**4. Conclusion**

One of the important features of the SCE modeling approach is the ability to change parameters of the potential functions that are used to describe interactions between elements to calibrate model representations of biomechanical properties of a particular type of a cell directly using experimental data. More specifically, the SCE model can be used to perform in silico bulk rheology experiments on a single cell in order to scale the parameters such that the passive biomechanical properties of each cell are independent of the number of elements used to represent each cell [117]. As a result, SCE simulation output captures the underlying biomechanical properties of the real biological system being studied and remains relevant regard-

As indicated in Fletcher et al. [84], computational experiments follow a creepstress protocol in which a constant extensile force is applied to the end of an SCE cell whose opposite end is fixed. Before the extensile force is released, the strain is measured as the extension of the cell in the direction of the force relative to its initial linear size. In silico estimates of the viscoelastic properties of cells modeled using the SCE approach have been shown in many biological applications to agree with in vitro rheology measurements [117, 118]. This indicates that the simple phenomenological dynamics of the SCE modeling approach are enough to capture low to intermediate responses of cytoskeletal networks over short timescales (10s) [118]. Over longer timescales (100 s), cells respond actively to external stresses by undergoing cytoskeletal remodeling, and this phenomenon can be incorporated into the SCE modeling approach by inserting and removing subcellular elements of a cell

The generalized Morse potential functions implemented in SCE modeling approach are commonly used in physics and chemistry to model intermolecular interactions [119] and in biology to represent volume exclusion of neighboring regions of the cytoskeleton [94, 95, 120–125]. While it is difficult to associate specific potential functions directly with specific cytoskeletal components of cells, computational studies of bulk properties at the tissue level have suggested that the precise functional form of the potential used in modeling has a small impact on

The subcellular element (SCE) modeling approach has been successful in modeling mechanical properties of individual cells as well as their components and

multicellular tissue as well as describing cellular interactions with mediums such as the extracellular matrix and fluids [76, 84, 86, 89, 90, 95, 96, 111, 117, 120, 122, 124– 127]. The general modeling framework was initially developed by Newman et al. [89] for simulating the detailed dynamics of cell shapes as an emergent response to mechanical stimuli. Recent applications of the SCE modeling approach show that it is flexible enough to model additional diverse biological processes such as intracellular signaling [122], cell differentiation [94], and motion of cells in fluid [124]. To date, the SCE has not been widely used to study biological processes outside the area of epithelial morphogenesis. Christley et al. [122] developed a model of epidermal growth on a basal membrane that incorporates cell growth through the addition on new elements and division by redistributing a cell's group of elements between two new daughter cells. This mode was a novel implementation because it was coupled to a subcellular gene network representing intercellular Notch signaling. The SCE modeling framework has also been coupled to a fluid flow model to simulate the attachment of platelets to blood vessel walls [124]. Using the SCE framework for modeling individual platelets in simulations provides a detailed

determining individual cell impact on the emerging properties of growing

less of the choice of the number of elements used in the model.

in regions under high or low stress [86].

*Apolipoproteins,Triglycerides and Cholesterol*

overall system dynamics [97, 117].

**82**

**3.6 Application of SCE models in biology**

In this chapter, we described the medical and scientific importance of studying misfolded protein diseases as well as provided a broad description of several classes of mathematical models that can be used to further investigate the underlying mechanisms governing protein aggregation and propagation in a multicellular system. In Section 2, we gave an overview of different modeling frameworks that have been developed and validated for studying protein aggregation in both yeast and mammalian systems. In Section 3, we described in detail various cell-based modeling frameworks that have been successfully used to gain insight about the impact of individual cell behaviors on macroscale properties of tumors, developmental tissues, and yeast colonies. However, our main goal in this chapter was to familiarize the reader with each class of model in order to facilitate further discussion about how these two types of models could be combined to develop a more complete representation of prion disease dynamics within an actively growing and dividing yeast colony.

While current models of protein aggregation in yeast have successfully provided further insight into important mechanisms driving prion disease dynamics (i.e., conversion and fragmentation), they have been unable to recreate a number of important physiological characteristics including variable phenotype induction rates that result in sectored phenotypes among a single yeast colony (see Section 1 for details). One reason for this may be that a large number of models representing intracellular dynamics of protein misfolding diseases were developed for studying protein aggregation dynamics in isolation and these models disregard the contribution of individual cell behaviors within the growing yeast colony as a possible mechanism governing prion disease dynamics. A major open question in prion biology is to understand how prion aggregates spread between cells within a whole colony or tissue. Experimental observations such as sectoring provide compelling data that transmission mechanisms other than what is addressed by current aggregation-only-models must play a role in the presence and persistence of prion disease phenotypes in yeast colonies, i.e., processes such as conversion, fragmentation, nucleation, and even enzyme-mediated fragmentation alone cannot entirely explain the spread of prion disease throughout a yeast colony. Thus, in order to test hypotheses about the impact of individual cell behaviors on the spread of misfolded proteins, it is necessary to develop a novel modeling framework.

Developing a modeling framework for investigating prion disease dynamics within an entire yeast colony is challenging because it requires capturing the physical processes on an individual cell level that determine colony growth (i.e., budding and variable cell cycle length) as well as capturing the interplay of individual cell processes with protein aggregation dynamics (i.e., asymmetric protein distribution at the time of division, persistence of diseased phenotype/aggregate that was given to daughter while it grows to begin a new cell cycle, lineage-dependent protein propagation). Several models have already been developed to investigate physical mechanisms controlling patterns of yeast colony growth such as cell division

polarity, mother-daughter size asymmetry, and cell-cell adhesion via budding of the new daughter cell [92, 93, 105]. Jönsson and Levchenko developed an offlattice, center-based model in which cells are modeled as elastic spheres of variable size [92]. Their work showed that cell growth inhibition by neighboring cells and polar division growth patterns were the most significant factors driving appreciable differences in the shape and size of yeast colony development. In addition, Wang et al. built an off-lattice, center-based model that incorporates key biological processes in yeast cell colonies including budding, mating, mating type switch, changes in cell cycle length and cell size due to aging, and cell death [93]. The main results of their work include proposed mechanisms for how budding patterns in yeast cells affect colony growth. Both of these studies serve as excellent starting points for developing a modeling framework that includes the physical processes and individual cell behaviors of yeast cells that govern colony growth.

One way forward would be to extend current cell-based models for yeast colony growth by combining them with a detailed model of protein aggregation dynamics as well as implementing specific model components that capture the interplay of individual cell behaviors with protein aggregation dynamics. Specifically, the new modeling framework would need to include asymmetric protein distribution at the time of division and varied cell cycle lengths to represent how growth of a new daughter cell impacts persistence of aggregates that were inherited at time of division and track protein concentration through different cell lineages to investigate the impact of lineage on colony phenotypes in yeast. In the model, prion dynamics would need to be simulated within each individual cell using models of intracellular dynamics described in Section 2. In addition, using a center-based model would allow for the spatial arrangement of cells to account for the effect of biophysical properties such as increased adhesion between a mother and daughter cell during budding.

Preliminary simulations (**Figure 7**) show that individual cell behaviors indeed impact yeast colony structure as well as protein aggregation dynamics in a growing yeast colony. For our preliminary framework, we extended current cell-based models for yeast colony growth by combining them with a simplified model of protein aggregation dynamics as well as implementing asymmetric protein distribution at the time of division and strong adhesion between mother and daughter cells during budding. These are two cell behaviors we hypothesize to play an important role in protein aggregation dynamics in yeast colonies. Our simulations begin with a single founding cell that divides to give birth to the first daughter. All cells continue to grow and divide throughout the simulation and stop once the colony reaches about 1500 cells total. Successive daughter cells of the founder cell are enumerated in the order of their division. For example, the first daughter is the cell born after the first division of the founder cell, the second daughter is the cell born after the second division of the founder cell, and so on.

sectored prion phenotypes in yeast colonies in addition to serving as a tool for

*The number of aggregates per cell is represented by different colors given in the key below.*

*Impact of individual cell behaviors on yeast colony structure and protein aggregation. In our preliminary simulations, we extended current cell-based modeling approaches for yeast colony growth by combining them with a simplified model of protein aggregation dynamics along with asymmetric protein distribution at cell division and strong adhesion between mother and daughter cells during budding. Simulations begin with a single founder cell and stop after 15 hours when the colony contains about 1500 cells total. (A) Example of subcolony structure for each of nine daughter cells born from the original founder (black cell). Each sub-colony is depicted with a different color indicated below. (B) We show in white the fifth-generation sub-colony corresponding to the fifth daughter of the founder cell and all of its offspring. (C) Example of the interplay between protein aggregation dynamics and individual cell behaviors in a simulation leading to a sectored colony. Protein aggregation dynamics for individual cells were modeled using a logistic growth function with a strong Allee effect which results in stable extinction of protein aggregates in the sub-colony of an early daughter cell.*

*Multi-Scale Mathematical Modeling of Prion Aggregate Dynamics and Phenotypes in Yeast…*

Although our preliminary studies appear promising, there is still a large gap to fill in terms of developing models that consider underlying microscopic processes of protein aggregation together with macroscopic properties of the environment in which they are taking place. Indeed, as mentioned in Section 2 there have been recent efforts to model the impact of tissue structures on the spread of protein misfolding diseases in mammalian systems [63, 64]. However, most of the largescale models for protein aggregation lack details of components at the microscopic scale that would allow for the study of the interplay of different temporal and spatial scales. We hope that future modeling studies will begin to incorporate both scales in order to test hypotheses about mechanisms that can explain unresolved experimental data and yield new strategies for treating protein misfolding diseases.

future hypothesis generation and testing.

*DOI: http://dx.doi.org/10.5772/intechopen.88575*

**Figure 7.**

**85**

**Figure 7A** shows sub-colony structure for each of nine daughters cells born from the original founder cell of the colony represented in black. The first daughter subcolony (grayish black) occupies a larger area than any of the other sub-colonies. In addition, the average location of each successive generation is located closer to the boundary (**Figure 7B**). Protein aggregation dynamics were modeled using a logistic growth function with a strong Allee effect. **Figure 7C** shows the number of aggregates per cell in a simulation that resulted in a sectored colony. The interplay between protein aggregation within each cell and individual cell behaviors results in complete loss of aggregates in an early daughter cell, and subsequently, the subcolony generated by that daughter remains free of aggregates throughout the simulation. These results suggest that the unified model may have the potential to predict mechanisms underlying experimentally observed phenomena such as

*Multi-Scale Mathematical Modeling of Prion Aggregate Dynamics and Phenotypes in Yeast… DOI: http://dx.doi.org/10.5772/intechopen.88575*

#### **Figure 7.**

polarity, mother-daughter size asymmetry, and cell-cell adhesion via budding of the new daughter cell [92, 93, 105]. Jönsson and Levchenko developed an offlattice, center-based model in which cells are modeled as elastic spheres of variable size [92]. Their work showed that cell growth inhibition by neighboring cells and polar division growth patterns were the most significant factors driving appreciable differences in the shape and size of yeast colony development. In addition, Wang et al. built an off-lattice, center-based model that incorporates key biological processes in yeast cell colonies including budding, mating, mating type switch, changes in cell cycle length and cell size due to aging, and cell death [93]. The main results of their work include proposed mechanisms for how budding patterns in yeast cells affect colony growth. Both of these studies serve as excellent starting points for developing a modeling framework that includes the physical processes and individ-

One way forward would be to extend current cell-based models for yeast colony growth by combining them with a detailed model of protein aggregation dynamics as well as implementing specific model components that capture the interplay of individual cell behaviors with protein aggregation dynamics. Specifically, the new modeling framework would need to include asymmetric protein distribution at the time of division and varied cell cycle lengths to represent how growth of a new daughter cell impacts persistence of aggregates that were inherited at time of division and track protein concentration through different cell lineages to investigate the impact of lineage on colony phenotypes in yeast. In the model, prion dynamics would need to be simulated within each individual cell using models of intracellular dynamics described in Section 2. In addition, using a center-based model would allow for the spatial arrangement of cells to account for the effect of biophysical properties such as increased adhesion between a mother and daughter cell during

Preliminary simulations (**Figure 7**) show that individual cell behaviors indeed impact yeast colony structure as well as protein aggregation dynamics in a growing yeast colony. For our preliminary framework, we extended current cell-based models for yeast colony growth by combining them with a simplified model of protein aggregation dynamics as well as implementing asymmetric protein distribution at the time of division and strong adhesion between mother and daughter cells during budding. These are two cell behaviors we hypothesize to play an important role in protein aggregation dynamics in yeast colonies. Our simulations begin with a single founding cell that divides to give birth to the first daughter. All cells continue to grow and divide throughout the simulation and stop once the colony reaches about 1500 cells total. Successive daughter cells of the founder cell are enumerated in the order of their division. For example, the first daughter is the cell born after the first division of the founder cell, the second daughter is the cell

**Figure 7A** shows sub-colony structure for each of nine daughters cells born from the original founder cell of the colony represented in black. The first daughter subcolony (grayish black) occupies a larger area than any of the other sub-colonies. In addition, the average location of each successive generation is located closer to the boundary (**Figure 7B**). Protein aggregation dynamics were modeled using a logistic growth function with a strong Allee effect. **Figure 7C** shows the number of aggregates per cell in a simulation that resulted in a sectored colony. The interplay between protein aggregation within each cell and individual cell behaviors results in complete loss of aggregates in an early daughter cell, and subsequently, the subcolony generated by that daughter remains free of aggregates throughout the simulation. These results suggest that the unified model may have the potential to predict mechanisms underlying experimentally observed phenomena such as

ual cell behaviors of yeast cells that govern colony growth.

*Apolipoproteins,Triglycerides and Cholesterol*

born after the second division of the founder cell, and so on.

budding.

**84**

*Impact of individual cell behaviors on yeast colony structure and protein aggregation. In our preliminary simulations, we extended current cell-based modeling approaches for yeast colony growth by combining them with a simplified model of protein aggregation dynamics along with asymmetric protein distribution at cell division and strong adhesion between mother and daughter cells during budding. Simulations begin with a single founder cell and stop after 15 hours when the colony contains about 1500 cells total. (A) Example of subcolony structure for each of nine daughter cells born from the original founder (black cell). Each sub-colony is depicted with a different color indicated below. (B) We show in white the fifth-generation sub-colony corresponding to the fifth daughter of the founder cell and all of its offspring. (C) Example of the interplay between protein aggregation dynamics and individual cell behaviors in a simulation leading to a sectored colony. Protein aggregation dynamics for individual cells were modeled using a logistic growth function with a strong Allee effect which results in stable extinction of protein aggregates in the sub-colony of an early daughter cell. The number of aggregates per cell is represented by different colors given in the key below.*

sectored prion phenotypes in yeast colonies in addition to serving as a tool for future hypothesis generation and testing.

Although our preliminary studies appear promising, there is still a large gap to fill in terms of developing models that consider underlying microscopic processes of protein aggregation together with macroscopic properties of the environment in which they are taking place. Indeed, as mentioned in Section 2 there have been recent efforts to model the impact of tissue structures on the spread of protein misfolding diseases in mammalian systems [63, 64]. However, most of the largescale models for protein aggregation lack details of components at the microscopic scale that would allow for the study of the interplay of different temporal and spatial scales. We hope that future modeling studies will begin to incorporate both scales in order to test hypotheses about mechanisms that can explain unresolved experimental data and yield new strategies for treating protein misfolding diseases.

*Apolipoproteins,Triglycerides and Cholesterol*

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