An Introduction to Systems Biology and Quantitative Systems Pharmacology

3.20 An Introduction to Systems Biology and Quantitative Systems Pharmacology BC Gomes, Pharmacometrics Modeling and Simulation, Novartis, Cambridge, MA, United States Ó 2017 Elsevier Ltd. All rights reserved.

3.20.1 Introduction 3.20.2 Definitions 3.20.2.1 Systems Biology 3.20.2.2 Simulation Versus Prediction 3.20.2.3 Multiscale Modeling 3.20.2.4 Metabolic Control Analysis 3.20.2.5 Pathway Modeling 3.20.2.6 Integrative Biology 3.20.2.7 Physiologically Based Pharmacokinetics 3.20.2.8 Quantitative Systems Pharmacology 3.20.3 Mathematical Formalisms 3.20.3.1 Boolean Modeling 3.20.3.2 Petri Nets 3.20.3.3 Agent-Based Modeling 3.20.3.4 Bayesian Network Models 3.20.3.5 ODE and PDE Models 3.20.4 A Detailed Discussion of a Systems Biology and Quantitative Systems Model 3.20.5 Summary References Relevant Websites

3.20.1

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Introduction

At the heart of modern scientific progress is a reductionist belief that biology can be understood by close examination of the individual components of the whole. This idea was first suggested by Rene Descartes (1596–1650). “He formulated the notion that complex situations can be analyzed by reducing them to manageable pieces, examining each in turn, and reassembling the whole from the behavior of the pieces.”1 By contrast, systems biology seeks to comprehensively understand the time-dependent changes in a network of interactions in order to describe the integration of all the molecular or cellular components. As such, systems biology lies at the confluence point of three streamsdbiology, computer simulation, and high-throughput screening. As a tool of pharmaceutical drug discovery, the goal of systems biology is to find the most efficacious point(s) of intervention to change the pathophysiological state of the biological process (the system) in order to return it to a more healthy condition. From a practical point of view, in academic and pharmaceutical industry labs, assays are usually created to test the effect of a single property at a time. For example, the concentration of a cytokine is varied and the amount of binding to its receptor is followed. After some calculation, this yields a binding affinity. A new candidate antagonist of that receptor can be tested to determine the strength of inhibition of that cytokine/receptor interaction, and hence provides a simple measure of the potency of a possible therapeutic. If a misguided investigator were to merely dose the same compound to an individual with a disease associated with this cytokine/receptor pair, and to ask whether the potential drug was effective in ameliorating the disease, he would find his questionable ethics and flawed experimental design justifiably criticized. We all “know” that successful drug discovery requires many steps (compound screening, chemical and/or biological refinement of candidates, animal models, safety assessment, dosing evaluation, etc.). And yet, in the modern pharmaceutical industry, at each step of this discovery and development process, different groups of individuals, with different focused skill sets, attempt to provide clear analysis by reducing the complexity of the system to a simple, human-comprehendible and variable-limited output. To be fair, this form of methodological reductionism2 is usually the result of a sensible approach to the limitations of the human mind. Most people would not be able to understand assays that tested multiple parameters simultaneously. Perception and visualization of trends of multidimensional data can be difficult. However (properly programmed) computers have no difficulty processing high-dimensional data. The availability of sufficient computing power and specialized modeling software is one of the influences that gave rise to systems biology. In fact, the notion of systems biology arose from the discipline of systems analysis and control theory, where self-monitoring circuits adapt outputs to varied inputs through amplification and feedback loops. The other major driver of the growth of systems biology was an increasing appreciation that a deeper comprehension of systems of metabolic pathways, genomic control, cellular signaling, etc., was available when biological processes were observed at larger scales than single, isolated reactions.

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This is the result of the concept of “emergent properties.” In the flocking of birds (or the schooling of fish, the coordinated migration of locust, and other complex systems), behavior of the flock is not the property of one leader, the master bird. Instead, all the birds move as if there is a linkage between individuals that supersedes an individual’s actions. This linkage is an emergent property of the system. The demonstration that the sum is greater than the parts is the “holy grail” of systems biology. This same type of behavior is recapitulated at many levels in humans and other organisms. Each cell in a multicellular organism is constantly receiving myriad cues from neighboring cells within an organ and more distant organs through signaling molecules, direct contact with circulating sensor cells (e.g., T and B cells), or enervations from nerve cells. The cell processes this information and integrates it with its own current state, metabolic demands and specialized functions, and responds with alterations in transcriptional output and activation of regulatory networks within the cell and, in turn, communicates with the rest of the organism by initiating release of its own signaling molecules. No single cell is the “master switch” for any process in a human, and yet complex, emergent behaviors are observed in the regulation of metabolism, the growth or decline in subpopulations of cells, the maintenance of a beating heart, our immune response to invading infectious agents, and even truly astonishing supracellular things like thought. But the discovery of emergent properties goes beyond the purely observational when it can be practically applied, as it now is, in drug discovery. Understanding the origins of an emergent property, such as a feedback loop in a signaling pathway can lead to a drug that targets a pathway with greater precision than one directed at a single component enzyme. Historically, one can always point to earlier and earlier harbingers and innovative people that led to the eventual creation of systems biology. It is the nature science to build on the work of others. Alan Hodgkin and Andrew Huxley are often cited as being instrumental in the founding of systems biology, though this discipline was not given that name at the time. Their success in creating a mathematical model of the generation and propagation of nerve impulses gave rise to other new attempts to apply mathematical principals to the description of fundamental physiological processes. By modeling the ion (Kþ and Naþ) conductances from a giant nerve fiber (see Fig. 1), they were able to construct a model that well mimicked the previously described, all-or-none action

Fig. 1 (A) Hodgkin and Huxley represented the plasma membrane as a number of parallel circuitsdone each for the movement of sodium, potassium, and other ions. In the equation shown, the G terms stand for conductances of the ions, E is the resting potential of the membrane, and ENa, EK, and EI are the electrical potentials of each ionic circuit. The terms n and m represent the activation of potassium and sodium ion transport, respectively, while h stands for the inhibition of sodium transport. The standard methods of circuit analysis generated the equation shown. (B) The simple model of the conductances allowed for the comparison between the simulated (upper curve) and the measured biological response (lower curve). The model correctly predicted the magnitude and wave form of the propagation of an action potential, as well as the refractory period that occurs after the firing of the axon.

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potential.3 It should be noted that this mathematical model was formulated before the details of ion channel firing were fully envisioned. This highlights the idea that useful and innovative modeling does not require knowledge of every molecular and physical event in the processdonly the important ones. Subsequently, Denis Noble built on the Hodgins–Huxley model to describe the action and pacemaker potential from the Purkinje fibers of the heart.4 There has been a continuous building on these early works, culminating in the recent mathematical description of heart failure.5 This paper describes how heart failure is a multiscale process that propagates from the modulation of sarcomere number, giving rise to alteration in muscle cell architecture. At the organ level, these changes then affect ventricular wall thickening and electrical properties of the heart. Another formative force for systems biology was the rapid evolution of molecular biology. In the 1960s, molecular biology exploded with the basic understanding of the structure of genetic controls. The most thorough of these was the description of the lac operon.6 It was realized, almost immediately, that these genetic processes were akin to the monitor-adapt-output cycles of mathematically approachable electronic circuitry. Labs began to construct reporter gene systems, genetic toggle switches, and simple memory, etc.7 Hence, they had created biological analogues of many of the components of circuits, which are amenable to modeling.8 By modularizing these control processes, they made it evident that biology is driven by recurring themes and, in turn, made in silico modeling of these processes tractable.

3.20.2

Definitions

Recently, systems biology has fragmented into many sub disciplinesdsystems immunology,9,10 systems genetics,11 systems oncology,12,13 systems pharmacology,14 etc. Even these more focused systems approaches vary widely in methodology. The following is a nonexhaustive explanation of some of the terms and forms of systems biology.

3.20.2.1

Systems Biology

Systems biology is somewhat difficult to define because different people use the term to mean different things. The term “systems biology” was first used in 1968 with a fairly strict mathematical and logic description,15 but has evolved to encompass virtually any mathematical approach to elucidate biology. All these treatments share an in silico computational component and all are directed at some level of biology, but the methods and focus of this work varies greatly. Systems biology attempts to provide a quantitative and comprehensive view of some biological phenomenon by the use of a collection of connected equations. It has been used to describe biological topics as diverse as predator/prey population dynamics,16 circadian rhythms,17 cell cycle regulation,18 the complement cascade,19 DNA damage repair by surveillance driven by p53,20 and many others. Systems biology seeks to find the underlying foundation for emergent properties in the biology under study. Mathematical formalisms will be discussed below, but all methods create a mapda topological organization of all the relevant components. Secondly, they have a linkage among elements that describes a time-dependent strength of interaction. In most cases, this means that rate constants are applied that determine the velocity of conversion of one species into another. Finally, the model ascribes initial concentrations for all species at the start of the simulation. The systems can be complex or simple, abstract or described in great detail, and often are nonlinear. They are dynamical (at each point in time there is a vector with a specified value for each species in the system). The concentration of the species and the relative contribution to the overall output of the model will mostly be changing throughout the simulation. To complete a molecular-level systems model, it is helpful to understand the full list of component parts (proteins, peptides, metabolites, etc.), as well as their concentrations. It is then a natural progression that systems biology has surged in usage with the completion of the human genome project (and the genome of species that are used in animal models) as well as advances in proteomics (e.g., mass spectrometric methods). Once all the pieces are elucidated, it becomes easier to create the network connections that make up a systems biology model. Thus, the definition of systems biology given by Leroy Hood, one of the leading voices in the human genome project as well as systems approaches, is most relevantd“Systems biology studies biological systems by systematically perturbing them (biologically, genetically, or chemically); monitoring the gene, protein, and informational pathway responses; integrating these data; and ultimately, formulating mathematical models that describe the structure of the system and its response to individual perturbations.”21 In the drug discovery area, it is a reasonable question to ask, “What does a pharmaceutical scientist gain by having a systems model of the healthy and diseased states that cannot be found by mass screening of drug candidates versus protein binding assays?” Firstly, it unlikely that binding or inhibition assays for all proteins that underlie the disease could be established in a timely manner (even if all the responsible proteins were known). Secondly, this kind of reductionism would discard all the interaction information in the crosstalk interrelationships of the network. The idea of “one disease–one target” is likely wrong for all but a few, enzyme deficiency disorders. Also, though it is an area of systems modeling that is in its infancy, systems biology does not necessarily uncover a single point of intervention, but often finds that synergistic effects are found by combinations of drugs.22 A good example of the increased understanding of combination therapies provided by systems modeling is the work of Bown et al.23 Using the previously described model of the PI3K/PTEN/AKT and RAF/MEK/ERK signaling pathways they simulated the development of druginduced resistance to a first anticancer agent, followed by treatment with a second drug to correct for the resistance. Another example of modeling combination therapies is in the Nitric Oxide–cGMP pathway.24 The authors investigated intervention at various points in the pathway as well as in combination. They give a conclusion that optimal therapy, when measured as maximizing cGMP accumulation, would be observed by intervention at three points in the pathway. This was confirmed by experimentation. A good mathematical discussion of how to evaluate synergy of combined treatment has been done.25

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3.20.2.2

491

Simulation Versus Prediction

Some models seek to mimic the observed outputs (objective function) of known biology. The previously mentioned mathematical treatment by Hodgkin–Huxley simulates nerve conduction in an axon. All models start with this kind of approach because it allows the modeler to construct (most of) the critical components and links in a way that is verifiable from known data. However, particularly valuable models go beyond simulation to further predict undocumented behavior of the system. Experimental confirmation of nonobvious predictions builds the confidence in the model. These predictive models contribute to scientific progress by allowing the formation of new hypotheses for further understanding of the underlying biology. In the pharmaceutical setting, when models make strong, testable predictions, drug discovery is accelerated.

3.20.2.3

Multiscale Modeling

To some extent, most systems approaches use known information about component parts to find more holistic descriptions of biological processes. Thus, these models cross scales of structure or extend the scale of time predictions. For example, individual kinetic parameters and species concentrations are linked together to produce a simulation of a signaling network. The emergent properties of this linked system might give a picture of the dynamic relationships that occur at a cellular level in terms of signal and response. In turn, the summation of many specialized cells or organelles, for example, Islet cells in the pancreas,26 mitochondria in the heart,27 or endothelial cells in intestine28 might exhibit an organism-level response of growth, correct cardiac rhythm, or ionic homeostasis in the blood. Does this understanding of multiscale processes lead to better drug discovery? To quote from a paper by Sanofi scientist, “. aiming for the discovery of novel targets increasing the glucose-dependent insulin secretion, we have established a mathematical model of the pancreatic beta cell. The ‘virtual pancreatic beta-cell’ has been generated by capturing all relevant processes in this cell type by combining three published models describing elements in the control of glucose-triggered insulin secretion in the beta-cell (GPCR/cAMP signaling, b-cell electrophysiology and calcium handling, insulin granule exocytosis). The resulting model contains numerous enzymes, transporters and ion-channels involved in glucose stimulated insulin secretion. The ‘virtual pancreatic b-cell’ is currently used to predict the best drug targets or target combinations for increasing the glucosedependent insulin secretion.”29

3.20.2.4

Metabolic Control Analysis

This is an area of modeling that can see its direct antecedents in the mathematical analysis of single enzyme reactions. This area was pioneered by Westerhoff.30 It is mostly used for looking at metabolic pathways. Earlier models had focused on synthesis and degradation pathways in which it was suggested that there was a single rate-limiting enzymatic conversion step, which was then studied in detail to understand regulation of the whole pathway. Metabolic Control Analysis seeks to understand the fluxes of metabolites through the whole collection of enzymes, in order to give a more refined view of the enzymatic ensemble. This technique is used for finding inhibitors that exploit metabolic differences as seen in disease-causing microorganisms or cancer cells.31

3.20.2.5

Pathway Modeling

One particularly well-worked area of systems biology is pathway modeling. This approach seeks to build mathematical models of cell-signaling networks.32 The modeling requires setting the concentration and rates of each enzyme in the pathway, as well as the strength of any feedback and feedforward loops. This usually requires a certain amount of parameter optimization, that is, fitting of unknown (or poorly characterized) parameters. This is done by fitting a range of parameter values to a known, experimentally derived, output graph. For example, a fluorescent assay might measure the production of an intermediate of the pathway. The concentration or rate of an enzyme with unknown values is determined by varying the rate constant or protein concentration until a reasonable fit to the data is obtained. This value is then entered into the pathway. In this way, the MAPK33,34 pathway and its connection to EGFR35,36 signaling have been modeled extensively. The topology of the pathway had been previously determined and most of the phosphorylations and dephosphorylations had been parameterized and feedback loops had been mapped. This has led to insights about control of the pathway, which in turn, has led to better drug design37 for this pathway, which is inappropriately regulated in breast cancer and other neoplasms. Many pathways have been modeled in this way, including the Wnt,38 JAK/STAT,39 PI3 kinase,40 NF-kB,41 Notch,42 TGF-b,43,44 and multiple others.

3.20.2.6

Integrative Biology

Integrative biology is every bit as difficult to define as systems biology. It stresses cross-discipline studies to unify a view of biological phenomenon into a cohesive whole.45 Integrative biology emphasizes biology as information, thus bringing insights from genomic, physics, chemistry, etc., to provide a clearer view of the area of study. Many papers in integrative biology highlight taxonomy and comparative biology. Others stress a hybridization of bioinformatics and -omics that rely heavily on clustering of data and finding statistical means to understand informational content that underlies the organizational connectedness of interacting proteins, genes, etc. It has been suggested that systems biology is a subset of integrative biology since systems biology also gives priority to combining computational and biological disciplines.46 Systems approaches face a challenge from “big data.” The system under

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study in a model, might be large, but does not contain influences from all other proteins and isoforms. Therefore, at some point integrative biology demarcates the most influential players in a system and allows systems modelers to draw operational boundaries for their studies. Quickly interrogating huge data files systematically, and incorporating the relevant pieces of them into a systems model has not been achieved (yet).

3.20.2.7

Physiologically Based Pharmacokinetics

PBPK is a specialized systems approach to drug distribution.47 Each organ in the body is essentially treated as a compartment with physiology-based pharmacokinetics (PBPK) is a specialized systems approach to drug distribution specified input and output of flows, a volume and (sometimes) a measure of drug metabolism, and binding capacity.48 These models can be quite useful to track the amount of a dosed drug that reaches the tissue in which the pathophysiology is located. They can also aid in understanding the half-life of a pharmaceutical and hence the required dosing regimen to be efficacious. Mostly, PBPK models have been used for molecules of low-molecular weight, though as more information becomes available on the physiological properties of different classes of macromolecules, they are also being studied in PBPK models.49 In general, macromolecules are dosed intravenously, so absorption from the gut does not need to be addressed in the model. Absorption of orally delivered, small molecule drugs from the gut can be a very complex process,50 and is idiosyncratic for each drug, but in combination with experimentally derived data, predictions of drug disposition can be helpful. While little information at the molecular level is usually present in these models, they can also have utility in informing more mechanistic models. An example of a physiologically informed simulation, with a significant degree of molecular detail is a model of calcium homeostasis and bone remodeling.51 In this model, the authors mathematically explain the delicate balance of calcium intake from the gut, kidney reabsorption/excretion of phosphate and calcium mediated by multiple endocrine mechanisms, and the constant bone loss and growth driven by the osteoclasts and osteoblasts. This model allows hypothesis testing in an area of intense pharmaceutical industry interest, due to competition for safe and effective osteoporosis drugs.

3.20.2.8

Quantitative Systems Pharmacology

Quantitative systems pharmacology (QSP; also called systems pharmacology, systems clinical pharmacology, and mechanistic physiological modeling, pharmacologically-orientated systems biology) is an enhancement to the more standard PKPD (pharmacokinetics pharmacodynamics) simulation. PKPD modeling is the traditional modeling found in the pharmaceutical industry and academia. PKPD links the dose of a drug to an observed time-dependent appearance and clearance of the drug (PK) and correlates this with a desired physiological effect (PD) or the appearance of an undesirable side effect (toxicology).52 While PKPD models are very valuable for aiding in dose prediction, cross species dose calculation, and drug safety assessment, they only rarely are based on understanding the basic biological phenomenon that they describe. To be fair, understanding basic biology is not the goal of PKPD models. Rather, they are mathematical constructs to give vital information that is necessary in registration of drugs with regulatory authorities. The models are descriptive and empirical. But to more firmly attach the PKPD framework to the physical characteristics of the pharmaceutical, it was a natural evolution that empirical PKPD models began to add some kinetic and mechanistic details to produce, what is known by the title, semimechanistic PKPD models. At early times in this migration to semimechanistic models, these simulations were standard PKPD models with details about drug/target binding added.53 However, some of these models have become much more expansive (e.g., see references 54–57). To continue the evolution of this area, the idea of adding “some” mechanistic detail soon gave rise to adding sufficient information to build systems models.58 However, systems models have now entered the regulatory arena.59 The FDA advisory board recently used a QSP model to evaluate the dosage for a recombinant human parathyroid hormone, for the long-term treatment of hypoparathyroidism. The FDA committee suggested that, based on the model, dosing needed to be further refined to avoid the safety concern of hypocalciuria. This might be a harbinger of greater QSP modeling involvement in FDA oversight of pharmaceutical safety and efficacy. At this point, it would seem that systems biology and QSP are overlapping in goals and methodology. However, true to its name, QSP, remains focused on drug discovery, while systems biology describes a broader study of any biology, pharmaceutically related or not. Also QSP has found a niche in translational medicine,60,61 the part of the drug discovery process that moves the new pharmaceutical agent from its research foundations in binding assays and early animal models of efficacy and preliminary toxicology onto the threshold of clinical trials, where human dose prediction and dosing regimen become paramount, that is, the “bench-to-bedside” bridge. One of the most ambitious and enlightening models was an immune response model62 (Fig. 2). This model simulates the reaction to infection. It shows reactions at the protein level (e.g., cytokines) as well as at the cellular level. It predicts the kinetic changes in T and B cell subsets, macrophages, dendritic cells, and the clearance of antigens. This model, when fully validated, will allow for evaluation of drugs that alter the immune response. Systems pharmacology is many times linked to pathways models. This is because many of the same signaling pathways are reoccurring themes in many disease states.63 There is an underlying belief that models from one disease can be “recycled” for use in other pathology. The problem with this assumption is that the same pathway is used differently in different cells or that the concentration of reactants varies in different tissues. The network of crosstalk or influence from other pathways is different throughout the organism. This is a source of on target toxicology and is hard to predict. Systems pharmacology has been suggested to be the future deliverer of personalized medicine.64 In theory, a general model of a disease process could be parameterized with information from each patient. This hypothetical model would then choose a drug

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Fig. 2 A fully integrated immune response model. (Adapted from Palsson, S.; Hickling, T. P.; Bradshaw-Pierce, E. L.; Zager, M.; Jooss, K.; O’Brien, P. J.; Spilker, M. E.; Palsson, B. O.; Vicini, P. The Development of a Fully-Integrated Immune Response Model (FIRM) Simulator of the Immune Response Through Integration of Multiple Subset Models. BMC Syst. Biol. 2013, 7, 95.) The model seeks to show the immune response (in this case the effects on cellular reactions and cytokine production) to tumors, bacterial infections, etc. Abbreviations: AB, antibody; B, naïve B-cells; BA, activated B-cells; BM, memory B-cells; BP, plasma B-cells; DEBRIS, tumor cell debris; IDC, immature dendritic cells; MA, activated macrophages; MAPC, antigen-presenting macrophages; MDC, mature dendritic cells; MI, infected macrophages; MR, resting macrophages; PE, extracellular bacteria; PI, intracellular bacteria; T, naïve T-cells; TC, cytotoxic T-cells; TCP, cytotoxic precursor T-cells; TH2, T-helper 2 cells; THP, helper precursor T-calls; TH1, T-helper 1 cells; Treg, regulatory T-cells.; TUMOR, tumor mass. In panel A is part of the topology for the immune response in the lung/tumor. Panel B represents the model simulation of the time course of subsets of T cells after an intracellular bacterial infection. The model actually computes time courses for all the cells, cytokines, and antigens represented in the model.

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(or mixture of drugs), dose, and dosing regimen for that person. But this is (at present) wishful thinking. QSP has considered the heterogeneity in drug action (epigenetic factors, environmental influences, proteomic variability, the individual’s reaction to the physicochemical properties of the drug, etc.)65 and it seems that without a massive computational approach, this will not be possible. However, this is a very complex system made of contributions from many disciplines. After all, even the prediction of an individual’s toxicology is still not possible.66 However, in science, today’s impossible is tomorrow’s attainable.

3.20.3

Mathematical Formalisms

As discussed above, systems biology seems dissectible into subdisciplines, based on the specific areas of biology addressed (systems toxicology, systems ecology, systems neuroscience, systems medicine, etc.). It can also be partitioned by the form of mathematics used to solve the complex network (rules-based modeling, process algebras, lattice-based cellular automata, Boolean analysis, ordinary differential equation-based models, etc.). The following is a brief description of some of the techniques used. The list is not exhaustive and the explanations are somewhat oversimplified. Additionally, many hybrid approaches exist. There is no single method of modeling that works under all conditions. Rather, the optimal choice of modeling algorithm is dependent on what questions need to be answered and what starting information is available. In the real world, modelers must also consider the audience for their workdsome modeling methods are easier to visualize and explain to others with less modeling background. All modeling formalisms have limitations, strengths, and assumptions, and modelers must choose to balance these and other considerations.

3.20.3.1

Boolean Modeling

Boolean modeling67,68 is, in some ways, the simplest modeling method that is applied to some of the most complex systems. It has been realized that the idea of discrete pathway models is an abstraction that does not match the cellular reality. For example, an MAP kinase pathway is easy to draw, with each enzyme having distinct, digitizable input and output kinetics, reactants, and products. But, at the next level of complexity, this pathway is modulated by signaling contact with other pathways (which are, in turn also regulated). Where do we draw the boundary between pathways in the network? Can this very complex system be parameterized? Many methods, simply practical or bounded by assumptions, try to address this conundrum. One approach is Boolean modeling,69 which simplifies the problem by limiting the number of states that each component in the system can have. The modeling usually proceeds through two stages (see Fig. 3A). The first stage establishes the architecture of system. This is usually completed by information from the literature or pathway data banks. The second step is to train the system from data. Some connections are strengthened while others are eliminated due to low contribution to the measured output. Each molecular entity (DNA, RNA, enzyme, etc.) in the system is set to active or inactive (in some forms of Boolean modeling they have three statesdhigh, medium, or low). It does this by determining the state of the components based on the sum of positive and negative (stimulatory and inhibitory) inputs at each node of the model, at a given time point. No intermediate states are allowed, only on or off. The model is then advanced one time unit and the states of nodes are reassessed. This leads to a time course for each node, for the length of the simulation. Because of the limited number of states for each node in the system, Boolean modeling is considered qualitative or semiquantitative. This type of reduced-complexity model is very useful for finding global steady states (called attractors) of the network, which can be, for instance, the homeostatic state or a disease state. It is obvious that ascribing only a binary status to each species does not mimic the observed behaviors of cells, tissues, or organisms. To address this limitation, a number of approaches have extended Boolean modeling, such as probabilistic approaches,70 and the use of fuzzy logic.71,72 Boolean modeling has found application in yeast budding,73 oncogenic pathway regulation,74,75 and T cell differentiation.76 In this last publication, the authors were able to demonstrate that a minimal transcriptional control network composed of the proteins GATA-3, T-BET, Bcl6, FoxP3, and RORyt could explain the cell fate attainment and plasticity of CD4þ T cells including Th0-, Th1-, Th2-, Th17-, Tfh-, Th9-, iTreg-, and Foxp3independent T regulatory cells. The interconversion of some of these cell types and the lack of interconversion of others would be extraordinarily difficult to model with other types of mathematical formalisms.

3.20.3.2

Petri Nets 77

Petri nets are graphical and computational representations of biological networks that, in their simplest form, represent species (formally called places) and reactions (formally called transitions) that can model multiple, concurrent processes (see Fig. 3B). Arcs (directed arrows) represent potential interactions and have specified weights.78 This gives a visually intuitive picture that is akin to metabolic pathway maps, but with a time-dependent flow. Tokens, added to the model, represent molecules or concentrations. Thus, a given reaction (transition) fires if the concentration (number of tokens) reaches a threshold level and if there is sufficient capacity in the receiving specie. Also, molecule de novo synthesis (gain in token number) and decay (loss of token number) can be represented as transitions without input or output arcs, respectively.79 Using petri nets, it is possible to get a dynamic picture of the system as it responds to stressors or modulators. This method has been used to model signal transduction induced by the proinflammatory cytokines, IL-1b and TNF-a80 and the hypoxia response driven by HIF-1a.81 In this last publication, the authors attempt to unravel the difficult hypoxia tolerance system that is activated in tumor. Petri nets were used to show the core regulatory net for this system. More importantly, they were able to identify the key node that allows for the switch-like behavior of the system.

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Fig. 3 (A) A simplified Boolean model. In the first step the model structure identifies entities, relationships, and connections. This structure is then trained with the biological data. For example, some stimulus is supplied to the system. The measured experimental time courses for the components (circles) are measured. Connections that allow the observed path outputs are given stronger weights and those that do not contribute are eliminated. (B) A simple Petri net for the binding of a cytokine and subsequent phosphorylation of the receptor. Circles represent species and boxes represent reactions. The dark circle within the circles represents tokens, which are interpreted as units of that species (number of molecules or concentrations).

3.20.3.3

Agent-Based Modeling82

In an agent-based model, a grid is created where each box (or cube in three-dimensional models) represents an independent entity. To make this easier to understand, let us assume that this entity is a cell. Within this cell a series of rules are established that create something like a decision tree. The rules can be complex or simple (in fact, in hybrid models the rules can be a mixture of discrete states and continuous variables). All the individual agent cells within the array are given a set of initial conditions and then apply their rules. The individual cell then “decides,” based on a preordained, triggering threshold to either wait or create some output (e.g., growth or secretion of substance that is a proliferative stimulus). Within the lattice, each cell only interacts with a limited set of surrounding cells, that is, the output of a cell is not distributed to all agents in the grid, but rather only to those in the limited contact environment of the single cell. The neighboring semiautonomous cells respond to that “secreted stimuli” by applying their own set of (identical) rules. Each agent is integrating the response of their neighbors and reacting by making a state decision. Agent-based models are particularly appealing in oncology because tumor cells expand or undergo apoptosis as a response to cues from their microenvironments (oxygen levels, nutrient availability, proinflammatory cytokines, etc.) and their own internal rule set based on the somatic mutations that cause malignant aberrations (chromosomal rearrangements to place growth signals under the control of active promoters, loss of contact inhibition receptors, loss of apoptotic pathways, upregulation of secreted proteases to allow tissue infiltration, etc.). In this way, the overexpressed EGFR signaling pathway was used as a decision rule set to study neoplasm growth in nonsmall cell lung cancer83 and brain glioblastoma.84 Cell death, proliferation, and migration in cancer were modeled by an agent-based algorithm,85 and the oncogenic cellular responses to DNA damaging compounds were also studied.86 In this last publication, the authors attempt to understand the processes that underlie how tumor cells respond to chemotherapeutics and radiation therapy. At the level of single cells, the processes of repair of damaging agents are somewhat understood, and yet the tumor acts as if there are emergent properties that are not solely under the control of these mechanisms. Part of this finding is due to the cells of the tumor being asynchronous. This agent-based model used probabilistic methods to predict tumor cell survival curves due

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to the population dynamics that result from the underlying DNA damage and repair processes. In other works, the study of cell cycle control in cancer, under the influence of checkpoint inhibitors, has benefited from agent-based models.87 Agent-based models have also been used to model colonies of microorganisms, such as in the establishment of antibiotic resistant biofilms,88 and to study Ebola virus epidemiology.89

3.20.3.4

Bayesian Network Models90

Like Petri Nets, Bayesian network analysis is a graph-based system. Unlike ordinary differential equation (ODE) modeling, which is deterministic, Bayesian network models are probabilistic. In other words, given the same set of initial conditions, ODE models will always give the same results. By contrast, given a data set, Bayesian models yield the likely probability of an ensemble of connections being found. Bayesian network models often start with large data sets, which are prone to noise. Bayesian network models look to uncover the statistical connections between a set of objects. In the case of signaling networks, the objects could be, for an example, phosphorylated proteins, activated enzymes, or genes.91 The connections that are uncovered can be simple two object physical interactions, for example, protein A binds to protein B, or protein A and B are in a product/precursor relationship. Alternatively, the relationship could be a complex, functional, and previously unknown relationship between many objects (e.g., protein A leads to a downregulation of protein B via a transcriptional suppression mechanism). Bayesian network models not only tolerate large data sets but require sizable collections to detect subtle relationships. The model returns the conditional probability that a particular set of connections explains the data. Therefore, Bayesian network analysis allows the modeler to rank possible network topologies. Often in signaling networks, the connections are already drawn into convenient pathway structures of seemingly fixed architecture of nodes and edges (e.g., the Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway maps92). Bayesian network analysis can test the likelihood of the connections that are drawn into this kind of map.93 This kind of Bayesian network analysis was used to assess the relative strengths of several different regulatory mechanisms for IL-12 in CD4þ T cells.94 Biomarkers for oral squamous cell carcinoma were identified by use of this technique.95 Also research based on microarrays was used in a Bayesian network to identify connections between ten receptor tyrosine kinases and two sites from Src kinase.96

3.20.3.5

ODE and PDE Models

Ordinary differential equation modeling is, by far, the most common mathematical methodology in systems biology and QSP because this approach is powerful and versatile. Ordinary differential modeling can supply a continuous and quantitative view of the dynamics of a system. In other words, the concentration of every species in the model is tracked over the time course of the simulation. If the topology and parameters are correctly specified, then the results of the model are accurate predictions. However, this is far from easy. To build ODE models, the initial concentrations of every species in the model and the rate constants of every reaction in the model must be known. For all but the simplest of models, the parameters are usually determined by experiments or derived from literature values. The values of literature-derived parameters are often difficult to use in a model. This is because the various parameters are inconsistent with respect to the choice of cells, tissues, animal species, and variations in experimental design. Later, we will discuss how modelers address some of these concerns. When a system shows spatial heterogeneity, that is, concentrations varies in both time and position, partial differential equation models are employed. For example, to model the growth and invasion of tumors, a partial differential equation model is desirable,97,98 or to understand the contribution of diffusion in cell signaling, PDE modeling is essential.99 In the context of modeling, the ordinary differential equations are simply rate expressions that reflect the formation and consumption for each species in the system. Therefore, each species in the system has a differential equation that subtracts the sum of rates that consume the species from those that form it. For example, the equilibrium interconversion of A and B: kforword

A%B kreverse

This would yield two ODE equations (one for each species). dA ¼ kreverse B  kforward A dt dB ¼ kforward A  kreverse B dt In the case of irreversible reactions, there may be no formation terms (e.g., the first-order decay of a species) or no consumption terms (e.g., the zeroth order synthesis of a species). These equations usually follow the conventions of mass action chemical reactions. However, more complex behavior such as Michaelis–Menten kinetics (hyperbolic reactions that exhibit saturation), Hill equation behavior (sigmoidal concentration dependencies and other more complex behaviors), and inhibition of the various kinds of kinetics, observed in standard enzyme assays can be simply reformulated into mass action formats,100 though one must be careful that the assumptions of the original equation are understood (e.g., steady-state assumptions or the requirement that substrate concentration greatly exceeds levels of enzymes). Mass action is not the only kind of rate equations that can be incorporated

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into models. Since ODE modeling is similar to chemical kinetics, any physical or chemical rate can be used in the model. For example, diffusion rates into tissues and tumors are routinely modeled using Fick’s law of diffusion. When the ODE formalism is applied to multistep reactions, the same methodology is employed that was shown above for a one step “system.” The summation of formation and consumption terms for particular specie may grow quite large if that specie is used as a reactant, or produced as a product, in many steps in the network, however, the procedure remains the same. When the reaction scheme is simple, as for example, the binding of a ligand to a receptor or a straightforward enzyme-catalyzed reaction, the ODE equations can be solved analytically by simple (though often tedious) algebraic manipulations. However, for most signaling networks, pathway models, or multiscale models of proteins, tissues and cells, which usually exhibit nonlinear behaviors, this kind of exact solution is not practical or possible. Under these conditions, the numerical solution of the set of ODE or PDE equations is done by discretizing the time component (and space component for PDE models). In other words, initial values for the concentrations of all the species are used to simultaneously recalculate their values at a very small time step away from time zero. The results of that calculation are then used in the next small time interval to generate the next set of concentrations, and the process is repeated. In this manner, a seemingly continuous trajectory is produced for each system species. Many ODE solvers are available to do this type of calculation. MatLab has several ODE solvers available (ode15s, ode45, etc.) and open source solvers are available on the Internet at https://sourceforge.net. As mentioned above, as the complexity of the ODE model increases, more care must be applied to parameterizing the model. Poor estimates of initial values can cascade through the model to produce spurious outputs. However, it is likely that some uncertainty will creep into most models that are large enough to have value. This uncertainty is further exacerbated if the model contains novel proteins, for which little information has been derived. One solution to the parameterization dilemma is “big data.” In theory, the measurement of the concentration of “protein X” in a wide range of settings (e.g., measuring the same protein in the same cell type in many healthy people and the same protein in the same cell type in a disease state) could produce a relevant range of values to apply to the model. Unfortunately, most big data databases are not organized in this way or use RNA expression data, which does not correlate with protein levels under many conditions.101 High-throughput proteomics by mass spectrometry is getting much closer to being able to deliver this kind of critical data. But even that approach can be hampered by biological heterogeneity (e.g., tumor cells exhibit great heterogeneity both between patients and within a single person102–104). Also, this approach only focuses on the concentration of reactants, not the rate constants. A second option is to do a sensitivity analysis on the model. All values are held constant except one, which is varied over a small range and the effect on the model is calculated. The same procedure is repeated for all parameters in the system. In this way, a modeler can determine which parameters in the model are uniquely sensitive to change, and whose exact values can cause large variation in modeling outputs. Those parameters with the most influence might need further experimental verification. This method also has flaws. If parameters synergize or are not mutually orthogonal, the “vary one at a time” method may not work. Also sensitivity analysis is notoriously sensitive to initial conditions, so being grossly incorrect about some important parameter in the model, might give erroneous sensitivity analysis results. There are global sensitivity analysis approaches that sample the huge parameter space and that bypass the problems of most single parameter methods.105,106 Validated and easy to use global sensitivity analysis will be a boon to modelers but are not generally available. Often, when a sensitivity analysis is done, a particular parameter that was hard to uncover from the literature, proves to be not sensitive. The model output may not be sensitive to this parameter and hence the precision with which it is known (within a reasonable range) does not change the results of the model. The last method to address a truly unknown value within the model is use parameter optimization. This requires an experimentally verified objective function for the system. To choose a very simple example, assume that the system under examination is the same protein A to protein B interconversion that was used above. Assume the time courses of A conversion to B and B to A are independently known. Further assume the initial concentrations of A and B are available and the forward rate constant is known, but the reverse rate constant is uncertain. Given this highly simplified example, the computer modeling software can be asked to vary the unknown rate constant until the time courses are fit. In this way, the fit rate constant can then be used in the model. In more complicated (more realistic) cases, unknown or uncertain parameters can be established by this method, but if the number of these unknowns is too high, the reliability of the model is questionable. In practice, modelers often face the problem of unknown parameters and use parameters from closely related species or protein family members to get a rough approximation for the unknown values as a starting point for further refinement. In addition to the pathway models mentioned in the “Definitions” section of this paper, ODE models have shown success in many systems. To name only a few, ODE models have been used to understand gene regulatory networks,107,108 to explore the development of autoimmunity through regulation of Treg cells,109 to unravel the complex dynamics of the immune system’s response to tumors,110 and to investigate the progression of Type 1 diabetes during b cell death.111

3.20.4

A Detailed Discussion of a Systems Biology and Quantitative Systems Model

In this section, the goal is to look at an example of a computational model in more detail, in order to give the reader a better understanding of the capabilities and limitations of modeling. The example is an ODE-based antibody drug conjugate (ADC) model. An ADC is a combination of an antibody, a drug payload, and a chemical linker that attaches the two other parts and has unique properties to release the toxin only when inside a cell.112–115 ADC molecules are, currently, highly effectively employed in cancer therapy.116–118 However not all ADC molecules are successful.119 To aid in the design of safe and effective ADC drugs, an ordinary differential equation model of the underlying biology of these drugs, was created.120 Probably this model best fits under the heading

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of a QSP model, as it was created to predict pharmacometric parameters. No model, no matter how detailed, can hope to get right all the details of any therapeutic. The human “system” is just too complex. However, from a practical standpoint, getting a validated model to correctly predict enough detail to design more successful drugs has been achieved with this model. To be an efficacious drug, an ADC must complete a series of rather stringent steps. ADC molecules are administered as an intravenous infusion into the blood. Therefore the potential drug must first traverse the capillary wall. With most reagents of the molecular weight and shape of an antibody, crossing the endothelial cell layer that makes up the blood vessels is tissue dependent. With some notable exceptions (such as brain and eye, for example), the concentration of an antibody-based drug in the tissue is about 5–15% of the blood concentration.121 Tumors are thought to be somewhat more leaky than normal tissue due to tortuous capillary architecture in the neoplasms.122 Hence tumor penetration by ADC drugs should be somewhat higher than normal tissue. Aberrant growth also produces lymphatic vessels that are not correctly connected. Hence there is little or no lymphatic drainage from tumors. This results in higher retention of ADC drugs, compared to that seen in normal tissues. The model must account for these differences. After leaving the blood vessels, the ADC is in the interstitial space of the tumor, where it encounters tumor cells, which express the target of the antibody moiety of the ADC. All tissues are arranged so that no cell is more than about three cells away from a blood vessel. This is due to the need for efficient transfer of nutrients and oxygen from the blood. Tumor cells can resist hypoxia due to the upregulation of the HIF-1a system, which results in some tumor cells being more distant from the capillary wall. This biology provides a radial symmetry to the cells of the tumor that allows the model to treat the capillary as if it is a long cylinder with cells projecting out from the center. This allowed simplification of the model, and is based on previous work.123 It also means that cells proximal to the capillary see the highest concentration of ADC, while cells distal to the capillary see less drug (see below). The ADC then binds to the tumor cell. In some cases, the ADC might be designed as an agonist of the receptor target and then it internalizes relatively rapidly. However, even in the case where the ADC does not trigger internalization, receptors on the surface of cells spontaneously internalize with a slower rate. This internalization carries the ADC reagent (and the receptor target) into a lipid bilayer enclosed endosome. The endosome is acidified by proton pumps. Depending on the composition of the chemical linker, some payloads are released during this step. In other chemical linkers, the intact ADC moves to the lysosome. In the lysosome, a number of proteases release the payload by proteolytic degradation of the antibody and linker. The payload has one of three fates if it is a mitotic toxin. It can accumulate if the cell is not in mitosis, or it can move to the nuclear machinery if the cell is in mitosis, or it can leave the cell either by diffusion or through the action of drug pumps. If the cell is in mitosis the payload binds tubulin and prevents its polymerization, which halts the synthesis of the mitotic spindles that are necessary for transit through mitosis. Freezing the cell in mitosis causes the cell to initiate a program of cell death, apoptosis. All of the preceding steps are necessary to make an effective drug, and the model was created to give a picture of the best blend of properties to achieve this goal. The model can be thought of as being divided into a number of modules (see Fig. 3). One component of the model computes the diffusion of the ADC reagent out of the capillary and across the tumor cells. It determines how much of the administered dose gets to the desired tissue (the tumor), and follows the fate of the payload (either delivery of the toxin to the tumor cells or the payload that is released from the tumor back into the interstitial space) (see Fig. 4A). This section also keeps track of how far the ADC must penetrate the tumor since different tumors are vascularized to different extents. Another section of equations explores the parameters for binding of the antibody moiety to its target receptor, the internalization kinetics of the ADC/receptor complex, rates of processing in the endosome or lysosome, and release of the payload by cleavage of the chemical linker (Fig. 4B). The third module of the model follows tumor dynamics (see Fig. 2C). Since the payload of the ADC (in the current state of the field) is usually a mitotic toxin. The payload will only act to kill the tumor if a neoplasm cell is in mitosis, and if the internal concentration of the payload exceeds a threshold level (the maximum tolerated dose, MTD). Thus the model must also track the proportion of cells that are in mitosis at any time point and decide if the internal cellular concentration is in excess of the MTD. The model then eliminates those cells that meet these criteria. Finally, this module of the model follows the tumor doubling rate, since the drug is only effective if the tumor is killed faster than it grows. A number of the predictions of the model have been verified by experimental results. For example, the model predicts that, if the affinity of the antibody portion of the molecule for its target is too high, the drug will be trapped by the first cell of the tumor that is proximal to the tumor capillaries (especially if the tumor target is expressed at high levels), and will hence not diffuse effectively and will not markedly reduce the tumor mass. However, too weak of an interaction between ADC and target also limits utility. This has been verified by experiments.124 Antibody moiety affinity is an important ADC consideration. Many ADC candidates are created by taking a preexisting antibody and coupling it to the payload. Preexisting antibodies are often those that have shown efficacy as drugs by themselves. These antibodies often exhibit high affinities. From the discussion above, it should be clear that high-affinity antibodies (especially those directed against high abundance targets) may not be optimal. The model predicts an optimal affinity for the antibody or suggests ways in which the drug can be adapted to compensate for binding that is too tight. The model shows that there is a delicate balance between several design properties of ADC reagents in addition to affinity. For example, molecular weight (and shape) of the reagent must be large enough to avoid rapid clearance by the kidneys, but not be so large that penetration of the capillary wall and tumor diffusion is unlikely. Antibody-based reagents have long half-lives because there is FcRn recycling, which is mediated by the Fc region of the immunoglobin. Molecules of less than about 50,000 Da molecular weight are cleared by the kidney (if they do not bind to larger molecules). Therefore, making hybrids of small and large molecules gives superior PK compared to small molecule drugs, but making ADC based on smaller parts of the IgG molecule can have some advantages.125 There are further balances between tumor properties and ADC characteristics addressed by the model. Lower tumor intercapillary distances leads to rapid transport of the ADC, however, highly vascularized tumors (e.g., head and neck squamous cell carcinomas) tend to double at an accelerated rate versus tumors with poor vascularization (e.g., pancreatic ductal cancers). Another critical balance derives from

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Fig. 4 A representation of the three conceptual modules in the antibody drug conjugate example. (A) The distribution of the ADC from the blood into the tumor. Because of the sink caused by the most capillary-proximal cells, a gradient of concentration is generated. (B) The kinetics of ADC binding, internalization, release of payload, and distribution of the toxin. (C) The module that keeps track of tumor dynamics, killing, and/or growth.

the loss of cleaved payload from the cell. On the one hand, this lowers the effective internal concentration of the drug, but it also promotes “bystander killing”126 and increased tumor penetration. In theory, the model can be parameterized with tumor variables and, separately, with healthy tissue characteristics. This, in principle, allows for the generation of an in silico therapeutic index, though this avenue has not yet been extensively explored. Properly applied, this model allows the pharmaceutical drug designer a way to objectively choose ADC properties that best match with the tumor characteristics. The model can reduce the number of design/test/redesign cycles that are necessary to create an ADC candidate. Hopefully, this example demonstrates that ODE modeling can easily accommodate most kinds of problems that occur in deterministic models. Instead of only handling sequential steps associated with pathway modeling, this model allows the incorporation of the treatment of many biological processesddiffusion, cell cycle monitoring, and tumor dynamics, as well as binding and internalization kinetics.

3.20.5

Summary

This article has given the reader a beginner’s guide to understanding systems approaches to biology and pharmaceutical discovery. Systems biology or QSP (or any of the other, myriad names this work is called) is at an important point in its development. The concomitant evolution of mass sequencing and even more massive biological data collection set out to give a complete components list of human biology, but has often left scientists to drown in vast information overload. Reductionism has been left behind for the very practical reason that there is just too much to study “one at a time.” There seem to be more contradictions than grand unifiers of biological knowledge. This is where systems biology will play a crucial role. The need to quickly examine new hypotheses about the nature of biological processes and to suggest the most impactful experiments to test them is fertile ground for the growth of systems biology. The closely related field of QSP applies the same toolbox to finding new drugs and describing the optimal properties of those drugs to achieve the best chance for successful amelioration of disease. Systems biology or QSP are an amalgam of biology, chemistry, physics, and computer sciences. The English biochemist Fredrick Gowland Hopkins said “.where sciences meet, there growth occurs.” The adoption of systems approaches in the pharmaceuticals industry is slow. Drug discoverers have been very successful without systems biology and systems pharmacology. Why change? The main driver of the growth in systems approaches

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is the realization that biology is complexdpatients vary in response to drugs; the same pathways giving rise to one output in one cell type gives a completely different outcome in another; arrangements of signaling networks are subverted by disease; and heterogeneity in toxicology is difficult to understand. Aristotle wrote, “The chief forms of beauty are order and symmetry and definiteness which the mathematical sciences demonstrate in a special degree.” But maybe Aristotle was not a systems thinker because biology is messy and systems studies thrive on complexity.

See also: 2.18. Systems Biology: Methods and Applications. 2.19. Systems Chemical Biology: Integrating Medicinal Chemistry and Systems Biology for Drug Discovery.

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Relevant Websites [1]. [2]. [3]. [4]. [5].

https://www.systemsbiology.orgdThis the website of the Institute for Systems Biology. http://www.systems-biology.orgdThis is the website is a great portal for tools, utilities, pointers to relevant journals, etc. https://sysbio.med.harvard.edudHarvard’s Department of Systems Biology. http://csbi.mit.edu/dMIT’s program in computational and systems biology. http://www.systemscenters.orgdThe National Institute for General Medical Sciences sponsored website for National Centers of Systems BiologydA collection of pointers to funded research in various branches of systems studies. [6]. http://biomodels.caltech.edudBioModels Database.