How and why to build a mathematical model: A case study using prion aggregation

Abstract

Biological systems are inherently complex, and the increasing level of detail with which we are able to experimentally probe such systems continually reveals new complexity. Fortunately, mathematical models are uniquely positioned to provide a tool suitable for rigorous analysis, hypothesis generation, and connecting results from isolated in vitro experiments with results from in vivo and whole-organism studies. However, developing useful mathematical models is challenging because of the often different domains of knowledge required in both math and biology. In this work, we endeavor to provide a useful guide for researchers interested in incorporating mathematical modeling into their scientific process. We advocate for the use of conceptual diagrams as a starting place to anchor researchers from both domains. These diagrams are useful for simplifying the biological process in question and distinguishing the essential components. Not only do they serve as the basis for developing a variety of mathematical models, but they ensure that any mathematical formulation of the biological system is led primarily by scientific questions. We provide a specific example of this process from our own work in studying prion aggregation to show the power of mathematical models to synergistically interact with experiments and push forward biological understanding. Choosing the most suitable model also depends on many different factors, and we consider how to make these choices based on different scales of biological organization and available data. We close by discussing the many opportunities that abound for both experimentalists and modelers to take advantage of collaborative work in this field.

Keywords: computational biology; differential equation; enzyme kinetics; law of mass action; mathematical methods; mathematical modeling; numerical analysis; prion disease; protein aggregation.

Figure 1.

Interplay between experiments and mathematical models. Mathematical models can be used to test hypotheses, probe changes in parameters, generate predictions, and design new experiments.

Figure 2.

Building a model of enzyme substrate kinetics. Left, a conceptual diagram illustrating the key biochemical species important in the system along with their interactions. Middle, explicit description of the biochemical reactions represented from the diagram. Right, a mathematical model, a system of ordinary differential equations, describing the rate of change of each biochemical species.

Figure 3.

The law of mass action. The relationship of the rate of a reaction to the concentrations of the chemical species that are involved in the reaction is given by the law of mass action. This law states that the rate of a reaction is proportional to the product of the concentrations of the reactants.

Figure 4.

Diagram to differential equations. Left, a diagram depicting the key players (circle, monomer; blocks, aggregate) and their interactions. Middle, the set of biochemical equations that govern the interactions between the key biochemical players monomer X and aggregates of size i = y_i. (Note that because of the assumption of a minimum stable nucleus size of n₀, our form of the fragmentation equation depicted is correct only if i ≥ n₀, j ≥ n₀). Right, differential equation model schematic depicting the temporal evolution of the concentration of monomer (x(t)) and aggregates of each size i (y_i(t)). See Section 3 for more information.

Figure 5.

Temporal kinetics of prion aggregation. We plot the temporal evolution of the concentration of monomer x(t) and protein in aggregates Z(t) (left) and the number of aggregates Y(t) (right) for “theoretical” choice of parameters. We note that although prion aggregates are present at the initial condition, there is a long delay before we see a significant decrease in the healthy protein. In B, the conversion rate is doubled, causing the time it takes for the system to reach steady state to be cut in half.

Figure 6.

Multiscale nature of biological systems. A diagram depicts the multiscale nature of prion disease dynamics. A, biochemical scale. Shown are the key biochemical players and their interactions. Protein is synthesized at a rate α, monomers are converted into aggregates at a rate β, and fragmentation of aggregates occurs at a rate γ. B, subcellular scale. The process of conversion, fragmentation, synthesis, and degradation of protein aggregates occurs inside individual cells. The presence of molecular chaperones, protein degradation factors, and other substances varies inside each individual cell and can impact the rate of biochemical interactions. C, multicellular scale. Division in yeast occurs through budding, a process during which protein and aggregates are segregated between the mother and daughter cell. Modeling individual cell behaviors makes it possible to study the interplay of molecular, subcellular, and multicellular phenomena. D, colony/tissue scale. At the largest scale, prion disease in yeast manifests as different phenotypes at the colony level (white, disease; red, disease-free), the most interesting of which is sectored colonies where sections of cells within a majority diseased colony lose all of their aggregates and become disease-free (84). Moreover, multiscale models hope to connect important components at the biochemical scale with observed spatiotemporal patterns in colony phenotypes. In mammalian neurodegenerative diseases, disease phenotypes are observed at the level of an organ (–86) (2).