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Mathematical Modeling

Computational analyses of measurements, mechanisms, and models

Kevin A. Janes

Genetics, cell biology, and biochemistry have been instrumental in identifying and organizing signaling proteins within biological pathways and networks (1). These networks are now known to be so complex that it is no longer feasible to predict by intuition how cells will respond to molecular perturbations (2). Encouragingly, this same wealth of knowledge makes it increasingly feasible to understand biological systems by modeling signaling pathways computationally (3).

The Modeling program of the CDP Center is based on the philosophy that the type of computation should be tailored to the specific biological question of interest (4). To remain biologically realistic, we argue that mathematical models of signaling networks can only be as detailed as the current understanding of the networks themselves (5). Based upon this requirement for prior biological knowledge, we are pursuing three categories of models in the CDP Center: 1) “mechanistic models”, which use ordinary differential equations (ODE’s) to encode the aggregate mechanistic understanding of signaling networks; 2) “model-based models”, which explore the input-output properties of well-defined signaling subnetworks within mechanistic models; and 3) “data-driven models”, which analyze primary experimental data to reveal new mechanistic connections between proteins and pathways. All three classes of models are focused on the signaling networks that control the decision to die or to survive in the presence of prodeath cytokines or prosurvival growth factors.

The most widely recognized signaling pathway activated by prodeath cytokines involves the apoptotic caspases (6). The CDP Center has developed mechanistic models of caspase activation induced by tumor necrosis factor (TNF), TNF-related apoptosis-inducing ligand (TRAIL), and Fas. Each of these models was designed to address questions specific to the death-receptor family. For example, a model of the TNF-activated death-inducing signaling complex (DISC) was used to investigate the distinct roles of membrane-associated and internalized DISC (7) toward apoptosis (8). Downstream of DISC signaling, a separate model of TNF- and TRAIL-induced caspase activation examined how the mitochondrial branch of the caspase network rapidly mediates the apoptotic phenotype in single cells (9). A Fas signaling model was developed to distinguish the molecular mechanisms of type I vs. type II apoptosis (10) in different cell lines (11). In addition to death-receptor signaling, we have also constructed a model of extracellular-regulated kinase (ERK) and Akt survival signaling induced by the epidermal growth factor (EGF) receptor and its related ErbB family members (12). This model was originally designed to study how ErbB dimerization contributes to the sensitivity of different tumor cell lines to anticancer ErbB inhibitors. Unexpectedly, the ErbB model also revealed the importance of protein phosphatase localization in downregulating ERK and Akt. Mechanistic models store our aggregate biological knowledge in a tractable way and are used to identify which proteins and pathways are most critical for mediating cell responses.

Within larger-scale mechanistic models, local networks can be interrogated by computational techniques other than ODE’s. Often, these types of model-based models provide unique insights into network function. At the CDP Center, several model-based approaches are being applied to different cell decision subnetworks. One prototypical signaling module is the mitogen-activated protein kinase (MAPK) pathway culminating in ERK activation (13). Using network optimization approaches (14), we are exploring different objective functions (e.g., response time, signal amplitude) to identify the selection pressures driving the evolved MAPK architecture. For the apoptotic caspase subnetwork, dynamical systems theory is being applied to estimate the initial protein concentrations that predispose cells to apoptotic and viable caspase activity patterns (15). Finally, thousands of simulations from the Fas signaling model (11) are being organized by decision tree analysis (16) to suggest the molecular-level variations that result in distinct kinetics of caspase activation. Together, our model-based analyses provide a firm theoretical grounding to explore how rate constants and protein concentrations affect the behavior of mechanistic models and, by extension, of cells.

For many cell-decision networks, there is simply not enough information about the signaling proteins and reactions to construct a believable mechanistic model. These more poorly characterized systems are best tackled experimentally and thus require modeling strategies that extract information from high-throughput quantitative measurements. We have adopted a probabilistic approach that deduces from single-cell measurements the most likely signaling network connecting measured intracellular proteins (17). This so-called Bayesian network recapitulated many known signaling pathways and correctly identified new dependencies between well-studied kinases. In parallel, we have pursued statistical approaches using partial least squares regression to link measured signaling proteins to measured cell-death responses (18). A partial least squares model of TNF, EGF, and insulin signaling accurately predicted 12 apoptotic responses from 19 dynamic measurements characterizing the network (19). Both data-driven approaches provide a compact, quantitative, and intuitive representation of thousands of signaling measurements. This substantially aids interpretation and hypothesis generation, providing a basis for identifying new biological pathways that can then improve the mechanistic models described earlier.

Computational modeling is tightly connected with the other efforts at the CDP Center. For instance, here I have not discussed modeling of the transport equations within microfabricated devices, since it is inseparable from the Microsystems program. Within the Modeling program, each computational scientist interacts with experimentalists on a weekly basis, at minimum, and it is most common for collaborators to meet multiple times per week. Furthermore, several of the described models have been developed by experimentalists themselves. Thus, in addition to it scientific pursuits, the CDP Center is also training a new generation of collaborative, interdisciplinary scientists at the interface of biology and computation. By applying diverse analytical approaches toward important biological questions, the Modeling program of the CDP Center strives to reveal the molecular logic underlying cellular decisions implicated in disease (20).


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  8. Gaudet, S., Schoeberl, B., Lauffenburger, D. A., and Sorger, P. K. (2005), in preparation
  9. Albeck, J. G., Burke, J. M., Lauffenburger, D. A., and Sorger, P. K. (2005), in preparation
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  11. Hua, F., Cornejo, M. G., Cardone, M. H., Stokes, C. L., and Lauffenburger, D. A. (2005) Effects of Bcl-2 levels on Fas signaling-induced caspase-3 activation: molecular genetic tests of computational model predictions. J Immunol, in press.
  12. Schoeberl, B., Pace, E. A., Nielsen, U. B., Lauffenburger, D. A., and Sorger, P. K. (2005), in preparation
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  15. Aldridge, B., Haller, G., Sorger, P. K., and Lauffenburger, D. A. (2005) Direct Lyapunov exponent analysis enables parametric study of transient signaling dynamics characterizing cell phenotypic behavior. in preparation
  16. Hautaniemi, S., Kharait, S., Iwabu, A., Wells, A., and Lauffenburger, D. A. (2005) Modeling of signal-response cascades using decision tree analysis. Bioinformatics 21, 2027-2035
  17. Sachs, K., Perez, O., Pe’er, D., Lauffenburger, D. A., and Nolan, G. P. (2005) Causal protein-signaling networks derived from multiparameter single-cell data. Science 308, 523-529
  18. Janes, K. A., Kelly, J. R., Gaudet, S., Albeck, J. G., Sorger, P. K., and Lauffenburger, D. A. (2004) Cu-e-signal-response analysis of TNF-induced apoptosis by partial least squares regression of dynamic multivariate data. J Comput Biol 11, 544-561
  19. Janes, K. A., Albeck, J. G., Gaudet, S., Sorger, P. K., Lauffenburger, D. A., and Yaffe, M. B. (2005) A predictive systems model of signaling identifies a molecular basis set for cytokine-induced apoptosis. In preparation
  20. Rudin, C. M., and Thompson, C. B. (1997) Apoptosis and disease: regulation and clinical relevance of programmed cell death. Annu Rev Med 48, 267-281

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