Mathematical and Computational Modeling in Complex Biological Systemsdownloads.hindawi.com/journals/bmri/2017/5958321.pdf · 2019-07-30 · ReviewArticle Mathematical and Computational

Review ArticleMathematical and Computational Modeling inComplex Biological Systems

Zhiwei Ji1 Ke Yan2 Wenyang Li3 Haigen Hu4 and Xiaoliang Zhu1

1School of Information amp Electronic Engineering Zhejiang Gongshang University 18 Xuezheng Road Hangzhou 310018 China2College of Information Engineering China Jiliang University 258 Xueyuan Street Hangzhou 310018 China3Chongqing Key Laboratory of Oral Diseases and Biomedical Sciences and College of StomatologyChongqing Medical University Chongqing 400016 China4Institute of Computer Vision College of Computer Science and Technology Zhejiang University of TechnologyHangzhou 310023 China

Correspondence should be addressed to Zhiwei Ji jzw18hotmailcom

Received 16 October 2016 Revised 20 December 2016 Accepted 16 January 2017 Published 13 March 2017

Academic Editor Xin Gao

Copyright copy 2017 Zhiwei Ji et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The biological process andmolecular functions involved in the cancer progression remain difficult to understand for biologists andclinical doctors Recent developments in high-throughput technologies urge the systems biology to achieve more precise modelsfor complex diseases Computational and mathematical models are gradually being used to help us understand the omics dataproduced by high-throughput experimental techniquesTheuse of computationalmodels in systems biology allows us to explore thepathogenesis of complex diseases improve our understanding of the latent molecular mechanisms and promote treatment strategyoptimization and new drug discovery Currently it is urgent to bridge the gap between the developments of high-throughputtechnologies and systemic modeling of the biological process in cancer research In this review we firstly studied several typicalmathematical modeling approaches of biological systems in different scales and deeply analyzed their characteristics advantagesapplications and limitations Next three potential research directions in systems modeling were summarized To conclude thisreview provides an update of important solutions using computational modeling approaches in systems biology

1 Introduction

In recent years the area of precision medicine was signifi-cantly promoted by the rapid development of next generationsequencing which implies lower cost and higher throughput[1 2] In the meanwhile high-throughput mass spectrumwas widely used to measure the protein expression andposttranslational modification and then generated variouskinds of proteomic and metabolic data [3] In additiongeneral public databases and platforms such as GEO TCGAand ENCODE also provide data for analysis and knowledgediscovery [4] Systems biology which usesmultiomic data fordeep analyses and predictions potentially provides insightsof the mechanisms of complicated diseases particularly asvarious cancers in human [5ndash7]

At present people are more interested in the discoveryof new drugs for cancer therapy even though molecular

and cell biology had greatly improved our understandingof many diseases in past decades The essential linkagebetween basic science and effective treatment was lostwhich is the inference and analysis of biological networks[8] Computational or mathematical modeling of biologicalsystems at multiple scales is an effective way to discovernew drugs for cancer therapy in clinic In the intracellularscale these networks explain how cells regulate signaling ormetabolic pathways to respond the external perturbationsor drug treatment [9] In the intercellular scale cell-cellcommunication networks reflect how different cell typescommunicate through various ligands to promote tumorgrowth metastasis and angiogenesis [10] In the tissue scalehow these ligands distribute and diffuse in the 3D tumorspace was also valuable to be studied [11] With the advanceof high-throughput technology systems biology devel-oped rapidly however the development of mathematical

HindawiBioMed Research InternationalVolume 2017 Article ID 5958321 16 pageshttpsdoiorg10115520175958321

2 BioMed Research International

ODE system

Petri net

Booleannetwork

Linearprogramming

Cell-Cell interaction networks 3D tumor growth

Tissue scale Intercellular scaleIntracellular scale

Signaling pathways

Gene regulatory networks

Metabolic pathways

ODE systems in ABM

Gene arrayRNA-Seq

Proteomicswestern blot Metabolics

Ligand production

Ligand binding

Gene array Cell viabilityHampE images

Animal model

Simulate dynamic of intracellular events Simulate cell-cell communications Liganddrug diffusiontumor angiogenesis

Blood vessel network

Functions

Typicalmodels

Biologicalnetwork(systems)

Data used inmodeling

PDE systemODE system

Stochastic model in ABM Stochastic model in ABM

Cytokine assay

Probabilisticmodel

Dirichlet boundary

Dynamical diffusion of ligands or drugs

Figure 1 The whole picture of the systemic modeling approaches introduced in this work

modeling approaches suffers from new biological questions[12]

In this review we firstly studied several well-establishedsystems modeling approaches of biological networks such asordinary differential equations Petri net Boolean networkand linear programming Secondly we summarized the typ-ical modeling studies for the cell-cell communications (suchas tumor-stromal interactions tumor-immune interactionsand stromal cell lineage process) in the heterogeneous tumormicroenvironment that is agent-based modelThirdly threepotential directions of multiscale modeling in systems biol-ogy were deeply discussed We believe that this work canprovide a big picture of systemicmodeling in systems biologyas well as promoting the development of precision medicinein the near future

2 Several Classical SystemicModeling Approaches

With the development of the high-throughput experimenttechnologies (such as gene microarray RNA-seq mass spec-trometry and metabolic profiles) computational and math-ematical modeling of biological processes provides deepinsights of the complex cellular systems [13] Researchersbuilt various computational models to elucidate the com-plex behaviors of cancers such as tumor progression drugresistance and immune inert It is well-known that bioin-formatics is data-driven [14 15] However systems biology ishypothesis-driven [16 17] since we often generate a testablehypothesis based on small-scale experimental observations

and then construct a systemicmodel based on this hypothesisto obtain mechanistic insights In this study we mainlyfocus on several classic systemic modeling approaches andtheir applications in current cancer research These popularmodeling approaches can simulate the dynamic changes ofregulatory networks (signaling pathways andmetabolic path-ways) tumor growth and its microenvironments such asordinary differential equations (ODEs) [10] Boolean network[18] Petri nets [19] linear programming (LP) basedmodel [920] agent-basedmodel [11] and the system biologymodelingapproach considering genetic variation [21] We presentthese models in Figure 1 Although there are many availablereverse-engineering [22] algorithms for the inference of generegulatory networks [23] such as ARACNe [22] and MINDy[24] we omitted them in this review since they are bettersuited to be categorized in the field of bioinformatics

21 ODE-Based Modeling With the rapid development ofcomputer performance ordinary differential equation (ODE)based approaches are widely used for continuous dynamicmodeling in complex biological systems [25] ODE-basedmethods represent the interactions among various biologicalmolecules (such as protein kinases or metabolites) whichreflect the time-varying effects of biological processes [26]Based on the different biological hypotheses the currentODE-based methods can be categorized into three types thelaw of mass action [27 28] Hill function [29] andMichaelis-MentenKinetics [30]The choice of a specificmethod dependson the biological questions or the experimental data Here weillustrate how to use these kinetic approaches to describe thebiochemical reactions

BioMed Research International 3

Law of Mass Action The law of mass states that the reactionrate is proportional to the probability of the collision ofthe reactants This probability is also proportional to theconcentration of reactants to the power of their molecularityand the number of them entering the specific reaction[31] For example a reaction between 119860 119861 and 119862 can berepresented as

119860 + 119861119870ON997888997888997888rarrlarr997888997888997888119870OFF

119862 (1)

By the definition of mass action law we can derive theconcentration change over time of the above two reactants(119860 and 119861) and one product (119862) by the following ODEs

119889 [119860]119889119905 =

119889 [119861]119889119905 = 119870OFF [119862] minus 119870ON [119860] [119861]

119889 [119862]119889119905 = 119870ON [119860] [119861] minus 119870OFF [119862]

(2)

Hill Function In the ODE models of signaling pathways Hillfunctions are generally used to represent a proteinrsquos activationor inhibition which are induced by their upstream parentalnodes In biochemistry the binding from a ligand towardsa large molecule can be enhanced if there is also anotherligand binding to it which is called cooperative binding Foreach protein involved in the signaling pathways the dynamicchanges of its expression can be described by Hill functionsas shown in the following formula

119889119910119889119905 =

119872

sum119894=1

119891+ (119909119894) +119873

sum119895=1

119891minus (119909119895) minus 119910 lowast 119889119910 (3)

119891plusmn (119909) = 119896119909119910119909plusmn119899119909

119867plusmn119899119909119909 + 119909plusmn119899119909 (4)

where 119910 is the concentration of activated protein 119909119894 (119894 =1 2 119872) is the 119894th protein which activates protein 119910 and119909119895 (119895 = 1 2 119873) represents the 119895th protein which inhibitsprotein 119910 In formula (4) 119891plusmn(119909) is the activating profile (+)or inhibiting profile (minus) induced by protein 119909 respectively119896119909119910 represents activating rate (+) or inhibiting rate (minus) 119867119909is the microscopic dissociation constant and 119899119909 is the Hillcoefficient 119889119910 is the degradation rate of protein 119910

Michaelis-Menten Kinetics When a signaling reaction iscatalyzed by an enzyme (kinase or phosphatase that isnot consumed or produced) it may form a temporarycomplex with substance in the reaction For such reactionthe Michaelis-Menten Kinetics can be used to describe thereaction rate under the key assumption of quasi-steady-stateapproximation while enzyme concentration is much lowerthan the substrate concentration and the enzyme is notallosteric [11] TheMichaelis-Menten Kinetics is expressed as

V = 119881max119878119870119898 + 119878

(5)

where119870119898 is the Michaelis constant

Generally ODE systems are suitable for modeling small-scale networks since there are many parameters need to beestimated If the network scale is large parameter estimationwill lead to high computational cost and the predictionaccuracy of model may decrease Therefore searching ofthe optimal parameters for an ODE system is a challengingquestion Intelligent algorithms with heuristic search suchas Genetic Algorithm (GA) or Particle Swarm Optimization(PSO) are often used as a heuristic strategy to obtain theparameters in ODE functions [28 32] In addition scattersearch potentially find solutions of a higher average qualitythan GA [33 34] Moreover Stochastic Ranking EvolutionStrategy (SRES) is incorporated in some computationalstrategies for parameter estimation in biological modelsincluding signaling pathways and gene regulation networkssuch as SBML-PET (Systems Biology Markup Language-based Parameter Estimation Tool) [35] and libSRES (C libraryfor Stochastic Ranking Evolution Strategy for parameterestimation) [36]

ODE-based approaches have been used in the underlyingmechanisms of complicated diseases in intracellular andintercellular levels Peng et al defined a series of ODE-basedapproaches with the law of mass action to creatively simulatethe intracellular pathways and obtain important biologicaldiscoveries [37ndash39] For example they developed an ODE-basedmodeling approach to comparably assess the inhibitioneffects of single or combined treatment of drugs on NFKBpathway in multiple myeloma cell and further predict thesynergism of drug combinations [38] Shao et al proposedODEs with Hill functions to study the signaling network sig-natures by integrating both therapeutic and side effects Thismodel was used to screen 27 kinase inhibitors for optimaltreatment concentration [29] Sun and colleagues designedan ODE-based model with Michaelis-Menten for modelingthe antiapoptotic pathways in prostate cancer and illustratedthe molecular mechanisms of psychological stress signalingin therapy-resistant cancer [30] Furthermore ODEs werealso successfully applied to describe the dynamic changesof metabolites in the small-scale metabolic reaction systems[40]

In particular ODEs have begun to be used to modelthe cell-cell interactions [10 32] For example Peng andcoworkers developed a novel ODE system to understandhow the cell-cell interactions regulate multiple myelomainitiating cell fate [32] The results from this dynamic systemmay be potentially useful for understanding mechanism ofcancer stem cells development Peng et al proposed a mul-tiscale multicomponent mathematical model to explore theinteractions between prostate tumor and immune microen-vironment using ODE-based strategy in both intercellularand intracellular levels [10] This study highlights a poten-tial therapeutic strategy in effectively managing prostatetumor growth and provides a framework of systems biol-ogy approach in studying tumor-related immune mecha-nism

Together all above works indicate that ODE-based mod-els are suitable for modeling the continuous changes ofkinetics in small-scale intracellular or intercellular networks[41 42]


2

Hydrogen

Oxygen

Reaction 1

2

Water

(O2)

(H2)

(H2O)

(a)

2

Hydrogen

Oxygen

Reaction 1

2

Water

(O2)

(H2)

(H2O)

(b)

Figure 2 An example of Petri nets (selected from literature [43]) (a) shows the initial marking before firing the enable transition t (b) showsthe marking after transition labeled reaction 1 fires

22 Petri Net-Based Modeling Approaches Petri net (PN)developed by Petri in 1962 was a graphical mathematicalmodeling tool applicable to a wild range of technical systems[43] Recently increasing number of studies are involved inthe utility of PNs in systems biology such as modeling ofsignaling pathways metabolic pathways and gene regulatorynetwork [19 44 45] A PN is a directed weighted bipartitegraph consisting of two types of nodes places and transitionsAs shown in Figure 2 places are represented by circles andtransitions are represented by boxes Through transitionfirings the source influences the number of tokens assignedto the target called token-count A place that has an outgoingarc towards a transition 119905 is known as input place of 119905 aplace that has an incoming arc from a transition 119905 is outputplace of 119905 Arcs are labeled with weights that represent theminimum tokens required by input places to enable thetransition When a transition fires it removes a token fromeach place connected to it by inputting arcs and adds a tokento each place connected to it by output arcs Therefore aPetri net can be defined as a 5-tuple (119875 119879 1198651198821198720) where119875 = 1199011 1199012 119901119898 is the set of places and 119898 is the totalnumber of places 119879 = 1199051 1199052 119905119899 is the set of transitionsthe set of arcs 119865 = (119875times119879)cup (119879times119875) and119882 119865 rarr 119873 119873 beingthe set of natural numbers is called theweighting function Inorder to simulate a dynamic process a number of tokens areassigned to each place indicating the quantitative propertyThis assignment of tokens to all the places represents thesystem states which is called a marking The initial markingof a PN is a mapping1198720 119875 rarr 0 1 2 From the initialstate 1198720 changes in the system are simulated by executingthe PN and a series states can be obtained (11987211198722 )

For biological molecular network modeling we usuallyconsider a token to be a unit of weight of a molecule Aplace-transition or transition-place connection is made by aweighted arc (directed edge) designating how much of theinput places (reactants) are required to produce tokens forthe output places (products) in a reaction A transition canonly fire when it is enabled meaning that each of its inputplaces has at least one token in the current marking Thetransition may fire (reaction occurs) afterwards If transition

119905 when fired on a marking1198721 produces marking1198722 thenwe write 1198721 | 119905 gt 1198722 Obviously this notation can beextended to represent the effect of firing a series of transitions120590 = (1199051 1199052 119905119895)

As a type of dynamic modeling approach PN was mainlyused to simulate not only signaling pathway networks [46 47]but also gene regulatory networks [48] In a signaling Petrinet its aim is to predict signal flow through a cell-specificnetwork in experimental conditions [49] Each place isdenoted to the activated signaling protein and each transitionis associated with a unique phosphorylation event

In fact PN is a discrete dynamicmodeling strategy whichsimulates signaling network with multiple states of placesAfter the topology of a network is determined PN is suitableto analyze the global property of the system characterizedas concurrent asynchronous distributed parallel nonde-terministic and stochastic As a novel systemic modelingstrategy to describe the biological systems with graphicalnotation PN can be used at multiple levels of abstractionand accommodate timing information Therefore it forms alanguage that allows the automatic generation of a specificsimulation However the obvious disadvantage of PNs is thatthe conception of PNs is too primitive so that the graphicalrepresentation may become too complex for analysis Hencedeveloping optimized PN models to reconstruct the specificbiological networks is still an important topic

23 Boolean Modeling Approaches In a Boolean networkeach node is described with binary states which are denotedby 1 and 0 corresponding to for example activationinactivation of a protein respectively The time variable isconsidered to be discrete The future state of a node ateach time step is determined by the current states of allits input nodes (parents) through a Boolean function Foreach Boolean function 119865 it can be represented as a mapping119861 0 1119896 rarr 0 1 This mapping denotes that the out-come of a node was determined by its 119896 parent nodes Andthe mapping 119861 also can be represented as a truth table[13] These Boolean functions are usually expressed together


V1

V2 V3

V4

(a)

A

B

C

D

Node

V2lowast= B1(V1 V3) = V1 AND (NOT V3)

V3lowast= B2(V1 V2) = (NOT V1) OR V2

V4lowast= B3(V2 V3) = V2 AND V3

Boolean function

mdash

(b)

1101 1110 1011 1000

0000 0010

(c)

Figure 3 An example of Boolean network (a) Network topological structure (b) the definition of Boolean functions (c) state transition ofBoolean network

with the logical operators including AND OR and NOTGiven a Boolean network including 119899 nodes there are 119899Boolean variables (1205901 1205902 120590119899) and 119899 Boolean functions(1198611 1198612 119861119899) There are generally two types of strategiesfor updating the network states synchronization [50] andasynchronization [51] In the synchronous pattern the statesof all the nodes are updated simultaneously as shown in thefollowing formula

120590119894 (119905 + 1) = 119861119894 (1205901198941 (119905) 1205901198942 (119905) 120590119894119896119894 (119905)) (6)

The asynchronous pattern can be expressed by the followingformula

120590lowast119894 (119905 + 1) = 119861119894 (1205901198941 1205901198942 120590119894119896119894 ) (7)

Formula (6) indicates that the state of node 120590119894 at time point119905 + 1was determined by the combination of the states of its 119896119894parent nodes at time point 119905 However the input variables onthe right side of formula (7) might be at different time points

The state of a Boolean network at each time step canbe expressed as a vector whose elements represent the stateof all the nodes at that time step By updating the statesof nodes at each time step the network states vary overtime which is called ldquostate transitionrdquo in BN [9 52] In factBoolean network modeling is between static network modeland continuous network model (such as ODEs) which is

especially suitable for large-scale network and is reputed forits high efficacy [9] An example of Boolean network is shownin Figure 3 to elaborate the above conceptions Figure 3(a)shows the topological structure of a Boolean network whichincludes four nodes 1198811 1198812 1198813 and 1198814 and their states aredefined by three Boolean variables 1205901 1205902 1205903 and 1205904 ThreeBoolean functions are listed in Figure 3(b) Figure 3(c) showsthe state transition graph Given a starting state the networkwill eventually converge to a steady state which is calledldquoattractorrdquo [52] According to Figure 3 we can easily concludethat the analysis of Boolean network always depends on anassumption the network structure is determined in advance

Generally speaking Boolean dynamic modeling of reg-ulatory network follows three steps (1) reconstructing thenetwork (2) identifying Boolean functions from the networktopological structure (3) analyzing the dynamics of thesystem with or without node perturbations As a parameter-free model it works efficiently even for large-scale networks

During the last decade researchers carried out manystudies of reconstruction of gene regulatory networks usingBoolean networkmodeling [18 53] which reflects the genericcoarse-grained properties of large genetic network Thehypothesis for the Boolean networks as models of generegulatory networks is that during regulation of functionalstates the cell exhibits switch-link behavior This hypothesisis important for cells to transfer from one state to anotherduring a complex biological process after the cells received


the external stimulations or perturbations [54] Drsquohaeseleeret al discussed the way to cluster coexpression profilesto infer large-scale gene regulatory network from high-throughput gene expression assays [55ndash57] Moreover theintrinsic properties of Boolean network such as stability[58 59] robustness [60] and fragility [51] for gene regulatorynetworks were deeply studied which speeded up the Booleannetwork development and applications in more areas Inaddition Boolean network modeling approaches have beenimproved in other fields Zhao and Ouyang et al proposedan algorithm for inferring gene regulatory networks by usingmodified Boolean network from a time series dataset [61ndash63] However Boolean network only provides a very limitedquantitative insight in biological systems due to their inherentqualitative nature of state and time In order to overcome thedeterministic rigidity of BNs probabilistic Boolean network(PBN) was introduced for the modeling of gene regulatorynetworks [64ndash67] Generally PBNs combine the rule-basedmodeling of Boolean networks with uncertainty principlesas described by Markov chains [68] Modeling with PBNsprovides a quantitative understanding of biological systemssuch as interactive effects between genes or average activitiesof certain genes given by steady-state probabilities [69] PBNsthus have been widely applied to various biological processes[64 66 70 71]

In recent years Boolean dynamic modeling begins to beapplied to signal transduction networks analysis Andersonet al proposed an approach for integrating gene set enrich-ment methods with Boolean dynamic modeling to revealthe induction of a densely connected network of cellular(TFs) and molecular (ligands) signaling upon influenza virusinfection of dendritic cell [72] Kaderali and colleagues alsoinferred signaling pathways from gene knockdown datausing Boolean networks with probabilistic Boolean thresholdfunctions [73] PBNs were also applied to study the crosstalkrelevancy in a given signaling pathways or simulate theoutcome with external perturbations [67 74]

Since many years ago Boolean network was widelyapplied in the studying of network modeling and analysishowever the key signaling pathways for some diseases orcancers are still unknown which limits the use of BN insuch cases Although the topological structure of signalingpathways can be determined with the prior knowledge thespecific pathways of some cancer cells are still unclear Basedon above questions Saez-Rodriguez et al developed a noveldiscrete logic model to infer cell-specific pathways [75]The proposed model represents the topological structure ofsignaling pathway as Boolean network (two states of nodes)Based on the experimental observations on a part of proteinnodes this model infers an optimal subnetwork from theoriginal generic pathway map as the cell-specific pathwaysfor further predictionsThis is the first work to implement theinference and optimization of cell-specific pathways using theconception of Boolean or discrete networks

24 Linear Programming Approaches Linear programming(LP) as an important subject in mathematics first appearedin the 1950s [76] Linear programming is the problem ofmaximizing or minimizing a linear function over a convex

polyhedron specified by linear and nonnegativity constraintsSimplistically linear programming is the optimization of anoutcome based on some set of constraints using a linearmathematical model In general linear programmingmodelscan be categorized into four types (1) Integer Programming(IP) defining a part of or all variables as integers (2)Binary Linear Programming (BLP) denoting all variables asbinary numbers [9] (3)Mixed Integer Programming (MIP)constraining that only a part of variables are integer [77] Inaddition nonlinear programming is also very common in thereal world which refers to the mathematical programmingwhich has nonlinear constraints or objective function [78]such asMixed Integer Quadratic Programming (MIQP) [79]

In recent years LP-based approaches were applied in thereconstruction of gene regulatory networks [80] metabolicnetworks [20] and inference of cell-specific signaling net-work [9 81 82] In the following paragraphs we will summa-rize how LP is utilized to model these molecular networks

Orth et al creatively proposed a novel computationalapproach for the genome-scale metabolic network recon-structions that is called flux balance analysis (FBA) [20] Themetabolic networks contain all knownmetabolic reactions inan organism and the genes that encode the enzyme of eachreaction [83 84] The fundamental theory of FBA is that theflowing energy between input and output should be balancedand FBA makes it possible to predict the growth rate ofan organism or the production rate of a biotechnologicallyimportant metabolite FBA-based approaches reconstructlarge-scale networks and LP is an effective strategy to solvethis kind of optimization problem [20] The general workflow of FBA is shown in Figure 4 According to the conceptof FBA a metabolic network reconstruction consists of alist of stoichiometrically balanced biochemical reactions andeach reaction is controlled by an enzyme (Figure 4(a)) Thereconstruction is converted into a mathematical model byforming a stoichiometric matrix 119878 in which each row rep-resents a metabolite and each column represents a reaction(Figure 4(b)) At steady state of the network the flux of eachreaction is given by 119878 sdot 119881 = 0 which defines a combinationof linear equations (Figure 4(c)) By defining an objectivefunction the linear programming with a set of constraintsidentifies the solution vector 119881 in a subspace (Figures 4(d)and 4(e)) On one hand FBA provides a way to reconstructmetabolic network in the scale of entire genome On theother hand most FBA applications do not consider the ther-modynamic realizability Hoppe was the first to put forwarda method by including metabolite concentrations in fluxbalance analysis [85] They demonstrated the usefulness oftheir method for assessing critical concentrations of externalmetabolites preventing attainment of ametabolic steady state

Another very important aspect of the LP-based strategiesis that those strategies were developed to infer cell-specificsignaling pathways with experimental proteomic data [9 8182 86] The rational of this type of approaches is that therelationships (states) between a child node and its connectedparental nodes were defined by a set of constraints Accord-ing to the experimental observations of a part of nodessome redundant edges in the network can be detected andthen removed in the process of optimization The inferred


Genome-scale metabolic reconstruction

Representing metabolic reactions and constraints

Mass balance defininga system of linear equations

Objective function

Linear constraints Optimize Z

Optimal solution

Allowable subspace

Original space

(a)

(b)

(c)

(d)

(e)Calculating fluxes that optimize Z

B + 2C rarr D

Reaction 1

Reaction 2

Reaction n

V1

V2

Vn

VbiomassVglucoseVoxygen

= 0

Fluxes V

lowast

Met

abol

ites

Stoichiometric matrix S

Reactions

Biom

ass

Glu

cose

Oxy

gen

minus1

1

1

minus1

minus2

1

minus1

minus1

1 2 middot middot middot nABCD

m

etc

Z = c1 lowast 1 + c2 lowast 2 + c3 lowast 3 + middot middot middot

1 1 1

2 2 2

3 3 3

A harr B + C

minusV1 + middot middot middot = 0

V1 minus V2 + middot middot middot = 0

V1 minus 2V2 + middot middot middot = 0

V2 + middot middot middot = 0

Figure 4 Flux balance analysis for metabolic network reconstruction (selected from literature [20])

cell-specific signaling network therefore generally becomes asubgraph of the generic pathway map Mitsos was the firstresearcher to propose an ILP-based approach for inferringcell-specific signaling pathways with phosphoproteomic dataand identify drug effects via pathway alterations [82] InMitsosrsquos approach the protein nodes were represented asBoolean states (ldquoactivatedrdquo or ldquoinactivatedrdquo) which is similarto the Boolean networkThe innovation is that the edges weredefined by Boolean variables and the states of connectednodes were constrained by mathematical rules Thereforegiven the observations of a part of nodes in the network thestates of all the nodes and links can be predicted with thedefined constraints and the inconsistent edges are removedfrom the topological structure of the generic pathwaysHowever the constraint system developed byMitsos can onlyaddress very simple topological structures of singling path-ways and the generalization of their method is so limitedWe proposed an improved Boolean Integer Programmingbased approach to overcome this limitation and definedfour linking patterns of connected proteins for modelingthe complex signaling networks [9] However Boolean states

defined in above approaches have obvious insufficiency forthe variations of phosphor-signals under different condi-tions Thus we further proposed a novel discrete modelingstrategy (DILP) to represent relative changes of phosphor-proteins with three states (0 minus1 and 1 denote ldquono changerdquoldquodownregulationrdquo and ldquoupregulationrdquo) and the edges arestill with binary states DILP was then applied to inferosteoclasts-mediated myeloma cell-specific pathways undernormoxic and hypoxic condition on time series proteomicdata and finally revealed that how OC-myeloma interactionin a hypoxic environment affects myeloma cell growth anddrug response [81] In summary as a type of parameter-freemethod LP-based model provides an innovative strategy formodeling large-scale molecular networks

25 Agent-BasedModel of Biological Systems An agent-basedmodel (ABM) is another class of computational models forsimulating the actions and interactions of autonomous agentswith a view of assessing their effects on the system as awhole [87] In systems biology ABM is usually used to modeltumor growth (drug response) and angiogenesis in the cancer


microenvironment Each cell type in this system is consideredas an agent and the complicated cell-cell communicationsare achieved through some secreted ligands [88ndash94] Theimplement of ABM is usually based onMarkov ChainMonteCarlo By ABM modeling we can stimulate the process ofstromal cell lineage tumor growth and angiogenesis in thecellular and tissue level and the process of signal transductionin the molecular level Some typical research works aboutABMmodeling were summarized as follows

Currently there are two types of agent-based models2D ABM [95] and 3D ABM [96] Solovyev et al proposeda two-dimensional agent-based model of ischemia-inducedhyperemia and pressure ulcer formation This model defineda 2D space of pressure ulcer formation and simulated theinteractions among skin cells inflammatory cells and bloodvessels through the cytokines TNF120572 and TGF120573 [95]

3D ABM models in systems biology were often designedto mimic the dynamic changes of tumor tissues and variousinteractions with other types of cells (stromal cells tumorcells and immune cells) in the heterogeneous microenviron-mentTherefore 3DABMmodels potentially integrate eventsat different spatial and temporal scales For the spatial scalesthe cell behaviors such as tumor cell migration and invasionand effector cell-induced clearance of target cells can besimulated in the 3D space For the temporal scales short-termintracellular signaling dynamics medium-term cell divisionand apoptosis long-term drug response and tumor growthalso can be modeled Su et al established an ABM modelusing the Markov Chain Monte Carlo approach to simulatethe effects of SDF1 induced chemophysical communicationsamongMICs and BMSCs onmyeloma cell growth and exam-ine whether the biophysical properties of myeloma nichesare druggable with two representative drugs [96] This studyprovided a typical approach to simulate the process of tumor-stromal cell lineage and intracellular signal transduction withHill function Particularly we further developed a hybridmultiscale agent-based model (HABM) that combines anODE system and agent-based model [97] The ODE systemwas used tomodel the dynamic changes of intracellular signaltransductions and the ABM is used to model the cell-cellinteractions between stromal cells cancer cells and immunesystem It is the first work to study systemic modeling oftumor growth and immune response within an integrated3D model (Figure 5) In addition tumor progression isrelated to angiogenesis therefore it is necessary to integratevascularization into ABM model to reflect the endothelialcells interaction with cancer cells through some key factorssuch as VEGF [98] Wang et al proposed an ABM modelthat integrates the angiogenesis into tumor growth to studythe response of melanoma cancer under combined drugtreatment [99] The diffusions of ligands or drugs in thetumormicroenvironments are always simulated by PDE [97]

To consider the real situation in tumor microenviron-ment multiscale ABM modeling tries to simulate the cellmigration in tissue level cell-cell communication throughligands in intercellular level and dynamical signal transduc-tion in intracellular level in an integrated system [11 100]For example Sun et al developed a multiscale ABM modelto study cell responses to growth factors released from a

Cell B

Cell A

Cell D

Ligand

Cell

Figure 5 A general framework of agent-based model

3D biodegradable porous calcium phosphate bone scaffoldSunrsquos model reconstructed the 3D bone regeneration systemand examined the effects of pore size and porosity on boneformation and angiogenesis

Although current ABM models are able to simulate thetissues organs and microenvironment closely enough to thereal situation the variability of model outcomes should beconsideredTherefore replicates are necessary to calculate themodelsrsquo average performance Uncertainty analysis shouldalso be used to represent the variability of the model results

26 Integrating Genetic Variation to Infer Cancer NetworksPreviously the systemic modeling of intracellular pathwaynetworks depends on an assumption that all cells in atumor tissue share the same pathways and the data (egwestern blot and gene array profiles) used in the processof modeling reflects the average expression of moleculesThe heterogeneous genetic variations occurred in tumorcell population are not considered in the existing modelsHow to integrate the information of genetic variations in thesystemic modeling work is now a new topic AlQuraishi et alfirst proposed a multiscale statistical mechanical frameworkintegrated genomic binding and structural data to predictthe effects of specific mutations on PPI networks and cancer-related pathways [21] Based on the concept of Hamiltonianthey modeled how the mutations in SH2 domains inducednetwork alterations and the experimental results validated theproposed model We believe this interesting topic will attractresearchersrsquo wide attention

3 Current Three Hot Directions inSystems Biology

Cancer is a genetic disease driven by mutations in keygenes that lead to uncontrolled growth and abnormal cellbehavior However the fact that tumor is living in a complexheterogeneous microenvironment drives the tumor progres-sion as well as treatment resistances Understanding theinterplay between homeostasis heterogeneity and evolutionin cancer progression is currently hot topics Multiscalecomputational modeling has strong potential to bridge the


gap between precision medicine and translational systemsbiology in which quantitative metrics and data guide patientcare through improved stratification diagnosis and therapy

31 Dynamics of Cell-Cell Interactions in Tumor ProgressionThe reciprocal relationship between cancer cells the hostimmune system and the tumor microenvironment evolvesduring the cancer progression [101] How these dynamicand unstable interactions prevent or drive tumor initiationand progression is not well understood At present someresearchers start to focus on predictive and testable hypothe-ses of how dynamic cell-cell (tumor-tumor [96] tumor-stroma [102] tumor-immune [103] and immune-stroma[104]) communication affects cancer development and theresponses to various therapies Due to the complex nature ofthese interactions mathematical and computational modelsare ideal tools to elucidate them and make predictions thatare profitable for both experiment and novel therapeuticapproaches [105ndash108] Su et al proposed an agent-basedmodel to mimic the multiple myeloma cancer cell lineageprocess and the interactions between myeloma initial cell(MIC) and bone marrow stromal cells [96] The com-putational model simulated the myeloma progression anddrug resistance driven by various cell-cell communicationsSimilarly to understand the mechanism of multiple cell-cell interactions involved in circulating tumor cell adhesionUppal et al used an ABM of Early Metastasis (ABMEM)to dynamically represent the hypotheses of essential stepsinvolved in circulating tumor cell adhesion and interac-tion with other circulating cells examine their functionalconstraints and predict effects of inhibiting mechanismsThe results show that the ABMEM successfully captures theessential interactions of the whole process and allows insilico iterative characterization and invalidation of proposedhypotheses regarding this process in conjunction with invitro and in vivo models [109] Furthermore both the innateand adaptive immune response have been demonstrated toinduce tumor cell death [110 111] Conversely cancers areassumed that have developed adaptive strategies to evadeimmune attack However systemic modeling of tumor for-mation growth (development and progression) and immunefunctions (including macrophage CTL and Treg) withinan integrated computational system was still rarely studiedTherefore modeling the interactions of cancer cells stromalcells and immune cells in its microenvironment may poten-tially improve our understanding of tumor growth immunetolerance and drug resistance [97]

32 Systems-Level Analyses of the Role of the Heterogeneityand the Microenvironment Usually researchers consider atumor to be a heterogeneous population containing poten-tially many distinct cellular phenotypes (variations of cell-specific traits such as cell-cell adhesion migration speed andproliferation rate) and via proliferation each cell had a smallchance to mutate to one of these phenotypes in a randommanner A number ofmodeling strategies have suggested thatthe local microscopic heterogeneity can vary wildly in tumorsuch as blood vessel cellular density and metabolism [112113] In another aspect genetic heterogeneity (eg mutation)

within a tumor continues to be of great interest to the cancercommunity [114] A major question in biology is how toconnect genotype with phenotype We believe that onlythrough the integration of computationalmodels with carefulexperimentation the gene-centric and microenvironment-centric views of cancer progression can be bridged

Furthermore tumor microenvironment (mE) is tempo-rally and spatially heterogeneous due to the variations inblood flow resulting in local fluctuations of nutrients (egO2) growth factors [115] extracellular matrix and othercellular populations [116] The mE is considered to play acrucial role in driving the evolution of aggressive tumorphenotypes however the problem of how the mE modulatesthis heterogeneous and drives the behavior of the tumor cellpopulation is still far away from being fully understandableby researchers [116] Picco et al developed a 2D hybriddiscrete-continuum cellular model to investigate the role ofenvironmental context in the expression of stem-like cellproperties through in silico simulation of ductal carcinoma[101] They demonstrated that variations in environmentalniche can produce cancers independent of genetic changesin the resident cells Also Ji et al investigated the effectsof oxygen heterogeneous distribution in bone marrow onmyeloma progression via osteoclast-myeloma cell interac-tions and proposed an LP-basedmodel to infer OC-mediatedmyeloma cell-specific signaling pathways under hypoxia andnormoxia [81]The abovemodeling studies guide us to realizethe therapies targeting the mE which may offer an alternativecancer prevention strategy

More recently quantitative measurement technologieshave facilitated the collection of chemical molecular struc-tural interactome and localization datawithin and across cellpopulations in the tumor microenvironment The systemsmodeling and analyses using multiomic data are potentiallyto predict cell behaviors and translate important informationacross space and time

33 Systems Biology Aided Clinical Trial Design Precisionmedicine requires integrating patient-specific characteristicsinto knowledge gained preclinical studies such as differ-ences in multicellular or multiclonal drug response stag-gered temporal dosing schedules and dynamic prediction ofeffective combination therapies Systems biology synthesizesdata obtained from individual reductionist perspectivesfocuses on construction of integrated holistic models ofdeterminants of biological responses and offers an excitingopportunity by which to identify potential therapeutic targets[117] Currently methylation marks transcript abundanceand miRNA profiles are integrated with SNP results fromGWAS to better explain the genome complexity as it mayrelate to pathophenotype and drug treatment response It isa good way to take clinically relevant cells from clinical trialparticipants treat those cultured cells with the relevant drugfrom the clinical trial and determine transcript abundancein the cell as a function of drug treatment response The datacan be further integrated with GWAS data from the trial toperforma regression of SNPon transcript abundance in orderto identify regulatory variants that can be utilized for pathwaymodeling or network analysis More recently using a systems


biology strategy Laaksonen et al demonstrated the treatmenteffects of high dose simvastatin on nonhepatic tissues as wellas a different profile of the effects of atorvastatin on suchtissues They described that the further understanding of theimpact of their lipidomic findings has the potential to lead toindividualized drug and dose selection [118] Another exampleof the potential benefits of a systemic modeling approach isdrug combination therapies [10] Expansion of the catalog ofdrug interactions using a systemicmodeling approach incor-porating pharmacogenomics and computational biology hasthe potential for optimizing pharmacotherapeutic schemes inthe future

Recent developments in molecular analysis and bioin-formatics make targeted treatment feasible however moreefficient multifaceted clinical trial designs are still neededThe studies on the utilization of systems biology approachesin clinic may help us to refine experimental measurementsand improve decision-making about therapies for clinicaltrial planning and ultimately personalized therapy Artemovet al presented a novel computationalmodel for predicting anoptimal personalized treatment for cancer patients based onhigh-throughput gene expression signature of the individualtumor samples The effectiveness of this model was validatedby using clinical trials data [119] Kim et al proposed anODE-based computational framework to implement virtual phase119894 trials in cancer using an experimentally calibrated mathe-matical model of melanoma combination therapy which canreadily capture observed heterogeneous clinical outcomesand be used to optimize clinical trial design [120] Lawleret al suggested potential solutions in precision medicinefor clinical trial design that balanced the cost and value todeliver cost-effective cancer care [121]Thebenefits of Lawlerrsquosapproach can be transferred directly to the patients [121]

4 Discussion

The development of cancer study is a complex multiscalebiological process in which genetic mutations occur ata subcellular level and manifest themselves as functionalchanges at the intracellular intercellular and tissue scale Sig-nificant developments of integrating mathematical modelingapproaches with experimental data provide deep insight ofthe mechanisms that provoke cancer initiation progressionand drug resistance Although various computational modelshave been developed in cancer research studies the currentchallenge is that the knowledge about the mechanistic detailsof many biological systems is still rare On the other handthe real biological systems aremuchmore complicated so thatcurrent established mathematical approaches are insufficientto describe all the details in these systems

In this review we firstly provide a methodology overviewof mathematical and computational modeling in systemsbiology which was well studied to elucidate how com-plex behaviors of biological systems are This category ofapproaches includes ordinary differential equations (ODEs)Petri net (PN) Boolean network (BN) Integer Linear Pro-gramming (ILP) for intracellular network modeling andagent-based model (ABM) for cell-cell communication inintercellular level ODE systems are well-suited to describe

continuous processes that can be approximated as well-mixed systemODE-basedmodeling is often applied to small-scale intracellular network or cell-cell interaction network insystems biology Boolean network a typical discrete mod-eling approach uses well-defined connectivity informationto study the global and dynamical properties of networkBoolean modeling of biological network realizes the statetransition of the computational system among discrete timepoints by defining the states of nodes and edges as binaryvariables [122] Petri net as a discrete dynamic model iswidely used in biological network modeling PN stimulatesthe dynamic changes of kinase in signaling pathways asODEsand represents the state transition between discrete timepoints as BN The linear programming based approach is aninnovative strategy to model large-scale molecular networks[81] LP-based method is parameter-free works efficientlyin the optimization of networks and can be combined withODEs to develop a two-stage hybrid model for signalingpathway reconstructionThe LP-based method was first usedto simplify the generic pathway map to obtain cell-specificpathways Then the ODE-based method was applied onthe inferred cell-specific network to analyze the dynamicchanges of each kinase Finally the agent-basedmodel and itsusage in biological systems are introduced [100] Agent-basedsimulations monitor the actions of a large number of simpleagents in order to observe their aggregated behavior ABM isnot only used to model 2D computational systems [95] (suchas skin injury and inflammation) but also used to model 3Dtumor growth with its microenvironment [96 100]

5 Parameter Estimation Sensitivity andUncertainty Analyses

51 Parameter Estimation Most mathematical models insystems biology face three troubles highly nonlinear modelsa large number of parameters for approximation and thescarce information content of the available experimental dataHence there is a demand for global optimization methodsthat are capable of estimating the parameters efficiently(Figure 6(a))

ODE is widely used to model signaling pathways andmetabolic pathways [29 123] The parameters in ODE sys-tems were often optimized by heuristic search algorithmssuch as Genetic Algorithm (GA) and Particle Swarm Opti-mization (PSO) [39 99] The classic algorithm of GA or PSOusually plunges into local optima therefore some advancedstrategies about the global search of parameters in large-scale optimization are required such as an enhanced scattersearch algorithm for parameter estimation in large-scalesystems biology models [124 125] Moreover StochasticRanking Evolution Strategy (SRES) is incorporated into somecomputational tools for parameter estimation jobs [36 126]For example libSRES [36] is a free C library for StochasticRanking Evolution Strategy for parameter estimation whichis suitable to use under opens source environment

However ABM is a computational model for the stochas-tic process simulating behaviors and interactions betweenautonomous agents where the outcome might be fluctuateeven though the parameters were fixed Generally parameter


Model

Result P1

P2

P3

(a)

Model

Parameter perturbations

Result viability

Sensitivity analysis

Uncertainty analysis

P1

P2

P3

(b)

Figure 6 Uncertainty and sensitivity analyses ofmodel output (a)The baseline of themodel (b) the framework of uncertainty and sensitivityanalysis Based on the changes of each parameter within a range UA is firstly used to analyze the viability of model results And then SA isused to identify which parameters are responsible for the result viability

tuning is a common way to determine the parameters inABM such as Su et alrsquos work [96] We suggest that theresearchers may manually determine the parameters in ABMmodel by tuning and ensure that the simulation outcomes areclose to experiment data For each condition it is necessaryto implement ABM model over hundred times and thedynamic changes of each cell population can be representedby averaging the results of all the times By using this strategythe predicted results of ABMmodel are reliable and stable

52 Sensitivity Analysis Sensitivity analysis (SA) of param-eters provides valuable insights of the parameters thatare responsible for the variability of model outputs (seeFigure 6(b)) Local and global sensitivity analysis approachesare commonly applied in systems biology [127] Taking theODE system as an example some local sensitivity analysesoften increase or decrease only one parameter with a smallrange at a time to learn the impact of small perturbationson the model outputs [29 38] On the other hand globalsensitivity analysis is used to investigate the effects of simul-taneous parameter variations over large but finite rangesIt also explores the effects of interactions between param-eters [128 129] The selection of proper sensitivity analysisapproaches depends on the specific biological models and theexperimental data

53 Uncertainty Analysis Uncertainty analysis (UA) evalu-ates how the variability of parameters propagates throughthe model and affects the output values [130] Differentfrom SA which evaluates how parameter variability con-tributes to model output the objective of UA is to quantify

the distribution of results given uncertain parameters (seeFigure 6(b)) In the process of UA it requiresmultiplemodelsto run where parameter values are randomly chosen fromtheir respective distributions The selection of the samplingmethods used to perform UA is essential The quasi-randomsampling that generates samples more uniformly over theentire parameter space is widely used [131] In UA the meanrepresents the central tendency of the stochastic process andthe variance summarizes the variability of the model output

Secondly we summarized several hot topics of systemsbiology in cancer researchModeling the dynamics of cell-cellinteractions in the process of tumor progression provides sig-nificant insight into the mechanisms of cancer developmentin the complicated microenvironment System-level analysesof the role of the heterogeneity discussed the differencesbetween patients with the same type of cancer the differentcancers for the same patient the behaviors of various celltypes among the tumor tissue and the different genomicchanges among the same type of cells Obviously there arestill many complicated biological processes and phenomenonthat are not explored or understood by human

In summary in order to simulate and represent morecomplicated biological systems the current modeling ap-proaches introduced in this review are still limited Anotherlimitation is that we did not further discuss the studiesof miRNA-mRNA interactions although there are somesystems biology approaches reported in the literature whichare related to functional genomics based analysis [132] Weexpected that in the near future the effects of genomicinformation (eg mutation and alternative splicing) on thecell states or cell behaviors can bewellmodeled Furthermore


the techniques in computer science and mathematics mayprovide some new theoretical models for systems biology

Conflicts of Interest

The authors declare that they have no conflicts of interest

Authorsrsquo Contributions

Zhiwei Ji and Ke Yan contributed equally to this work

Acknowledgments

This work was supported by the National Science Foundationof China (nos 61602431 and 61374094) and the Founda-tion of Zhejiang Provincial Department of Education (no1120KZ0416255) This study was also supported by the Foun-dation of Talentrsquos Startup Project in Zhejiang GongshangUniversity (no 1120XJ2116016)

References

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[87] M Wang B Cribb A R Clarke and J Hanan ldquoA genericindividual-based spatially explicit model as a novel tool forinvestigating insect-plant interactions A Case Study Of TheBehavioural Ecology Of Frugivorous Tephritidaerdquo PLOS ONEvol 11 no 3 Article ID e0151777 2016

[88] X Tong J Chen H Miao T Li and L Zhang ldquoDevelopmentof an agent-basedmodel (ABM) to simulate the immune systemand integration of a regressionmethod to estimate the keyABMparameters by fitting the experimental datardquo PLoS ONE vol 10no 11 Article ID e0141295 2015

[89] P Mina K Tsaneva-Atanasova and M D Bernardo ldquoEntrain-ment and control of bacterial populations an in silico study overa spatially extended agent based modelrdquo ACS Synthetic Biologyvol 5 no 7 pp 639ndash653 2016

[90] J Walpole J C Chappell J G Cluceru F Mac Gabhann V LBautch and S M Peirce ldquoAgent-based model of angiogenesissimulates capillary sprout initiation in multicellular networksrdquoIntegrative Biology (United Kingdom) vol 7 no 9 pp 987ndash9972015

[91] D Guo K C Li T R Peters B M Snively K A Poehlingand X Zhou ldquoMulti-scale modeling for the transmissionof influenza and the evaluation of interventions toward itrdquoScientific Reports vol 5 article no 8980 2015

[92] C S Borlin V Lang A Hamacher-Brady and N R BradyldquoAgent-based modeling of autophagy reveals emergent regu-latory behavior of spatio-temporal autophagy dynamicsrdquo CellCommunication and Signaling vol 12 no 1 article no 56 2014


[93] J Dutta-Moscato A Solovyev Q Mi et al ldquoA multiscale agent-based in silico model of liver fibrosis progressionrdquo Frontiers inBioengineering and Biotechnology vol 2 article no 18 2014

[94] S M Mniszewski C A Manore C Bryan S Y Del Valle andD Roberts ldquoTowards a hybrid agent-based model for mosquitoborne diseaserdquo in Proceedings of the Summer Computer Simu-lation Conference (SCSC rsquo14)mdashPart of the Summer SimulationMulticonference (SummerSim rsquo14) Monterey Calif USA July2014

[95] A Solovyev Q Mi Y-T Tzen D Brienza and Y VodovotzldquoHybrid equationagent-based model of ischemia-inducedhyperemia and pressure ulcer formation predicts greaterpropensity to ulcerate in subjects with spinal cord injuryrdquo PLoSComputational Biology vol 9 no 5 Article ID e1003070 2013

[96] J Su L ZhangW Zhang et al ldquoTargeting the biophysical prop-erties of the myeloma initiating cell niches a pharmaceuticalsynergism analysis using multi-scale agent-based modelingrdquoPLoS ONE vol 9 no 1 Article ID e85059 2014

[97] Z Ji J Su D Wu et al ldquoPredicting the impact of combinedtherapies on myeloma cell growth using a hybrid multi-scaleagent-based modelrdquo Oncotarget vol 8 no 5 pp 7647ndash76652017

[98] J T Daub and R M H Merks ldquoCell-based computationalmodeling of vascular morphogenesis using Tissue SimulationToolkitrdquo Methods in Molecular Biology vol 1214 pp 67ndash1272015

[99] J Wang L Zhang C Y Jing et al ldquoMulti-scale agent-basedmodeling on melanoma and its related angiogenesis analysisrdquoTheoretical Biology and Medical Modelling vol 10 article 412013

[100] X Sun L Zhang H Tan J Bao C Strouthos and X ZhouldquoMulti-scale agent-based brain cancer modeling and predictionof TKI treatment response incorporating EGFR signalingpathway and angiogenesisrdquo BMC Bioinformatics vol 13 article218 2012

[101] N Picco R Gatenby and R Anderson ldquoNiche-driven stem cellplasticity and its role in cancer progressionrdquo IEEE Transactionson Biomedical Engineering vol 64 no 3 pp 528ndash537 2017

[102] J G Scott A B Hjelmeland P Chinnaiyan A R A Andersonand D Basanta ldquoMicroenvironmental variables must influenceintrinsic phenotypic parameters of cancer stem cells to affecttumourigenicityrdquo PLoS Computational Biology vol 10 no 1Article ID e1003433 2014

[103] M Robertson-Tessi R J Gillies R A Gatenby and A RA Anderson ldquoNon-linear tumor-immune interactions arisingfrom spatial metabolic heterogeneityrdquo BioRxiv 2016

[104] J Escamilla S Schokrpur C Liu et al ldquoCSF1 receptor targetingin prostate cancer reverses macrophage-mediated resistance toandrogen blockade therapyrdquo Cancer Research vol 75 no 6 pp950ndash962 2015

[105] A L Bauer T L Jackson and Y Jiang ldquoA cell-based modelexhibiting branching and anastomosis during tumor-inducedangiogenesisrdquo Biophysical Journal vol 92 no 9 pp 3105ndash31212007

[106] P Macklin and J S Lowengrub ldquoA new ghost celllevel setmethod for moving boundary problems application to tumorgrowthrdquo Journal of Scientific Computing vol 35 no 2-3 pp266ndash299 2008

[107] A R A Anderson ldquoA hybrid mathematical model of solidtumour invasion the importance of cell adhesionrdquoMathemati-cal Medicine and Biology vol 22 no 2 pp 163ndash186 2005

[108] P K Newton J Mason K Bethel et al ldquoSpreaders and spongesdefine metastasis in lung cancer a markov chain monte carlomathematical modelrdquo Cancer Research vol 73 no 9 pp 2760ndash2769 2013

[109] A Uppal S C Wightman S Ganai R R Weichselbaum andG An ldquoInvestigation of the essential role of platelet-tumorcell interactions in metastasis progression using an agent-basedmodelrdquoTheoretical Biology andMedicalModelling vol 11 article17 2014

[110] C M Koebel W Vermi J B Swann et al ldquoAdaptive immunitymaintains occult cancer in an equilibrium staterdquo Nature vol450 no 7171 pp 903ndash907 2007

[111] T OrsquoSullivan R Saddawi-Konefka W Vermi et al ldquoCancerimmunoediting by the innate immune system in the absence ofadaptive immunityrdquoThe Journal of Experimental Medicine vol209 no 10 pp 1869ndash1882 2012

[112] T Alarcon H M Byrne and P K Maini ldquoA cellular automatonmodel for tumour growth in inhomogeneous environmentrdquoJournal of Theoretical Biology vol 225 no 2 pp 257ndash274 2003

[113] G Powathil M KohandelMMilosevic and S SivaloganathanldquoModeling the spatial distribution of chronic tumor hypoxiaimplications for experimental and clinical studiesrdquo Computa-tional and Mathematical Methods in Medicine vol 2012 ArticleID 410602 11 pages 2012

[114] M S Lawrence P Stojanov P Polak et al ldquoMutational het-erogeneity in cancer and the search for new cancer-associatedgenesrdquo Nature vol 499 no 7457 pp 214ndash218 2013

[115] W Qiao W Wang E Laurenti et al ldquoIntercellular networkstructure and regulatory motifs in the human hematopoieticsystemrdquo Molecular Systems Biology vol 10 no 7 article 7412014

[116] M Robertson-Tessi R J Gillies R A Gatenby and A RA Anderson ldquoImpact of metabolic heterogeneity on tumorgrowth invasion and treatment outcomesrdquo Cancer Researchvol 75 no 8 pp 1567ndash1579 2015

[117] T D Querec R S Akondy E K Lee et al ldquoSystems biologyapproach predicts immunogenicity of the yellow fever vaccinein humansrdquoNature Immunology vol 10 no 1 pp 116ndash125 2009

[118] R Laaksonen M Katajamaa H Paiva et al ldquoA systems biologystrategy reveals biological pathways and plasma biomarker can-didates for potentially toxic statin-induced changes in musclerdquoPLoS ONE vol 1 article e97 2006

[119] A Artemov A Aliper M Korzinkin et al ldquoA method forpredicting target drug efficiency in cancer based on the analysisof signaling pathway activationrdquo Oncotarget vol 6 no 30 pp29347ndash29356 2015

[120] E Kim V W Rebecca K S Smalley and A R AndersonldquoPhase i trials inmelanoma a framework to translate preclinicalfindings to the clinicrdquo European Journal of Cancer vol 67 pp213ndash222 2016

[121] M Lawler and R Sullivan ldquoPersonalised and precision medi-cine in cancer clinical trials panacea for progress or Pandorarsquosboxrdquo Public Health Genomics vol 18 no 6 pp 329ndash337 2015

[122] A R A Anderson and M A J Chaplain ldquoContinuous anddiscrete mathematical models of tumor-induced angiogenesisrdquoBulletin of Mathematical Biology vol 60 no 5 pp 857ndash8991998

[123] N A Finn and M L Kemp ldquoPro-oxidant and antioxidanteffects of N-acetylcysteine regulate doxorubicin-induced NF-kappa B activity in leukemic cellsrdquo Molecular BioSystems vol8 no 2 pp 650ndash662 2012


[124] A F Villaverde J A Egea and J R Banga ldquoA cooperativestrategy for parameter estimation in large scale systems biologymodelsrdquo BMC Systems Biology vol 6 article no 75 2012

[125] M Rodriguez-Fernandez M Rehberg A Kremling and JR Banga ldquoSimultaneous model discrimination and parameterestimation in dynamic models of cellular systemsrdquo BMC Sys-tems Biology vol 7 article no 76 2013

[126] Z Zi ldquoSBML-PET-MPI a parallel parameter estimation tool forSystems BiologyMarkup Language basedmodelsrdquoBioinformat-ics vol 27 no 7 pp 1028ndash1029 2011

[127] Z Zi ldquoSensitivity analysis approaches applied to systems biol-ogymodelsrdquo IET Systems Biology vol 5 no 6 pp 336ndash346 2011

[128] T Sumner E Shephard and I D L Bogle ldquoA methodology forglobal-sensitivity analysis of time-dependent outputs in systemsbiology modellingrdquo Journal of the Royal Society Interface vol 9no 74 pp 2156ndash2166 2012

[129] M Rodriguez-Fernandez and J R Banga ldquoSensSB a softwaretoolbox for the development and sensitivity analysis of systemsbiology modelsrdquo Bioinformatics vol 26 no 13 pp 1675ndash16762010

[130] J Vanlier C A Tiemann P A J Hilbers and N AW van RielldquoAn integrated strategy for prediction uncertainty analysisrdquoBioinformatics vol 28 no 8 pp 1130ndash1135 2012

[131] A Saltelli P Annoni I Azzini F Campolongo M Ratto and STarantola ldquoVariance based sensitivity analysis of model outputDesign and estimator for the total sensitivity indexrdquo ComputerPhysics Communications An International Journal and ProgramLibrary for Computational Physics and Physical Chemistry vol181 no 2 pp 259ndash270 2010

[132] A Najafi M Tavallaei and S M Hosseini ldquoA systems biologyapproach for miRNA-mRNA expression patterns analysis innon-small cell lung cancerrdquo Cancer Biomarkers vol 16 no 1pp 31ndash45 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of


Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology


Molecular Biology International

GenomicsInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


BioinformaticsAdvances in

Marine BiologyJournal of



Signal TransductionJournal of


BioMed Research International

Evolutionary BiologyInternational Journal of



Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Genetics Research International


Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational



Enzyme Research



Microbiology


ODE system

Petri net

Booleannetwork

Linearprogramming

Cell-Cell interaction networks 3D tumor growth

Tissue scale Intercellular scaleIntracellular scale

Signaling pathways

Gene regulatory networks

Metabolic pathways

ODE systems in ABM

Gene arrayRNA-Seq

Proteomicswestern blot Metabolics

Ligand production

Ligand binding

Gene array Cell viabilityHampE images

Animal model

Simulate dynamic of intracellular events Simulate cell-cell communications Liganddrug diffusiontumor angiogenesis

Blood vessel network

Functions

Typicalmodels

Biologicalnetwork(systems)

Data used inmodeling

PDE systemODE system

Stochastic model in ABM Stochastic model in ABM

Cytokine assay

Probabilisticmodel

Dirichlet boundary

Dynamical diffusion of ligands or drugs

Figure 1 The whole picture of the systemic modeling approaches introduced in this work

modeling approaches suffers from new biological questions[12]

In this review we firstly studied several well-establishedsystems modeling approaches of biological networks such asordinary differential equations Petri net Boolean networkand linear programming Secondly we summarized the typ-ical modeling studies for the cell-cell communications (suchas tumor-stromal interactions tumor-immune interactionsand stromal cell lineage process) in the heterogeneous tumormicroenvironment that is agent-based modelThirdly threepotential directions of multiscale modeling in systems biol-ogy were deeply discussed We believe that this work canprovide a big picture of systemicmodeling in systems biologyas well as promoting the development of precision medicinein the near future

2 Several Classical SystemicModeling Approaches

With the development of the high-throughput experimenttechnologies (such as gene microarray RNA-seq mass spec-trometry and metabolic profiles) computational and math-ematical modeling of biological processes provides deepinsights of the complex cellular systems [13] Researchersbuilt various computational models to elucidate the com-plex behaviors of cancers such as tumor progression drugresistance and immune inert It is well-known that bioin-formatics is data-driven [14 15] However systems biology ishypothesis-driven [16 17] since we often generate a testablehypothesis based on small-scale experimental observations

and then construct a systemicmodel based on this hypothesisto obtain mechanistic insights In this study we mainlyfocus on several classic systemic modeling approaches andtheir applications in current cancer research These popularmodeling approaches can simulate the dynamic changes ofregulatory networks (signaling pathways andmetabolic path-ways) tumor growth and its microenvironments such asordinary differential equations (ODEs) [10] Boolean network[18] Petri nets [19] linear programming (LP) basedmodel [920] agent-basedmodel [11] and the system biologymodelingapproach considering genetic variation [21] We presentthese models in Figure 1 Although there are many availablereverse-engineering [22] algorithms for the inference of generegulatory networks [23] such as ARACNe [22] and MINDy[24] we omitted them in this review since they are bettersuited to be categorized in the field of bioinformatics

21 ODE-Based Modeling With the rapid development ofcomputer performance ordinary differential equation (ODE)based approaches are widely used for continuous dynamicmodeling in complex biological systems [25] ODE-basedmethods represent the interactions among various biologicalmolecules (such as protein kinases or metabolites) whichreflect the time-varying effects of biological processes [26]Based on the different biological hypotheses the currentODE-based methods can be categorized into three types thelaw of mass action [27 28] Hill function [29] andMichaelis-MentenKinetics [30]The choice of a specificmethod dependson the biological questions or the experimental data Here weillustrate how to use these kinetic approaches to describe thebiochemical reactions



119860 + 119861119870ON997888997888997888rarrlarr997888997888997888119870OFF

119862 (1)


119889 [119860]119889119905 =

119889 [119861]119889119905 = 119870OFF [119862] minus 119870ON [119860] [119861]

119889 [119862]119889119905 = 119870ON [119860] [119861] minus 119870OFF [119862]

(2)


119889119910119889119905 =

119872

sum119894=1

119891+ (119909119894) +119873

sum119895=1


119891plusmn (119909) = 119896119909119910119909plusmn119899119909

119867plusmn119899119909119909 + 119909plusmn119899119909 (4)



V = 119881max119878119870119898 + 119878

(5)







2

Hydrogen

Oxygen

Reaction 1

2

Water

(O2)

(H2)

(H2O)

(a)

2

Hydrogen

Oxygen

Reaction 1

2

Water

(O2)

(H2)

(H2O)

(b)









V1

V2 V3

V4

(a)

A

B

C

D

Node




Boolean function

mdash

(b)

1101 1110 1011 1000

0000 0010

(c)



120590119894 (119905 + 1) = 119861119894 (1205901198941 (119905) 1205901198942 (119905) 120590119894119896119894 (119905)) (6)


120590lowast119894 (119905 + 1) = 119861119894 (1205901198941 1205901198942 120590119894119896119894 ) (7)



















Objective function


Optimal solution

Allowable subspace

Original space

(a)

(b)

(c)

(d)


B + 2C rarr D

Reaction 1

Reaction 2

Reaction n

V1

V2

Vn


= 0

Fluxes V

lowast

Met

abol

ites


Reactions

Biom

ass

Glu

cose

Oxy

gen

minus1

1

1

minus1

minus2

1

minus1

minus1


m

etc


1 1 1

2 2 2

3 3 3

A harr B + C














Cell B

Cell A

Cell D

Ligand

Cell


















4 Discussion









Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology



119860 + 119861119870ON997888997888997888rarrlarr997888997888997888119870OFF

119862 (1)


119889 [119860]119889119905 =

119889 [119861]119889119905 = 119870OFF [119862] minus 119870ON [119860] [119861]

119889 [119862]119889119905 = 119870ON [119860] [119861] minus 119870OFF [119862]

(2)


119889119910119889119905 =

119872

sum119894=1

119891+ (119909119894) +119873

sum119895=1


119891plusmn (119909) = 119896119909119910119909plusmn119899119909

119867plusmn119899119909119909 + 119909plusmn119899119909 (4)



V = 119881max119878119870119898 + 119878

(5)







2

Hydrogen

Oxygen

Reaction 1

2

Water

(O2)

(H2)

(H2O)

(a)

2

Hydrogen

Oxygen

Reaction 1

2

Water

(O2)

(H2)

(H2O)

(b)









V1

V2 V3

V4

(a)

A

B

C

D

Node




Boolean function

mdash

(b)

1101 1110 1011 1000

0000 0010

(c)



120590119894 (119905 + 1) = 119861119894 (1205901198941 (119905) 1205901198942 (119905) 120590119894119896119894 (119905)) (6)


120590lowast119894 (119905 + 1) = 119861119894 (1205901198941 1205901198942 120590119894119896119894 ) (7)



















Objective function


Optimal solution

Allowable subspace

Original space

(a)

(b)

(c)

(d)


B + 2C rarr D

Reaction 1

Reaction 2

Reaction n

V1

V2

Vn


= 0

Fluxes V

lowast

Met

abol

ites


Reactions

Biom

ass

Glu

cose

Oxy

gen

minus1

1

1

minus1

minus2

1

minus1

minus1


m

etc


1 1 1

2 2 2

3 3 3

A harr B + C














Cell B

Cell A

Cell D

Ligand

Cell


















4 Discussion









Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


2

Hydrogen

Oxygen

Reaction 1

2

Water

(O2)

(H2)

(H2O)

(a)

2

Hydrogen

Oxygen

Reaction 1

2

Water

(O2)

(H2)

(H2O)

(b)









V1

V2 V3

V4

(a)

A

B

C

D

Node




Boolean function

mdash

(b)

1101 1110 1011 1000

0000 0010

(c)



120590119894 (119905 + 1) = 119861119894 (1205901198941 (119905) 1205901198942 (119905) 120590119894119896119894 (119905)) (6)


120590lowast119894 (119905 + 1) = 119861119894 (1205901198941 1205901198942 120590119894119896119894 ) (7)



















Objective function


Optimal solution

Allowable subspace

Original space

(a)

(b)

(c)

(d)


B + 2C rarr D

Reaction 1

Reaction 2

Reaction n

V1

V2

Vn


= 0

Fluxes V

lowast

Met

abol

ites


Reactions

Biom

ass

Glu

cose

Oxy

gen

minus1

1

1

minus1

minus2

1

minus1

minus1


m

etc


1 1 1

2 2 2

3 3 3

A harr B + C














Cell B

Cell A

Cell D

Ligand

Cell


















4 Discussion









Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


V1

V2 V3

V4

(a)

A

B

C

D

Node




Boolean function

mdash

(b)

1101 1110 1011 1000

0000 0010

(c)



120590119894 (119905 + 1) = 119861119894 (1205901198941 (119905) 1205901198942 (119905) 120590119894119896119894 (119905)) (6)


120590lowast119894 (119905 + 1) = 119861119894 (1205901198941 1205901198942 120590119894119896119894 ) (7)



















Objective function


Optimal solution

Allowable subspace

Original space

(a)

(b)

(c)

(d)


B + 2C rarr D

Reaction 1

Reaction 2

Reaction n

V1

V2

Vn


= 0

Fluxes V

lowast

Met

abol

ites


Reactions

Biom

ass

Glu

cose

Oxy

gen

minus1

1

1

minus1

minus2

1

minus1

minus1


m

etc


1 1 1

2 2 2

3 3 3

A harr B + C














Cell B

Cell A

Cell D

Ligand

Cell


















4 Discussion









Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology














Objective function


Optimal solution

Allowable subspace

Original space

(a)

(b)

(c)

(d)


B + 2C rarr D

Reaction 1

Reaction 2

Reaction n

V1

V2

Vn


= 0

Fluxes V

lowast

Met

abol

ites


Reactions

Biom

ass

Glu

cose

Oxy

gen

minus1

1

1

minus1

minus2

1

minus1

minus1


m

etc


1 1 1

2 2 2

3 3 3

A harr B + C














Cell B

Cell A

Cell D

Ligand

Cell


















4 Discussion









Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology





Objective function


Optimal solution

Allowable subspace

Original space

(a)

(b)

(c)

(d)


B + 2C rarr D

Reaction 1

Reaction 2

Reaction n

V1

V2

Vn


= 0

Fluxes V

lowast

Met

abol

ites


Reactions

Biom

ass

Glu

cose

Oxy

gen

minus1

1

1

minus1

minus2

1

minus1

minus1


m

etc


1 1 1

2 2 2

3 3 3

A harr B + C














Cell B

Cell A

Cell D

Ligand

Cell


















4 Discussion









Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology






Cell B

Cell A

Cell D

Ligand

Cell


















4 Discussion









Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology












4 Discussion









Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology




4 Discussion









Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


Model

Result P1

P2

P3

(a)

Model


Result viability



P1

P2

P3

(b)














Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology







Acknowledgments


References

















































































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology





















































































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


















































































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


















































Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology


















Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology








Volume 2014

Zoology






















Advances in

Virolog y



Volume 2014




Enzyme Research



Microbiology

Documents

Mathematical and Computational Modeling in Complex Biological Systemsdownloads.hindawi.com/journals/bmri/2017/5958321.pdf · 2019-07-30 · ReviewArticle Mathematical and Computational