Accepted Manuscript Title: A unified approach to computational drug discovery Author: Chih-Yuan Tseng Jack Tuszynski PII: DOI: Reference:
S1359-6446(15)00275-5 http://dx.doi.org/doi:10.1016/j.drudis.2015.07.004 DRUDIS 1649
To appear in: Received date: Revised date: Accepted date:
17-10-2014 13-6-2015 9-7-2015
Please cite this article as: Tseng, C.-Y., Tuszynski, J.,A unified approach to computational drug discovery, Drug Discovery Today (2015), http://dx.doi.org/10.1016/j.drudis.2015.07.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights: Current strategies for drug discovery are reviewed We propose that drugs can be better discovered through an entropic inductive inference-based unified approach Examples from different stages in drug discovery are presented to demonstrate the concept and use of the proposed unified approach
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A unified approach to computational drug discovery Chih-Yuan Tseng1,3,* and Jack Tuszynski1,2 1
Department of Oncology, University of Alberta Edmonton, AB T6G 1Z2, Canada Department of Physics, University of Alberta Edmonton, AB T6G 1Z2, Canada 3 MDT Canada, Edmonton, AB, Canada *
Corresponding author: Tseng, C.Y. ([email protected]).
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Keywords: Maximum entropy; inductive inference; drug discovery; target identification; compound design; pharmacokinetics; pharmacodynamics.
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Teaser: Entropic inductive inference is the foundation of an information-based unified approach to computational drug discovery.
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It has been reported that a slowdown in the development of new medical therapies is affecting clinical outcomes. The FDA has thus initiated the Critical Path Initiative project investigating better approaches. We review the current strategies in drug discovery and focus on the advantages of the maximum entropy method being introduced in this area. The maximum entropy principle is derived from statistical thermodynamics and has been demonstrated to be an inductive inference tool. We propose a unified method to drug discovery that hinges on robust information processing using entropic inductive inference. Increasingly, applications of maximum entropy in drug discovery employ this unified approach and demonstrate the usefulness of the concept in the area of pharmaceutical sciences.
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PACS codes: 02.50.Tt; 87.10.Vg[s1] Drug discovery applies multidisciplinary approaches to either experimentally or computationally (or a combination of both) identify lead compounds with therapeutic applications to various diseases. In this context, one faces an issue that information required to ‘discover’ drugs is always insufficient because of the inherent complexity of biological systems. Recently, it has been reported that a slowdown in the development of new medical therapies is adversely affecting patient outcomes [1–3]. The FDA has thus initiated a long-term project: the Critical Path Initiative, to assist researchers in their investigations aimed at a design of better approaches to drug discovery [4]. In this review, we provide an overview of the drug discovery process and argue that this field will significantly benefit from a unified approach based on the maximum entropy principle. Among numerous methods applied in drug discovery, we believe that the maximum entropy method could provide an appropriate approach because it is designed to tackle problems for which information available is inherently insufficient [5]. However, as we elaborate on later, it is surprising that only a limited number of applications of entropybased methods to various topics in drug discovery can be found in the literature. One of the reasons might be that the development of the maximum entropy method as a tool for inductive inference was conducted over a roughly similar time span when computational rational drug discovery was being developed. Researchers paid more attention to the foundation of the maximum entropy method and its applications in fields such as astronomy, physics and engineering rather than in the life sciences. We believe that the time has now come to usher the maximum entropy methods into the drug discovery area. The progress of the method of maximum entropy in terms of inductive inference has enabled clarification on unified approaches to computational drug discovery. Here, we discuss not only how the maximum entropy principle is applied to solving problems of significance in drug discovery but also how it is an attempt to provide an alternative but a unified approach when compared with standard methods. The foundation of this unified approach is what we later refer to as the entropic inference scheme. It is our hope that the illustration presented here reveals advantages of this scheme and the way of utilizing it to improve and accelerate drug discovery.
Conventional strategies in rational drug discovery The central idea of the function of a drug is to inhibit (or sometimes activate) functions of biological targets (proteins, enzymes or DNA) in the human body to prevent, stall or cure diseases that are related to abnormal activities of such biological targets. The conventional wisdom employed to ‘discover’ such a drug includes two main categories: compound design and investigation of its drug-like properties. Compound design includes two subjects: identification of biological targets and design of small molecules to bind to active sites of biological targets. The drug-like properties category includes two subjects: screening of these molecules to determine candidates that satisfy specific properties to act as a drug and demonstration of the desired pharmacokinetics and pharmacodynamics of these agents. Additional filtering of a set of candidate compounds can[s2] involve the prediction of potential side-effects of the drugs and their metabolites. Compound design
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Target identification. The first stage in drug discovery is to identify biological targets that are responsible for causing abnormal physiological activities in biological systems. In the past, the complexity of biological systems has limited the concept of drug design to the so-called single-drug–single-target paradigm. However, there is increasing evidence that this paradigm fails to yield better drugs with satisfactory effects expected to treat various diseases [1–3]. The recent blossoming of systems biology further demonstrates that this paradigm is inadequate because of the enormous complexity of signaling, regulatory and/or metabolic networks involved in the emergence and stability of disease states [6–9]. In other words, there can be multiple targets in the networks involved in cellular state regulation. Therefore these studies further suggest the existence of a better paradigm the central idea of which is rooted in the multiple-target–multiple-drug principle [6,7]. Systems-biology-based approaches, therefore, were devised to assess which targets are likely to be key factors to inhibit abnormal regulation caused by diseases. The method proposed by Yang et al. [7] is one specific example illustrating this point. Yang et al. [7] proposed to perturb the network and optimize it toward the desired state using a Monte-Carlo-simulated annealing optimization algorithm. By comparing diseased and normal network states in simulations, one can assess which targets are most likely to be responsible for the disease states as potential drug targets.
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Determination of target structures. Once key biological targets have been identified, one needs to determine their tertiary (3D) structures to design small molecules that can specifically and selectively bind to active sites located on them in order to regulate their functions. Several wet-lab techniques including NMR, electron- and X-raycrystallography have been the primary tools for this purpose. If target structures cannot be determined through NMR, electron- or X-ray- crystallography techniques one can still apply the computational homology modeling method to construct a reliable approximation to the tertiary structure of the target molecule provided a primary sequence is known and a reasonably similar template structure exists [10]. The homology modeling method involves applications of several bioinformatic tools including protein sequence analysis and structure analysis (it consists of comparison, construction, equilibration and validation). The reader is referred to ref. [10] for more details. Drug design strategy. Screening chemical libraries is a common approach to discover drug candidates. The size and structural diversity of the library of chemical compounds used in the discovery process are key factors to the successful identification of potential lead compounds that can bind specifically to molecular targets. However, the chemical space covering organic molecules is astronomically large [11]. It is highly unlikely that one can efficiently discover lead compounds through screening compound libraries without introducing very efficient searching algorithms that cover a diverse and large part of the chemical space. In the past three decades, virtual screening approaches based on integration of various computational methods have been developed to address this issue [12,13]. Basically, these approaches consist of three ingredients: (i) the creation of a chemical library; (ii) search for an optimal ligand–receptor-binding mode through docking algorithms; and (iii) evaluation of the binding affinity of each ligand examined with the receptor chosen as the target. Accordingly, three criteria are required to identify compound candidates successfully. First, the chemical library needs to contain large
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numbers of diverse chemical structures. Second, conformational search algorithms need to be able to screen all major binding modes within reasonable compute time. Third, an appropriate scoring function needs to be utilized to evaluate the binding preference of specific compounds correctly. AutoDock software is one of the methods commonly used in virtual screening approaches. It utilizes a grid-space method for rapid binding energy calculations, a Lamarckian genetic algorithm for increasing efficiency and accuracy in conformational searches and a semi-empirical free energy force field to evaluate the binding free energy more precisely [14]. In spite of the advances made in the screening techniques and energy calculation methods, virtual screening is still facing the issue of small sizes and insufficient structural diversity of the chemical compound libraries used in these endeavors. Consequently, fragment-based approaches have been proposed to provide an alternative. Ludi is one example of computer-aided design [15] whereas SAR by NMR is an example of in-vitro-based design [16]. In silico and in vitro methods are rooted in a similar concept. It starts with identifying molecular fragments that can bind to potential pockets in molecular targets to then link those fragments with appropriate linkage fragments to form a single molecule that has the highest binding affinity. This approach dramatically reduces the conformational space and increases the structural variety of molecules. However, combinatorial explosion in terms of the generation of a huge number of resultant compounds from a limited number of fragments is the bottleneck of this type of design.
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Drug likeness The compound design strategies discussed above do not guarantee that compounds discovered are drug candidates. In order for a compound to be considered a valid drug candidate, it needs to possess drug-like properties. Several drug-like properties that are normally introduced in the drug discovery process as screening criteria are discussed below.
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Lipinski’s Rule of Five. In 1997, Lipinski et al. [17] reported an intensive review of the physicochemical criteria for orally administrated active compounds that have poor solubility and permeability. They found that when a compound has a molecular weight (MWT) larger than 500 Da, LogP greater than 5, more than five H-bond donors or more than ten H-bond acceptors then the compound is highly likely to exhibit poor absorption. Namely, the compound is unlikely to be absorbed into the blood circulation system and reach active sites. The above listed criteria are known as Lipinski’s Rule of Five, which has become a gold standard and a first filtering step in screening medicinal chemistry compounds. Applying these criteria as filters for libraries of potential medicinal chemistry compounds helps to screen out compounds designed for oral administration that have a higher chance of being absorbed. Pharmacokinetics: ADMET. Even though medicinal chemistry compounds might possess good absorption properties, this does not guarantee that such compounds can be considered as drugs. After absorption, as a drug, a molecule must be able to reach active sites that are believed to be the sources causing the disease in question and bind to target proteins to regulate their functions with the highest efficacy. To achieve a desired efficacy, the amount of the drug entering active sites will need to be maintained at a
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sufficiently high level after this drug has undergone various types of biotransformations such as intestinal and liver metabolism. In the meantime, the toxicity caused by off-target interactions needs to be reduced to a minimum. Although there are no quantitative criteria defined based on these pharmacokinetic properties, this provides a qualitative guideline to increase the probability that the screened compounds can achieve desired pharmacokinetic properties. With the advance of chemoinformatics and bioinformatics, the ADMET Predictor software from Simulation Plus [18] is a good example of software that uses computational (artificial intelligence) methods to predict ADMET of a given set of compounds.
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Pharmacodynamics. The final criterion we address here is the pharmacodynamics of the tested compounds. This describes the response of the body to the drug and the mechanisms of drug action. As is well known, the goal of designing a drug is to produce desired physiological responses in the body given that a specific amount of the drug has been administered over a particular amount of time. Therefore, the knowledge of the pharmacodynamics of compounds certainly provides crucial information for selecting promising compounds as drugs. Unfortunately, one cannot directly conduct in vivo experiments without identifying lead compounds. One solution is to utilize in-vitro–invivo correlation (IVIVC) prediction, if such a correlation exists to predict in vivo effects based on prior in vitro studies [19,20]. Searching a better way to establish IVIVC has become one of the most important subjects of study in the pharmaceutical industry to improve on drug design and optimize drug formulation.
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Entropy-based inductive inference Two reasoning methods have been utilized in developing theories to interpret phenomena we observe in nature and to make predictions about them. They have been applied to methodologies developed in drug discovery without exception. The first is deduction. It allows us to draw conclusions when sufficient information is available. The second method is variously called inductive logic, inductive inference, probable inference or just inference. It allows us to reason when the available information is insufficient for deduction. It is called either ‘inference’ (for the cases that we make estimates of quantities when we do not know enough to deduce their values) or ‘induction’ (when we generalize from special cases) [21]. When dealing with complicated biological systems, which involve many-body interactions at a microscopic level, complicated regulating protein–protein networks at a mesoscopic level or genetic populations at a macroscopic level, it is common to find that we do not have sufficient knowledge to understand these kinds of systems in detail. We normally rely on inductive inference to deduce the most preferable solutions to the problems related to these systems based on available information. Before we direct our attention back to drug discovery, we discuss the mathematical tools required for inductive inference. The central idea involved in these tools hinges on the Bayesian interpretation of probability and the rules of probability theory. The former treats probability as a measure of our state of knowledge or degree of belief about the system of interest rather than a frequency of occurrence of an event. Cox demonstrated that this type of probability can be manipulated by the rules of subtraction, multiplication and addition given by the standard probability theory [22,23].
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The state of knowledge. Based on the Bayesian interpretation of probability, Laplace’s principle of insufficient reasoning and Cox’s axiomatic approach, Jaynes first showed that the method of maximum entropy (denoted here by MaxEnt) is a reasoning tool to assign probabilities on the basis of limited information about the systems of interest with the least bias [24,25]. The preferred probability distribution of the system in a specific state is the one that maximizes the entropy of the system subject to the constraint of a limited amount of information available. Note that entropy is defined by the well-known formula in Equation 1:
S p pi ln pi
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[Equation 1]
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where pi is the probability of the system in state i and the sum is over all states the system can adopt. Here, the information about the system available is in the form of constraints as described by Equation 2[s3]:
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O piOi , i
[Equation 2]
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which represents the expectation value of the observed quantity Oi (e.g. it can be the binding energy of compounds or the expression level of target proteins). The probability distribution pi represents our state of knowledge regarding the system in state i. In summary, Jaynes’ pioneering work [s4]established a mathematical framework for determining the most preferable inference, namely the probability one can estimate according to the information about the system in hand. It is worth stressing that the meaning of entropy is beyond a physical quantity for measuring the degree of randomness or information. It is a reasoning tool as well.
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Maximum-entropy-based inference scheme. The next breakthrough was to establish a tool beyond MaxEnt for updating our state of knowledge regarding the system when new information is acquired. Shore and Johnson, Skilling[s5] and Caticha all followed the same reasoning as Jaynes and proposed that the method of maximum entropy (denoted here by ME) can also be utilized as a tool for updating probability distributions based on newly acquired information [26–29]. They showed that the key to updating probability distributions is through relative entropy rather than the entropy used in MaxEnt (Equation 3):
S p, m pi ln pi i
mi
[Equation 3]
where mi a prior is a probability distribution regarding the system in state i before any new information is acquired. Caticha further demonstrated that ME unifies the two methods, MaxEnt and Bayesian methods, into a single inference scheme [30]. In summary, based on this scheme, inductive inference is made by first resolving the following two issues. First, what is a prior and hence what information is relevant to the
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system of interest. Second, what is the relevant application of the method of maximum entropy. Because ME allows arbitrary prior and constraints, one can expect that this general scheme can be applied to an arbitrary set of problems as long as information acquired to investigate these problems is in the form of constraints.
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General inference scheme. As discussed above, the most probable inference made by ME is the one that maximizes relative entropy (Equation 3). However, the question remains: to what extent are less favorable inferences ruled out by the maximum entropy principle. Information geometry (IG) offers a straightforward tool to answer this question [31]. IG introduces the concept of manifolds for characterizing and manipulating information [32,33]. Probability distributions then can be simply treated as points on the information manifold, which is defined based on the Fisher–Rao metric [32]. Therefore, one can quantify preferences for all probability distributions, inferences, through a distribution of corresponding points on the information manifold [31]. Utilizing ME and IG together then provides a general scheme to manipulate information and make inductive inference on the manifold [31]. Specifically, one will not only make the most favorable inference but one also has the knowledge regarding the preference of less favorable inferences. In summary, the general inference scheme or the entropic inference scheme is a reasoning framework to manipulate the state of knowledge and make the most plausible inference for the system under investigation. Under this scheme, the principle of maximum entropy fundamentally defines the way information is to be processed. MaxEnt, ME and IG are appropriate mathematical tools to manipulate information.
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An entropic unified approach to computational drug discovery
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Logic and implementation In this section we discuss the logic behind the proposed unified approach to drug discovery and the way of implementing this approach in real pharmaceutical applications. The next section will then introduce three examples to illustrate it.
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Logic. As has been shown previously, we need to introduce specific methods and tools to solve problems at different stages in drug discovery. Although these methods lead to acceptable results, this still poses a debatable issue. Namely, because all kinds of information required are interconnected, the question remains as to whether we should tackle each stage of drug discovery independently or simultaneously. In either way, we also need to face a problem: because characteristics of each stage are different how are we going to appropriately utilize different types of information to determine the most preferable solution? This suggests a necessity for unified approaches to improve the discovery process. Yet, owing to the complexity involved in drug discovery, it seems impossible to develop unified approaches to integrate fundamental principles at different stages in drug discovery. At the present time, systems biology provides the best concept and strategy toward establishment of a foundation for a unified approach [8,9]. However, as Berg pointed out even with the advances made in systems biology, several issues still hinder the drug discovery process [9]. We argue that systems biology like other areas impacting
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drug discovery merely provides additional information at the systems level. Systems biology still does not directly tackle drug discovery process. By recognizing the complexity inherent in physiological systems from the cellular to the whole body level, the overview of different stages of drug discovery above suggests that a fundamental principle governing drug discovery hinges on information processing. Specifically, the success of drug discovery depends on the way of utilizing all kinds of information from subcellular compound–target interactions to cellular regulation mechanisms to the organ- and body-level pharmacokinetic properties of a compound to make the most plausible inference about drug candidates. From the information theory point of view, this way is independent of either goodness of the virtual screening approach, completeness of regulation networks or correctness of pharmacokinetics prediction. This way involves logic and the principles of objectively manipulating the available information. The entropic inference scheme is exactly designed to provide a unique and robust recipe for developing a unified approach for processing all kinds of information involved in drug discovery and to suggest the most preferable drug candidates as long as information can be formulated in terms of constraint equations of the type shown in Equation 2. This development requires two ingredients: (i) asking the relevant question and (ii) collecting relevant information. Given these two ingredients, the maximum entropy principle will be the fundamental principle guiding the discovery process.
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Relevant questions. The goal is to turn deterministic sound problems into probabilistic ones by asking questions relevant to the problem of interest. Those relevant questions are the cornerstone of a unified approach. The overview of drug discovery described above suggests that the most relevant and general question one should ask is: what is the probability distribution of a mixture of compounds that regulates a disease state of a biological entity and transforms into another specific state? Rather than asking which targets, compounds and their combinations should be selected with the objective of achieving a maximum potency. As was discussed in previous sections, this regulation involves a series of key properties including target identification, compound design, sufficiency of drug-like properties and optimal pharmacokinetic and pharmacodynamic behavior. Namely, this probability distribution is likely to be a joint probability distribution that codifies information regarding these properties. Determining the preferred joint probability distribution suggests a preferred mixture of compounds. Based on the concept of conditional probability, we can further ask a series of sub-questions specifically intended to answer this general question properly. The first sub-question to be asked is regarding target identification: what is the probability distribution of specific biological targets such as proteins being at specific abnormal or disease states? The second sub-question to be asked is regarding compound design: given the answer to the first question, what is the probability distribution of a set of compounds interacting with those biological targets? The third sub-question to be asked is also related to compound design. It involves the sufficiency of drug-like properties and optimal pharmacokinetics: given the answer to the second question, what is the probability distribution of a set of compounds being delivered into active sites that contain those biological targets? Finally, the fourth sub-question is regarding pharmacodynamics: given the answer to the third question, what is the probability
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distribution of specific combinations of compounds regulating disease state to a specific state?
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Relevant information. The key to answering the relevant questions hinges on collecting relevant information and utilizing the right principles to process the information. In the framework of the entropic inference scheme, the principle of maximum entropy is the only fundamental principle that leads to unique information processing with the least bias from subjective influences. Therefore, under the entropic inference scheme it shifts endeavors in discovering drugs from developing better theories and methods to collecting information relevant to answering those questions systematically. Unfortunately, seeking out relevant information still relies on empirical knowledge and a method of trial and error. Success of the unified approach to drug discovery will heavily depend on the ways to overcome this information collection bottleneck.
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Mathematical formulation Cox has shown that the state of knowledge can be manipulated by the same rules that are used in probability theory [22,23]. Based on these rules and supposing the answers to questions raised above can be found, we can derive mathematical relations among them. These relations are illustrated in Figure 1 and this also represents a fundamental information flow required to make the most plausible inference to discover drug candidates. For the sake of simplicity in derivation: we define D as a specific disease; T as possible targets; C as medicinal compounds; L as lead compounds; IT as information required to identify targets; IC as information required to determine compounds; and IKD as information from pharmacokinetic and pharmacodynamic studies. Our ultimate goal is to determine the probability distribution of a mixture of lead compounds L that regulates a disease state of a biological entity to another specific state PL CTDI C I T I KD , given information of compounds C that can bind to targets T responsible for disease D having desired pharmacokinetic and pharmacodynamic properties IKD. According to an associativity constraint, if a state of knowledge can be calculated in two different ways these two must agree, the joint [s6]probability distribution of having lead compounds L, all possible compounds C, potential targets T, disease of interests D and all information IC, IT and IKD can be written in two ways (Equations 4,5): [s7] PLCTDI C I T I KD P LCTDI C I T I KD PI KD PI KD LCTDI C I T PLCTDI C I T [Eqn 4] The left-hand term of the second equality is the product of the probability distribution of having information IKD, PI KD and the conditional probability distribution of having L,
C, T and D given IKD, PLCTDI C I T I KD . The second equality gives:
PI KD LCTDI C I T PLCTDI C I T I KD PLCTDI C I T PI KD PLCTDI C IT LLCTDI C I T I KD
[Equation 5]
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where LLCTDI C I T I KD PI KD LCTDI C I T PI KD is the likelihood function. Because (Equation 6): PLCTDI C I T I KD P L CTDI C I T I KD PCTDI C I T I KD
[Equation 6]
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we have Equation 7:
LLCTDI C I T I KD [Equation 7] PCTDI C I T I KD We apply associativity again to the joint probability PLCTDI C I T and obtain Equation 8:
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PLCTDI C I T PL CTD PC TDI C P T DIT PD
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PL CTDI C I T I KD PLCTDI C I T
[Equation 8]
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Therefore, substituting Equation 8 into Equation 7 gives Equation 9:
PL CTDI C I T I KD PL CTD PC TDI C PT DI T PD
LLCTDI C I T I KD PCTDI C I T I KD
PL CTD PC TDI C PT DI T PD
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PL CTD PC TDI C PT DI T PD LˆLCTDI C I T I KD
[Eqn 9]
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This equation reiterates our reasoning to discover efficacious drugs. It indicates determination of the conditional probability distribution of a mixture of lead compounds L that regulates disease state of a biological entity to a specific state required. We have, first, PT DI T , the conditional probability distribution of targets that are responsible for
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disease D and satisfy information IT, second, PC TDI C , conditional probability distribution of compounds that bind to targets given T and D and information Ic, and, third, PL CTD , the prior probability distribution of lead compounds that regulate disease state to specific state given C, T and D before acquiring information IKD. Furthermore, the term: LˆLCTDI C I T I KD LLCTDI C I T I KD PCTDI C I T I KD is the overall likelihood function to be determined through experimental data and empirical knowledge. In summary, Equation 9 represents the mathematical foundation of the unified approach to drug discovery. Given Equation 9, the remaining tasks are to determine each conditional probability distribution using the maximum entropy principle and the appropriate likelihood function according to the information acquired. Case study examples Although the method of maximum entropy has been introduced to drug discovery as either a tool or a reasoning tool for developing methods to solve problems of relevance, those applications did not completely exhaust the potential beyond its original usage. In the following, we go through different stages in drug discovery, which have been tackled
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through an entropy-based method, in terms of an entropic inference scheme to illustrate the original use and demonstrate that those applications are merely parts of the proposed unified approach. Specifically, we discuss three important topics: (i) target identification, PT DI T ; (ii) compound design, PC TDI C ; and (iii) pharmacokinetics and
pharmacodynamics, PL CTDI C I T I KD .
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Target identification As mentioned above, the relevant question regarding target identification is: what is the probability distribution of specific biological targets such as proteins or enzymes being at a specific abnormal or disease state PT DI T ? To answer it, we need to acquire information, IT, relevant to characteristics of the disease state. Several aspects of information, as shown below, have been studied that can be considered relevant in this connection. For example, Yang et al. defined the activity of a target based on the reaction rate to characterize the corresponding regulation network [7]. Fuhrman et al. considered the temporal gene expression patterns of targets, which are either genes that underwent a disease process or a normal phenotypic change [34]. Tseng et al. utilized compound– target binding energy, which are shown to correlate target protein expression level changes when cells are treated by chemotherapeutic compounds [35]. In their work, Fuhrman et al. merely utilized the Shannon entropy (Equation 1) to quantify information codified in p i , a probability distribution of target gene expression being at a specific level [30]. The entropy value S p i pi log pi indicates variation
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in a temporal series of gene expression data that describe a certain biological process. For genes with no variation in the expression level, a zero entropy value was assigned and hence they are considered to contain no information. Namely, these genes are not responsible for triggering the processes in question. Therefore, candidate drugs for these targets will correspond to maximum entropy. Unfortunately, because their goal was to introduce entropy as a measuring rather than an inductive inference tool, these authors focused on defining probability distribution p i through gene expression data in hand and empirical knowledge regarding target activity and interpretation of gene expression. Therefore, the determination of target candidates is subjected to the quality of data and bias inherent in this empirical knowledge. By contrast, the work of Tseng et al. [35] provides an example that demonstrates that we can go beyond the use of entropy as a measuring tool as in the work of Fuhrman et al. [34], [s8]even though these two examples tackle target identification from different angles. In terms of the entropic inference scheme, the goal is to determine the most preferable probability distribution of targets being at specific expression levels based on the information available. Therefore, we can then infer potential target candidates through this distribution. In the work of Tseng et al. [35] [s9]information that relates cytotoxicity and the binding energy of compounds was analyzed. This can be formulated in terms of a constraint equation (Equation 10): M
RTe log IC50 Pi Gi i 1
[Equation 10]
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where refers to compounds, R is the gas constant, Te is the absolute temperature, IC50 is the drug concentration that inhibits 50% of proliferating cancer cells and Gi is the
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e Gi Z
[Equation 11]
where the partition function Z i 1 e M
Gi
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Pi mi
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binding energy between target i and compound By maximizing relative entropy (Equation 3), subject to this constraint equation, they determined the preferred probability distribution of target i having an expression level after it was treated by compound to be that given by Equation 11:
and the Largrangian multiplier are
determined from Equation 10. Suppose we consider priors mi to be a probability
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distribution of the target protein expression level before any treatments involving chemotherapy compounds. Therefore, given information regarding the disease D=DCancer and IT =IBindingenergy, candidates for targets selected for optimal treatments can be determined by P T DCancer , I Binding energy Pi .
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Compound design Next, we aim to answer the following question: given the answer to the first question, what is the probability distribution of a set of compounds interacting with those biological targets? Here, we only focus on two common methods, virtual screening and a fragment-based approach, for illustration.
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Virtual screening. Under the entropic inference scheme, the first (chemical library) and third (scoring function) criteria mentioned above[s10] merely represent the fundamental information required in virtual screening approaches. The second criterion, searching for optimal binding modes of compounds, then can be treated as an information processing guideline. The efficiency and accuracy of this step will depend on the methods of information processing. The genetic algorithm is one such example. The algorithm borrows the concept of a genome evolution process to search conformations of complexes involving targets and chemical compounds. Chang et al. [36] have shown a better alternative, namely MEDock. Although MEDock does not completely follow the entropic inference scheme for conformational searches, it simply utilizes the maximum entropy principle as a decision-making guideline in search processes. Similarly to the question we pose here, the fundamental question asked in MEDock is: what is the probability of finding the deepest energy valley in the ligand–target interaction energy landscape? Maximum entropy suggests the preferable direction to update the initial guess of a binding mode (described by an almost
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uniform distribution) to an optimal mode (a localized distribution around the global energy minimum).
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Fragment-based drug design. The second example for compound design involves utilizing the entropic inference scheme to design aptamers. Aptamers are short nucleic acid sequences typically identified through an experimental technique: systematic evolution of ligands by exponential enrichment (SELEX) [37,38]. Because of their strong and specific binding through molecular recognition, aptamers are promising tools in molecular biology and have therapeutic and diagnostic clinical applications [37–40]. Unfortunately, some limitations of the SELEX technique have seriously slowed down the progress of discovering aptamers [41]. With the help of entropic inductive inference, a fragment-based approach has been developed to design aptamers given the structure of the target of interest [41]. The concept behind the fragment-based approach is to ask the question: given the structural information about the target, what is the preferred probability distribution of having an aptamer that is most likely to interact with the target? The solution involves acquisition of two types of information: first, regarding the relevant information for determining the choice of the first nucleotide, the so-called seed; and second, given the preferred first nucleotide, the relevant information for determining the next nucleotide. The approach first determines the preferred probability distribution of the first nucleotide Pnt1 Qi , q1 j
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P Q , q H Q , q
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H Total Qi , q1 j
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that probably interacts with the target. Because the information that is relevant to the interaction between the target and the first nucleotide is the total free energy of this complex, it is reasonable to consider the expected total energy of the target denoted by ‘tar’ and the first nucleotide denoted by ‘nt1’ (Equation 12): 1
nt1
i
j
1
Total
i
j
[Equation 12]
1
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Qi , q j
as preliminary information in hand, where the Hamiltonian of the target–first-nucleotide complex is H Total Qi , q1j H tar Qi H nt1 q1 j H tar - nt1 Qi , q1j H solvent and
Q R , P stands for i 1,..., N , i.e. the atomic spatial coordinates and momenta of the N -atom target and q r , p stands for the j 1,..., N atomic spatial coordinates and momenta of the first N -atom nucleotide. Furthermore, H Q is the total internal energy of the target ‘tar’, H q is the total internal energy of the first Q , q denotes the interaction energy of the target and the first nucleotide ‘nt1’, H i
i
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Q
1
Q
1 j
j
1
nt1
j
nt1
tar
i
1
nt1
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nucleotide and H solvent represents the solvation energy. Afterward, the approach keeps updating the probability distribution when more nucleotides are added to the previous one. Again, the total energy of the target and all
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nucleotides (the previous one and the added one) provides a reasonable choice for new information relevant to the interactions between targets and additional nucleotides. For example, the information acquired in the second step is the expected total energy of the target and two-mer nucleotides (Equation 13):
P Q , q , q H Q , q , q 1
nt2
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j
2 j
1
Total
i
2 j
j
[Equation 13]
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H Total Qi , q1 j , q2j
1
Qi ,q j
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where q2j r2j , p2j stands for the j 1,..., N nt 2 atomic spatial coordinates and
momenta of the second N nt 2 -atom nucleotide. Maximizing the relative entropy (Equation
S Pnt2 Pnt1
1
Pnt2 Qi , q1 j , q2j log
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14):
Pnt2 Qi , q1j , q2j
2
Pnt1 Qi , q1 j
[Equation 14]
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Qi , q j ,q j
we then determine the probability distribution of two-mer nucleotides interacting with the target, Pnt2 Qi , q1 j , q2j , which is updated from Pnt1 Qi , q1 j . Continually using Equation 3, the maximum entropy allows us to determine to what extent this update is sufficient and a sequence of nucleotides of maximum length L with probability distribution P Qi , q1...j L that is likely to bind to the target can be determined. This probability
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distribution is equivalent to PC TDI C where C q1...j L the spatial coordinates of the L-
mer sequence, T Qi the spatial coordinates of the target responsible for the disease D
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and I C total energy as information required for compound design. The method has been applied to the design of aptamers which has been computationally and experimentally validated to bind specifically to targets such as thrombin [41], phosphatidylserine [42] and galectin-3 (under experimental confirmation). Pharmacokinetics and pharmacodynamics Drug absorption rate estimation. Regarding pharmacokinetics, the relevant question is: given the answer to the second question about compounds, what is the probability distribution of a set of compounds being delivered into active sites that contain those biological targets? Here, we only focus on the very first step of distributions of orally administrated drugs: absorption, for illustration purposes. The absorption of drugs involves several processes including dissolution, gastric and intestinal release and plasma disposition. Several methods have been developed for predicting drug absorption. One traditional way is to utilize a statistical approach to deduce models from training sets for absorption prediction [43]. Their success relies on the selection of an appropriate training set and statistical approaches. Another approach is based on a mechanistic model such as the well-known mass balance approach [44], a multicompartment model [45] and an integration of these two [46]. This approach requires only in vitro acquired knowledge of
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drug permeability through the gut membrane and diffusion properties without conducting any clinical studies. However, the complexity of the absorption process introduces many uncertainties in linking permeability and the absorption rate. Based on this knowledge, the relevant question for the first step of drug distribution can be rephrased as: given the answer to the second question, what is the probability distribution of a set of compounds being absorbed at a specific rate? At present, to the best of our knowledge, there is surprisingly only one maximum entropy method developed to estimate drug absorption rates based on the knowledge of the drug disposition kinetics and plasma concentration profiles obtained from subjects who receive drugs intravenously and orally [47]. Although this method was designed to analyze plasma concentration data and lacks a predictive power because of the need for clinical data, it still sheds important light on the development of a mechanistic model that is more appropriate than a mass-balance theory-based approach commonly used now. The crux in Charter and Gull’s approach [47] is inductive inference based on the Bayes theorem. The question asked is: what is the probability distribution of a drug being absorbed given experimental plasma concentration profile data and mechanistic processes such as diffusion relevant to the absorption process? Charter and Gull [47] argued that utilizing inductive inference provides a correct mechanism to tackle uncertainty in the absorption process and noise in the data. Furthermore, because the method incorporates mechanistic kinetics relevant to the absorption process, there is no need for additional preprocessing of experimental data for analysis and an estimate of the absorption rate. Although Charter and Gull’s method [47] was designed to analyze experimental data to estimate the absorption rate, we argue that one can apply the same approach to develop probabilistic kinetics and replace traditional deterministic kinetics. The method introduced in Usansky and Sinko [46] could provide a way to integrate probabilistic kinetics into multicompartment models. Therefore, it could lead to a more appropriate description of complex absorption processes.
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Prediction of the response to multidrug combinations. Regarding pharmacodynamics, the relevant question is: given the answer to the third question about a preferred set of compounds, what is the probability distribution of specific combinations of compounds regulating disease state to a specific state? As mentioned above, the multiple-drug– multiple-target paradigm becomes a new trend in modern drug discovery. However, statistical mechanics has clearly demonstrated that many-body problems are intractable without introducing appropriate approximations because of the involvement of complicated many-body interactions. When one attempts to investigate synergism and antagonism of a mixture of multiple drugs through drug–drug and drug–target interactions, it is inevitable to face the same difficulty as in statistical mechanics descriptions of many-body systems. Using empirical knowledge, Chou [48] proposed a generic approximation method to study synergism and antagonism for mixtures of multiple drugs. The idea is to approximate complicated many-body interactions by single-drug effects and pair-wise interactions. Furthermore, mechanistic equations such as the mass–action law and the equilibrium law are fundamental relations used to describe dose–response relations. Alternatively, Wood et al. [49] utilized an entropy scheme to develop a mechanismindependent method for predicting effects of mixtures of multiple drugs.
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In the work of Wood et al., the crux of the matter in terms of entropic inference scheme is to ask the question: what is the joint probability distribution of combinations of drug pairs that generate observed responses? Again, according to the maximum entropy principle and cell growth data, the expectation value of the growth rate g i iPpair r ri when cells are treated by a known compound Ci and the expectation
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value of the growth rate g ij i j Ppair r ri r j when cells are treated by known
Ppair r
i ri J ji ri ri i
i
Z
[Equation 15]
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compounds Ci and Cj simultaneously, with the preferred probability distribution for a pair of drugs Lpair, lead to the expected cell growth rate in Equation 15:
Because compounds and targets are known in this case, PT DI T and PC TDI C are 1.
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Namely, Ppair r is equivalent to P L pair CDTI Cell growth or
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PL pair CDT Lˆ L pair CDTI Cell growth , which represents the probability distribution of an optimal pair of drugs given information regarding compounds C, disease D and cell growth data I Cell growth g i , g ij . One can then estimate the expected response for multiple drugs through this probability distribution and determine an optimal pair of drugs. Because Largrangian multipliers i and J ij are determined through the use of cell
growth data g i and g ij , respectively, there is no need for any mechanistic relations and
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this avoids intractable problems due to many-body interactions. However, the main shortcoming of this method is that these parameters do not lend themselves to any obvious physiological interpretation. Yet this method still provides information for inferring relationships between the response and pair-wise interactions. Although the method was originally proposed to investigate the effects of multidrug combinations in bacterial growth, the entropic framework is independent of biological systems. Therefore, the same framework can be applied to systems other than bacteria. Concluding remarks We have presented a promising direction based on the entropic inference scheme paving a way toward a unified approach to the computational drug discovery. As we have addressed in each subject dealt with in this paper, the key to solving the problems emerging in the framework of entropic inference scheme is to ask the relevant questions first. Subsequently, one needs to seek information relevant to the problem. Equation 9, PL CTDI C I T I KD , provides the mathematical foundation required to integrate all relevant information to answer the questions posed. The principle of maximum entropy leads to the most honest inference one can make to solve the problem. Although the topics in drug discovery discussed here were originally tackled using different aspects of entropy, we demonstrated that they are merely applications of the entropic inference scheme. We can conclude that the principle of maximum entropy is the central concept
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that governs the choice of the right drug targets, the structures of proper regulators and appropriate combinations thereof that lead to optimal pharmacokinetics and pharmacodynamics. We can expect to see more applications of this entropic inference scheme in the future. For example, one can apply a similar strategy used in aptamer design to develop a fragment-based approach to the design of small molecules (or peptides) to bind to specific targets based on information such as that relevant to interactions between chemical fragments (functional groups) and targets and the ways of linking these fragments into stable chemical molecules or molecular complexes. In pharmacodynamics, the entropic inference scheme will definitely provide a least biased approach to analyze and model in vitro cell response studies, in which one sometimes cannot obtain sufficient data owing to biological or experimental limitations. In particular, one can build the most preferred model given insufficient cellular response data. We hope this entropic inference scheme can attract more attention in the future and lead to the establishment of a complete unified approach to drug discovery.
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Acknowledgments
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C.Y.T. is grateful for many valuable discussions regarding pharmacokinetics and pharcomcodynamics with Dr Y.K. Tam. J.A.T. acknowledges funding for this research from NSERC (Canada). Author contributions
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Conflicts of interest
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The authors contributed equally to the presented mathematical and computational framework and the writing of the paper
The authors declare no conflict of interest.
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References
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47. Charter, M.K. and Gull, S.F. (1991) Maximum entropy and drug absorption. J. Pharmacokinet. Biopharm. 19, 497–520 48. Chou, T-C. (2006) Theoretical basis, experimental design, and computerized simulation of synergism and antagonism in drug combination studies. Pharmacol. Rev. 58, 621–681 49. Wood, K. et al. (2012) Mechanism-independent method for predicting response to multidrug combinations in bacteria. Proc. Natl. Acad. Sci. U. S. A. 109, 12254–12259
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Figure 1. [s13]Information flow based on entropic inductive inference for drug discovery. Blue circles represent three products: T for target candidates, C for library of compounds and L for lead compounds. Red box represents information required to determine the three products. Blue box represents methods used to determine the products. Red arrows denote direction of information feeding and blue arrows simply indicate actions. The mathematical relation of PT DI T , PC TDI C and PL CTDI KD P L CTDI C I T I KD is given in Equation 9.
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Author biographies
Chih-Yuan Tseng Dr C-Y. Tseng is currently working at MDT Canada (http://www.mdtcanada.ca/), which he co-founded. He is also a research scientist at Sinoveda Canada. He received his PhD in physics from State University of New York at Albany. At MDT Canada, he focuses on the foundation and applications of entropic inference in biomedical sciences. Particularly, his research interests include information physics, protein folding dynamics, biological signal analysis, aptamer design, drug discovery methods in cancer, pharmacokinetic and pharmacodynamic modeling and simulation. At Sinoveda Canada, he focuses on the development of theoretical and computational modeling for pharmacokinetics and pharmacodynmaics.
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Jack Tuszynski Dr Jack Tuszynski is a Fellow of the National Institute for Nanotechnology of Canada. He is an Allard Chair and Professor in the Department of Oncology at the University of Alberta and a Professor in the Department of Physics. He received his MS with a distinction in physics from the University of Poznan (Poland). He received his PhD in physics from the University of Calgary. He has published over 360 peer-reviewed papers, over 50 conference proceedings, ten book chapters and ten books. Jack Tuszynski heads a multidisciplinary team creating ‘designer drugs’ for cancer chemotherapy using a computational biophysics methods.
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S1359-6446(15)00275-5 http://dx.doi.org/doi:10.1016/j.drudis.2015.07.004 DRUDIS 1649
To appear in: Received date: Revised date: Accepted date:
17-10-2014 13-6-2015 9-7-2015
Please cite this article as: Tseng, C.-Y., Tuszynski, J.,A unified approach to computational drug discovery, Drug Discovery Today (2015), http://dx.doi.org/10.1016/j.drudis.2015.07.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights: Current strategies for drug discovery are reviewed We propose that drugs can be better discovered through an entropic inductive inference-based unified approach Examples from different stages in drug discovery are presented to demonstrate the concept and use of the proposed unified approach
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A unified approach to computational drug discovery Chih-Yuan Tseng1,3,* and Jack Tuszynski1,2 1
Department of Oncology, University of Alberta Edmonton, AB T6G 1Z2, Canada Department of Physics, University of Alberta Edmonton, AB T6G 1Z2, Canada 3 MDT Canada, Edmonton, AB, Canada *
Corresponding author: Tseng, C.Y. ([email protected]).
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Keywords: Maximum entropy; inductive inference; drug discovery; target identification; compound design; pharmacokinetics; pharmacodynamics.
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Teaser: Entropic inductive inference is the foundation of an information-based unified approach to computational drug discovery.
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It has been reported that a slowdown in the development of new medical therapies is affecting clinical outcomes. The FDA has thus initiated the Critical Path Initiative project investigating better approaches. We review the current strategies in drug discovery and focus on the advantages of the maximum entropy method being introduced in this area. The maximum entropy principle is derived from statistical thermodynamics and has been demonstrated to be an inductive inference tool. We propose a unified method to drug discovery that hinges on robust information processing using entropic inductive inference. Increasingly, applications of maximum entropy in drug discovery employ this unified approach and demonstrate the usefulness of the concept in the area of pharmaceutical sciences.
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PACS codes: 02.50.Tt; 87.10.Vg[s1] Drug discovery applies multidisciplinary approaches to either experimentally or computationally (or a combination of both) identify lead compounds with therapeutic applications to various diseases. In this context, one faces an issue that information required to ‘discover’ drugs is always insufficient because of the inherent complexity of biological systems. Recently, it has been reported that a slowdown in the development of new medical therapies is adversely affecting patient outcomes [1–3]. The FDA has thus initiated a long-term project: the Critical Path Initiative, to assist researchers in their investigations aimed at a design of better approaches to drug discovery [4]. In this review, we provide an overview of the drug discovery process and argue that this field will significantly benefit from a unified approach based on the maximum entropy principle. Among numerous methods applied in drug discovery, we believe that the maximum entropy method could provide an appropriate approach because it is designed to tackle problems for which information available is inherently insufficient [5]. However, as we elaborate on later, it is surprising that only a limited number of applications of entropybased methods to various topics in drug discovery can be found in the literature. One of the reasons might be that the development of the maximum entropy method as a tool for inductive inference was conducted over a roughly similar time span when computational rational drug discovery was being developed. Researchers paid more attention to the foundation of the maximum entropy method and its applications in fields such as astronomy, physics and engineering rather than in the life sciences. We believe that the time has now come to usher the maximum entropy methods into the drug discovery area. The progress of the method of maximum entropy in terms of inductive inference has enabled clarification on unified approaches to computational drug discovery. Here, we discuss not only how the maximum entropy principle is applied to solving problems of significance in drug discovery but also how it is an attempt to provide an alternative but a unified approach when compared with standard methods. The foundation of this unified approach is what we later refer to as the entropic inference scheme. It is our hope that the illustration presented here reveals advantages of this scheme and the way of utilizing it to improve and accelerate drug discovery.
Conventional strategies in rational drug discovery The central idea of the function of a drug is to inhibit (or sometimes activate) functions of biological targets (proteins, enzymes or DNA) in the human body to prevent, stall or cure diseases that are related to abnormal activities of such biological targets. The conventional wisdom employed to ‘discover’ such a drug includes two main categories: compound design and investigation of its drug-like properties. Compound design includes two subjects: identification of biological targets and design of small molecules to bind to active sites of biological targets. The drug-like properties category includes two subjects: screening of these molecules to determine candidates that satisfy specific properties to act as a drug and demonstration of the desired pharmacokinetics and pharmacodynamics of these agents. Additional filtering of a set of candidate compounds can[s2] involve the prediction of potential side-effects of the drugs and their metabolites. Compound design
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Target identification. The first stage in drug discovery is to identify biological targets that are responsible for causing abnormal physiological activities in biological systems. In the past, the complexity of biological systems has limited the concept of drug design to the so-called single-drug–single-target paradigm. However, there is increasing evidence that this paradigm fails to yield better drugs with satisfactory effects expected to treat various diseases [1–3]. The recent blossoming of systems biology further demonstrates that this paradigm is inadequate because of the enormous complexity of signaling, regulatory and/or metabolic networks involved in the emergence and stability of disease states [6–9]. In other words, there can be multiple targets in the networks involved in cellular state regulation. Therefore these studies further suggest the existence of a better paradigm the central idea of which is rooted in the multiple-target–multiple-drug principle [6,7]. Systems-biology-based approaches, therefore, were devised to assess which targets are likely to be key factors to inhibit abnormal regulation caused by diseases. The method proposed by Yang et al. [7] is one specific example illustrating this point. Yang et al. [7] proposed to perturb the network and optimize it toward the desired state using a Monte-Carlo-simulated annealing optimization algorithm. By comparing diseased and normal network states in simulations, one can assess which targets are most likely to be responsible for the disease states as potential drug targets.
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Determination of target structures. Once key biological targets have been identified, one needs to determine their tertiary (3D) structures to design small molecules that can specifically and selectively bind to active sites located on them in order to regulate their functions. Several wet-lab techniques including NMR, electron- and X-raycrystallography have been the primary tools for this purpose. If target structures cannot be determined through NMR, electron- or X-ray- crystallography techniques one can still apply the computational homology modeling method to construct a reliable approximation to the tertiary structure of the target molecule provided a primary sequence is known and a reasonably similar template structure exists [10]. The homology modeling method involves applications of several bioinformatic tools including protein sequence analysis and structure analysis (it consists of comparison, construction, equilibration and validation). The reader is referred to ref. [10] for more details. Drug design strategy. Screening chemical libraries is a common approach to discover drug candidates. The size and structural diversity of the library of chemical compounds used in the discovery process are key factors to the successful identification of potential lead compounds that can bind specifically to molecular targets. However, the chemical space covering organic molecules is astronomically large [11]. It is highly unlikely that one can efficiently discover lead compounds through screening compound libraries without introducing very efficient searching algorithms that cover a diverse and large part of the chemical space. In the past three decades, virtual screening approaches based on integration of various computational methods have been developed to address this issue [12,13]. Basically, these approaches consist of three ingredients: (i) the creation of a chemical library; (ii) search for an optimal ligand–receptor-binding mode through docking algorithms; and (iii) evaluation of the binding affinity of each ligand examined with the receptor chosen as the target. Accordingly, three criteria are required to identify compound candidates successfully. First, the chemical library needs to contain large
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numbers of diverse chemical structures. Second, conformational search algorithms need to be able to screen all major binding modes within reasonable compute time. Third, an appropriate scoring function needs to be utilized to evaluate the binding preference of specific compounds correctly. AutoDock software is one of the methods commonly used in virtual screening approaches. It utilizes a grid-space method for rapid binding energy calculations, a Lamarckian genetic algorithm for increasing efficiency and accuracy in conformational searches and a semi-empirical free energy force field to evaluate the binding free energy more precisely [14]. In spite of the advances made in the screening techniques and energy calculation methods, virtual screening is still facing the issue of small sizes and insufficient structural diversity of the chemical compound libraries used in these endeavors. Consequently, fragment-based approaches have been proposed to provide an alternative. Ludi is one example of computer-aided design [15] whereas SAR by NMR is an example of in-vitro-based design [16]. In silico and in vitro methods are rooted in a similar concept. It starts with identifying molecular fragments that can bind to potential pockets in molecular targets to then link those fragments with appropriate linkage fragments to form a single molecule that has the highest binding affinity. This approach dramatically reduces the conformational space and increases the structural variety of molecules. However, combinatorial explosion in terms of the generation of a huge number of resultant compounds from a limited number of fragments is the bottleneck of this type of design.
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Drug likeness The compound design strategies discussed above do not guarantee that compounds discovered are drug candidates. In order for a compound to be considered a valid drug candidate, it needs to possess drug-like properties. Several drug-like properties that are normally introduced in the drug discovery process as screening criteria are discussed below.
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Lipinski’s Rule of Five. In 1997, Lipinski et al. [17] reported an intensive review of the physicochemical criteria for orally administrated active compounds that have poor solubility and permeability. They found that when a compound has a molecular weight (MWT) larger than 500 Da, LogP greater than 5, more than five H-bond donors or more than ten H-bond acceptors then the compound is highly likely to exhibit poor absorption. Namely, the compound is unlikely to be absorbed into the blood circulation system and reach active sites. The above listed criteria are known as Lipinski’s Rule of Five, which has become a gold standard and a first filtering step in screening medicinal chemistry compounds. Applying these criteria as filters for libraries of potential medicinal chemistry compounds helps to screen out compounds designed for oral administration that have a higher chance of being absorbed. Pharmacokinetics: ADMET. Even though medicinal chemistry compounds might possess good absorption properties, this does not guarantee that such compounds can be considered as drugs. After absorption, as a drug, a molecule must be able to reach active sites that are believed to be the sources causing the disease in question and bind to target proteins to regulate their functions with the highest efficacy. To achieve a desired efficacy, the amount of the drug entering active sites will need to be maintained at a
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sufficiently high level after this drug has undergone various types of biotransformations such as intestinal and liver metabolism. In the meantime, the toxicity caused by off-target interactions needs to be reduced to a minimum. Although there are no quantitative criteria defined based on these pharmacokinetic properties, this provides a qualitative guideline to increase the probability that the screened compounds can achieve desired pharmacokinetic properties. With the advance of chemoinformatics and bioinformatics, the ADMET Predictor software from Simulation Plus [18] is a good example of software that uses computational (artificial intelligence) methods to predict ADMET of a given set of compounds.
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Pharmacodynamics. The final criterion we address here is the pharmacodynamics of the tested compounds. This describes the response of the body to the drug and the mechanisms of drug action. As is well known, the goal of designing a drug is to produce desired physiological responses in the body given that a specific amount of the drug has been administered over a particular amount of time. Therefore, the knowledge of the pharmacodynamics of compounds certainly provides crucial information for selecting promising compounds as drugs. Unfortunately, one cannot directly conduct in vivo experiments without identifying lead compounds. One solution is to utilize in-vitro–invivo correlation (IVIVC) prediction, if such a correlation exists to predict in vivo effects based on prior in vitro studies [19,20]. Searching a better way to establish IVIVC has become one of the most important subjects of study in the pharmaceutical industry to improve on drug design and optimize drug formulation.
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Entropy-based inductive inference Two reasoning methods have been utilized in developing theories to interpret phenomena we observe in nature and to make predictions about them. They have been applied to methodologies developed in drug discovery without exception. The first is deduction. It allows us to draw conclusions when sufficient information is available. The second method is variously called inductive logic, inductive inference, probable inference or just inference. It allows us to reason when the available information is insufficient for deduction. It is called either ‘inference’ (for the cases that we make estimates of quantities when we do not know enough to deduce their values) or ‘induction’ (when we generalize from special cases) [21]. When dealing with complicated biological systems, which involve many-body interactions at a microscopic level, complicated regulating protein–protein networks at a mesoscopic level or genetic populations at a macroscopic level, it is common to find that we do not have sufficient knowledge to understand these kinds of systems in detail. We normally rely on inductive inference to deduce the most preferable solutions to the problems related to these systems based on available information. Before we direct our attention back to drug discovery, we discuss the mathematical tools required for inductive inference. The central idea involved in these tools hinges on the Bayesian interpretation of probability and the rules of probability theory. The former treats probability as a measure of our state of knowledge or degree of belief about the system of interest rather than a frequency of occurrence of an event. Cox demonstrated that this type of probability can be manipulated by the rules of subtraction, multiplication and addition given by the standard probability theory [22,23].
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The state of knowledge. Based on the Bayesian interpretation of probability, Laplace’s principle of insufficient reasoning and Cox’s axiomatic approach, Jaynes first showed that the method of maximum entropy (denoted here by MaxEnt) is a reasoning tool to assign probabilities on the basis of limited information about the systems of interest with the least bias [24,25]. The preferred probability distribution of the system in a specific state is the one that maximizes the entropy of the system subject to the constraint of a limited amount of information available. Note that entropy is defined by the well-known formula in Equation 1:
S p pi ln pi
cr
[Equation 1]
i
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where pi is the probability of the system in state i and the sum is over all states the system can adopt. Here, the information about the system available is in the form of constraints as described by Equation 2[s3]:
an
O piOi , i
[Equation 2]
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which represents the expectation value of the observed quantity Oi (e.g. it can be the binding energy of compounds or the expression level of target proteins). The probability distribution pi represents our state of knowledge regarding the system in state i. In summary, Jaynes’ pioneering work [s4]established a mathematical framework for determining the most preferable inference, namely the probability one can estimate according to the information about the system in hand. It is worth stressing that the meaning of entropy is beyond a physical quantity for measuring the degree of randomness or information. It is a reasoning tool as well.
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Maximum-entropy-based inference scheme. The next breakthrough was to establish a tool beyond MaxEnt for updating our state of knowledge regarding the system when new information is acquired. Shore and Johnson, Skilling[s5] and Caticha all followed the same reasoning as Jaynes and proposed that the method of maximum entropy (denoted here by ME) can also be utilized as a tool for updating probability distributions based on newly acquired information [26–29]. They showed that the key to updating probability distributions is through relative entropy rather than the entropy used in MaxEnt (Equation 3):
S p, m pi ln pi i
mi
[Equation 3]
where mi a prior is a probability distribution regarding the system in state i before any new information is acquired. Caticha further demonstrated that ME unifies the two methods, MaxEnt and Bayesian methods, into a single inference scheme [30]. In summary, based on this scheme, inductive inference is made by first resolving the following two issues. First, what is a prior and hence what information is relevant to the
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system of interest. Second, what is the relevant application of the method of maximum entropy. Because ME allows arbitrary prior and constraints, one can expect that this general scheme can be applied to an arbitrary set of problems as long as information acquired to investigate these problems is in the form of constraints.
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General inference scheme. As discussed above, the most probable inference made by ME is the one that maximizes relative entropy (Equation 3). However, the question remains: to what extent are less favorable inferences ruled out by the maximum entropy principle. Information geometry (IG) offers a straightforward tool to answer this question [31]. IG introduces the concept of manifolds for characterizing and manipulating information [32,33]. Probability distributions then can be simply treated as points on the information manifold, which is defined based on the Fisher–Rao metric [32]. Therefore, one can quantify preferences for all probability distributions, inferences, through a distribution of corresponding points on the information manifold [31]. Utilizing ME and IG together then provides a general scheme to manipulate information and make inductive inference on the manifold [31]. Specifically, one will not only make the most favorable inference but one also has the knowledge regarding the preference of less favorable inferences. In summary, the general inference scheme or the entropic inference scheme is a reasoning framework to manipulate the state of knowledge and make the most plausible inference for the system under investigation. Under this scheme, the principle of maximum entropy fundamentally defines the way information is to be processed. MaxEnt, ME and IG are appropriate mathematical tools to manipulate information.
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An entropic unified approach to computational drug discovery
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Logic and implementation In this section we discuss the logic behind the proposed unified approach to drug discovery and the way of implementing this approach in real pharmaceutical applications. The next section will then introduce three examples to illustrate it.
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Logic. As has been shown previously, we need to introduce specific methods and tools to solve problems at different stages in drug discovery. Although these methods lead to acceptable results, this still poses a debatable issue. Namely, because all kinds of information required are interconnected, the question remains as to whether we should tackle each stage of drug discovery independently or simultaneously. In either way, we also need to face a problem: because characteristics of each stage are different how are we going to appropriately utilize different types of information to determine the most preferable solution? This suggests a necessity for unified approaches to improve the discovery process. Yet, owing to the complexity involved in drug discovery, it seems impossible to develop unified approaches to integrate fundamental principles at different stages in drug discovery. At the present time, systems biology provides the best concept and strategy toward establishment of a foundation for a unified approach [8,9]. However, as Berg pointed out even with the advances made in systems biology, several issues still hinder the drug discovery process [9]. We argue that systems biology like other areas impacting
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drug discovery merely provides additional information at the systems level. Systems biology still does not directly tackle drug discovery process. By recognizing the complexity inherent in physiological systems from the cellular to the whole body level, the overview of different stages of drug discovery above suggests that a fundamental principle governing drug discovery hinges on information processing. Specifically, the success of drug discovery depends on the way of utilizing all kinds of information from subcellular compound–target interactions to cellular regulation mechanisms to the organ- and body-level pharmacokinetic properties of a compound to make the most plausible inference about drug candidates. From the information theory point of view, this way is independent of either goodness of the virtual screening approach, completeness of regulation networks or correctness of pharmacokinetics prediction. This way involves logic and the principles of objectively manipulating the available information. The entropic inference scheme is exactly designed to provide a unique and robust recipe for developing a unified approach for processing all kinds of information involved in drug discovery and to suggest the most preferable drug candidates as long as information can be formulated in terms of constraint equations of the type shown in Equation 2. This development requires two ingredients: (i) asking the relevant question and (ii) collecting relevant information. Given these two ingredients, the maximum entropy principle will be the fundamental principle guiding the discovery process.
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Relevant questions. The goal is to turn deterministic sound problems into probabilistic ones by asking questions relevant to the problem of interest. Those relevant questions are the cornerstone of a unified approach. The overview of drug discovery described above suggests that the most relevant and general question one should ask is: what is the probability distribution of a mixture of compounds that regulates a disease state of a biological entity and transforms into another specific state? Rather than asking which targets, compounds and their combinations should be selected with the objective of achieving a maximum potency. As was discussed in previous sections, this regulation involves a series of key properties including target identification, compound design, sufficiency of drug-like properties and optimal pharmacokinetic and pharmacodynamic behavior. Namely, this probability distribution is likely to be a joint probability distribution that codifies information regarding these properties. Determining the preferred joint probability distribution suggests a preferred mixture of compounds. Based on the concept of conditional probability, we can further ask a series of sub-questions specifically intended to answer this general question properly. The first sub-question to be asked is regarding target identification: what is the probability distribution of specific biological targets such as proteins being at specific abnormal or disease states? The second sub-question to be asked is regarding compound design: given the answer to the first question, what is the probability distribution of a set of compounds interacting with those biological targets? The third sub-question to be asked is also related to compound design. It involves the sufficiency of drug-like properties and optimal pharmacokinetics: given the answer to the second question, what is the probability distribution of a set of compounds being delivered into active sites that contain those biological targets? Finally, the fourth sub-question is regarding pharmacodynamics: given the answer to the third question, what is the probability
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distribution of specific combinations of compounds regulating disease state to a specific state?
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Relevant information. The key to answering the relevant questions hinges on collecting relevant information and utilizing the right principles to process the information. In the framework of the entropic inference scheme, the principle of maximum entropy is the only fundamental principle that leads to unique information processing with the least bias from subjective influences. Therefore, under the entropic inference scheme it shifts endeavors in discovering drugs from developing better theories and methods to collecting information relevant to answering those questions systematically. Unfortunately, seeking out relevant information still relies on empirical knowledge and a method of trial and error. Success of the unified approach to drug discovery will heavily depend on the ways to overcome this information collection bottleneck.
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Mathematical formulation Cox has shown that the state of knowledge can be manipulated by the same rules that are used in probability theory [22,23]. Based on these rules and supposing the answers to questions raised above can be found, we can derive mathematical relations among them. These relations are illustrated in Figure 1 and this also represents a fundamental information flow required to make the most plausible inference to discover drug candidates. For the sake of simplicity in derivation: we define D as a specific disease; T as possible targets; C as medicinal compounds; L as lead compounds; IT as information required to identify targets; IC as information required to determine compounds; and IKD as information from pharmacokinetic and pharmacodynamic studies. Our ultimate goal is to determine the probability distribution of a mixture of lead compounds L that regulates a disease state of a biological entity to another specific state PL CTDI C I T I KD , given information of compounds C that can bind to targets T responsible for disease D having desired pharmacokinetic and pharmacodynamic properties IKD. According to an associativity constraint, if a state of knowledge can be calculated in two different ways these two must agree, the joint [s6]probability distribution of having lead compounds L, all possible compounds C, potential targets T, disease of interests D and all information IC, IT and IKD can be written in two ways (Equations 4,5): [s7] PLCTDI C I T I KD P LCTDI C I T I KD PI KD PI KD LCTDI C I T PLCTDI C I T [Eqn 4] The left-hand term of the second equality is the product of the probability distribution of having information IKD, PI KD and the conditional probability distribution of having L,
C, T and D given IKD, PLCTDI C I T I KD . The second equality gives:
PI KD LCTDI C I T PLCTDI C I T I KD PLCTDI C I T PI KD PLCTDI C IT LLCTDI C I T I KD
[Equation 5]
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where LLCTDI C I T I KD PI KD LCTDI C I T PI KD is the likelihood function. Because (Equation 6): PLCTDI C I T I KD P L CTDI C I T I KD PCTDI C I T I KD
[Equation 6]
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we have Equation 7:
LLCTDI C I T I KD [Equation 7] PCTDI C I T I KD We apply associativity again to the joint probability PLCTDI C I T and obtain Equation 8:
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PLCTDI C I T PL CTD PC TDI C P T DIT PD
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PL CTDI C I T I KD PLCTDI C I T
[Equation 8]
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Therefore, substituting Equation 8 into Equation 7 gives Equation 9:
PL CTDI C I T I KD PL CTD PC TDI C PT DI T PD
LLCTDI C I T I KD PCTDI C I T I KD
PL CTD PC TDI C PT DI T PD
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PL CTD PC TDI C PT DI T PD LˆLCTDI C I T I KD
[Eqn 9]
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This equation reiterates our reasoning to discover efficacious drugs. It indicates determination of the conditional probability distribution of a mixture of lead compounds L that regulates disease state of a biological entity to a specific state required. We have, first, PT DI T , the conditional probability distribution of targets that are responsible for
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disease D and satisfy information IT, second, PC TDI C , conditional probability distribution of compounds that bind to targets given T and D and information Ic, and, third, PL CTD , the prior probability distribution of lead compounds that regulate disease state to specific state given C, T and D before acquiring information IKD. Furthermore, the term: LˆLCTDI C I T I KD LLCTDI C I T I KD PCTDI C I T I KD is the overall likelihood function to be determined through experimental data and empirical knowledge. In summary, Equation 9 represents the mathematical foundation of the unified approach to drug discovery. Given Equation 9, the remaining tasks are to determine each conditional probability distribution using the maximum entropy principle and the appropriate likelihood function according to the information acquired. Case study examples Although the method of maximum entropy has been introduced to drug discovery as either a tool or a reasoning tool for developing methods to solve problems of relevance, those applications did not completely exhaust the potential beyond its original usage. In the following, we go through different stages in drug discovery, which have been tackled
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through an entropy-based method, in terms of an entropic inference scheme to illustrate the original use and demonstrate that those applications are merely parts of the proposed unified approach. Specifically, we discuss three important topics: (i) target identification, PT DI T ; (ii) compound design, PC TDI C ; and (iii) pharmacokinetics and
pharmacodynamics, PL CTDI C I T I KD .
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Target identification As mentioned above, the relevant question regarding target identification is: what is the probability distribution of specific biological targets such as proteins or enzymes being at a specific abnormal or disease state PT DI T ? To answer it, we need to acquire information, IT, relevant to characteristics of the disease state. Several aspects of information, as shown below, have been studied that can be considered relevant in this connection. For example, Yang et al. defined the activity of a target based on the reaction rate to characterize the corresponding regulation network [7]. Fuhrman et al. considered the temporal gene expression patterns of targets, which are either genes that underwent a disease process or a normal phenotypic change [34]. Tseng et al. utilized compound– target binding energy, which are shown to correlate target protein expression level changes when cells are treated by chemotherapeutic compounds [35]. In their work, Fuhrman et al. merely utilized the Shannon entropy (Equation 1) to quantify information codified in p i , a probability distribution of target gene expression being at a specific level [30]. The entropy value S p i pi log pi indicates variation
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in a temporal series of gene expression data that describe a certain biological process. For genes with no variation in the expression level, a zero entropy value was assigned and hence they are considered to contain no information. Namely, these genes are not responsible for triggering the processes in question. Therefore, candidate drugs for these targets will correspond to maximum entropy. Unfortunately, because their goal was to introduce entropy as a measuring rather than an inductive inference tool, these authors focused on defining probability distribution p i through gene expression data in hand and empirical knowledge regarding target activity and interpretation of gene expression. Therefore, the determination of target candidates is subjected to the quality of data and bias inherent in this empirical knowledge. By contrast, the work of Tseng et al. [35] provides an example that demonstrates that we can go beyond the use of entropy as a measuring tool as in the work of Fuhrman et al. [34], [s8]even though these two examples tackle target identification from different angles. In terms of the entropic inference scheme, the goal is to determine the most preferable probability distribution of targets being at specific expression levels based on the information available. Therefore, we can then infer potential target candidates through this distribution. In the work of Tseng et al. [35] [s9]information that relates cytotoxicity and the binding energy of compounds was analyzed. This can be formulated in terms of a constraint equation (Equation 10): M
RTe log IC50 Pi Gi i 1
[Equation 10]
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where refers to compounds, R is the gas constant, Te is the absolute temperature, IC50 is the drug concentration that inhibits 50% of proliferating cancer cells and Gi is the
cr
e Gi Z
[Equation 11]
where the partition function Z i 1 e M
Gi
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Pi mi
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binding energy between target i and compound By maximizing relative entropy (Equation 3), subject to this constraint equation, they determined the preferred probability distribution of target i having an expression level after it was treated by compound to be that given by Equation 11:
and the Largrangian multiplier are
determined from Equation 10. Suppose we consider priors mi to be a probability
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distribution of the target protein expression level before any treatments involving chemotherapy compounds. Therefore, given information regarding the disease D=DCancer and IT =IBindingenergy, candidates for targets selected for optimal treatments can be determined by P T DCancer , I Binding energy Pi .
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Compound design Next, we aim to answer the following question: given the answer to the first question, what is the probability distribution of a set of compounds interacting with those biological targets? Here, we only focus on two common methods, virtual screening and a fragment-based approach, for illustration.
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Virtual screening. Under the entropic inference scheme, the first (chemical library) and third (scoring function) criteria mentioned above[s10] merely represent the fundamental information required in virtual screening approaches. The second criterion, searching for optimal binding modes of compounds, then can be treated as an information processing guideline. The efficiency and accuracy of this step will depend on the methods of information processing. The genetic algorithm is one such example. The algorithm borrows the concept of a genome evolution process to search conformations of complexes involving targets and chemical compounds. Chang et al. [36] have shown a better alternative, namely MEDock. Although MEDock does not completely follow the entropic inference scheme for conformational searches, it simply utilizes the maximum entropy principle as a decision-making guideline in search processes. Similarly to the question we pose here, the fundamental question asked in MEDock is: what is the probability of finding the deepest energy valley in the ligand–target interaction energy landscape? Maximum entropy suggests the preferable direction to update the initial guess of a binding mode (described by an almost
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uniform distribution) to an optimal mode (a localized distribution around the global energy minimum).
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Fragment-based drug design. The second example for compound design involves utilizing the entropic inference scheme to design aptamers. Aptamers are short nucleic acid sequences typically identified through an experimental technique: systematic evolution of ligands by exponential enrichment (SELEX) [37,38]. Because of their strong and specific binding through molecular recognition, aptamers are promising tools in molecular biology and have therapeutic and diagnostic clinical applications [37–40]. Unfortunately, some limitations of the SELEX technique have seriously slowed down the progress of discovering aptamers [41]. With the help of entropic inductive inference, a fragment-based approach has been developed to design aptamers given the structure of the target of interest [41]. The concept behind the fragment-based approach is to ask the question: given the structural information about the target, what is the preferred probability distribution of having an aptamer that is most likely to interact with the target? The solution involves acquisition of two types of information: first, regarding the relevant information for determining the choice of the first nucleotide, the so-called seed; and second, given the preferred first nucleotide, the relevant information for determining the next nucleotide. The approach first determines the preferred probability distribution of the first nucleotide Pnt1 Qi , q1 j
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P Q , q H Q , q
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H Total Qi , q1 j
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that probably interacts with the target. Because the information that is relevant to the interaction between the target and the first nucleotide is the total free energy of this complex, it is reasonable to consider the expected total energy of the target denoted by ‘tar’ and the first nucleotide denoted by ‘nt1’ (Equation 12): 1
nt1
i
j
1
Total
i
j
[Equation 12]
1
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Qi , q j
as preliminary information in hand, where the Hamiltonian of the target–first-nucleotide complex is H Total Qi , q1j H tar Qi H nt1 q1 j H tar - nt1 Qi , q1j H solvent and
Q R , P stands for i 1,..., N , i.e. the atomic spatial coordinates and momenta of the N -atom target and q r , p stands for the j 1,..., N atomic spatial coordinates and momenta of the first N -atom nucleotide. Furthermore, H Q is the total internal energy of the target ‘tar’, H q is the total internal energy of the first Q , q denotes the interaction energy of the target and the first nucleotide ‘nt1’, H i
i
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Q
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Q
1 j
j
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nt1
j
nt1
tar
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nt1
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tar - nt1
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nucleotide and H solvent represents the solvation energy. Afterward, the approach keeps updating the probability distribution when more nucleotides are added to the previous one. Again, the total energy of the target and all
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nucleotides (the previous one and the added one) provides a reasonable choice for new information relevant to the interactions between targets and additional nucleotides. For example, the information acquired in the second step is the expected total energy of the target and two-mer nucleotides (Equation 13):
P Q , q , q H Q , q , q 1
nt2
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2 j
1
Total
i
2 j
j
[Equation 13]
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H Total Qi , q1 j , q2j
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Qi ,q j
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where q2j r2j , p2j stands for the j 1,..., N nt 2 atomic spatial coordinates and
momenta of the second N nt 2 -atom nucleotide. Maximizing the relative entropy (Equation
S Pnt2 Pnt1
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Pnt2 Qi , q1 j , q2j log
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14):
Pnt2 Qi , q1j , q2j
2
Pnt1 Qi , q1 j
[Equation 14]
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Qi , q j ,q j
we then determine the probability distribution of two-mer nucleotides interacting with the target, Pnt2 Qi , q1 j , q2j , which is updated from Pnt1 Qi , q1 j . Continually using Equation 3, the maximum entropy allows us to determine to what extent this update is sufficient and a sequence of nucleotides of maximum length L with probability distribution P Qi , q1...j L that is likely to bind to the target can be determined. This probability
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distribution is equivalent to PC TDI C where C q1...j L the spatial coordinates of the L-
mer sequence, T Qi the spatial coordinates of the target responsible for the disease D
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and I C total energy as information required for compound design. The method has been applied to the design of aptamers which has been computationally and experimentally validated to bind specifically to targets such as thrombin [41], phosphatidylserine [42] and galectin-3 (under experimental confirmation). Pharmacokinetics and pharmacodynamics Drug absorption rate estimation. Regarding pharmacokinetics, the relevant question is: given the answer to the second question about compounds, what is the probability distribution of a set of compounds being delivered into active sites that contain those biological targets? Here, we only focus on the very first step of distributions of orally administrated drugs: absorption, for illustration purposes. The absorption of drugs involves several processes including dissolution, gastric and intestinal release and plasma disposition. Several methods have been developed for predicting drug absorption. One traditional way is to utilize a statistical approach to deduce models from training sets for absorption prediction [43]. Their success relies on the selection of an appropriate training set and statistical approaches. Another approach is based on a mechanistic model such as the well-known mass balance approach [44], a multicompartment model [45] and an integration of these two [46]. This approach requires only in vitro acquired knowledge of
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drug permeability through the gut membrane and diffusion properties without conducting any clinical studies. However, the complexity of the absorption process introduces many uncertainties in linking permeability and the absorption rate. Based on this knowledge, the relevant question for the first step of drug distribution can be rephrased as: given the answer to the second question, what is the probability distribution of a set of compounds being absorbed at a specific rate? At present, to the best of our knowledge, there is surprisingly only one maximum entropy method developed to estimate drug absorption rates based on the knowledge of the drug disposition kinetics and plasma concentration profiles obtained from subjects who receive drugs intravenously and orally [47]. Although this method was designed to analyze plasma concentration data and lacks a predictive power because of the need for clinical data, it still sheds important light on the development of a mechanistic model that is more appropriate than a mass-balance theory-based approach commonly used now. The crux in Charter and Gull’s approach [47] is inductive inference based on the Bayes theorem. The question asked is: what is the probability distribution of a drug being absorbed given experimental plasma concentration profile data and mechanistic processes such as diffusion relevant to the absorption process? Charter and Gull [47] argued that utilizing inductive inference provides a correct mechanism to tackle uncertainty in the absorption process and noise in the data. Furthermore, because the method incorporates mechanistic kinetics relevant to the absorption process, there is no need for additional preprocessing of experimental data for analysis and an estimate of the absorption rate. Although Charter and Gull’s method [47] was designed to analyze experimental data to estimate the absorption rate, we argue that one can apply the same approach to develop probabilistic kinetics and replace traditional deterministic kinetics. The method introduced in Usansky and Sinko [46] could provide a way to integrate probabilistic kinetics into multicompartment models. Therefore, it could lead to a more appropriate description of complex absorption processes.
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Prediction of the response to multidrug combinations. Regarding pharmacodynamics, the relevant question is: given the answer to the third question about a preferred set of compounds, what is the probability distribution of specific combinations of compounds regulating disease state to a specific state? As mentioned above, the multiple-drug– multiple-target paradigm becomes a new trend in modern drug discovery. However, statistical mechanics has clearly demonstrated that many-body problems are intractable without introducing appropriate approximations because of the involvement of complicated many-body interactions. When one attempts to investigate synergism and antagonism of a mixture of multiple drugs through drug–drug and drug–target interactions, it is inevitable to face the same difficulty as in statistical mechanics descriptions of many-body systems. Using empirical knowledge, Chou [48] proposed a generic approximation method to study synergism and antagonism for mixtures of multiple drugs. The idea is to approximate complicated many-body interactions by single-drug effects and pair-wise interactions. Furthermore, mechanistic equations such as the mass–action law and the equilibrium law are fundamental relations used to describe dose–response relations. Alternatively, Wood et al. [49] utilized an entropy scheme to develop a mechanismindependent method for predicting effects of mixtures of multiple drugs.
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In the work of Wood et al., the crux of the matter in terms of entropic inference scheme is to ask the question: what is the joint probability distribution of combinations of drug pairs that generate observed responses? Again, according to the maximum entropy principle and cell growth data, the expectation value of the growth rate g i iPpair r ri when cells are treated by a known compound Ci and the expectation
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value of the growth rate g ij i j Ppair r ri r j when cells are treated by known
Ppair r
i ri J ji ri ri i
i
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[Equation 15]
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compounds Ci and Cj simultaneously, with the preferred probability distribution for a pair of drugs Lpair, lead to the expected cell growth rate in Equation 15:
Because compounds and targets are known in this case, PT DI T and PC TDI C are 1.
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Namely, Ppair r is equivalent to P L pair CDTI Cell growth or
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PL pair CDT Lˆ L pair CDTI Cell growth , which represents the probability distribution of an optimal pair of drugs given information regarding compounds C, disease D and cell growth data I Cell growth g i , g ij . One can then estimate the expected response for multiple drugs through this probability distribution and determine an optimal pair of drugs. Because Largrangian multipliers i and J ij are determined through the use of cell
growth data g i and g ij , respectively, there is no need for any mechanistic relations and
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this avoids intractable problems due to many-body interactions. However, the main shortcoming of this method is that these parameters do not lend themselves to any obvious physiological interpretation. Yet this method still provides information for inferring relationships between the response and pair-wise interactions. Although the method was originally proposed to investigate the effects of multidrug combinations in bacterial growth, the entropic framework is independent of biological systems. Therefore, the same framework can be applied to systems other than bacteria. Concluding remarks We have presented a promising direction based on the entropic inference scheme paving a way toward a unified approach to the computational drug discovery. As we have addressed in each subject dealt with in this paper, the key to solving the problems emerging in the framework of entropic inference scheme is to ask the relevant questions first. Subsequently, one needs to seek information relevant to the problem. Equation 9, PL CTDI C I T I KD , provides the mathematical foundation required to integrate all relevant information to answer the questions posed. The principle of maximum entropy leads to the most honest inference one can make to solve the problem. Although the topics in drug discovery discussed here were originally tackled using different aspects of entropy, we demonstrated that they are merely applications of the entropic inference scheme. We can conclude that the principle of maximum entropy is the central concept
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that governs the choice of the right drug targets, the structures of proper regulators and appropriate combinations thereof that lead to optimal pharmacokinetics and pharmacodynamics. We can expect to see more applications of this entropic inference scheme in the future. For example, one can apply a similar strategy used in aptamer design to develop a fragment-based approach to the design of small molecules (or peptides) to bind to specific targets based on information such as that relevant to interactions between chemical fragments (functional groups) and targets and the ways of linking these fragments into stable chemical molecules or molecular complexes. In pharmacodynamics, the entropic inference scheme will definitely provide a least biased approach to analyze and model in vitro cell response studies, in which one sometimes cannot obtain sufficient data owing to biological or experimental limitations. In particular, one can build the most preferred model given insufficient cellular response data. We hope this entropic inference scheme can attract more attention in the future and lead to the establishment of a complete unified approach to drug discovery.
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Acknowledgments
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C.Y.T. is grateful for many valuable discussions regarding pharmacokinetics and pharcomcodynamics with Dr Y.K. Tam. J.A.T. acknowledges funding for this research from NSERC (Canada). Author contributions
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Conflicts of interest
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The authors contributed equally to the presented mathematical and computational framework and the writing of the paper
The authors declare no conflict of interest.
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References
1. Sams-Dodd, F. (2005) Target-based drug discovery: is something wrong? Drug Discov. Today 10, 139–147 2. Szuromi, P. et al. (2004) Rethinking drug discovery. Science 303, 1795 3. Food and Drug Administration (2004) Innovation and Stagnation: Challenge and Opportunity on the Critical Path to New Medical Products. Available at: http://www.fda.gov/downloads/ScienceResearch/SpecialTopics/CriticalPathInitiative/ CriticalPAthOpportunitiesReports/ucm113411.pdf 4. Food and Drug Administration (2010) Critical Path Initiative: Report on Projects Receiving Critical Path Support. Available at: http://www.fda.gov/downloads/ScienceResearch/SpecialTopics/CriticalPathInitiative/ UCM249262.pdf 5. Tseng, C-Y. and Tuszynski, J.A. (2014) Maximum entropy in drug discovery. Entropy 16, 3754–3768
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Figure 1. [s13]Information flow based on entropic inductive inference for drug discovery. Blue circles represent three products: T for target candidates, C for library of compounds and L for lead compounds. Red box represents information required to determine the three products. Blue box represents methods used to determine the products. Red arrows denote direction of information feeding and blue arrows simply indicate actions. The mathematical relation of PT DI T , PC TDI C and PL CTDI KD P L CTDI C I T I KD is given in Equation 9.
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Author biographies
Chih-Yuan Tseng Dr C-Y. Tseng is currently working at MDT Canada (http://www.mdtcanada.ca/), which he co-founded. He is also a research scientist at Sinoveda Canada. He received his PhD in physics from State University of New York at Albany. At MDT Canada, he focuses on the foundation and applications of entropic inference in biomedical sciences. Particularly, his research interests include information physics, protein folding dynamics, biological signal analysis, aptamer design, drug discovery methods in cancer, pharmacokinetic and pharmacodynamic modeling and simulation. At Sinoveda Canada, he focuses on the development of theoretical and computational modeling for pharmacokinetics and pharmacodynmaics.
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Jack Tuszynski Dr Jack Tuszynski is a Fellow of the National Institute for Nanotechnology of Canada. He is an Allard Chair and Professor in the Department of Oncology at the University of Alberta and a Professor in the Department of Physics. He received his MS with a distinction in physics from the University of Poznan (Poland). He received his PhD in physics from the University of Calgary. He has published over 360 peer-reviewed papers, over 50 conference proceedings, ten book chapters and ten books. Jack Tuszynski heads a multidisciplinary team creating ‘designer drugs’ for cancer chemotherapy using a computational biophysics methods.
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Figure 1
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