JigCell Model Connector: building large molecular network models from components

Transcription

1 Simulation Applications JigCell Moel Connector: builing large molecular network moels from components Simulation: Transactions of the Society for Moeling an Simulation International 2018, Vol. 94(11) Ó The Author(s) 2018 DOI: / journals.sagepub.com/home/sim Thomas C Jones Jr 1, Stefan Hoops 3, Layne T Watson 1, Alia Palmisano 1,2, John J Tyson 2 an Cliffor A Shaffer 1 Abstract The growing size an complexity of molecular network moels makes them increasingly ifficult to construct an unerstan. Moifying a moel that consists of tens of reactions is no easy task. Attempting the same on a moel containing hunres of reactions can seem nearly impossible. We present the JigCell Moel Connector, a software tool that supports large-scale molecular network moeling. Our approach to eveloping large moels is to combine smaller moels, making the result easier to comprehen. At the base, the smaller moels (calle moules) are efine by small collections of reactions. Moules connect together to form larger moules through clearly efine interfaces, calle ports. In this work, we enhance the port concept by efining three types of ports. An output port is linke to an internal component that will sen a value. An input port is linke to an internal component that will receive a value. An equivalence port is linke to an internal component that will both receive an sen values. Not all moules connect together in the same way; therefore, multiple connection options nee to exist. Keywors computational systems biology, hierarchical moel composition, JigCell, moeling tool, SBML, software 1. Introuction The functions of a living cell are controlle by macromolecular interactions. These complex interactions between genes an proteins can be mappe as regulatory networks. In an effort to unerstan the ynamic properties of these networks, mathematical moels of the biochemical reactions are often constructe. 1 3 During this process, moelers have the ifficult task of specifying reaction etails between species connecte in these complex regulatory networks. Moeling a system accurately is an iterative process involving frequent changes to the unerlying network. 4 Once a moel is rafte, the equations can be analyze an simulate to escribe the ynamical behavior of the regulatory network. 2 These computational results can then be compare to existing experimental ata, an if inconsistencies arise, the moel can be moifie. Once a moel has been teste against existing experimental ata, it can be use to make preictions that might guie the irection of future experiments. 5 If further experiments uncover inconsistencies, then the moel can be moifie again. As molecular biologists iscover more information about how gene an protein interactions affect cell physiology, the size an complexity of the mathematical moels ten to grow. Constructing these moels is becoming more ifficult. Historically, moelers have been limite in the scope of the behavior being moele by the complexity of larger moels. For these reasons, moelers have a nee to eal with increasingly complex moels, which require new moeling approaches. Hierarchical moel composition is a moeling technique where, instea of builing one large, complex moel, smaller moels are combine together to form a larger moel. By breaking a complex system into smaller parts, with clearly efine interactions between the parts, it can be more easily unerstoo. In this paper we present the JigCell Moel Connector (JCMC), a software tool to support hierarchical moel 1 Department of Computer Science, Virginia Tech, Blacksburg, VA, USA 2 Department of Biological Sciences, Virginia Tech, Blacksburg, VA, USA 3 Biocomplexity Institute, Virginia Tech, Blacksburg, VA, USA Corresponing author: Cliffor A Shaffer, Department of Computer Science, Virginia Tech, Blacksburg, VA 24061, USA. shaffer@cs.vt.eu.

2 994 Simulation: Transactions of the Society for Moeling an Simulation International 94(11) composition. In our tool, the smallest moels (calle moules) are efine by small collections of reactions. Moules connect together to form larger moules through clearly efine interfaces, calle ports. Moelers are able to regulate external access to internal components of a moule by utilizing ports. We implement three port types that allow moules to connect in ifferent ways. An output port is linke to an internal component that will sen a value to an external reference. An input port is linke to an internal component that will receive a value from an external reference. An equivalence port is linke to an internal component that will both receive an sen values from an external reference. Once a moel is create in the JCMC, it can be exporte to one of the existing stanar moel formats, at which point the moel can be simulate an analyze using other tools. Our goal is to evelop large moels in a moular way, making the result easier to comprehen. A major contribution of this paper over prior efforts to efine systems for builing moels in a moular fashion are new port an noe constructs that allow moules to connect in ifferent ways using the JCMC. The remainer of this paper is organize as follows. First, previous work on hierarchical moel composition stanars an tools is reviewe. Next, we present the JCMC environment an its components. Then the ifferent types of ports an their impact on a moel are iscusse. Last, best practices for constructing hierarchical moels, conclusions, an future work are presente in the form of a case stuy. 2. Backgroun In this section, we iscuss the basic concepts of hierarchical moeling, an a stanar format that enables the sharing of moels. We also review previous attempts to create hierarchical moeling tools Hierarchical moel composition Ranhawa an colleagues 6 8 escribe moel composition as another approach to create large moels from smaller moels. The smaller moels become submoels of a larger compose moel. Composition involves escribing how components from ifferent submoels interact with one another, without changing the inner workings of each submoel. The interaction escriptions are store in the overarching compose moel. Here, large moels are simply collections of submoels, an can be organize in a hierarchical fashion. Unlike fusion, moel composition is a reversible process. If the interaction escriptions are remove, then the original submoels can be recovere. Ranhawa an colleagues 8,9 characterize moel aggregation as a restricte form of moel composition. Here, they efine a moule as a collection of moel components. Figure 1. Moel aggregation. A moule also inclues a specification for preetermine ports. A port is a link to an internal moel component, such as a species or parameter. Therefore, a moule is a submoel with ports. The ports of a moule form an interface, which only allows access to specific components within the moule. The process of grouping moel components an assigning ports is referre to as moularization, as shown in Figure 1. The ifference between moel composition an moel aggregation lies in the restrictions place on access to the internals of the moules. In moel composition, any component of a submoel coul be reference in a larger compose moel. The lack of informationhiing interfaces is consiere a white-box coupling approach. 10 In moel aggregation, only a component linke to a port can be reference in a larger compose moel. Incorporating information-hiing interfaces is consiere a black-box coupling approach. 10 Moules are then connecte together by their interface ports. With moel aggregation, moelers can buil larger moels in a controlle manner. Ranhawa an colleagues 6,8,9 also presente the concept of moel flattening. Moel flattening converts compose or aggregate moels to their flattene versions. The interaction etails of the compose or aggregate moels are use as instructions uring the flattening process. The result is a single large (flat) moel, which is equivalent to fusing the submoels. The flat moel is in a stanar format that can be rea by existing software tools for the purpose of running simulations an further analysis. The flattening process loses the hierarchical an other relationships between the various moules, yieling a set of reaction equations.

3 Jones et al The SBML stanar Systems Biology Markup Language (SBML) is a format that represents systems biology moels electronically. Such a stanar allows for the sharing an collaboration of moels. SBML Level 1 11 was introuce in A moel efine in SBML can consist of many components, such as compartments, species, reactions, an parameters. The interactions between components in the moel are also efine in SBML. SBML is not esigne with the goal of being an easily human-reaable format. Moelers are not expecte to write their moels by han in SBML. Instea, software tools are expecte to rea an write the format. Since SBML supports the concept of efining moels as consisting of many components, it alreay inclues many of the features necessary for hierarchical moeling. For those hierarchical moeling features that o not exist within the SBML stanar, extra information about a component can be store in annotations. Annotations can be thought of as comments in an SBML file. These comments can be associate with any SBML component, such as a reaction, species, or parameter. Software that processes stanar SBML moels can also be augmente to process this aitional information carrie in the annotations. Therefore, annotations can be use by software evelopers to inclue application-specific ata. The latest version of SBML, Level 3 Version 1 Core, 12 was release in This version of SBML incorporates new features that support hierarchical moel efinitions. Such feature extensions are referre to as packages. Two SBML packages relevant to component-base moeling, comp an layout, are escribe next The SBML comp package. The latest version of the SBML comp package 13 was release in This package allows instances of moels to be incorporate as submoels within a moel. The moel structure is extene to inclue a list of submoels an a list of ports. A submoel is an instance of a moel efinition, which in turn is a complete, self-containe moel. Moel efinitions (instantiate as submoels) are locate either in the list of internal moel efinitions or the list of external moel efinitions. An internal moel efinition is store within the SBML file. An external moel efinition is a placeholer that specifies the location of an external file containing the moel efinition. This external file can be on the local machine or available on the internet. Ports allow moels to interact with other moels through a esignate interface. A port references some component within the moel that is being explicitly expose, such as a species or parameter. These extene features of comp enable hierarchical moel composition in SBML The SBML layout package. The latest version of the SBML layout package 14 was release in The layout package allows components of a moel to be represente graphically, along with their positions. To o so, the moel structure is extene to inclue a list of layouts. A layout can store information specifying the graphics representing some or all components of the SBML moel. These graphics are referre to as glyphs in the layout package. A compartment, species, or reaction can be represente by a CompartmentGlyph, SpeciesGlyph, or ReactionGlyph, respectively. A GeneralGlyph can be use to represent parts of a moel that are not specifie in the Level 3 Version 1 Core, such as a submoel from comp. A glyph stores information pertaining to the location an imension of a graphical object. A glyph oes not inclue information escribing the shape, color, or style of a graphical object; it is left up to the software tool reaing the layout to isplay such etails. These extene features of layout enable moel visualization in SBML, an these features are use heavily by JCMC Relate tools There are numerous software tools available for the moeling an simulation of molecular networks. For example, Antimony is a moel efinition language that can be use to create, import, an combine moels in a moular way. 15 However, Antimony is text-base an only provies limite support for importing/exporting moels using SBML comp. Similar to Antimony, Genetic Engineering of living Cells (GEC) is a text-base formal language. 16 GEC allows interactions between proteins an genes to be expresse in a logical manner. GEC programs can be use to create moels in a moular way. However, GEC oes not support exporting moels using SBML comp. COPASI is a tool use to moel, simulate, an analyze biochemical networks. 17 Its graphical user interface offers many features such as stochastic an eterministic simulation methos, parameter estimation, an ata visualization. COPASI is an excellent tool for creating a single moel, an it provies support for importing/exporting stanar SBML (Level 3). However, COPASI lacks features to support hierarchical moeling an SBML comp. TinkerCell is a tool that supports hierarchical moeling. 18 The graphical user interface lets users create moules that can be connecte together to form larger moels. TinkerCell oes not have a notion of port types an oes not support SBML output. The JigCell suite of tools can be use to moel, simulate, an analyze biochemical networks. 7,9,19 23 Previous iterations of JigCell have inclue a Moel Builer, Aggregation Connector, Run Manager, Comparator, an Parameter Estimation Toolkit. The Moel Builer is use to create an eit reactions, species, an other moel properties in a tabular format. The Aggregation Connector is use to combine moels in a moular way. The Run

4 996 Simulation: Transactions of the Society for Moeling an Simulation International 94(11) Manager an Parameter Estimation Toolkit are use to efine simulation properties an etermine unknown parameter values within the moel. The Comparator is use to compare the moel simulations with experimental results. The JigCell suite provies support for importing/exporting stanar SBML (Level 2). JigCell Multistate Moel Builer (JC-MSMB) is a tool that supports the moeling of biochemical networks. 24 The graphical user interface buils on the tabular spreasheet format use by Vass an colleagues. 22,23 JC-MSMB reuces the complexity of moel creation by introucing a new syntax to escribe multistate species. This syntax requires fewer reactions to represent complex molecular systems. The tool has many eiting support features such as flexible autocompletion an consistency checks to assist users uring the moel-creation process. It provies support for importing/exporting SBML (Level 3). However, JC-MSMB lacks features to support hierarchical moeling an SBML comp. The current work on JCMC can be viewe as a new generation beyon the JigCell Aggregation Connector, working in connection with the JigCell Multistate Moel Builer. ibiosim is a tool for the moeling, analysis, an esign of genetic circuits. 25 In synthetic biology, genetic circuits can be use to esign an construct networks to implement a particular cellular function. 26 Although primarily esigne for genetic circuits, it can be use to stuy biological networks as well. Its graphical user interface can be use to create, import, an combine moels in a moular way. ibiosim offers multiple simulation methos, moel analysis, an ata visualization, along with support for importing/exporting hierarchical moels using SBML comp. Both JigCell s Aggregation Connector an ibiosim offer support for SBML an hierarchical moeling. However, JigCell lacks the features of ifferent port types. ibiosim oes offer input an output port types, but oes not support equivalence ports Moel simulation Once a moel has been create, it is typically use for simulation an further analysis. There are multiple approaches to simulate a hierarchical moel. One option is to first flatten the moel. Once flattene, copies of all submoels have been instantiate an replacements an eletions have been applie, the moel is no longer in its hierarchical form. After flattening, the moel is simulate. COPASI is an example of a tool that uses this approach. 17 Another option is to simulate the moel in its hierarchical form. One such algorithm is hssa. 27 Instea of instantiating copies of each submoel, hssa instantiates a single copy of each unique submoel an re-uses them as neee. Replacements an eletions are mae as they are Figure 2. Three panels an the menu bar. encountere. ibiosim is an example of a tool that uses this approach JigCell Moel Connector We next escribe the JCMC in etail. In orer to better unerstan the purpose an key features of the JCMC, it helps to first have an overview of the system s user interface. At this point, the reaer nee not worry too much about the unerlying meaning of the various components that are presente here. This will be iscusse later Interface The JCMC interface consists of three panels, shown in Figure TreeView. The left panel contains the TreeView. This isplays the hierarchical relationships between components of the moel, somewhat like a traitional file foler isplay. An example is shown in Figure 3. A moule can be selecte from the TreeView using the left mouse button, an it will be highlighte. Double-clicks (using the left mouse button) will expan/collapse the selecte moule. After selecting a moule, the user can a or remove submoules (using the Moule menu) DrawingBoar. The right panel contains the DrawingBoar. This isplays the graphical view of a moule, its submoules, an any connections among them, as shown in Figure 4. The currently loae moule is calle the container moule. In Figure 4, the container moule is name RegulationExample. Submoules can be move

5 Jones et al. 997 Figure 4. DrawingBoar. Figure 5. MoelBuiler. Figure 3. TreeView. insie of the container moule. In Figure 4, Ch1 an CycB are submoules. If ports exist, they are isplaye on the container moule an submoules. Connections between ports, visible variable noes, an equivalence noes are also shown. Examples of these components are presente in later sections MoelBuiler. The bottom panel is the MoelBuiler, shown in Figure 5. The MoelBuiler in JCMC is actually a version of the JC-MSMB, 24 which was previously implemente by our group. The version of JC-MSMB use by JCMC is a subset of the complete moel eitor because JCMC oes not support multistate species. JC-MSMB has a tabular spreasheet interface that isplays the etails of a moule. Attributes such as reactions, species, parameters, an events can be moifie. 4. Components 4.1. Container moule The container moule is the moule currently loae into the DrawingBoar. The TreeView panel shows the container moule s name in bol font. The MoelBuiler panel isplays the moule efinition for the container moule. A moule efinition contains a moule s etaile information, such as reactions, species, parameters, an events. The DrawingBoar isplays the container moule an any submoules, ports, or connections in the moule Submoule A submoule is simply a moule containe within another moule. The TreeView panel lists a submoule uner the container moule to which it belongs. In the DrawingBoar, a submoule can be move an resize within the bouns of its container moule. A submoule s information is liste as: \ Definition Name. \ Submoule Name. Definition Name correspons to the name of the moule efinition. A moule efinition contains etaile information, such as reactions, species, parameters, an events. Submoule Name correspons to the name of a specific instantiation of the moule efinition. In Figure 4, submoule Ch1 is an instantiation of moule efinition PhosDephos. Similarly, submoule CycB is an instantiation of moule efinition SynDeg. A single moule efinition can be instantiate multiple times. We show examples of this in later sections. A submoule s etaile information (reactions, species, parameters, events, etc.) is not liste in the MoelBuiler panel because the container moule s information is isplaye instea. However, a submoule s information can be previewe in the MoelBuiler panel. Each submoule has a button in the top left-han corner. When this button is clicke, the information for that submoule will be

6 998 Simulation: Transactions of the Society for Moeling an Simulation International 94(11) Figure 6. Submoule information preview. isplaye in the MoelBuiler. An example is shown in Figure 6. After the button for Ch1 is clicke, the template information for Ch1 is isplaye in the MoelBuiler pane. Notice that the tables are graye-out in Figure 6. This is because the information is a preview only, an cannot be moifie. To moify the information, the user woul loa the submoule as the container moule. Submoules can be ae an remove using the Moule menu. Removing a submoule will remove the selecte moule, all of its submoules, an any connections associate with other moules. 5. Technical contributions As iscusse in Section 2, Ranhawa an colleagues 8,9 efine moel aggregation as a restricte form of moel composition. With aggregation, moelers can regulate external access to internal components of a moule by efining ports. A port allows an internal component to be reference in a larger compose moel. Moules can then be connecte together by their interface ports to buil larger moels in a controlle manner. However, not all moules are the same. Internal components linke to ports o not necessarily serve the same purpose for every moule. Not all moules connect together in the same way; therefore, multiple connection options nee to exist. Our primary contribution to hierarchical moeling is a richer variety of port types for connecting moules. In this section, we present these ifferent port an noe constructs. We also iscuss how these connections can impact a moel Ports Ports expose internal components of a moule so that they can be reference from outsie of that moule. A port can be linke to either a species or a moule quantity. Once create, the ports collectively form an interface to the moule, with external access to a moule s internal components being regulate by the interface. Moules can be connecte together by their interfaces to buil larger moels. Previous software tools support ports, 7,9,21 an ports are a part of the SBML comp package. 13 However, this prior work treats all ports the same. Internal components linke to ports o not necessarily serve the same purpose for every moule. When the port mechanism gives no information as to how an internal component is use, moelers have no way to iscern a component s purpose within a moule. So there exists a nee for ifferent port types. The port types supporte by JCMC are as follows. An output port is linke to an internal component that will sen a value to an external reference. The component linke to the port may be moifie insie the moule, but the component is not meant to be moifie outsie the moule. Consier the scenario in which a species is synthesize in a moule an then use as a transcription factor outsie of the moule. An output port is appropriate because the species is not moifie outsie of the moule. A etaile example is presente in Section 6.1. Output ports are represente as triangles on the ege of moules. They are oriente so the arrowhea points out of the moule. An input port is linke to an internal component that will receive a value from an external reference. The component linke to the port is not meant to be moifie within the moule. Consier the scenario in which a rate constant for a reaction within a moule has a value etermine outsie of the moule. An input port is appropriate because the rate constant is only use in calculations for the reaction, an not moifie insie the moule. A etaile example is presente in Section 6.1. Input ports are represente as triangles on the ege of moules. They are oriente so the arrowhea points into the moule. An equivalence port is linke to an internal component that will both receive an sen values from an external reference. The component linke to the port may be moifie insie an outsie the moule. Consier the scenario in which a species is synthesize in one moule an phosphorylate in another moule. An equivalence port is appropriate because the species is moifie in both moules. A etaile example is presente in Section 6.2. An equivalence port is represente as a iamon on the ege of a moule. In the DrawingBoar panel, ports are isplaye on moule bounaries. In the MoelBuiler panel, ports are liste uner the Ports tab (shown in Figure 7). The list is populate with ports from the container moule an ports from any submoules in the container moule. When a port is selecte in the DrawingBoar panel, the Ports tab is

7 Jones et al. 999 Figure 7. Ports tab. Figure 8. Visible variable create with a connection. isplaye an the corresponing port is highlighte in the MoelBuiler panel. Each port has three properties: Ref Name: the species or moule quantity reference by the port; Port Type: the type of port; Port Name: the name of the port. Port aitions or removals can only happen to the container moule. To moify the ports of a submoule, the user must first loa the submoule as the container moule Noes A noe allows connections to occur between moule ports. The type of noe use epens on the type of ports connecte Visible variable noe. A visible variable noe is automatically create when a connection is mae between input or output ports of two moules. Figure 8 shows two submoules after a connection has been mae, a visible variable was create, an the new variable was ae in the MoelBuiler panel. Another way to create a visible variable noe is to right-click the active moule an select Show Variable. Once selecte, a pop-up winow will appear with a rop-own box that contains a list of all the species an moule quantities in the moule. The user will select a variable an click A, then a visible variable noe will be create in the DrawingBoar panel. A visible variable noe can have at most one incoming connection. A single incoming connection lets the noe receive values. A visible variable noe can have multiple outgoing connections. The outgoing connections are use to sen values. Table 1. Rules for submoule (source) to submoule (target) connections Target submoule port Input Output Equivalence Source Input Invali Invali Invali Output Vali Invali Invali Equivalence Vali Invali Vali Equivalence noe. An equivalence noe is create automatically when a connection is mae between an equivalence port an any other port in the DrawingBoar panel. When create, the new variable is ae in the MoelBuiler panel. An equivalence noe can have multiple connections. Since values are both sent an receive, there is no istinction between incoming an outgoing connections Connections A set of connections link moules together. Connections can occur between the ports of ifferent moules, visible variable noes, an equivalence noes. The rules for connections are liste in Tables 1, 2, an 3. A connection can be create by ragging a line from a vali source to a vali target. Attempting to create an invali connection will result in a warning message, an no connection will be create. 6. Port, noe, an connection effects In this section we explore the effect that ifferent ports, noes, an connections have on a moel. In the equation notation use below, variables are represente by strings,

8 1000 Simulation: Transactions of the Society for Moeling an Simulation International 94(11) Table 2. Rules for container moule (source) to submoule connections (target) Target submoule port Input Output Equivalence Source Input Vali Invali Invali Output Invali Invali Invali Equivalence Vali Invali Vali Table 3. Rules for submoule (source) to container moule (target) connections Target container moule port Input Output Equivalence Figure 9. Synthesis an egraation moule SynDeg. Source Input Invali Invali Invali Output Invali Vali Invali Equivalence Invali Invali Vali multiplication is enote by 3, an ifferentiation by t ½nameŠ. (Throughout this paper, the notation ½nameŠ refers to the concentration of species name.) 6.1. Input an output ports We begin with a simple synthesis an egraation moule. Figure 9 shows how a synthesis an egraation moule woul look in JCMC. The reactions are isplaye in the MoelBuiler panel at the bottom. The rate of synthesis is etermine by rate constant k0 an transcription factor F. The rate of egraation is etermine by mass action kinetics with rate constant k1. The ynamics of species R in moule SynDeg are as follows: ½RŠ =(k03½fš) (k1 3 ½RŠ): t ð1þ Figure 9 also shows that moule quantities k0 an k1 are connecte to input ports. Because k0 an k1 are connecte to input ports, they can receive values from external connections. Note that k0 an k1 are not moifie within the moule SynDeg, they are only use for the computations of other variables. Figure 10 shows a phosphorylation an ephosphorylation moule in JCMC. The rate of phosphorylation is etermine by mass action kinetics with rate constant kp, while the rate of ephosphorylation is etermine by mass action kinetics with rate constant kh. The species ynamics of moule PhosDephos are efine as follows: ½SŠ = (kp 3 ½SŠ)+(kh 3 ½SPŠ), t ð2þ Figure 10. Phosphorylation an ephosphorylation moule PhosDephos. ½SPŠ =(kp 3 ½SŠ) (kh 3 ½SPŠ): t ð3þ Figure 11 isplays Moel01, where S an R are submoules. Submoule R is an instantiation of SynDeg, shown in Figure 9. Submoule S is an instantiation of PhosDephos, shown in Figure 10. Species TF, moule quantity ks, an moule quantity g are isplaye as visible variable noes. Submoule S has an output port linke to species SP an submoule R has input ports linke to species F, moule quantity k0, an moule quantity k1. Noe ks is connecte to the k0 port on submoule R. This means that k0 will receive the value of ks. To accomplish this, k0 will be replace by ks in submoule R. The same will happen with k1 ang. The replacement is not immeiate, but will occur when the entire moel is flattene. There is one connection from submoule S s SP port to noe TF, an another from noe TF to submoule R s F

9 Jones et al Figure 11. Output port example. port. Submoule S s SP port will sen its internal value to noe TF. Noe TF will then sen the value to submoule R s F port. Finally, the internal species F in submoule R will receive the value. When the moel is flattene, TF will replace SP in submoule S an F in submoule R. The species ynamics after flattening are as follows: ½SŠ = (kp 3 ½SŠ)+(kh 3 ½TFŠ), t ð4þ ½TFŠ =(kp 3 ½SŠ) (kh 3 ½TFŠ), t ð5þ ½RŠ =(ks 3 ½TFŠ) (g 3 ½RŠ): t ð6þ Equation (3) has been replace by Equation (5) an TF has replace SP in Equation (2) to form the upate Equation (4). Similarly, TF has replace F in Equation (1) to form the upate Equation (6) Equivalence port Figure 12 shows the MoelBuiler panel for three ifferent moules. Figure 12(a) shows the reaction etails for the synthesis an egraation of species X in moule SynDeg. The rate of synthesis is etermine by ks an the rate of egraation is etermine by mass action kinetics with rate constant g. The ynamics of species X in moule SynDeg are efine as follows: ½X Š = ks (g 3 ½X Š): t ð7þ Figure 12(b) shows the reaction etails for the phosphorylation an ephosphorylation of species Y in moule PhosDephos. The rate of phosphorylation is etermine by mass action kinetics with rate constant kp. The rate of ephosphorylation is etermine by mass action kinetics Figure 12. Moule reaction information. with rate constant kh. The species ynamics of moule PhosDephos are efine as follows: ½Y Š = (kp 3 ½YŠ)+(kh 3 ½YPŠ), t ð8þ ½YPŠ =(kp 3 ½Y Š) (kh 3 ½YPŠ): t ð9þ Figure 12(c) shows the reaction etails for the association an issociation of species Comp in moule AssocDissoc. The rate of association is etermine by mass action kinetics with rate constant ka. The rate of issociation is etermine by mass action kinetics with rate constant k. The species ynamics of moule AssocDissoc are efine as follows: ½ZŠ = (ka 3 ½ZŠ 3 ½WŠ)+(k 3 ½CompŠ), t ð10þ ½WŠ = (ka 3 ½ZŠ 3 ½WŠ)+(k 3 ½CompŠ), t ð11þ ½CompŠ =(ka 3 ½ZŠ 3 ½WŠ) (k 3 ½CompŠ): t ð12þ Figure 13 shows Moel02, where X, Y, an Comp are submoules. Submoule X is an instantiation of SynDeg, submoule Y is an instantiation of PhosDephos, an submoule Comp is an instantiation of AssocDissoc. Species A is isplaye as an equivalence noe. Submoule X has an equivalence port linke to species X, submoule Y has an equivalence port linke to species Y, an submoule Comp has an equivalence port linke to species Z. Each of these equivalence ports are connecte to noe A in Moel02. When the moel is flattene for simulation, A will replace X in submoule X, Y in submoule Y, an Z in submoule Comp. The species ynamics after flattening are as follows:

10 1002 Simulation: Transactions of the Society for Moeling an Simulation International 94(11) Figure 13. Equivalence port example. ½YPŠ =(kp 3 ½AŠ) (kh 3 ½YPŠ), t ð13þ ½WŠ = (ka 3 ½AŠ 3 ½WŠ)+(k 3 ½CompŠ), t ð14þ ½CompŠ =(ka 3 ½AŠ 3 ½WŠ) (k 3 ½CompŠ), t ð15þ t from X in submoule X zfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflffl{ ½AŠ = ks (g 3 ½AŠ) from Y in submoule Y zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ (kp 3 ½AŠ)+(kh 3 ½YPŠ) from Z in submoule Comp zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ (ka 3 ½AŠ 3 ½WŠ)+(k 3 ½CompŠ) : ð16þ Equation (13) is an upate version of Equation (9), where A has replace Y. Similarly, A has replace Z in Equations (11) an (12) to form the upate Equations (14) an (15). Notice that Equation (16) is an aggregate of Equations (7), (8), an (10). Since noe A is connecte to equivalence ports, values are both sent an receive, an therefore information from each connection is kept. SBML, we ecie to use SBML annotations an Systems Biology Ontology (SBO) terms. 28 The annotation stores the variable type an variable name by using the tags vtype an refname. The SBO term stores the port type. SBO terms can be use an rea by other software tools, such as ibiosim. 25 SBML Syntax in the Supplemental Material shows an example of two ports using the comp package. The layout package oes not have a specific glyph to represent ports. It oes support a GraphicalObject, which can be use to store general information about an object. We ecie to use GraphicalObjects combine with annotations to escribe the port layout. The GraphicalObject stores the position of each port. The aitional annotation stores the port type, variable type, variable name, variable i, an the name of the moule in which the port is locate. SBML Syntax in the Supplemental Material shows the graphical information for two ports using the layout package Noes. Visible variable noes an equivalence noes are not efine in the comp package. However, a noe always represents either a species or a moule quantity. Since both species an moule quantities are present in SBML, we inclue noe information with an annotation. SBML Syntax in the Supplemental Material shows an example of a visible variable noe ClbS an an equivalence noe ClbM. The annotation stores the variable name an noe type. The layout package oes have a SpeciesGlyph to represent species, but it oes not have a specific glyph to represent moule quantities. Since a noe can be a species or a moule quantity, we ecie to use GraphicalObjects combine with annotations to escribe their layout. SBML Syntax in the Supplemental Material shows an example of the graphical information for visible variable noe ClbS an equivalence noe ClbM using the layout package. Similar to ports, the GraphicalObject stores the position of each noe. The annotation stores the variable name an variable type SBML syntax We iscusse the SBML stanar in Section 2.2. JCMC is able to store the information that escribes a hierarchical moel by using the SBML comp package. The submoules in JCMC can be store as moel efinitions within the SBML file. In this section we explain how the port an noe constructs are store in SBML Ports. Ports are inclue in the comp package. However, the ifferent port types introuce in Section 5.1 are not. In orer to store this extra information in Connections. In SBML, there is no efinition for a connection. Instea, the comp package introuce the notion of replacements. A replacement allows one component to replace another. Replacements are the interaction etails use as instructions if the moel is flattene. Most connections in JCMC are represente as replacements in SBML. All of our replacements are top-own, which makes them easy to follow an they o not require any rules for resolution. It is important to note that the topown approach applies to initial values as well. After flattening, initial values may nee to be reassigne before simulation.

11 Jones et al Figure 14. Moel of cell-cycle control in buing yeast Case stuy In this section, we will emonstrate the features of JCMC by builing a complex biological moel. Barik an colleagues 29 publishe a moel of yeast cell-cycle regulation, consisting of 58 species an 220 reactions. We will reconstruct this moel with a more efficient approach by utilizing moules. We will show how submoules connect together an the role that ports play in the process Biological moel Figure 14 shows a wiring iagram of the biological moel from the work of Barik an colleagues. 29 Species are represente by the labele shapes. Chemical reactions are represente by soli arrows an enzymatic activities are represente by ashe arrows. Reversible bining reactions are represente by T-shape arrows with balls on the cross-bars. For clarification purposes, Figure 14 only isplays some of the major regulatory interactions containe in the moel. For example, the synthesis an egraation reactions for Whi5, SBF, Ch1, Net1, Hbf, Hi5, an Ht1 are not shown. The moel by Barik an colleagues 29 captures the molecular controls of cell-cycle events, incluing the initiation of DNA synthesis (by ClbS) an of mitosis (by ClbM), an exit from mitosis, incluing cell ivision (by Cc14). When a mother cell ivies, the volume of the cell an the number of molecules of each chemical species within the cell are evenly split between the two resulting aughter cells. (For simplicity, we are ignoring the fact that buing yeast cells ivie asymmetrically.) In the moel, the event of cell ivision is triggere by ClbM. When the concentration of ClbM rops below 12 nm, the cell will ivie evenly. Figure 15. Moular moel of cell-cycle control in buing yeast The hierarchical moel First, we will introuce a transcription an translation moule that will appear multiple times in our moel. Next, we will moularize the moel (Figure 15) an buil up each moule iniviually. Then, we will connect the moules together to form the final hierarchical moel. Finally, we will valiate the hierarchical moel by comparing simulation results with the original moel. In the following escriptions, variables refer to the number of molecules Transcription an translation coupling. From Figure 14 we can see that some of the regulatory functions in the moel are similar. The mechanisms regulating ClbM, ClbS, an Cln3 appear to follow the same pattern. The synthesis of the protein is epenent upon the synthesis of its mrna. This is calle transcription an translation coupling. Since it occurs multiple times in the moel, we can buil it as a reusable, generic moule. Figure 16 shows moule TTCoupling, short for transcription an translation coupling. The MoelBuiler panel at the bottom isplays the four reactions in the moule that escribe the synthesis an egraation of mrna an protein X. Moule quantities ksm, gm, ksx, an gx are the constants that etermine the rates of the four reactions. Protein X is linke to an output port an the rate constants are linke to input ports. The ports allow external proteins an rate constants to connect to the moule an utilize the interior transcription an translation reaction structure. We will see how this is one in later sections. There are also two more input ports for species V an ClbM. These species are use to calculate when the concentration of ClbM falls below 12.5 nm, which is when cell ivision occurs. Since V is the volume of the cell an

12 1004 Simulation: Transactions of the Society for Moeling an Simulation International 94(11) Figure 16. Transcription an translation coupling moule TTCoupling. phosphorylate states, for a total of 11 phosphorylation states. In this moel, only the unphosphorylate state of Ch1 is active. ClbM affects many parts of this moel. Figure 17 isplays moule Step01. Step01 contains submoules Ch1Reg an ClbMTT. Visible variable noes Ch1 an.ch1pt. receive connections from Ch1Reg s output ports Ch1a an Ch1i. Ch1 an Ch1Pt are use to calculate moule quantity gbmt. The relationship between Ch1, Ch1Pt, an gbmt is not explicitly shown on the DrawingBoar. The calculation efining gbmt is viewable in the Moule Quantities tab of the MoelBuiler panel (not shown in the figure). Visible variable noes ksmbm, gmbm, ksbmt, an gbmt connect the moule quantities to ClbMTT s input ports ksm, gm, ksx, an gx. Visible variable noe ClbM connects the species to ports on both submoules as well as moule Step01. Noe ClbM receives its value from submoule ClbMTT, where ClbM is regulate. Noe ClbM then sens its value to submoule Ch1Reg, where ClbM influences the phosphorylation of Ch1. Noe ClbM is also connecte to Step01 s output port ClbM, so it can be reference by other parts of the moel. Step 01 has three input ports connecte to visible variable noes. These noes are then connecte to submoules, where their values can be use for calculations. Details for the submoules in Step01 can be foun in the Supplemental Material. Figure 17. Step01 moule in JCMC. ClbM is the number of ClbM molecules, they both must be inclue to calculate the concentration of ClbM. When the cell ivies, most species are ivie in half. To accomplish this, an event is use. The event calculates the concentration of ClbM to etermine when the cell ivies. When the cell ivies, the species numbers within the moule are reassigne appropriately. An event like this occurs in most moules, which is why most moules require V an ClbM. In the TTCoupling moule, species X an mrna are reassigne to half their current values, assuming that the protein an mrna molecules are ivie equally between the two aughter cells ClbM regulation. Step 1 of the moel will consist of the interactions containe in area A of Figure 15. The regulation of Ch1 an ClbM play big roles in step 1. Ch1 has 1 unphosphorylate state as well as Cc14 regulation. Step 2 of the moel will consist of the interactions containe in area B of Figure 15. In step 2, the active phosphorylation states of Net1 combine with Cc14 to form the RENT complex. Ht1 causes ephosphorylation of both Net1 an RENT phosphorylate states. Figure 18 isplays moule Step02. Step02 contains submoules Ht1Reg, Cc14Reg, Net1Reg, an RENTReg. Visible variable noe Ht1 connects the species to three ifferent submoules. Noe Ht1 receives a value from submoule Ht1Reg, where it is regulate. Noe Ht1 sens its value to submoules Net1Reg an RENTReg, where it promotes ephosphorylation. Equivalence noes for each of the Net1 phosphorylation states connect to Net1Reg an RENTReg because the species are moifie in both submoules. Equivalence noe Cc14 connects to submoules Cc14Reg an RENTReg. Noe Cc14 also connects to an output port on moule Step02, so it can be reference by other parts of the moel. Details for the submoules in Step02 can be foun in the Supplemental Material SBF regulation. Step 3 of the moel will consist of the interactions containe in area C of Figure 15. In step 3, the active phosphorylation states of SBF an Whi5 combine to form the Cmp complex. Hi5 is involve with the

13 Jones et al Figure 18. Step02 moule in JCMC. Figure 20. Final CellCycle moel in JCMC. Figure 19. Step03 moule in JCMC. ephosphorylation of Whi5 an Cmp, while Hbf is involve with the ephosphorylation of SBF. Figure 19 isplays moule Step03. Step03 contains submoules Hi5Reg, SBFReg, Whi5Reg, an CmpReg. Visible variable noe Hi5 connects the species to three ifferent submoules. Noe Hi5 receives a value from submoule Hi5Reg, where it is regulate. Noe Hi5 sens its value to submoules Whi5Reg an CmpReg, where it promotes ephosphorylation. Equivalence noes for each of the Whi5 phosphorylation states connect to Whi5Reg an CmpReg because the species are moifie in both submoules. Equivalence noe SBF connects to submoules SBFReg an CmpReg. Noe SBF also connects to an output port on moule Step03, so it can be reference by other parts of the moel. Step 03 has five input ports connecte to visible variable noes. These noes are then connecte to ifferent submoules, where their values can be use for calculations. Details for the submoules in Step03 can be foun in the Supplemental Material Final moel. We have create all of the moules necessary for our moel, so now it is time to put them together. A JCMC user woul start by importing all of the moules we have iscusse. After the moules are importe, the CellCycle moel contains submoules Step01, Step02, Step03, ClbSReg, an Cln3Reg. Figure 20 isplays the complete CellCycle moel. Visible variable noe ClbS connects the species to three submoules. Noe ClbS receives a value from submoule ClbSReg, where it is regulate. Noe ClbS sens its value to submoules Step01 an Step03, where it promotes phosphorylation. Visible variable noe Cc14 connects the species to three submoules as well. Noe Cc14 receives a value from submoule Step02, where it is regulate an part of the RENT complex. Noe Cc14 sens its value to submoules Step01 an Step03, where it promotes ephosphorylation. Visible variable noe Cln3 connects the species to two submoules. Noe Cln3 receives a value from submoule Cln3Reg, where it is regulate. Noe Cln3 sens its value to submoule Step03, where it promotes phosphorylation. Visible variable noe SBF connects the species to two submoules. Noe SBF receives a value from submoule Step03, where it is regulate an part of the Cmp complex. Noe SBF sens its value to submoule ClbSReg, where it promotes ClbS transcription. Visible variables noes ClbM an V are connecte to each of the submoules through either input or output ports. ClbM an V are present in every submoule because their values are use to calculate when cell ivision occurs Simulation results. To valiate the hierarchical moel, we compare its simulation results with results from the original moel. Figure 21 isplays the two moel simulations. We exporte the hierarchical moel into SBML format with the comp package using JCMC. Since the original

14 1006 Simulation: Transactions of the Society for Moeling an Simulation International 94(11) moel is a non-hierarchical moel, we ecie to use the software tool COPASI to flatten an simulate the hierarchical moel. 17 COPASI was able to flatten the moel by following the moule interaction etails store with SBML comp. Parameter values an initial concentrations were set as escribe by Barik an colleagues. 29 We compare the flattene hierarchical moel to the eterministic version of the original moel, in which the cell ivies symmetrically. The simulation results from the flattene hierarchical moel in Figure 21(b) match the results from the original moel in Figure 21(a). This was verifie by confirming that the time series simulation ata from both moels were ientical. 8. Conclusions an future work When builing a mathematical moel, templates can provie a compact an expressive way to re-use moel components. A template is characterize as a generic moule efinition. Templates o not hol information relate to one specific species. Instea, they contain general molecular mechanisms that are common in the regulatory network. A goo example is the transcription an translation coupling moule from the case stuy, shown in Figure 16. Here, species X is linke to an output port an the rate constants for the reactions are all linke to input ports. The ports let external entities utilize the interior transcription an translation reaction structure of the moule. A template can be instantiate multiple times to escribe the molecular behavior of ifferent species without neeing to be moifie. The esire species an rate constants can be connecte to the appropriate ports. This is one with ClbM, ClbS, an Cln3 in the case stuy. Alternatively, we coul have create iniviual moule efinitions for ClbM, ClbS, an Cln3. In this case, each moule efinition woul have containe the same reaction structure an we woul have neee to write 12 repetitive reactions. Using a template allowe us to avoi such inefficiencies. The ability to test a moel is important, an frequent testing shoul occur uring the moel-builing process. In software engineering, evelopment testing ensures components are correct as they are evelope. 30 Each component is teste iniviually before it is ae to the system. This allows errors to be iscovere, an hopefully fixe, early in evelopment. A similar approach shoul be taken when builing a mathematical moel. As moules are create, their inner reaction structures can be verifie using simulation tools such as COPASI. 17 As moules are connecte together, their connections can be verifie by checking the resulting equations. These tests shoul be one throughout the moel-builing process. Waiting until the en to test can make it extremely ifficult to fin the cause of an error. As molecular network moels continue to grow in size an complexity, traitional moeling practices are Figure 21. Time course simulation results. becoming obsolete. Builing complex moels in a controlle, organize manner is important. Without proper organization, complex moels become ifficult to unerstan. In orer to construct an comprehen such moels, we must evolve our moeling approach. In this paper, we have reviewe improve moeling approaches relate to hierarchical moel composition, along with software stanars that support this moeling approach. We propose enhancements to the way moules interact in moel aggregation. We introuce new port an noe constructs that enable multiple connection options for moules. We also escribe how ifferent connections can impact a moel. Our tool that supports hierarchical moeling is name the JigCell Moel Connector (JCMC). We escribe the JCMC interface an explaine how the components of a hierarchical moel are represente. We etaile how input, output, an equivalence ports form the moule interface. We also explaine how visible variable an equivalence noes are utilize in JCMC. Finally, we emonstrate the benefits of hierarchical moeling with JCMC by reconstructing a complex biological moel in a moular fashion. A possible enhancement to the JCMC woul be to incorporate multistate moeling. Currently, the JCMC only buils moels with single-state species. Multistate moeling can rastically reuce the size of moels containing species that have multiple states. Aing this enhancement to JCMC woul give moelers the opportunity to create complex multistate moels by connecting smaller submoules together. Another possible enhancement to the JCMC woul be to let any moel component link to a port. Currently, JCMC only links species an moule quantities to ports. SBML comp allows any moel component to link to a port. Aing this feature to JCMC woul allow for the import an export of reactions, events, an other moel components between moules. Disclaimer Alia Palmisano contribute to this manuscript as a follow up to work one while employe at Virginia Tech. The opinions