toward computational systems biology - duke

INTRODUCTION

Advances in experimental and computa-tional technologies for biosciences have beenrevolutionizing biological research. The lastseveral decades have witnessed the develop-ment and maturation of several remarkableexperimental techniques, such as DNAsequencing technique, DNA microarray (1),and large-scale two-dimensional protein gelelectrophoresis (2). Emerging from the applica-

tion of these technologies is a new mode ofbiology, the systems biology, which empha-sizes a holistic understanding of how biologi-cal systems function (3,4).

As demonstrated by the successful sequenc-ing of more than 1000 genomes of natural plas-mids, organelles, viruses and viroids, bacteria,plants, and animals, including mouse (5) andhuman (6,7), there are few, if any, technologicalhurdles in obtaining the genetic information ofvirtually any organism. In addition to itsimplications for practical applications, suchinformation promises to bring us closer to acomplete understanding on how the geneticinformation stored in a genome determines thebehaviors or characteristics (i.e., the phenotype)

Toward Computational Systems BiologyLingchong You*

Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125

Abstract

The development and successful application of high-throughput technologies are transform-ing biological research. The large quantities of data being generated by these technologies haveled to the emergence of systems biology, which emphasizes large-scale, parallel characterizationof biological systems and integration of fragmentary information into a coherent whole.Complementing the reductionist approach that has dominated biology for the last century, math-ematical modeling is becoming a powerful tool to achieve an integrated understanding of com-plex biological systems and to guide experimental efforts of engineering biological systems forpractical applications. Here I give an overview of current mainstream approaches in modelingbiological systems, highlight specific applications of modeling in various settings, and point outfuture research opportunities and challenges.

Index Entries: Systems biology; mathematical modeling; gene networks; deterministic simula-tion; stochastic simulation; biological databases.

*Author to whom all correspondence and reprintrequests should be addressed. E-mail: [email protected]

REVIEW ARTICLE

© Copyright 2004 by Humana Press Inc.All rights of any nature whatsoever reserved.1085-9195/04/40/167–184/$25.00

Cell Biochemistry and Biophysics 167 Volume 40, 2004

of an organism or a cell in a particular environ-ment. As shown in Fig. 1, however, muchremains to be done to really make the linkbetween a genome and the resulting pheno-type(s). Logical next steps in this odyssey are toidentify the genes in a genome and to determinetheir functions, in particular, by elucidating whatproducts these genes produce and how theseproducts interact with one another. For any par-ticular biological system, these downstreamanalyses can be orders of magnitude more com-plex than sequencing the genome. Indeed, theyrequire a wide spectrum of tools to characterizeindividual gene products by using biochemical,biophysical, or genetic techniques or a large set

of such molecular components by profiling geneexpression at the mRNA level—using DNAmicroarray (1)—or at the protein level—usingtwo-dimensional protein gels (2) or mass spec-trometry (8–10). In addition, other high-through-put techniques, such as yeast two-hybridanalysis (11), have been successfully applied tostudy interactions between gene products at alarge scale (12,13).

Yet, large-scale characterization of the com-ponents of biological systems is not the ulti-mate goal, and it should not be. Supposing thatsomeday we succeed in characterizing the func-tion of all genes in a cell by identifying all thegene products and constructing the interaction

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Cell Biochemistry and Biophysics Volume 40, 2004

Fig. 1. A new mode of biology research. Sequencing of various genomes has led to the questionas how these genomes “program” cellular behaviors, or the phenotype, in a particular environment.Different intermediate levels of understanding need to be established to make this connection (leftpanel). Many computational and experimental tools (right panel) are required in achieving suchunderstanding. Mathematical modeling will play a key role in integrating fragmentary informationinto a coherent whole. A fundamental goal of modeling is to explain existing data and to predictsystem behaviors. Every model should be examined in the context of experiment whenever possi-ble. Iterations of model prediction, comparison with experiment, and model revision in the light ofnew data and mechanisms will constantly improve our understanding of the underlying system.(See text for details.)

map among these gene products, will we havecompleted the link between the genome andcellular behaviors? Partly. Extensive as it is, allthis information is still local and fragmentary.Knowing what a cell is composed of and howeach component works does not necessarilymean understanding how the cell as a wholeworks. For example, understanding what partsa car is composed of and how each part worksdoes not mean that we would understand howthe car itself works. To be confident that weunderstand how the car works, we should beable to put the parts back together and demon-strate that the car works. Similarly, to under-stand how a cell function as a whole, we willneed to integrate our understanding of theparts and see whether or to what extent thesepieces of understanding will coherently predictcellular behaviors. For any system, the processof integrating and analyzing the information onindividual pieces can be loosely defined as“modeling.”

Regardless of its formulation, a model bydefinition represents an integration of theknowledge of the underlying system. It isbased on answers to questions such as: whatcomponents is the system composed of? Andhow do they interact? Note that the integrationof fragmentary information is a critical step forachieving global, system-level understanding.As discussed in detail in the section SpecificApplications, this integration process helps toreveal features not easily recognizable byexamining the constituent parts. For example,information regarding some individual piecesmay be inconsistent with other information,but we often will not know the inconsistencyuntil we put these pieces together. The funda-mental goal of a model is to explain existingdata and to predict system behaviors.Oftentimes, model predictions will deviatefrom the experiment, which calls for re-evalua-tion of the knowledge integrated into themodel or for additional experiments. Eitherway, our understanding will improve by goingthrough iterations of model prediction, com-parison between the model prediction andexperiment, and model refinement (see Fig. 1).

The wide appreciation and interest in model-ing biological systems is evidenced by the pub-lication of not only countless modeling studies,but also many excellent reviews that discussvarious aspects of modeling (4,14–23) (note “ABrief Guide to Recent Reviews” in ref. 17).Intending to give a comprehensive tutorial onmodeling in biology, here is provided anoverview of mainstream modeling approachesand discuss their application domains. Toachieve this goal, many details are omitted ofthese modeling approaches but instead theirconnections are highlighted between each otherso that they can be examined in a coherent man-ner. Focusing on kinetic models, recent applica-tions are discussed of modeling in enhancingour understanding of natural or engineered bio-logical systems. Although modeling is the focusissue of this review, I strive to put the discus-sions in a broader context of systems biology,emphasizing how modeling may facilitate sys-tem-level study or understanding of complexbiological systems. Finally, this article closeswith some future opportunities and challengesfor the mathematical modeling community.

MODELING WITH VARYINGRESOLUTIONS

Qualitative Models

Depending on its specific objectives, a modelmay involve details at different levels. One ofthe most profound biological models is thecentral dogma of molecular biology, whichdescribes the basic information transferprocess from DNA to RNA and from RNA toprotein (Fig. 2). Although omitting manydetails involved in the individual steps—forexample, the binding of RNA polymerases tothe DNA sequence during transcription—thecentral dogma summarizes decades ofendeavor by biologists that led to the elucida-tion of this process (24). Reducing complex bio-logical processes into one dimension, it hasserved as an elegant conceptual framework forthe establishment of molecular biology. As

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with all other models, the central dogma asdepicted in Fig. 2 only reflects partial truth,and it evolves. Decades of biological researchhas tremendously enriched the content of themodel—a more accurate representation of therelationship, although still omitting details,would include the replication of DNA, reversetranscription of RNA to DNA, replication ofRNA, and catalyzing role of the protein in allthese processes.

The central dogma exemplifies probably themost intuitive models that biologists havebeen using: diagrams. Diagrams have beenwidely used and will probably continue toprevail in biology textbooks and the literature.Recently, there have been significant efforts toformalize the conventions of drawing diagrammodels (25). Diagrams have served as tremen-dous visual aids in understanding the biologi-cal processes and for formulating newhypotheses to be tested experimentally. In aclosely related area, several groups have ana-lyzed large-scale metabolic networks basedsolely on the connectivity of network compo-nents (26–33). Unlike conventional diagramsthat involve perhaps dozens of components atmost, wiring of thousands of components(often proteins) in these large systems makeslittle sense to the naked eye. Yet computationalanalysis has provided insights into someglobal, usually topological properties of meta-bolic networks, such as degree of network con-nectivity, clustering of network components,and lengths of biochemical pathways (26–31).Recently, this statistical approach has beenapplied to reveal potential cellular motifs inlarge metabolic networks (32,33). Despite theirintuitiveness (excluding computer-generateddiagrams of large metabolic networks), the

drawback of diagrams is that they are descrip-tive only, and cannot predict in a quantitativeand sometimes not even in a qualitative fash-ion behaviors of a given system.

The simplest approach to characterizing thedynamics of biological networks is a Booleanmodel, which is often applied to studyinggene networks (34–36). In this paradigm, eachplayer (e.g., a gene) has two states, on and off;the system of interest is represented as a logicnetwork, and the dynamics describe howgenes interact to change one another’s statesover time. A Boolean model is advantageousin its simplicity and it does not requiredetailed data on how cellular componentsinteract. Despite their apparent simplicity,Boolean models can provide many insightsinto the qualitative behavior of the underlyingsystem. For instance, Kauffman has success-fully employed Boolean models to exploreself-organization phenomena and their impli-cations in evolution (37). Yet, Boolean modelsare often overly simplified and tend to giveambiguous predictions on system behaviors(38). For example, consider a simple circuit inwhich a component S negatively regulates itsown accumulation. In the Boolean formula-tion, this process can be modeled as: S(T + 1)= NOT S(T), where S(T) means the state of S attime step T. This model predicts that the levelof S will oscillate between 0 and 1, irrespec-tive of the initial condition. However, in real-ity, such a system often leads to a steady state(e.g., see ref. 39) unless there is a significanttime delay in the self-regulation. In addition,some ambiguity is evident in the simplemodel: the function “NOT” could mean that Sslows down its own synthesis or that S facili-tates its own degradation.

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Fig. 2. The central dogma of molecular biology. The central dogma states that genetic informationcan be perpetuated or transferred, but the transfer of information into protein is irreversible (151).

Quantitative Models

A more detailed and precise approach ofrepresenting a biological system is to treat it asa network of chemical reactions (Fig. 3). To ana-lyze the system, one needs the stoichiometry of

all the reactions involved. Using this formula-tion, one can gain deep insights into thesteady-state system behavior and characteris-tics of the network topology by merely analyz-ing the underlying stoichiometric matrix(40–42). A number of computational tech-



Fig. 3. A simple predator-prey system. (A) A diagram representation of the predator-prey sys-tem: The prey feeds on unspecified foods (represented by *) but is consumed by the predator. Thepredator dies and degenerates into unspecified products (represented by *). Thus the prey promotesthe increase in the predator population (indicated by +) but the predator facilitates the decrease inthe prey population (indicated by a –). (B) The system can be represented by a set of simple reac-tions (X = prey; Y = predator); each reaction is characterized by its stoichiometry and its kinetics.(C) A typical deterministic simulation result from an ODE model based on the reactions in (B). Thedifferential equations used for this simulation are:

.

(D) A typical simulation result using the Gillespie algorithm. Results in (C) and (D) were generatedusing Dynetica (i.e., a simulator of dynamic networks) (143). Levels of prey and predator areexpressed in numbers. Note how the stochastic simulation result resembles the deterministic simula-tion result in terms of qualitative behavior—oscillation—but drastically differs from the latter innumerical details. From the given initial condition, the stochastic simulation can generate completelydifferent dynamics from the deterministic simulation (not shown).

dX dY—– = k1X – k2XY; —– = k2XY – k3Ydt dt

niques, such as metabolic flux analysis (41,42)and flux balance analysis (43), have been suc-cessfully developed to assist such analysis.They have played an instrumental role in shap-ing the field of metabolic engineering, by pro-viding theoretical guidance for experimentalmanipulation of metabolic networks (44).Recently, these techniques have been success-fully employed to reveal the underlying struc-ture of metabolic networks by determining theelementary flux modes (45) and the null spacebase vectors (46), and in predicting steady-state metabolic capabilities of several modelorganisms, such as Escherichia coli (47,48) andHaemophilus influenzae (49).

However, it is often difficult to formulate instoichiometric models regulatory interactions(50), which are clearly ubiquitous in biologicalsystems. Moreover, because stoichiometricmodels lack the time-domain in their formula-tion, they are unable to predict the temporalevolution of biological systems. To make suchpredictions, a stoichiometric model needs besupplemented with detailed kinetic informa-tion. That is, one needs to specify how fast eachreaction is occurring, in addition to its stoi-chiometry (Fig. 3B). Compared with stoichio-metric models, the drawback of kinetic modelsappears obvious: construction of each modelwill require much more information, andkinetic data are usually much more difficult toacquire than the stoichiometry of a reactionnetwork. But the payoff of the added complex-ity is significant: with an appropriate kineticmodel, the modeler can gain much deeperinsight into the behavior of the system thanwith a stoichiometric model.

Usually kinetic models are represented as aset of ordinary differential equations (ODE)describing the rates of change for the interact-ing components (Fig. 3B). Solving these ODEs,often numerically, generates time courses ofthe interacting components. This ability to pre-dict dynamical behaviors is essential for ana-lyzing systems with rich temporal behaviors,such as circadian clocks. ODE-based kineticmodels are also termed deterministic because agiven initial condition will completely deter-

mine the temporal behavior of the underlyingsystems (within the errors of the numericalsolutions) (Fig. 3C). As to be discussed in thefollowing section, kinetic models can be for-mulated in a stochastic framework, which maygive finer details of system dynamics byrevealing random fluctuations in the numbersof each interacting component.

Application of kinetic models in biology has along history. For example, Lotka and Volterraindependently developed a kinetic modelnearly 80 years ago to describe dynamics of apredator-prey ecosystem (as cited in (51); see alsoFig. 3). Today, kinetic modeling continues toplay a major role in modern ecology (51,52). Incellular biology, decades of genetic, biochemical,and biophysical studies have generated a largeamount of experimental data that can be used todeduce reaction mechanisms and rate constantparameters, which in turn makes kinetic model-ing a preferable choice for describing cellularprocesses (16,19,23). This point is particularlyobvious for many well-characterized systems,including bacterial chemotaxis signaling net-works (53,54), developmental pattern formationin Drosophila (55–58), aggregation stage networkof Dictyostelium (59), viral infection (60–65), cir-cadian rhythms (66,67), E. coli stress responsecircuit(68), single E. coli cell growth (69), yeastcell-cycle control (70,71), and physiologicalprocesses (72–74).

Kinetic models may be coupled with equa-tions describing mass transport processes,leading to more complicated mathematicalmodels. For example, in modeling some bio-logical processes, it is necessary and feasible toaccount for not only reaction kinetics, but alsotransport of interacting components by diffu-sion. This approach is most commonly adoptedin models describing the spatiotemporaldynamics of small chemicals such as calcium(75–79) and cyclic adenosine monophosphate(80–83), or larger components (84–89) whosetransport can be treated as diffusion. Suchmodels often involve using partial differentialequations (PDE) in addition to ODEs. Currentefforts are under way to acquire detailed infor-mation on the distribution of molecules in the

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cell. Yet, for most intracellular gene expressionprocesses, experimental data on the spatialdomain are often too sparse to make sensiblePDE models. In addition, solving a PDE modeloften invokes much higher computational costfor the same number of interacting compo-nents. Thus, most current mathematical mod-els of gene regulation networks have ignoredspatial heterogeneity in a system. When it isessential to describe transport processes, theseprocesses may be approximated by first-orderreactions, which fall into the framework ofODE models (55,56,63).

Stochastic Formulation of Reaction Networks

Despite their broad applications, ODE-basedkinetic models are criticized for their implicitassumption of continuity in the concentrationsof interacting cellular species, particularly forintracellular processes. In particular, manyproteins are expressed at nanomolar levels,which correspond to only tens or hundreds ofmolecules per cell. In this scenario, the smallnumbers of the interacting species can lead tosignificant random fluctuations in the levels ofthese species. A good deal of experimentaldata has indeed demonstrated significantnoise in gene expression processes (90–94).Deterministic in nature, ODE models will failto predict such fluctuations. For this reason,some researchers have questioned the use ofdeterministic simulations in characterizing thebehaviors of biological systems, and suggestedusing stochastic simulations instead (95–97).

Several algorithms are available for carryingout stochastic simulations (98); the Gillespiealgorithm (99) is by far the most popular.Following a Monte Carlo procedure, theGillespie algorithm predicts the time evolutionof the system by determining when and inwhat order the next reaction is going to occur.This algorithm has a rigorous theoretical foun-dation, and is shown to give exact solution fora network of elementary reactions occurring ina well-stirred environment (99,100). It oftengenerates dynamics drastically different from

the prediction by deterministic simulations,particularly when some reactions have nonlin-ear terms in their rate expressions (101; see alsoFig. 3D). The application of stochastic simula-tions has led to speculations regarding theimplications of intracellular noise, in particu-lar, how it may be exploited in nature and howit can be effectively controlled (98). For exam-ple, it has been suggested that the intrinsicnoise might be exploited in generating pheno-typic diversity in a clonal population so thatthe population is more capable of survivingdifferent environments (102).

Although the Gillespie algorithm reveals sto-chastic fluctuations resulting from small molec-ular numbers, several outstanding questionsmake it unclear whether or to what extent it ismore appropriate than a deterministic approachin modeling cellular reaction networks. From apractical perspective, well-polished computa-tional techniques (e.g., bifurcation analysis—analysis of qualitative changes in the dynamicsof a system caused by the variation of some sys-tem parameters (103)) and software tools (e.g.,Xppaut at http://www.math.pitt.edu/~bard/xpp/xpp.html) are available for high-levelanalysis of system dynamics using ODEs,although such analysis is far from practicalusing the stochastic formulation (18,98). Also,stochastic simulations by the Gillespie algo-rithm are often much more time consumingthan deterministic simulations. In fact, the com-putation time of this algorithm approximatelyscales with the frequency of the reaction events:the more reactions there are, or the more mole-cules there are, the longer the computation willtake for a given simulated time span (99). Recentefforts have been made to improve the compu-tation efficiency of stochastic simulations, eitherby directly improving the efficiency of theGillespie algorithm while keeping its rigor (104),or by approximating the computation for fastreactions when separation of time scales is justi-fiable (105,106). Another alternative to exactsimulation by the Gillespie algorithm is to incor-porate fluctuations by explicitly including ran-dom variables in the differential equationsdescribing the system (107–111). This approach



results in a Langevin equation or a stochasticdifferential equation, leading to substantial gainin computation speed at the acceptable loss ofcomputation accuracy (109,110). A major advan-tage of this approach is that it ties the basis ofGillespie algorithm—a chemical master equa-tion—to a conventional, deterministic formula-tion of chemical kinetics. In fact, a chemicalmaster equation is shown to be asymptoticallyequivalent to the corresponding Langevin equa-tion under certain conditions (109). For moredetailed discussion on this topic, the reader isdirected to an excellent recent review (98).

Aside from the issue of computationalspeed, a fundamental question is to whatextent the simulated noise reflects the truenoise. For the Gillespie algorithm to be exact,the system must consist of elementary reac-tions only and it must be well-stirred (i.e., spa-tially homogeneous) (99,100). These criteriaare rarely met when modeling intracellularprocesses: the intracellular environment ishighly heterogeneous and many a reactionlumps multiple steps. For example, a reactionas basic as transcription of a single gene con-sists of several more basic steps, such as bind-ing of RNA polymerase to promoter region informing a closed complex, formation of anopen complex, and eventually transcriptionalelongation. Moreover, in addition to smallnumbers of interacting molecules, random-ness may stem from other factors, such as thecellular components not specified in the modeland even conformational changes of biologicalmacromolecules. From this standpoint, theGillespie algorithm, as well as other stochasticalgorithms, gives empirical predictions (whenapplied to describing intracellular processes),just as its ODE counterpart does. In fact, recentexperiments by Elowitz et al. demonstratedthat the total noise of gene expression consistsof both intrinsic noise and extrinsic noise, andthat the total noise does not correlate with theintrinsic noise (91). It is probable, but notproven, that the noise generated by theGillespie algorithm accounts for a major partof the intrinsic noise but not the extrinsic noisefor cellular processes. In all, one needs to take

caution in interpreting the fluctuations gener-ated by a stochastic simulation.

SPECIFIC APPLICATIONS

Because the majority of current mathemati-cal models of biological systems are kineticmodels, I will focus here on applications ofkinetic models only. The categorization of dif-ferent applications is somewhat arbitrary. Itshould also be noted that some of these appli-cations may be achieved by other types of themodeling as well, with potentially differentresolutions and outcomes.

Revealing Gaps in Current Understanding

By integrating current understanding of theunderlying system, a kinetic model can be usedto test the consistency in the experimental dataor mechanisms. Often times, model predictionswill deviate from the experiment. This discrep-ancy is not necessarily a negative thing; in fact,it can be very informative. It indicates the gapor hole in our knowledge (or at least theknowledge integrated into the model) and mayprovide guidance for future experimentation.

von Dassow et al. recently developed amathematical model of the segmentation net-work of Drosophila embryo development (55).Their initial model based on known networkconnectivity was unable to predict the segmen-tation pattern observed in experiments. To testwhether this discrepancy was due to inaccurateparameters, the authors randomly changed thekinetic parameters over a wide range of plausi-ble values and carried out simulations for eachparameter set. However, none of these parame-ter sets could generate the desired pattern. Thisdiscrepancy between simulation and experi-ment, the authors argued, indicates a gap in theunderstanding of the segmentation network.Specifically, if the network connectivity asimplemented in the initial model had been cor-rect, then at least some parameter sets shouldhave generated the desired pattern. To accountfor the discrepancy, the authors hypothesized

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additional connections, based on experimentalobservations in the literature, in their networkmodel and repeated their computational analy-sis. Interestingly, the added links indeedseemed to be an important missing piece: theirmodel now could generate the desired patternfor a significant portion of the parameter setsthey generated. Although these predictionsawait experimental verification, this work high-lights a key use of modeling: to identify incon-sistency in our knowledge and to suggestexperimentally testable hypothesis. It is worthnoting that a similar model was developed byReinitz et al. to describe pattern formation inDrosophila embryos (56–58). The strength ofthese studies lies in the extensive use ofdetailed experimental data to fit the parametersin the dynamic model and subsequent applica-tion of the model to explore specific roles of dif-ferent genes in forming embryonic patterns.

Based on more than four decades of litera-ture data, Endy et al. developed a detailedmodel of phage T7 (62). This model describesthe entire intracellular life cycle of the virusfrom the entry of viral genome to the produc-tion of viral progeny. This model was laterused to predict how the viral growth ratewould depend on the organization of geneticelements (such as genes, promoters, and tran-scription terminators) (112,113). Among otherresults, the simulation predicted that the T7growth rate overall would decrease as the T7polymerase gene (gene 1) was moved awayfrom the entering end of the T7 genome. Thismakes intuitive sense: the further downstreamT7 gene 1 is, the more delayed its expression.The delay in gene 1 expression subsequentlywould lead to a delay in the expression of themajority of T7 genes, thus slowing down T7growth. Interestingly, the simulation also pre-dicted that when gene 1 is immediately down-stream the early T7 promoters, the T7 growthrate would be higher than the wild-type valuebecause of the establishment of a positive feed-back for the production of T7 RNA poly-merase. However, experimental results withthree ectopic gene 1 mutants only verified theoptimality of the wild-type T7. The mutant

ecto1.7, which was predicted to grow fasterthan the wild type, turned out to grow muchmore slowly. One possible reason for this mis-match, the authors found, was that disruptionof the seemingly nonessential gene 1.7 (inecto1.7, gene 1 was inserted within the codingregion of gene 1.7) actually had significantquantitative effects on T7 growth due tounknown mechanisms.

Another possible reason for the discrepancyis the assumption that the host cell offersunlimited translation resources, such as ribo-somes and amino acids. In fact, with a morerealistic representation of the E. coli host (65),an extended T7 model was able to give moreaccurate predictions for the growth of ectopicgene 1 mutants (114). The revised simulationshowed that the positive feedback for produc-ing the T7 RNA polymerase was detrimental tooverall viral growth by causing unfavorabledistribution of limited protein synthesisresources. Moreover, the revised model alsopredicted that the phage T7 growth rate wouldbe faster in faster growing host cells, a predic-tion that was soon confirmed experimentally(65). In addition to suggesting possible mecha-nisms for the mismatch between the originalmodel and experiment, these series of workbased on the revised model highlighted theimportance of the host environment in thedevelopment of an organism, a factor ignoredin the original T7 model.

Characterizing Emergent Properties

Mathematical models are beginning toreveal the so-called emergent properties, whichare often difficult to grasp intuitively by exam-ining the constituent parts of these systems. Anoften-characterized emergent property isrobustness of a system. For experimental biolo-gists, robustness describes the stability of aphenotype in the presence of genetic and envi-ronmental variations, but the term often comeswith some ambiguity in its exact meaning andis difficult to quantify (115). In a well-definedmathematical model, however, robustness canbe quantified in a straightforward fashion: it



can be measured by the sensitivity of a systemfunction to variations in parameters.

Barkai and Leibler (53) carried out a detailedanalysis of how the output of a chemotaxis net-work responds to perturbations in the kineticparameters that define the network behavior.Their simulations demonstrated that key prop-erties of the network are robust to parametricperturbations. Admirably, key findings of thiscomputational study were later verified byexperimental work from the same group (116).The authors went on arguing that such robustbehavior might be a generic feature that is nec-essary to ensure proper functioning of a widevariety of biological systems (53). This argu-ment was echoed by Morohashi et al. (117),who contended that robustness to variationscould be used as a measure of the plausibilityof the models of a given system. Moreover, oth-ers have argued that many complex systems,including biological systems, often demon-strate “robust yet fragile” features (118,119).These arguments are gaining support from sev-eral recent modeling studies. In the previouslymentioned work by von Dassow et al. (55), the“remedied” version of the Drosophila segmen-tation network model demonstrated robustfeatures: the model was able to predict the cor-rect segmentation pattern for a wide range ofparameter settings (55). Later, the same groupfound that another system—the Drosophilaneurogenic network—also demonstrated sig-nificant robustness with respect to networkparameters (120). Using an extended phage T7model, You and Yin recently found that theability of phage T7 to survive was overallrobust to perturbations to its kinetic parame-ters that define its physiology, but sensitive toperturbations to the organization of geneticelements in the genome (121).

Analysis of robustness by modeling is still arelatively new area. Despite impressive pro-gresses made so far, open questions remain aswhether and to what extent results generatedfrom simulations reflect reality (122). A majorchallenge in addressing these questions is toproperly map kinetic parameters to the geno-type of a given biological system. Ideally, we

should be able to answer the question: whatmutations are required to achieve a certainamount of change in a given parameter?However, this mapping is difficult for manyparameters, especially when a mutation inone gene may affect multiple phenotypictraits (112,123).

Testing Complex Hypotheses

As models become more “realistic” by incor-porating increasingly detailed data andmechanisms, they may be treated as in silicoorganisms and used to explore applied or fun-damental questions that are beyond the under-lying system per se. In particular, such complexmodels can also be called upon to test hypothe-ses or theories that are difficult, expensive, oreven impossible to explore experimentally withcurrent technology. Note that the ability toquantitatively test complex hypotheses is animportant feature of sophisticated kinetic mod-els. Models of further abstraction, such as dia-grams or Boolean models, lack this ability,largely because of detachment between systembehaviors and physical parameters.

Thanks to the ease of using the phage T7model to create thousands of T7 mutants in sil-ico and to efficiently evaluate their fitness, Youand Yin were able to systematically characterizethe nature of the interactions among deleteri-ous mutations in terms of their effects on fitnessat the population level (123). Such genetic inter-actions play a major role in a variety of funda-mental biological phenomena, includingevolution of recombination, dynamics of fitnesslandscapes, and buffering of genetic variations,but their experimental characterization hasbeen hindered by the difficulty in generatingand quantifying a large number of mutants.From their simulation, You and Yin found thatthe nature of genetic interactions depended onthe growth environment for the organism, aswell as the severity of the deleterious muta-tions. Their results offered an intuitive explana-tion for the seemingly conflicting conclusionson the nature of genetic interactions from priorexperimental studies: the mutations tested in

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experiments may have been of differing sever-ity, and their interactions may have been testedin different environments.

Guiding Experimental Design of De Novo Gene Circuits

With the elucidation of a wide variety of bio-logical components, including genes and geneexpression regulation elements, there has beensoaring enthusiasm for building de novo or syn-thetic gene networks in the last several years(39,124–131). Designing and constructing suchsynthetic gene networks has great potential ingenerating genetic “gadgets” with novel appli-cations, as well as in improving our under-standing of how cellular components function.As discussed in detail in a recent review (126),the basic strategy of constructing such net-works follows roughly the same procedure:

1. Outline the basic network connectivity.2. Analyze the network behavior by mathe-

matical modeling.3. Implement the network experimentally.

Usually the adopted models are purpose-fully simplified to capture the qualitativebehavior of the designed system. In addition totime course simulations, bifurcation analysis isoften employed to examine how qualitativecharacteristics of the system dynamics maydepend on model parameters (103). So far, thisdeceivingly simple procedure has yielded onlya few successful products, including a negativefeedback control circuit (39), a toggle switch(128), a ring oscillator (or the “repressilator”)(129), a relaxation-type oscillator (132), and agenetic inverter (130). One of the major chal-lenges of constructing these engineered circuitsis that the behavior of the implemented circuitoften deviated significantly from the predictedbehavior. This is probably more an indicationof our lack of detailed understanding of howcellular components function than a failure ofthe approach. In fact, in a sense the success ofthis series of seminal work was safeguarded bythe proper use of modeling. For example, indesigning the repressilator, a simplified kinetic

model was used to highlight several designgoals for achieving the desired function—oscil-lations in protein concentrations (129). In con-structing the genetic inverter, Weiss used amathematical model to guide his efforts to“rationally debug” an initially nonfunctionalcircuit. When modeling fails, powerful experi-mental methods can come into play. As demon-strated by recent seminal work (133),directed-evolution techniques, which haveproven to be particularly powerful for optimiz-ing enzymes (134,135), are equally valuable forfine-tuning the function of de novo gene cir-cuits. In addition to fine-tuning circuit func-tions, this “design-then-mutate” approach mayalso play an important role in the developmentand refinement of quantitative models by offer-ing additional structure-function insights (136).

OUTLOOK: A MATURINGINFRASTRUCTURE FOR MODELING

With increasing appreciation of the meritof modeling for deeper understanding of bio-logical systems, I anticipate that two lines ofrelated research efforts—namely, the devel-opment of biologist-friendly modeling toolsand high-level databases harboring informa-tion on biomolecular interactions (includingpathway databases)—will dramatically facili-tate broader application of modeling in biol-ogy by providing an streamlined platform forstorage and communication of data and inte-grated models.

Software Tools

Despite its potential benefits for fundamentaland applied biological research, the applicationof mathematical modeling in biology has beenhindered by the lack of software tools to build,analyze, and visualize models, particularly forresearchers unfamiliar with programming andnumerical methods. But the situation is chang-ing. To address this issue, a number of programsthat aim to facilitate the model construction andanalysis have been developed in the last several



years. A partial list of these programs includesGepasi (137), DBsolve (138), E-Cell (139), VirtualCell (140,141), StochSim (97), Jarnac/Jdesigner(http://www.cds.caltech.edu/~hsauro/index.htm), Cellerator (142), and Dynetica(121,143).

Although different implementation strate-gies are adopted, all these programs strive toprovide an intuitive interface and versatilesimulation capabilities for the user. It is as yetdifficult to give an unbiased evaluation of allthese programs without detailed third-partybenchmark comparisons. Briefly, Gepasi andDBsolve focus on the analysis of biochemicaland metabolic networks. In addition to basictime-course simulations, these programs pro-vide additional modules to explore the prop-erties of metabolic networks. E-Cell aims toconstruct whole-cell models, and it has beenapplied to model a self-sustaining hypotheticcell (144) and a human erythrocyte (139).Virtual Cell is advantageous in that itaccounts for the diffusion of molecules inaddition to their reactions in describing cellu-lar processes. StochSim simulates the systemdynamics using an approximate stochasticalgorithm, which is more efficient but lessaccurate than the Gillespie algorithm. Janarcis an interactive and interpreted languagefor describing and modeling cellular net-works, and it can interact with JDesigner, atool for visual construction of these networks.Implemented using Mathematica as the backend, Cellerator introduces palette-driven,arrow-based notations to represent biochemi-cal reaction networks, and provides a mecha-nism to translate these representations intoODEs that can be solved by Mathematica.Dynetica integrates model construction, analy-sis, and network visualization into a unifiedmodeling framework; it is distinct from oth-ers in that (1) it allows time-course simula-tions using both deterministic algorithm(ODE-based) and the Gillespie algorithm (e.g.,see Fig. 3), and (2) it facilitates the constructionof genetic networks, where the majority ofreactions revolve around gene expression(121,143).

Although encouraging for biologists, animmediate issue raised by these diverse pro-grams is the lack of compatibility among them.To address this issue, there have been manyefforts toward developing standard modelinglanguages, particularly the SBML (SystemsBiology Markup Language) (145) (see alsohttp://www.cds.caltech.edu/erato) and theCellML (Cell Markup Language) (http://www.cellml.org), both of which are based on XML(http://www.xml.org), a popular structural datarepresentation language. It is foreseeable that inthe near future, many of these programs canshare models via such standard modeling lan-guages. If this is realized, the user can use any ofthe modeling software to build models and con-veniently switch to other tools if needed. Assuch, the user can take full advantage of differ-ent software packages with minimum efforts.

Databases of Biological Interactions

The usefulness of a model is often deter-mined by the quality of the underlying experi-mental data and mechanisms. This pointhighlights the importance of managing suchinformation. But what kind of informationshould we document? A good starting point isprobably the data on biomolecular interactionsand reaction pathways, because these types ofdata can be mapped into a model in a straight-forward fashion. In fact, an amazing number ofdatabases or knowledge environments havebeen developed along this line, many beingaccessible from the Internet. Notably amongthese are the AfCS-Nature Signaling Gateway(http://www.signalinggateway.org), the Bio-molecular Interaction Network Database (BIND,http://www.bind.ca/) (146), the Database ofInteracting Proteins (DIP, http://dip.doe-mbi.ucla.edu/) (147), the EcoCyc (http://www.ecocyc.org/) (148), the Kyoto Encyclopedia ofGenes and Genomes (KEGG, http://www.genome.ad.jp/kegg/) (149), and the SignalingTransduction Knowledge Environment (http://stke.sciencemag.org/).

The blooming efforts to collect and docu-ment experimental data on cellular processes

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reflect encouraging recent progresses in acquir-ing such data in large quantities. More impor-tantly, the documented data are beginning toform an information infrastructure for high-level data compilation and analysis. But muchchallenge is still ahead, especially if one wishesto make efficient use of such information to cre-ate mathematical models. In contrast to gene orprotein sequences, whose data structure isstraightforward to define, complexity of bio-logical interactions as well as different waysthat researchers perceive as interactions havehindered data representation (150). The diver-sity of data format, data quality, notations, andaccess interfaces will probably pose a hurdleeven greater than incompatible modeling lan-guages for communicating these interactiondata. To maximally benefit the research com-munity, the databases eventually need to con-verge into using common standard dataformats. Again, XML or XML-based standardlanguages may prove useful for addressingthe compatibility issue. In fact, XML hasalready being adopted in KEGG to representmetabolic pathways. A potential advantage ofdatabases built upon a standard representa-tion language is that the stored data can bereadily mapped into an integrated mathemati-cal model. In the long run, standardization ofdata representation languages may not onlyfacilitate the construction of the informationtechnology infrastructure per se, but also pro-vide guidance and incentive for experimentalbiologists to document data in a more consis-tent fashion.

ACKNOWLEDGMENT

I thank Edward Massaro for inviting me towrite this review. I gratefully acknowledgeJohn Yin and Frances Arnold for their adviceand guidance. Chris Rao and two anonymousreviewers provided many valuable commentsto the manuscript. Financial support is pro-vided by the Defense Advanced ResearchProjects Agency (DARPA) under Award No.N66001-02-1-8929.

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