*Cellular Modeling of Cancer-Do we have the tools?
*

**Abstract:**

Cancer research to date has largely focused on individual cellular subsystems, and the specific malfunction of the regulatory mechanisms (so-called "dysregulation") associated with each one. In an influential review, Hanahan and Weinberg (2000) identified a set of six distinct acquired functional capabilities-called "hallmarks"-that are common to virtually all types of human cancers, namely: (1) self-sufficiency in growth signals, (2) insensitivity to antigrowth signals, (3) evasion of apoptosis, (4) limitless replicative potential, (5) sustained angiogenesis, and (6) tissue invasion and metastasis. Such a categorization then suggests a systematic approach to studying and analyzing the mechanism of occurrence of an otherwise bewildering array of cancer types and tumor subtypes within specific organs. A detailed understanding of the molecular, biochemical and cellular mechanisms by which each "hallmark" trait is acquired should then provide a rational basis for developing efficacious protocols of targeted therapy for cancer treatment or tumor prevention.

It is clear, however, from Hanahan and Weinberg's review, and several other works as well (for example Downward, 1999), that (i) the emergence (or suppression) of any particular hallmark involves interactions between various associated cellular subsystems, and (ii) the dysfunction of a single subsystem can contribute to the emergence of several hallmarks simultaneously. For example, it is known that the dysregulation of the growth factor signaling protein Ras can lead to self-sufficiency in growth signals (Medema and Bos, 1993), the evasion of apoptosis (Downward, 1998), and angiogenesis (Kerbel et al., 1998). Also, Chen and Goeddel, 2002, have noted that the tumor necrosis factor, (TNF) signaling pathway interacts significantly with many others, including apoptosis and NF-kB (a transcription factor that regulates several genes that mediate tumorigenesis and metastasis). Specifically, it is known that enforced activation of NF-kB protects against apoptosis, while susceptibility to TNF-induced apoptosis increases in the absence of NF-kB. Similarly, the investigation reported in Schoeberl et al., 2001, shows how the single ligand TNF can elicit the activation of both apoptotic and pro-growth subsystems, noting, among other things, that TNF binding to TNF receptor 1 (TNFR1) (as opposed to TNFR2) can activate apoptosis through cytochrome C release and also antiapoptotic cellular responses. Thus, without accounting for potential modulating effects (synergistic or otherwise) caused by connectivities and interactions, isolated subsystem analysis may be inadequate and misleading.

**Mathematical Modeling: Approach and Preliminary Results**

It is becoming increasingly accepted that mathematical modeling provides an efficient way of obtaining truly quantitative insight into the behavior of these subsystems and their interactions. As a starting point for quantitative characterization of the dynamic behavior of cellular subsystems and the effect of known interactions on the emergence of cancer, we have focused first on the interaction between the growth factor and apoptotic signaling pathways, specifically via PI3-kinase (PI3K). It is known that PI3K, when activated by EGF (epidermal growth factor) binding to the EGF receptor (EGFR), activates protein kinase B (PKB), which phosphorylates and thereby inactivates the pro-apoptotic protein BAD (Downward, 1999). We have developed a model of this connected ensemble as follows: we modeled the connections between PI3K and PKB and between PKB and BAD by combining reasonable mechanistic assumptions and available literature data (Franke et al., 1997); we then used this to link a published model of EGFR signaling that included activation of PI3K (Moehren et al., 2002) with our modified and updated version of an apoptosis model that incorporates the pro-apoptotic effects of BAD (Fussenegger et al., 2000). Our presentation will highlight the model development and how we have used the resulting ensemble model to explore quantitatively how anti-apoptotic EGFR signaling may potentially interact with pro-apoptotic stress signals. We will discuss how this model provides a quantitative understanding of how other factors outside of the obvious apoptotic subsystem is implicated in acquiring the "evasion of apoptosis" trait. For example, our model predictions suggest, among other things, the somewhat surprising result that it might be possible for EGFR signaling to disrupt the apoptotic pathway, possibly as much as an hour after the initiation of stress signals. Our future efforts will be directed towards further mechanistic refinement of the apoptosis model and the PI3K-PKB-BAD links. We will also begin modeling interactions between EGFR and the apoptotic signaling occurring further downstream, particularly through the signaling protein Ras. Several models of EGFR signaling that include Ras activation have already been published (Bhalla and Iyengar, 1999; Schoeberl et al., 2002), leaving the majority of the modeling to the interaction of Ras with the apoptotic pathway. These models will then be used to explore the interaction of these pathways during normal function and during the dysfunction of specific signaling components.

**How appropriate are Differential Equations for Modeling Complex Biological Systems? **

Our ultimate goal is to use these models to generate experimentally testable hypotheses regarding novel therapeutic approaches that may not have been conceivable had the subsystems been studied in isolation. Clearly such models must be capable of predicting observable physiological responses with sufficient fidelity since the application and validation of the therapeutic recommendations will ultimately be carried out at the physiological level. Now, obtaining physiological responses from single cell models requires integrating up to the tissue level (as an ensemble collection of single cells), then from the tissue to the organ level, and from the organ to the organism's overall physiology. It is from this perspective that we now observe that there may be some potential problems with the mathematical modeling approach adopted thus far.

The models discussed above have employed differential equations to represent the dynamic behavior of the signaling pathways within a single cell. As more and of such pathways are incorporated and integrated, even if we have all the requisite detailed component information available (which is clearly not the case), the number of differential equations required to represent the complete networks could increase to the point where at least two potential problems become inevitable: (i) the computational effort required to produce numerical solutions in any reasonable amount of time becomes prohibitive; (ii) more importantly, our ability to analyze the models and gain insight from them is hampered by the sheer explosion of the number of parameters and variables involved. While keeping an eye on such potential problems for the single cell model one must not lose sight of the ultimate goal: the realistic predictions of observable physiological responses which ought to be based not just on a single cell but on an ensemble of cells, in turn organized appropriately into that portion of the organism that is of particular physiological interest. There appears to be a very real danger that differential equation based models could potentially run into a limit of usefulness.

Some primary questions for discussion:

1. Should we (or should we not) be concerned about these potential problems? |

2. Are there ways by which the differential equations based approach (that has served the scientific community for so long) be modified appropriately to cope with these potential problems? |

3. Has the time come for a serious attempt at developing alternative approaches to differential equations as the basis for developing models of complex networks of biological phenomena that span the entire spectrum from the molecular to the cellular to the tissue to the organ and to the organism, for high fidelity prediction of physiological responses? |

**References:**

Bhalla US, Iyengar R. Emergent properties of networks of biological signaling pathways. Science. 1999 Jan 15;283(5400):381-7.

Chen G., and Goeddel D.V., TNF-R1 Signalling: A Beautiful Pathway. Science, 2002 May 31; 296 :1634-1635

Downward J. Ras signalling and apoptosis. Curr Opin Genet Dev. 1998 Feb;8(1):49-54.

Downward J. How BAD phosphorylation is good for survival. Nat Cell Biol.1999 Jun;1(2):E33-5.

Franke TF, Kaplan DR, Cantley LC, Toker A. Direct regulation of the Akt proto-oncogene product by phosphatidylinositol-3,4-bisphosphate. Science. 1997 Jan 31;275(5300):665-8.

Fussenegger M, Bailey JE, Varner J. A mathematical model of caspase function in apoptosis. Nat Biotechnol. 2000 Jul;18(7):768-74.

Hanahan D, Weinberg RA. The hallmarks of cancer. Cell. 2000 Jan 7;100(1):57-70.

Kerbel RS, Viloria-Petit A, Okada F, Rak J. Establishing a link between oncogenes and tumor angiogenesis. Mol Med. 1998;4(5):286-95.

Medema RH, Bos JL. The role of p21ras in receptor tyrosine kinase signaling. Crit Rev Oncog. 1993;4(6):615-61.

Moehren G, Markevich N, Demin O, Kiyatkin A, Goryanin I, Hoek JB, Kholodenko BN. Temperature dependence of the epidermal growth factor receptor signaling network can be accounted for by a kinetic model. Biochemistry. 2002 Jan 8;41(1):306-20.

Schoeberl B, Eichler-Jonsson C, Gilles ED, Muller G. Computational modeling of the dynamics of the MAP kinase cascade activated by surface and internalized EGF receptors. Nat Biotechnol. 2002 Apr;20(4):370-5.

Schoeberl B, Gilles ED, P. Scheurich, M. Fotin, G. Mueller, and H. Wajant. A mathematical vision of TNF receptor interaction. Proceedings of the Second International Conference on Systems Biology. 158-167, 2001.

Primary Collaborators:

Daniel E. Zak*, Thomas G. Lombardo*, Boris N. Kholodenko+, Chen Su#

*Department of Chemical Engineering, University of Delaware, Newark, DE 19711

+Department of Pathology, Anatomy, and Cell Biology, Thomas Jefferson University, Philadelphia, PA 19107

#Eli Lilly and Company, Indianapolis, IN 46285