*2.2.2 Mammalian tissue and organismal-level models*

work on yeast cell division which has facilitated the cellular scale of prion dynamics.

One recent study took a different approach by considering the related aggregate recovery assay using a generation and aggregate structured population model [61]. (Once again, the experimental process is too complicated to fully describe, so we offer a simplified description.) Aggregate recovery assays monitor the number of aggregates in a typical cell in time. This is a two-stage experiment where all cells are driven to have (ideally) at most one aggregate in a single cell. Then the cells are released, and aggregates will thus amplify within each cell, and cells will continue to grow and divide as normal. In this assay, cells from the recovering population are sampled in time, and the number of aggregates counted according to the assay in the previous paragraph. Both for mathematical simplicity and because this experiment only monitors the number of aggregates and not their size of concentration of protein in the normal conformation, Banks et al. [61] considered a simplified model of intracellular dynamics where the number of aggregates in a single cell increased according to either an exponential or logistic growth equation. They developed a structured population model that separately tracked the number of aggregates in cells having undergone 0, 1, 2, etc. cell divisions since the start of the experiment. Although this model greatly simplified the intracellular aggregate dynamics, researchers were able to recapitulate know biological relationships between prion

At present, only two studies have considered both a detailed view of the intercellular aggregate dynamics and multicellular yeast population [27, 62]. Because these models were analytically intractable, both researchers developed a multi-scale stochastic simulation framework. Each cell in the population was modeled as a well-mixed compartment with discrete numbers of proteins in the normal conformation and in each possible aggregate size. The intracellular aggregate dynamics were forward simulated according to the Gillespie algorithm for sampling from the chemical master equation. During cell division aggregates were distributed between cells, and the lineage relationships between all cells were tracked. Tanaka et al. [62] used this stochastic multi-scale framework as a way to validate conclusions that drew from a system of ODEs they developed which considered population averages. Derdowski et al. [27] considered explicit comparisons with single-cell experiments in yeast. They used their model to conclude that in vivo

First, a number of models have emerged that allow for direct comparison between yeast "aggregate counting assays" [26, 57–60]. In such assays, an original cell is sampled, and and it is exposed to GdnHCl which is thought to severely reduce the fragmentation rate without altering the ability of cells to divide. Because the number of aggregates remains (relatively) constant while the number of cells increases exponentially, eventually each cell in the population is assumed to have at most one aggregate. When these resulting cells are restored to normal growth conditions (i.e., aggregate fragmentation is normal) and allowed to find their own independent colony, the number of *PSI*<sup>þ</sup> ½ � colonies observed is taken as reflective of the number of aggregates in the original cell. (For the purpose of narrative simplicity, we have greatly simplified both the experimental assay and its conclusions assay. Technically this assay counts the number of "propagons" in a cell, and here we assume that propagon is synonymous with prion aggregate although the true case is more complicated [6, 60].) Because the number of aggregates does not increase in most aggregate counting assays, this allows the intracellular processes to be greatly simplified. That is, the number of aggregates changes only through cell division when aggregates are distributed perhaps according to experimentally

We discuss two approaches taken to model this heterogeneity.

*Apolipoproteins,Triglycerides and Cholesterol*

observed biases in cell division.

strains.

**72**

In mammals, mathematical models have been developed to study the spread of prion (and more generally protein aggregation) in the brain and between organisms. In the study of Alzheimer's, researchers have leveraged knowledge about the connectivity between regions of the brain to develop a network diffusion model of disease progression [63, 64]. Remarkably, they discovered that characteristic patterns of atrophy in patients can be explained by eigenelements of their network model. However, as in the models of aggregate counting assays in yeast, they assumed a highly simplified model of protein dynamics. That is, they did not consider molecular scale processes of aggregation but only the diffusion of an agent through the brain and associated atrophy itself with higher concentrations of the agent. However, because the early phases of mammalian prion and neurodegenerative are poorly characterized in vivo, it is possible that their models are consistent with late stages of neurodegenerative disease.

Finally, because prion disease spreads between animals in the same population, epidemic models have been employed to study the population-level dynamics of these diseases. In particular, researchers have studied the spread of scrapie in flocks of sheep [65–67] and chronic wasting disease in elk [68, 69]. Although these models have been informative of the specific mechanisms which prion disease is likely to spread in these populations (mother to offspring, indirect environmental transmission), they again consider only the organismal scale and not the molecular mechanisms responsible for within the host aggregation dynamics.

Cell-based models, a type of agent-based simulation framework, offer the ability to consider complex intracellular dynamics, distinguish between individual cells, and track the spatial distribution of cells within a colony. In the next section, we discuss common cell-based model formulations with a focus on the mathematical foundation for these models as well as their contributions to particular biological domains.
