Mathematical Modeling As a Tool for Investigating Cell Cycle Control Networks
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Methods 41 (2007) 238–247 www.elsevier.com/locate/ymeth Mathematical modeling as a tool for investigating cell cycle control networks Jill C. Sible *, John J. Tyson Department of Biological Sciences, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0406, USA Accepted 11 July 2006 Abstract Although not a traditional experimental ‘‘method,’’ mathematical modeling can provide a powerful approach for investigating com- plex cell signaling networks, such as those that regulate the eukaryotic cell division cycle. We describe here one modeling approach based on expressing the rates of biochemical reactions in terms of nonlinear ordinary differential equations. We discuss the steps and challenges in assigning numerical values to model parameters and the importance of experimental testing of a mathematical model. We illustrate this approach throughout with the simple and well-characterized example of mitotic cell cycles in frog egg extracts. To facilitate new modeling efforts, we describe several publicly available modeling environments, each with a collection of integrated programs for math- ematical modeling. This review is intended to justify the place of mathematical modeling as a standard method for studying molecular regulatory networks and to guide the non-expert to initiate modeling projects in order to gain a systems-level perspective for complex control systems. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Cell cycle; Computational biology; Frog egg extracts; Mathematical modeling; Ordinary differential equations; Parameter estimation; Systems biology; Xenopus laevis 1. Introduction networks. Models organize a large body of experimental data, describe the fundamental behaviors of the system as Because a universal mechanism controlling DNA syn- a whole, bridge gaps where experimental data are missing, thesis, mitosis, and cell division underlies the growth, and drive hypothesis-building for the next round of exper- development, and reproduction of all eukaryotes, an imentation. The value of mathematical modeling in understanding of this molecular regulatory system is one describing and predicting the behavior of complex systems of the most important goals of modern cell biology. As has been well established in fields such as chemical engi- the complex network of cell cycle controls is uncovered, neering and meteorology, but its power has been underap- it becomes increasingly difficult to make reliable predic- preciated until recently in molecular cell biology. tions about how modification of one component affects Although mathematical models can be built to describe the system as a whole. However, such predictions are need- any signaling network, application of modeling tools to cell ed if we are to identify the host of mutations contributing cycle regulation is particularly well suited and timely. First, to cancer or find within the molecular network novel the data in this field are vast, both providing a large body targets for therapeutic intervention. Mathematical models of information to build comprehensive models and creating provide powerful tools for managing the complexity of the need for a tool to understand how these data fit togeth- the cell cycle control system and of other signaling er. Second, cell cycle signaling networks are modular, allowing models to be constructed in parts and then assem- * Corresponding author. Fax: +1 540 231 9307. bled and reassembled in various ways. Furthermore, many E-mail address: [email protected] (J.C. Sible). models of the network are comparable between different 1046-2023/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ymeth.2006.08.003 J.C. Sible, J.J. Tyson / Methods 41 (2007) 238–247 239 organisms (e.g., budding yeast and mammals) so that it is tively simple, many predictions of the model have been val- feasible to make relatively small changes to a model idated experimentally, and this module can be adapted to describing one particular system in order to apply it to more complex cell cycles both in Xenopus and other another. Thus, each new model need not be constructed eukaryotes. from scratch. Third, a reasonable amount of quantitative or semi-quantitative information can be extracted from 2. Approach the literature, facilitating the early phases of model build- ing. By modifying established protocols (described in the 2.1. Construction of a wiring diagram accompanying articles), additional quantitative data can be generated to improve parameter estimation and experi- The first step in the modeling process is to organize the mental validation of models. Finally, despite the wealth of known interactions of the relevant molecules into a concept detailed information on cell cycle molecules and their spe- map or wiring diagram. A simple diagram representing the cific interactions, we lack a systems-level perspective of this core machinery regulating mitotic transitions in Xenopus complex control network. Modeling can provide this per- cell-free egg extracts is depicted in Fig. 1. The diagram spective by helping to identify underlying regulatory prin- depicts each biochemical entity with a separate icon. For ciples. Where a specific experimental detail is missing, example, the phosphorylated form of Wee1 is represented modeling can serve as bridge, enabling progress in building by a modified version of the icon for unphosphorylated a systems-level view, and guiding the design and execution Wee1. Solid arrows indicate chemical transitions between of future experiments. states, and dotted arrows represent a modulating signal Although the term ‘‘mathematical modeling’’ encom- on a biochemical reaction (often the catalytic influence of passes a wide range of computational approaches applica- an enzyme). Because no universal conventions for building ble to cell cycle studies, we focus this review on one branch these diagrams exist, all symbols must be defined explicitly of modeling: The construction of ordinary differential and used consistently. Hopefully, contributors to this field equations (ODEs) to describe protein interaction networks will adopt a common symbolism in the near future, that regulate the cell cycle. This approach has a strong facilitating the exchange of information between wiring track record of yielding accurate, predictive and testable diagrams and their affiliated mathematical models. models [1–7], and in recent years, has been integrated well Despite the wealth of information regarding cell cycle with experimental methodologies [8–12]. We will illustrate signaling networks, gaps in knowledge may exist such each step of the modeling process with an example: The that portions of the diagram cannot be wired with cer- core signaling network that controls entry into and exit tainty. It is neither necessary nor advised to postpone from mitosis in frog (Xenopus laevis) egg extracts by regu- the modeling process indefinitely until all molecular lating the activity of the mitotic cyclin-dependent kinase, details of the wiring diagram are understood. Rather, Cdk1. We chose this example because the network is rela- the gaps in the wiring diagram can be filled with place- amino acids OFF Cyclin P APC IE Cdk1 P Cdc25 Cdc25 IE ON PC Cdk1 P Cdk1 P A P Cdk1 Cyclin Cyclin Wee1 Wee1 P degraded cyclin Fig. 1. Wiring diagram of the core mitotic cell cycle engine in Xenopus cell-free egg extracts. Central to the diagram is the active cyclin–Cdk1 complex (also referred to as MPF), with activating phosphorylation on Thr 161 indicated by the green ‘P’. Cyclin enters the system by de novo synthesis and then combines with Cdk1. The phosphorylation on Thr 161 is rapid and therefore not represented in the diagram or the mathematical equations. Active Cdk1 can be phosphorylated on Thr 14 and Tyr 15 to form inactive preMPF. The double arrows indicate the reversibility of this process. The inhibitory phosphates are represented by the red ‘P’; and the influences of the relevant kinase (Wee1) and phosphatase (Cdc25C) are indicated by the dotted arrows. MPF itself phosphorylates Cdc25 and Wee1, positively and negatively affecting their activities, respectively. These feedback loops are easily appreciated from the wiring diagram. MPF also participates in a negative feedback loop whereby it phosphorylates a component of the APC, which subsequently directs the polyubiquitination of cyclin, tagging it for degradation by the proteasome. Like cyclin synthesis, cyclin degradation is represented by a single unidirectional arrow because the process is irreversible. Although not indicated in the diagram, cyclin degradation by the APC applies equally to free cyclin monomers and to preMPF. IE indicates an intermediate component (see text). 240 J.C. Sible, J.J. Tyson / Methods 41 (2007) 238–247 holders. These placeholders can be generic ones for which where k1 is the rate constant for cyclin synthesis, k2 is a parameters will be selected empirically to model the behav- function that describes the activity of the APC in promot- ior of the system without representing the molecular mech- ing cyclin degradation, and k3 is the rate constant for asso- anism in detail. This approach allows the modeling effort to ciation of cyclin monomers and Cdk monomers to form continue in order to discover properties of the system as a MPF dimers. These dimers are phosphorylated on threo- whole as well as specific features of better-characterized nine 161 to make active MPF. Because this reaction is fast portions of the diagram. Alternatively, a more specific and essentially irreversible, we are justified in neglecting the placeholder can be created that makes some untested T161-unphosphorylated forms of MPF. assumptions about the molecular mechanism. These The rate equation for [MPF] is likewise based on mass- assumptions in the wiring diagram will be tested in simula- action kinetics: tions of the model to determine whether they are consistent d with known behaviors of the system. Eventually, the MPF k3 Cyclin Cdk k2 MPF dt ½ ¼ ½ ½ À ½ assumptions that work in silico must be tested experimen- k MPF k preMPF tally when appropriate methodologies and reagents become À wee½ þ 25½ available. Ideally, portions of the wiring diagram that are A similar equation pertains to [preMPF].