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.

Dissection of enzymatic kinetics and elucidation of detailed parameters based on the Michaelis‐Menten model. Kinetic and thermodynamic connections

“…An analogous phenomenon was previously described by Weber 42 as the "conformational drift of protein subunits" to explain the decrease in the free energy of association of monomers after protein dissociation by pressure or low temperature. The quantification of , which modulates the energy of ligand association/dissociation, involves a relationship between the rate constant and the initial substrate concentration, as expressed by Equations (15) to (17).…”

“…The parameter in these equations is obtained by fitting results from Figure 3 and is therefore not necessarily constant. Consequently Equations (15) to (17) cannot be used to calculate k as a function of s o , but serve to calculate from k and s o . Another point to be noted in these equations is that these parameters are obtained with respect to an initial substrate concentration of reference s 0i .…”

“…(D) Change in the conformational drift parameter (Δ E + S → ES , Δ ES → E + P and Δ S → P ) according to the Equations (24) and (25). The results shown in panels C and D were obtained using Equations (24) to (26) To relate the kinetic constants to the corresponding Gibbs free energy, the natural logarithmic is applied to both sides of Equations (15), (16) and (17) followed by multiplication by -RT (where R is the gas constant and T the temperature), which yields…”

Towards Generative Design of Computationally Efficient Mathematical Models with Evolutionary Learning

“…An extensive literature review shows many attempts for mathematical models design in the different fields [ 9 , 10 ]. In particular, the methods of the automated model design is highly valuable part of the various researches [ 11 ].…”

Prion Interaction with Normal Protein in Topological Changing Secondary Structure to Aggregation

“…Aggregates consist of groups of misfolding proteins and every discrete size should be larger than the critical nucleus size [3] . The aggregation and disaggregation process is dynamic.…”