4

Modeling with Compartments

Turning to other modeling domains, Lauffenburger proposed to the workshop participants a simple taxonomy of modeling according to what discipline and what goal are uppermost in the researcher’s mind:

  • Computer simulation. Used primarily to mimic behavior so as to allow the manipulation of a system that is suggestive of real biomedical processes;

  • Mathematical metaphor. Used to suggest conceptual principles by approximating biomedical processes with mathematical entities that are amenable to analysis, computation, and extrapolation; and

  • Engineering design. Used to emulate reality to a degree that provides real understanding that might guide bioengineering design.

Byron Goldstein, of Los Alamos National Laboratory, presented work that he thought fell under the first and third of these classifications. He described mathematical models used for studying immunoreceptor signaling that is initiated by different receptors in general organisms. He argued that general models could be effectively used to address detailed features in specific organisms.

Many important receptors—including growth factor, cytokine (which promotes cell division), immune response, and killer cell inhibitory receptors—initiate signaling through a series of four biological steps, each having a unique biological function. Building on work of McKeithan (1995) that proposed a generic model of cell signaling, Goldstein developed a mathematical model for T-cell receptor (TCR) internalization in the immunological synapse. Goldstein’s model takes different contact areas into account and was used to predict TCR internalization at 1 hour for the experiments in Grakoui et al. (1999).

To date, the major effort in cell signaling has been to identify the molecules (e.g., ligands, receptors, enzymes, and adapter proteins) that participate in various signaling pathways and, for each molecule in the pathway, determine which other molecules it interacts with. With an ever-increasing number of participating molecules being identified and new regulation mechanisms being discovered, it has become clear that a major problem will be how to incorporate this information into a useful predictive model.



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Making Sense of Complexity: Summary of the Workshop on Dynamical Modeling of Complex Biomedical Systems 4 Modeling with Compartments Turning to other modeling domains, Lauffenburger proposed to the workshop participants a simple taxonomy of modeling according to what discipline and what goal are uppermost in the researcher’s mind: Computer simulation. Used primarily to mimic behavior so as to allow the manipulation of a system that is suggestive of real biomedical processes; Mathematical metaphor. Used to suggest conceptual principles by approximating biomedical processes with mathematical entities that are amenable to analysis, computation, and extrapolation; and Engineering design. Used to emulate reality to a degree that provides real understanding that might guide bioengineering design. Byron Goldstein, of Los Alamos National Laboratory, presented work that he thought fell under the first and third of these classifications. He described mathematical models used for studying immunoreceptor signaling that is initiated by different receptors in general organisms. He argued that general models could be effectively used to address detailed features in specific organisms. Many important receptors—including growth factor, cytokine (which promotes cell division), immune response, and killer cell inhibitory receptors—initiate signaling through a series of four biological steps, each having a unique biological function. Building on work of McKeithan (1995) that proposed a generic model of cell signaling, Goldstein developed a mathematical model for T-cell receptor (TCR) internalization in the immunological synapse. Goldstein’s model takes different contact areas into account and was used to predict TCR internalization at 1 hour for the experiments in Grakoui et al. (1999). To date, the major effort in cell signaling has been to identify the molecules (e.g., ligands, receptors, enzymes, and adapter proteins) that participate in various signaling pathways and, for each molecule in the pathway, determine which other molecules it interacts with. With an ever-increasing number of participating molecules being identified and new regulation mechanisms being discovered, it has become clear that a major problem will be how to incorporate this information into a useful predictive model.

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Making Sense of Complexity: Summary of the Workshop on Dynamical Modeling of Complex Biomedical Systems To have any hope of success, such a model must constantly be tested against experiments. What makes this possible is the ability of molecular biologists to create experimental systems containing only small numbers of signaling molecules. Thus, separate parts of the model can be tested directly. Where are we at the moment in our attempt to build a detailed model of cell signaling? Goldstein has used deterministic and stochastic approaches to create the following detailed models of cell signaling: An algorithm has been created to generate the chemical rate equations that describe the dynamics of the average concentrations of chemical species involved in a generic signaling cascade. A stochastic model for the time dependence of the state concentrations has been developed, and it has been shown that the stochastic and deterministic formulations agree in the cases studied to date. A model has been created for the signaling cascade that is mediated by the immunoreceptor that plays a central role in allergic reactions. This model includes a bivalent ligand, a monovalent receptor, and the first two enzymes in the cascades, Lyn and Syk. Additional information on Goldstein’s modeling may be found at <http://www.t10.lanl.gov/profiles/Goldtein.html>. Moving from intracellular processes, Bruce Levin, of Emory University, presented some research that uses mathematical models to understand trends in antibiotic resistance, a serious public health concern worldwide. Levin is addressing the need to know what trends are and are not of serious importance. As an example, he noted that resistance to vancomycin (an antibiotic) increased from approximately 1 percent in 1989 to 16 percent in 1997. It does not necessarily follow, however, that this is a serious problem. Lipsitch et al. (2000) state as follows: Although it generally is assumed that use of a particular antibiotic will be positively related to the level of resistance to that drug . . . it is difficult to judge whether an intervention has been successful . . . . Mathematical models can provide such quantitative predictions, which naturally give rise to criteria for evaluating the interventions. Population dynamics can be examined with a compartment model, as shown in Figure 4-1. The compartments represent the disease state of the individual (S, susceptible; IS, immune/susceptible; IR, immune/resistant). The proportion p stands for those under treatment, and the parameters represent the rate of movement from one compartment to another. Based on such a model, one can calculate parameters such as basic reproductive numbers and then establish rates and conditions under which the percent of resistance will increase when a proposed treatment is applied. What is often observed in public health is that the rate of resistance changes as the efficacy of the treatment changes, with high efficacy corresponding to high resistance, and the rate of resistance increases more rapidly than it decreases. To further investigate how a host controls infection, Levin examined E. coli infection in mice, where the following threshold effect has been observed experimentally: While high doses of E. coli kill mice, lower doses can be brought under control. A differential equations model was developed that includes this threshold effect, and it was found to fit the data quite well. Levin’s results again illustrate one of the common themes of the workshop, that a mathematical model—built on a functional premise, even if simple, and verified with data—allows us to quantify biophysical processes in a way that can lead to valuable insight about the underlying structure of the processes.

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Making Sense of Complexity: Summary of the Workshop on Dynamical Modeling of Complex Biomedical Systems FIGURE 4-1 Compartment model for the epidemiology of the antibiotic therapy of prophylaxis. Figure courtesy of Bruce Levin.