Mathematical modeling of multi-drugs therapy: a challenge for determining the optimal combinations of antiviral drugs

In the current era of antiviral drug therapy, combining multiple drugs is a primary approach for improving antiviral effects, reducing the doses of individual drugs, relieving the side effects of strong antiviral drugs, and preventing the emergence of drug-resistant viruses. Although a variety of new drugs have been developed for HIV, HCV and influenza virus, the optimal combinations of multiple drugs are incompletely understood. To optimize the benefits of multi-drugs combinations, we must investigate the interactions between the combined drugs and their target viruses. Mathematical models of viral infection dynamics provide an ideal tool for this purpose. Additionally, whether drug combinations computed by these models are synergistic can be assessed by two prominent drug combination theories, Loewe additivity and Bliss independence. By combining the mathematical modeling of virus dynamics with drug combination theories, we could show the principles by which drug combinations yield a synergistic effect. Here, we describe the theoretical aspects of multi-drugs therapy and discuss their application to antiviral research.     

Parallelism,deep homology,and evo‐devo

Parallelism has been the subject of a number of recent studies that have resulted in reassessment of the term and the process. Parallelism has been aligned with homology leaving convergence as the only case of homoplasy, regarded as a transition between homologous and convergent characters, and defined as the independent evolution of genetic traits. Another study advocates abolishing the term parallelism and treating all cases of the independent evolution of characters as convergence. With the sophistication of modern genomics and genetic analysis, parallelism of characters of the phenotype is being discovered to reflect parallel genetic evolution. Approaching parallelism from developmental and genetic perspectives enables us to tease out the degree to which the reuse of pathways represent deep homology and is a major task for evolutionary developmental biology in the coming decades.     

Evaluation of experimental combined toxicity by use of dose-frequency curves: comparison with theoretical additivity as well as independence

Dose-frequency curves of toxic effects of a substance A were evaluated in the absence and in the presence of a fixed dose of a second substance B. Data were fitted by the curve-fitting program ALLFIT. Observed combined frequencies of A + B were compared statistically with the expected frequencies of additivity and (or) independence by the phi 2-square goodness-of-fit test. The theoretical dose-frequency curves expected for an additive response were obtained by a solely graphical procedure and the theoretical curves for independent effects were calculated from the effects of B and A at certain doses. In rotarod tests with trained mice, the combined deteriorating effect of ethanol and benzodiazepines were significantly over-additive. However, their lethal interaction appeared underadditive in mice. The lethal underadditive interaction of ethanol and phencyclidine (PCP) can be ascribed largely to independent actions of these compounds. Loss of righting reflex was additively enhanced by PCP, whereas PCP overadditively enhanced the effect of ethanol. The insecticidal action of the cholinesterase inhibitors malathion and parathion appeared additive and significantly different from independent interaction. A comparison of results from dose-response curves with isoboles showed good agreement. The method appears as an attractive alternative or as a complementary procedure to the isobolographic analysis. Combination experiments as described can be carried out and evaluated rather simply, with a minimum of expenditure and a maximum of information.     

A generalized response surface model with varying relative potency for assessing drug interaction

When multiple drugs are administered simultaneously, investigators are often interested in assessing whether the drug combinations are synergistic, additive, or antagonistic. Based on the Loewe additivity reference model, many existing response surface models require constant relative potency and some of them use a single parameter to capture synergy, additivity, or antagonism. However, the assumption of constant relative potency is too restrictive, and these models using a single parameter to capture drug interaction are inadequate to describe the phenomenon when synergy, additivity, and antagonism are interspersed in different regions of drug combinations. We propose a generalized response surface model with a function of doses instead of one single parameter to identify and quantify departure from additivity. The proposed model can incorporate varying relative potencies among multiple drugs as well. Examples and simulations are given to demonstrate that the proposed model is effective in capturing different patterns of drug interaction.     

What is parallelism?

Although parallel and convergent evolution are discussed extensively in technical articles and textbooks, their meaning can be overlapping, imprecise, and contradictory. The meaning of parallel evolution in much of the evolutionary literature grapples with two separate hypotheses in relation to phenotype and genotype, but often these two hypotheses have been inferred from only one hypothesis, and a number of subsidiary but problematic criteria, in relation to the phenotype. However, examples of parallel evolution of genetic traits that underpin or are at least associated with convergent phenotypes are now emerging. Four criteria for distinguishing parallelism from convergence are reviewed. All are found to be incompatible with any single proposition of homoplasy. Therefore, all homoplasy is equivalent to a broad view of convergence. Based on this concept, all phenotypic homoplasy can be described as convergence and all genotypic homoplasy as parallelism, which can be viewed as the equivalent concept of convergence for molecular data. Parallel changes of molecular traits may or may not be associated with convergent phenotypes but if so describe homoplasy at two biological levels-genotype and phenotype. Parallelism is not an alternative to convergence, but rather it entails homoplastic genetics that can be associated with and potentially explain, at the molecular level, how convergent phenotypes evolve.     

Parallelism, non-biotic data and phylogeny reconstruction in paleobiology

Webb, G.E. 1994 1015: Parallelism, non-biotic data and phylogeny reconstruction in paleobiology.
Many systematists equate parallelism and convergence. However, whereas convergence is relatively uncommon and easily recognized using divergent characters, parallelism is common but more difficult to recognize because divergent characters are less abundant. Cladists, in particular, equate homeomorphy with convergence and reject parallelism as a distinct concept. Unfortunately, cladistic parsimony analysis may not resolve most parallelism. Therefore, criteria for the a priori recognition and objective evaluation of parallelism are very significant. Non-biotic data (e.g., stratigraphic and geographic distribution) provide independent criteria for the construction of hypotheses of parallelism in cases where taxa (1) were geographically isolated during homeomorphic character-state transformations, (2) occurred with endemic faunas, and (3) evolved in similar environmental conditions as suggested by paleoecological data. Australian lithostrotionoid corals were long considered congeneric with European taxa. However, because of their geographic isolation, occurrence with endemic rugose corals and occurrence in similar depositional environments as European forms, they are now considered a homeomorphic clade, resulting from an extended sequence of parallel character-state transformations. The high degree of parallelism, combined with abundant symplesiomorphic characters, led to erroneous phylogenetic inferences when non-biotic data were excluded from analysis. Cladistics, homeomorphy, lithostrotionoid corals, parallelism, phylogeny .     

Dose-finding with two agents in Phase I oncology trials

We propose an adaptive two-stage Bayesian design for finding one or more acceptable dose combinations of two cytotoxic agents used together in a Phase I clinical trial. The method requires that each of the two agents has been studied previously as a single agent, which is almost invariably the case in practice. A parametric model is assumed for the probability of toxicity as a function of the two doses. Informative priors for parameters characterizing the single-agent toxicity probability curves are either elicited from the physician(s) planning the trial or obtained from historical data, and vague priors are assumed for parameters characterizing two-agent interactions. A method for eliciting the single-agent parameter priors is described. The design is applied to a trial of gemcitabine and cyclophosphamide, and a simulation study is presented.     

Limits on the computing power of biological systems

The theory of computational complexity and certain explicitly-stated hypotheses imply limitations on the information processing power of biological systems. Parallelism, special purpose organization, and analog mechanisms may provide speedup critical for life processes, but have little power in the face of exponential growth. We show that “polynomially simulatable” biological systems cannot exhibit dynamic behavior which produces the solution of an intractable problem. The argument implies that parallelism does not allow biological systems to defeat the exponential explosion, but rather is important because it allows polynomial time algorithms to be used more efficiently.     

Effect of dihydroxyanthraquinone (DHAQ) and radiation on the survival of cultured Chinese hamster ovary cells

Dihydroxyanthraquinone (DHAQ) is currently being tested as a cancer chemotherapeutic agent because of its structural similarity to Adriamycin (ADR) and other DNA-intercalating antibiotics. The interaction of DHAQ and ionizing radiation on the induction of cell lethality was investigated in Chinese hamster ovary cells in culture. In asynchronous populations of cells, DHAQ produced a slight enhancement of radiation-induced cell lethality as evidenced by changes in both shoulder and slope of the radiation dose-survival curves. However, DHAQ had no effect on either the extent or time course of recovery from sublethal radiation damage. In synchronous populations of cells treated at various times before or after selection in mitosis, the combination of DHAQ and radiation produced greater cell killing than that predicted based on simple additivity of effect, with a decided enhancement for cells treated during S phase. These results indicate that DHAQ is similar to other DNA-intercalating antibiotics in regard to the interaction with ionizing radiation to produce cell lethality.     

Reduction in opiate receptor reserve in morphine tolerant guinea pig ilea

Spare opiate receptors in the guinea pig ileum have been detected by the use of the opiate receptor alkylating agent beta-chlornaltrexamine (CNA). Treatment of the guinea pig ileum longitudinal muscle in vitro with low concentrations (less than 10nM) of CNA resulted in an irreversible parallel shift to the right of the normorphine log concentration response curve. With increasing concentration of the reagent, the agonist EC50 becomes progressively greater. Finally a point is reached at which the maximal agonist effect decreases, so that parallelism is no longer seen. The maximal parallel shift provides a measure from which one can estimate the spare receptor fraction that is present in untreated tissue. In ilea from normal guinea pigs, roughly 80-90% of the opiate receptors for normorphine were found to be spare. Even after the largest parallel shifts that could be achieved, the naloxone Ke value for antagonism was unchanged, indicating that normorphine acts through spare mu receptors. Ilea from guinea pigs made tolerant by chronic morphine pellet implantation were found to be more sensitive to the effects of CNA treatment; there was a reduction in the number of spare receptors for normorphine. It is suggested that the opiate spare receptor fraction is physiologically modulated to control neuronal sensitivity to opioid effect.     

A formal model for analyzing drug combination effects and its application in TNF-α-induced NFκB pathway

Background

Drug combination therapy is commonly used in clinical practice. Many methods including Bliss independence method have been proposed for drug combination design based on simulations models or experiments. Although Bliss independence method can help to solve the drug combination design problem when there are only a small number of combinations, as the number of combinations increases, it may not be scalable. Exploration of system structure becomes important to reduce the complexity of the design problem.

Results

In this paper, we deduced a mathematical model which can simplify the serial structure and parallel structure of biological pathway for synergy evaluation of drug combinations. We demonstrated in steady state the sign of the synergism assessment factor derivative of the original system can be predicted by the sign of its simplified system. In addition, we analyzed the influence of feedback structure on survival ratio of the serial structure. We provided a sufficient condition under which the combination effect could be maintained. Furthermore, we applied our method to find three synergistic drug combinations on tumor necrosis factor α-induced NFκB pathway and subsequently verified by the cell experiment.

Conclusions

We identified several structural properties underlying the Bliss independence criterion, and developed a systematic simplification framework for drug combiation desgin by combining simulation and system reaction network topology analysis. We hope that this work can provide insights to tackle the challenging problem of assessment of combinational drug therapy effect in a large scale signaling pathway. And hopefully in the future our method could be expanded to more general criteria.     

Application of the isobologram technique for the analysis of combined effects with respect to additivity as well as independence

Combined actions of two substances with similar effects are frequently expressed by pairs of doses that produce a fixed response, usually 50%, in so-called isobolograms (ED50 isobolograms). In addition to the dose scales in such graphs we propose the addition of effect scales, where possible, to indicate the effect at certain doses, e.g., the ED30. We further propose to construct isoboles for expected independent interaction, in addition to the additivity line, for which purpose a simple procedure is delineated. In practice, an independent isobole for 50% effect passes through the point formed by the ED30s of A and of B in ED50 isobolograms. Thus, the ED30s constitute the "zenith" of an independent isobole in ED50 isobolograms. It is shown that theoretical independent isoboles can either represent additive, overadditive, or underadditive interactions, depending on the steepness of the dose-response curves of the components. Hence, drugs with shallow dose-response curves exhibit overadditive independent effects, compounds with exponentially steep curves show additive independent interactions. Substances with very steep dose-response curves, producing lethal effects, exhibited marked underadditive effects which could be ascribed largely to an independent mechanism of action of the components. Hence, the inclusion of independent isoboles into conventional isobolograms provides new insights into the mechanisms of interactions and into the actions of the components. Interactions can thus be characterized better and more completely, and misinterpretations appear less likely than with conventional isoboles.     

On some aspects of parallel evolution in Chelicerata

A study is made of some aspects of parallel evolution in Chelicerata. Definitions are given of parallel evolution, convergence, homology and analogy. It is pointed out that the concept of parallel evolution (parallelism) is initially formed in an empirical way, and that a judgment must be based on formal criteria. Particular attention is paid to the rôle of gene regulation in parallel evolution, to the special case of convergence as a result of heterologous regulatory mechanisms, to parallel evolution in homonomous structures (and the superposition of parallelisms and divergences), and to parallelism in the evolution of characters used in higher classification.     

Substitution of calorigenic effects of noradrenaline and adrenaline and differences in their inhibition by propranolol.

The calorigenic effect of infused adrenaline and noradrenaline was measured in cold-acclimated rats. The slopes of the dose-response curves for the two catecholamines and the maxima of the curves were the same. The adrenaline dose-response curve showed a shift to the right, towards higher infusion doses, compared with the noradrenaline curve. Thermogenesis due to the two catecholamines was not additive throughout the whole range of doses used. In interaction with noradrenaline, propranolol caused a parallel shift of the dose-response curve to the right, whereas in interaction with adrenaline it depressed the maximum. The concept that the two catecholamines act via different regulation sites on a common thermogenetic effector is discussed.     

6-Aminonicotinamide-induced cleft palate in the mouse: the nature of the difference between the A/J and C57Bl/6J strains in frequency of response and its genetic basis.

Cleft palate induction by 6-aminonicotinamide (6-AN) was examined in the A/J and C57BL/6J strains of mice to determine the nature of the strain difference in frequency of cleft-palate response. Probit analysis of the cleft-palate response to dose of different genotypes revealed a family of linear and parallel dose-response curves. The genotypes differ only in dosage tolerance (log ED50) to 6-AN that is required for the cleft-palate response. No evidence for a maternal cytoplasmic effect on 6-AN-induced cleft palate was found under the conditions of the present study. When the difference is dosage tolerance to 6-AN between A/J and C57BL/6J was examined with a single dose and measured by differences in frequency of induced cleft palate on a probit scale, there was some departure from genetic additivity. There was an indication off dominance deviation of the F1 embryos in the direction of C57BL/6J.A3-locus, epistatic model is proposed to account for the difference in embryonic tolerance ot 6-AN-induced cleft palate. There was a suggestion of association with the brown (b) locus.     

Blockade of histamine-induced contractions of guinea pig ielum by beta-haloalkylamines.

In the longitudinal muscle strip of guinea pig ileum phenoxybenzamine (POB) produces a maximum parallel shift of 0.7 log units in the dose-response curve to histamine. In the presence of sodium thiosulfate in the wash fluid the parallel shift whith retention of maximum response increases to about 2 log units, and a similar value is obtained for Nethyl-N-(2-bromoethyl)-1-naphthylamine. The The agent N-ethyl-N- (2-chloroethyl)benzylamine produces a significantly smaller shift of dose-response curve of 1.53 log units before the maximum response becomes depressed. The receptor-specific depression of maximum response produced by higher doses of POB is reversed by sodium thiosulfate and by bovine serum albumin, while the parallel shift in dose-response curve is unaffected by both treatments. These findings may be explained by a hypothesis involving interaction of 2-haloalkylamines at two sites.     

Combined effects of vasoactive intestinal peptide and dopamine on adenylate cyclase in prolactin-secreting cells

VIP stimulates adenylate cyclase activity of male and female rat anterior pituitaries and human prolactinomas, while dopamine inhibits the enzyme activity of female rat pituitaries and prolactinomas. A dopamine inhibited cyclase can be detected also in male rats provided the enzyme activity is increased by VIP. The analysis of the dose-response curves for one agent (VIP or dopamine) in the absence or in the presence of the other indicates that the two agents exhibit a different pattern of interaction in the different systems. In fact, in female rat pituitaries and in human prolactinomas, the curves for dopamine±VIP and for VIP±dopamine were parallel, indicating that the two agents exherted their effects independently from one another. On the contrary, in male rat pituitaries, the curves were definitively non parallel, that is, the inhibitory effect of dopamine was greatly amplified by VIP. In no case was the apparent affinity (EC₅₀) of one agent modified by the presence of the other. It is concluded that two different modes of interaction between stimulatory and inhibitory neurohormones might exist at the level of adenylate cyclase from anterior pituitary cells.     

A Bayesian Approach to Dose–Response Assessment and Synergy and Its Application to In Vitro Dose–Response Studies

Summary In this article, we propose a Bayesian approach to dose–response assessment and the assessment of synergy between two combined agents. We consider the case of an in vitro ovarian cancer research study aimed at investigating the antiproliferative activities of four agents, alone and paired, in two human ovarian cancer cell lines. In this article, independent dose–response experiments were repeated three times. Each experiment included replicates at investigated dose levels including control (no drug). We have developed a Bayesian hierarchical nonlinear regression model that accounts for variability between experiments, variability within experiments (i.e., replicates), and variability in the observed responses of the controls. We use Markov chain Monte Carlo to fit the model to the data and carry out posterior inference on quantities of interest (e.g., median inhibitory concentration IC ₅₀ ). In addition, we have developed a method, based on Loewe additivity, that allows one to assess the presence of synergy with honest accounting of uncertainty. Extensive simulation studies show that our proposed approach is more reliable in declaring synergy compared to current standard analyses such as the median‐effect principle/combination index method ( Chou and Talalay, 1984 , Advances in Enzyme Regulation 22, 27–55), which ignore important sources of variability and uncertainty.