drugs, whose sites of action are in a parallel circuit relative to each other, 3) Two enhancer drugs, whose sites of action are in a series circuit relative to each other, and 4) Two enhancer drugs, whose sites of action are in a parallel circuit relative to each other.

An Excel spreadsheet, "circuits.xls," accompanies this manuscript in a supplement on the Anesthesia & Analgesia web site. Interested readers should download this spreadsheet and test the circuits to help understand their behavior. The mathematics were developed using Mathematica (Wolfram Research, Champaign, IL). Appendix D gives the derivation of the equations, which was performed using Mathematica. Appendix D and the Mathematica notebooks are available as Web supplements (www.anesthesia-analgesia.org) for interested readers.

Series Signal Processing Unit: Two Suppressors in Series

As explained in Appendix B, suppressors are likely to be arranged serially. The probability of a signal crossing N suppressors is

where P_un is the probability of passage past an unbound receptor, E is the efficacy of a single bound receptor suppressing signal propagation on a scale from 0 (no action) to 1 (complete suppression), and Su is the concentration of the suppressor as a fraction of the concentration associated with 50% receptor occupancy. Consider two different drugs, each of which suppresses signal propagation, arranged in series relative to each other. That would be the arrangement shown in the upper graph of Figure 3. The probability of passage is the probability that the signal would pass the assembly of receptors for suppressor A,

, times the probability the signal would pass the assembly of receptors for receptor B,

, namely

.

Notice that the receptor for suppressor A and the receptor for suppressor B have unique probabilities for passage of a signal past an unbound receptor (P_un,A and P_un,B), as well unique efficacies (E_A and E_B). In our analysis we will consider how the probability of passage past an unbound receptor and the intrinsic efficacy of a single bound receptor affect whether drugs are additive or synergistic in their interaction. For simplicity we assume that P_un,A = P_un,B, and E_A = E_B.

With the simplifying assumptions that P_un,A = P_un,B and E_A = E_B, suppressor A and suppressor B have identical behavior in this circuit, except that they need not exist in the same concentration. Given our simplifying assumptions, X units of suppressor A given alone has exactly the same effect as X units of suppressor B given alone. What happens if we give suppressors A and B concurrently? Let's call M the Multiplier on X, so that M x X units of suppressor A, given concurrently with M x X units of suppressor B produces exactly the same effect as X units of suppressor A alone or X units of suppressor B alone. If M = 0.5, then we have strict additivity, because 0.5 X units of A + 0.5 X units of B produces the same effect as 1.0 X of either drug alone. We have defined synergy as the sum of fractions being <90% of the normalized dose (i.e., X in this example) of either drug alone. If we can produce the target drug effect with M <0.45, then we have synergy, because 0.45 X units of drug A plus 0.45 X units of drug B gives 0.90 units of X, exactly at the border between synergy and additivity.

The effect of a single suppressor in this circuit, given at concentration Su, is

given that Su of the other drug is 0, reducing equation 6 to P^un_N. The effect of two suppressors, each given at a concentration of M Su, is

We are looking for the value of M such that these two effects are equal. Thus, we want to find M so that

=

The solution for M is shown in Appendix D, which is available in the online supplementary material.

Having solved for M, the fraction of two drugs that, given together, produces the same effect as either drug given alone, we can now examine the behavior of M as a function of:

1. Su, the concentration of the suppressor that produces the targeted effect,
   
2. P_un, the probability of a signal passing an unbound receptor,
   
3. E, the efficacy of a bound receptor in completely suppressing the signal, and
   
4. N, the number of suppressors in series at site A and site B.
   

If we could create graphs in five dimensional space we could capture the relationship in a single figure. Since this is not possible, Figure A4 explores the relationship as a series of three-dimensional graphs.

View larger version (46K): [in this window] [in a new window]   Figure A4 Suppressors in Series: Graphs exploring the behavior of a signal processing unit for two drugs that suppress signal propagation. The fundamental parameters defined in Appendix B for each suppressor are the probability of a signal passing a single unbound receptor (Pun), the efficacy of drug action (E), and the number of receptors in the circuit (N). The two circuits (i.e., one for each suppressor) are arranged in series. Pun, E, and N are assumed to be identical for each circuit. The targeted drug effect is that produced by Su units of one suppressor given alone (X axis), where the units of Su are the fraction of kd, the concentration associated with 50% receptor occupancy. M is the multiplier (Z axis), so that M · Su units of each suppressor, given together, produces the same effect as Su units of either suppressor given alone. Additivity occurs when M 0.45, and synergy occurs when M < 0.45. Pun (first row), E (second row), and N (third and fourth rows) appear on the Y axis of the graphs. The isobole on each graph separates the region of additivity (M> = 0.45) from the region of synergy (M < 0.45). Two dimensional graphs of just the isoboles can be downloaded from the Web Supplement, and appear in the Mathematica workbook as well.

In every graph in Figure A4, the Z axis is M, the multiplier on the concentration, Su, of a single suppressor such that two different suppressors, each at a concentration M Su, produces the same effect as either suppressor alone at a concentration of Su. In every graph the X axis is the concentration of suppressor producing the desire effect by itself. The Y axis is P_un (top row), E (second row) or N (third and fourth rows).

The most obvious relationship shown in A4 is that regardless of the value of P_un, N, and E, M approaches 0.5 (perfect additivity) for very small values of Su. Since Su = 1 is the concentration associated with 50% receptor occupancy, Figure A4 shows that regardless of the other parameters, two suppressor circuits, arranged in a signal processing unit in serial with respect to each other, will always show simple additivity if the drug effect of interest occurs at concentrations that are <1% of those associated with 50% receptor occupancy. This is not a surprising finding, as fractional receptor occupancy increases almost linearly with concentration at very low levels of receptor occupancy. On the other hand, if the drug effect of interest requires 50% receptor occupancy (Su = 1), then profound synergy is expected, regardless of the values of P_un, N, and E.

Looking at more subtle findings for this circuit, the top row of graphs shows that relationship between M and Su is independent of P_un for this circuit. The second row of graphs shows that for values of Su greater than 0.1, E, the intrinsic efficacy, influences whether the relationship is additive (values of E near 1) or synergistic (values of E <0.5). However, there is almost no change in the relationship as E varies from 0.001 to 0.5. The last two rows of graphs explore the influence of N, the number of suppressor receptors in series at Site A, and at Site B (i.e., N = N_A = N_B). There is virtually no influence of N on whether the relationship demonstrates synergy or additivity.

Thus when a signal processing unit consists of two suppressors arranged in series, the relationship between the two drugs can be synergistic or additive, depending on the concentration at which the drug exerts the effect of interest relative to the concentration associated with 50% receptor occupancy.

Series Signal Processing Unit: Two Suppressors in Parallel

Two suppressors in a parallel circuit have a more complex behavior. The effect of a single suppressor in this circuit, given at concentration Su, is

. The effect of two suppressors, each given at a concentration of M Su, is

. We are looking for the value of M such that these two effects are equal. Thus, we want to find M so that

= 1

. The solution for M is shown in Appendix D, which is available in the online supplementary material.

Figure A5 shows the relationship between M, Su, P_un, N, and E. Overall, we see the same pattern as in Figure A4. The relationship is additive for values of Su that are <1% of the concentration associated with 50% receptor binding, regardless of the other parameters. At concentrations higher than 0.01, the relationship is more likely to be synergistic. As seen for suppressors in series, P_un has little effect. Interestingly, the effect of E is the opposite of what was seen in Figure A4, in that values of E nearly 1 are associated with profound synergy. There is an effect of N, but it is modest compared to the effect of Su. It is interesting that several of the graphs are incomplete. At large values of N these signal processing units produce very small numbers for probability of signal propagation, typically in the range of 10^{– 20}. Such small numbers challenge the accuracy of the computation, particularly when the final calculation, M, involves the ratio of two very small numbers. The graphs become irregular, and then unsolvable, due to numeric instability when challenged by large values of N in combination with very low values of P_un and E.

View larger version (47K): [in this window] [in a new window]   Figure A5 Suppressors in Parallel: Graphs exploring the behavior of a signal processing unit for two drugs that suppress signal propagation. Unlike Figure A4, the two circuits are arranged in parallel.

Series Signal Processing Unit: Two Enhancers in Serial

This was the most challenging circuit to analyze. Looking at the top circuit in Figure 3, one might guess that if Site A and Site B are assemblies of enhancers arranged in parallel, and the probability of passage was at each site was

, then the probability of passing both sites, each with a concentration of En was

. This is not correct.

The reason is that at Site B the "incoming" probability is not one, but is the probability coming out of Site A, which is

. Every one of the parallel enhancer circuits sees this probability, not one, in the calculation for the probability effects of site B. This can be readily appreciated with the Excel spreadsheet "circuits.xls" provided with this manuscript as a supplement.

An interesting aspect of this is that enhancer A and enhancer B are not exchangeable. Enhancer B is more potent than enhancer A. Enhancers A and B also have different peak effects and different half-maximal effects, even when P_un, N, and E are identical.

As explained in Appendix D, this circuit can be analyzed as follows. First, propose an effect that occurs at a certain concentration of enhancer A. Because enhancer A is less potent than enhancer B, any effect produced by enhancer A can also be produced by enhancer B. The effect from En_A units of enhancer A is

(See appendix D in the online supplementary material).

Second, find the concentration of enhancer B, En_B, that produces the same effect. A closed form solution exists and is given in Appendix D. This establishes the concentrations of enhancers A and B that produce a given drug effect.

Third, find M, so that M · En_A units of enhancer A, given concurrently with M · En_B units of enhancer B, produces exactly the same drug effect as En_A units of enhancer A, or En_B units of enhancer B. There is no closed form solution for M. It must be solved numerically. Appendix D shows how the numeric solution was found with Mathematica.

Figure A6 shows the relationship between M and En_A, P_un, N, and E. As was seen for suppressors arranged in serial and parallel signal processing units, the most important variable for two enhancers arranged in a serial signal processing unit is En_A, the concentration associated with the effect of interest. The relationship is almost always additive for values of En_A that are <1% of the concentration associated with 50% receptor binding, although the graph in the upper left corner shows that this is not always the case. Figure A6 also demonstrates that when two enhancers are arranged in a serial circuit, M decreases (and hence synergy increases) increases as P_un decreases, and E increases. The effects of N are modest, and not easily characterized (e.g., the first figure in the third row shows decreasing M at both low and high values of N).

View larger version (47K): [in this window] [in a new window]   Figure A6 Enhancers in Series: Graphs exploring the behavior of a signal processing unit for two drugs that enhance signal propagation. The two circuits are arranged in series. Unlike the signal processing units examined in Figures A4, A5, and A7, in this circuit the behavior of the two drugs is not identical. Therefore, the drug effect is defined as the effect of Enhancer A (EnA).

View larger version (46K): [in this window] [in a new window]   Figure A7 Enhancers in Parallel: Graphs exploring the behavior of a signal processing unit for two drugs that enhance signal propagation. The two circuits are arranged in parallel. It is interesting to note that two enhancers in a parallel circuit are indistinguishable in this analysis from two suppressors in a serial circuit (Fig. A4).

The effect of a single enhancer in this circuit, given at concentration En, is

. The effect of two enhancers, each given at a concentration of M En, is

. We are looking for the value of M such that these two effects are equal. Thus, we want to find M so that 1–(1–P_un)^N

. The solution for M is shown in Appendix D in the online supplementary material.

Figure A7 shows the relationship between M and En, P_un, N, and E. Even though the function for two enhancers arranged in parallel is not the same as the function for two suppressors arranged in serial, Figure A7 is indistinguishable from Figure A4. As was true for the other circuits, the relationship is strictly additive if the effect of interest is observed at values of En that are <1% of the value associated with 50% receptor occupancy. The values of P_un and N have no effect on whether the interaction is additive or synergistic. Very high values of E are associated with a modest decrease in synergy.

Summary

Perhaps anticlimactically, all four simple circuits generated qualitatively similar results. When two suppressors or two enhancers are arranged into simple signal processing units, they demonstrate synergy when the drug effect of interest approximates or exceeds 50% receptor occupancy. However, if the drug effect occurs at concentrations much <50% receptor occupancy, then the drug effect is more likely to be additive.

	Appendix D: Derivations

Top Abstract Introduction Theoretical Development DISCUSSION APPENDIX A: TWO DRUGS... APPENDIX B: RECEPTORS ARRANGED... APPENDIX C: TWO DRUGS... Appendix D: Derivations REFERENCES

 
Appendix D involves several very large equations that are not easily typeset. It is available in the online supplementary material at www.anesthesia-analgesia.org.

	Footnotes

 
This article has supplementary material on the Web site: www.anesthesia-analgesia.org.

Accepted for publication March 27, 2008.

Dr. Steven L. Shafer, Editor-in-Chief, was recused from all editorial decision, related to this manuscript.

Dr. Flood is the wife of Dr. Shafer, Editor-in-Chief of Anesthesia & Analgesia. This manuscript was handled by James Bovill, former Section Editor of Anesthetic Pharmacology and Dr. Shafer was not involved in any way with the editorial process or decision.

	REFERENCES

Top Abstract Introduction Theoretical Development DISCUSSION APPENDIX A: TWO DRUGS... APPENDIX B: RECEPTORS ARRANGED... APPENDIX C: TWO DRUGS... Appendix D: Derivations REFERENCES

 

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