We investigated how this type of combinations leads to drug antagonism in the next section. We performed a theoretical analysis on the origins of synergy and antagonism in typical drug combination motifs.
First, we derived a criterion for judging synergy or antagonism from the definition of Loewe synergy. Based on this criterion, we have dissected the source of synergy for the simplest cases of serial combination and parallel combination, as well as the basic motif of antagonistic combination File S1. Inhibiting a positive feedback may produce a non-hyperbolic dose response relationship that is prone to antagonism.
We have provided a detailed discussion in File S1. Our results demonstrated the high relevance of drug synergy in small scale enzymatic pathways. Previously Jansen et al. Our work provides a theoretical basis for how their approach is successful. Feedback and crosstalk abound in these two pathways, but consistent with our results, they generally do not alter the qualitative feature of the combination, so that most combinations of targets inside these pathways are synergistic.
Thus, designing synergistic combinations targeting closely connected targets in many enzymatic pathway diseases is a promising strategy. Targeting unrelated drug targets in large networks often fail to show synergy, as exemplified by high throughput screening studies. Yeh et al. Well studied cases of drug antagonism are chiefly of this type, such as the combination of the antibiotics spiramycin and trimethoprim [15] , and the combination of dexamethasone and paclitaxel for lung cancer chemotherapy [33] — [35].
We observed no cases of suppressive drug antagonism in our modeling study: all the antagonistic cases we identified were buffering antagonisms one example shown in Fig. Therefore, our results suggest that in antagonistic drug combinations, drugs do not necessarily jeopardize the action of each other as commonly thought.
Instead, certain network topological arrangement of the drug targets, such as the motifs involving positive feedbacks in our findings, could naturally produce a buffering antagonism between the drugs. Whether such antagonism exists, and whether it contributes to clinically observed antagonism is another interesting topic for future investigations.
In contrast to previous ideas, we have shown that drug synergy or antagonism strongly depends on the underlying target-network topology for enzymatic systems. It is therefore useful to compile catalogs of synergistic or antagonistic combination motifs. Such catalogs can be exploited for rationally designing drug combinations, or multi-target drugs. For example, our calculations suggest that many motifs Figure 4 are highly consistent in showing synergistic behaviors.
If a disease-related biological network falls into one of the categories we found, then a combination could be safely proposed to be a synergistic combination.
Also, the observation that simple serial and parallel combinations tend to be mostly synergistic could be a guideline that can be readily applied to many scenarios involving signal transduction pathways similar to those shown in Fig. Previously, most synergistic combinations were proposed based on experience. Here we provide a rational and readily applicable approach toward synergistic drug combination design.
We need to stress that our results come from calculations on enzymatic networks, and other types of biological networks still need to be further studied. This work describes a comprehensive study of the combined effects of drugs in three-node enzymatic networks.
Drug synergy or antagonism was shown to be a property of target-related network topology. Several basic synergistic and antagonistic motifs were summarized and analyzed. Synergistic motifs could be classified into parallel, serial and mixed type combinations, whereas antagonistic combinations fall into one basic type involving positive feedbacks. Motifs described here can be used to design drug combinations in certain enzymatic contexts. Further work is warranted to clarify the effects of drug combinations on more complex biological networks.
Ministry of Science and Technology of China No. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. National Center for Biotechnology Information , U. PLoS One. Published online Apr 8. Patrick Aloy, Editor. Author information Article notes Copyright and License information Disclaimer. Competing Interests: The authors have declared that no competing interests exist.
Received Jan 23; Accepted Mar This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited. This article has been cited by other articles in PMC.
Abstract Drug combinations may exhibit synergistic or antagonistic effects. Introduction Drug combinations have been envisaged by many to be a promising approach to treat complex diseases such as cancer, inflammation and type 2 diabetes [1] — [3].
Methods Modeling three-node enzymatic networks To extensively model drug action in diverse conditions and elucidate the connection between network topology and drug interactions, we chose to first study small networks which could be thought of as simplifications of disease related networks. Open in a separate window. Figure 1. Modeling process to study drug combinations. Modeling drug action on three-node enzymatic networks One enzyme node C in Figure 1A from the three enzyme system was chosen as the output node, whose active form concentration in the stable steady state of the system was recorded to monitor the efficacy of drugs.
Figure 2. Distribution of percentage of synergistic cases under various parameter sets for all combinations studied. Figure 3. Distribution of CI t values for several combinations. Synergistic combinations are abundant in our model Figure 2 shows the distribution of percentage of solved cases being synergistic for all 33, drug combinations we have studied.
Figure 4. List of basic motifs that could result in drug synergy. Quantitative comparison of parallel and serial drug combination patterns As a quantitative measure, CI t Figure 1B can be used to judge the extent of synergy or antagonism.
Figure 5. Comparison of distributions of average CI t 's for parallel and serial combinations. Definitions of parallel and serial combinations are presented in the main text. Antagonistic combinations in our model Since most drug combinations we found were synergistic, the less common antagonistic cases in our model deserve special attention. Figure 6. Theoretical analysis of the origin of synergy and antagonism in typical motifs We performed a theoretical analysis on the origins of synergy and antagonism in typical drug combination motifs.
Discussion Synergy prevails in drug combinations targeting closely connected targets Our results demonstrated the high relevance of drug synergy in small scale enzymatic pathways. Figure 7. Basic structure of the growth factor signaling pathway, showing examples of serial and parallel combinations.
Possible source for buffering antagonism Yeh et al. Conclusion This work describes a comprehensive study of the combined effects of drugs in three-node enzymatic networks.
PDF Click here for additional data file. Acknowledgments N. As compared with the latter, the IA approach underestimates antagonism and overestimates synergism.
Hill green , Colby red and experimental blue mortality curves for pyrethroid mixtures, together with the Hill response surfaces cyan. Dashed lines indicate the respective EC 50 values.
Aryl hydrogen receptor AhR ligands were used to compare the toxic equivalence factor TEF approach and the more general GCA ansatz by [ 38 ] in predicting the expected effect of mixtures containing partial agonists or competitive antagonists. However, as the slope parameters from the fits are only marginally different from 1 c.
Table 3 , the differences between the GCA and Hill in the E max -, EC 50 -values and consequently in their response surfaces not shown are much smaller than the error bars of the experimental data, while the surface predictions are of comparable quality. As the classification of mixture effects in terms of interaction indexes is not an adequate means to describe this complex phenomenon, an analysis of responses by looking at the form of iso-effect levels, isoboles, is certainly more appropriate [ 10 , 14 , 16 ].
Deviations from straight lines were used to classify the effects as Loewe and Bliss agonistic or antagonistic. From a response surface view [ 3 , 33 , 39 ] isoboles are cuts of the response surface at defined effect levels and the corresponding contour plots are the easiest way to get a full picture for the whole range of dose-combinations. This facilitates the understanding of the fact that linear isoboles are the exception and not the rule and are not a general means of detecting synergism.
For critical discussions on the interpretation of the shape of isoboles see, e. In Fig. Response surfaces a — c and isoboles d — f for binary mixtures. Dashed lines denote the respective d 50 values.
This means that Bliss independent action tends to overestimate synergism at doses smaller than their median effect doses and to underestimate synergism at doses above. Some of these findings are illustrated in Fig. For ternary mixtures the pendants of isoboles are iso-surfaces, i. A visualization of effects resulting from more than three agents in one graphical object is hardly possible.
As shown for some fictitious ternary mixtures in Fig. In this case different maximum effects of the mixture partners do not affect planarity. Iso-surfaces of mixtures of agents having different slopes are non-planar. Shown on logarithmic a — c and linear d — f dose scales. This is shown for ternary anesthetic mixtures [ 43 ] as analyzed in great detail by [ 28 ] and [ 30 ].
For the present purpose the data for midazolam, propofol, and alfentanil were fitted to Hill dose-response curves assuming effect ranges from 0 no hypnosis to 1 full hypnosis. The characteristics of fits and of predictions for the mixtures are summarized in Table 4.
The corresponding iso-surfaces in Fig. As the Hill model provides only null-interaction surfaces, the size of the deviations of experimental data from the Hill prediction may give some hints on the presence of synergy. Deviation from planarity results from differing slopes of the individual anesthetics. Starting from a logistic PDE, analytical expressions for the response surfaces of n-component mixtures have been derived under the sole provision that each a.
No further assumptions are required. Deviations from this reference response in order to quantify synergism or antagonism should possibly be handled by some sort of perturbation theory. It can be applied to mixtures of compounds having different maximum effects and differing slope-parameters.
Many Loewe-additivity based approaches are found to be special cases of the Hill surface while Bliss IA is incompatible with the logistic ansatz.
The independent action model is frequently used, e. Its outcome should be checked by comparison with the Hill response surface for any relevant mixture ratio. In fact, under these conditions synergism postulated by MSM is an upper boundary for the synergism predicted by the Hill surface approach.
As a consequence for claiming synergistic effects at doses below the respective d 50 -values, special caution is required whenever the predictions from MSM and Hill approaches differ considerably.
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