**1. Introduction**

Prion proteins are most commonly associated with fatal progressive neurodegenerative diseases in humans and other mammals [1–4]. Intriguingly, all mammalian prion diseases are the result of a single host expressed protein, PrP. In mammalian prion disease, a misfolded form of PrP appears, and rather than being cleared by protein quality control mechanisms, this misfolded form persists and associates into aggregates. These aggregates then act as templates that convert normally folded protein to the misfolded form and may break into multiple aggregates further amplifying the conversion of normal protein [5, 6]. The initiating event of prion disease, the initial appearance of an infectious protein agent, can occur spontaneously, as in sporadic Creutzfeldt-Jakob disease [7]; because of a genetic mutation, as in fatal familial insomnia [8]; and through interaction with

another infected host from the same (as in scrapie [9], chronic wasting disease [10], or Kuru [11]) or different species, as in when Bovine spongiform encephalopathy ("mad cow" disease) is acquired by humans [12] or scrapie is spread from sheep to goats [13]. Further, some prion diseases can occur through multiple modes, such as Creutzfeldt-Jakob which has been associated with each of these modes of transmission [14]. More generally, the amyloid structure of prion aggregates is common to non-prion proteins associated with neurodegenerative disorders such as Alzheimer's, Parkinson's, and Huntington's disease, and evidence continues to suggest commonalities between such disorders and prions [15, 16], including their transmissibility [17]. At present, prion disease, and other amyloid disease such as Alzheimer's [18], has no effective methods for treatment or early detection. However, one potential source to gain insight into these pathogenic phenotypes is to study instances in biology where amyloids confer beneficial consequences, such as growth advantages in yeast and long-term memory storage in mammals [19], and therefore the amyloid dynamics can be studied absent of the disease state [16, 20].

Then, these protein aggregates must be transmitted to other cells. Note that in yeast, this transmission occurs between mother and daughter cells during division [26, 27], while in mammals, this transmission typically occurs between cells in a tissue [28, 29]. Finally, prion phenotypes are determined by examining populations of cells. In yeast, prion phenotypes are assayed at the level of a yeast colony, and mammalian prion phenotypes are typically distinguished by their patterns of neu-

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

that accurately represent all scales of the yeast prion system.

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Beyond mammalian prion disease, yeast prions offer a particularly unique opportunity to investigate and model the emergence and role of heterogeneity in cellular populations. In the past, our knowledge of individual cell behaviors was based on ensemble measurements that do not account for effects of cellular heterogeneity within a population. However, emerging technologies such as single-cell gene expression analysis, single-cell genome sequencing, and single-cell imaging technologies [32] provide an unprecedented opportunity to quantitatively measure heterogeneity at the single-cell level. Molecular and live imaging experiments investigating growth and development of multicellular systems provide rich data

Although there is growing appreciation for the multi-scale nature of mammalian and yeast prion phenotypes, the vast majority of mathematical models developed for studying such phenotypes typically focus on only one scale. In particular, the majority of mathematical model have focused on the isolated prion aggregate dynamics. While these models have been particularly useful in comparisons to in vitro prion aggregation, they are limited in their ability to insight to the in vivo systems. In a population of living cells, different cell behaviors such as growth, diffusion, and division are known to impact the abundances and concentrations of reactants and could have a large impact on protein aggregation or more specifically propagation of prion aggregates. Moreover, in the case of yeast, since the proteins are only transmitted during cell division [6] and yeast have an asymmetric cell cycle [30], a colony of yeast with a prion phenotype will exhibit considerable heterogeneity in prion aggregates and potentially other relevant cellular constitutes [27]. One particular yeast prion phenotype that is intimately linked to population heterogeneity and yet to be mathematically characterized is that of colony sectoring for the *PSI*<sup>þ</sup> ½ � yeast prion phenotype (**Figure 1(C)**). The *PSI*<sup>þ</sup> ½ � phenotype is associated with the color of a colony: a red colony indicates a ½ � *psi*� (prion-free) phenotype and *PSI*<sup>þ</sup> ½ � colony is in a shade of white or pink. The actual color displayed by the colony is associated with a loss of function of the normal Sup35 protein as in when in a prion aggregate the Sup35 monomers have greatly reduced normal function [31]. A sectored colony represents a case when cells in the colony display both ½ � *psi*� and *PSI*<sup>þ</sup> ½ � phenotypes. Such a colony is thought to appear when the founding cell contains prion aggregates, but at some point through division of cells and growth of the colony, some cells lose the ability to propagate the prion phenotype. Because spontaneous appearance of Sup35 prion aggregates is a rare event, any subsequent daughter cells will also lack prion aggregates. These sub-colonies associated with the loss of the prion phenotypes are thought to correspond to the red sectors. Since the number and pattern of red sectors is fairly similar under the same condition, this suggests an intimate connection between the lineage of ancestry of the cells which were the first to lose the ability to propagate the prion phenotype. Indeed a previous study that did not consider the spatial distribution of individual cells suggested that loss of prion aggregates for a particular variant of *PSI*<sup>þ</sup> ½ � occurred in a lineage-dependent fashion [27]. What is the mechanistic basis for this partial loss? Can these mechanisms be exploited in the context of mammalian prion diseases? Answering these questions requires developing mathematical frameworks

rological decay [14].

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

One promising candidate for probing the mechanisms behind prion-associated phenotypes is the yeast *S. cerevisiae* [21–23]. In yeast, prion proteins were first identified in the 1980s when interrogating the genetic basis for non-Mendelian phenotypes [21, 23]. Today, nearly a dozen harmless phenotypes in yeast are known to propagate through misfolded protein aggregates [6]. Unlike their mammalian counterparts, yeast prion phenotypes have been associated with many different proteins including Sup35 ( *PSI*<sup>þ</sup> ½ � phenotype), Rnq1 (½ � *PIN* phenotype), and Ure2 (½ � *URE*3 phenotype). As for mammalian prions, each of these phenotypes has multiple variants (or strains) which themselves are associated with different properties such as infectivity and phenotype presentation. However, unlike mammalian prion disease, which can take months of years to present, yeast prion phenotypes manifest quickly and, most remarkably, can be made to appear or disappear through mild experimental manipulations which do not impact the host cell [6, 16, 24, 25]. The underlying aggregate dynamics are the same regardless of organism. As shown in **Figure 1**, the establishment of prion phenotypes in yeast (and of course mammals) is an inherent multi-scale process. First, there is the biochemistry of the interactions between normal and misfolded protein. Then, there is the level of an individual cell where these protein aggregates interact not only with the cellular environment including molecular chaperones and protein degradation factors.

#### **Figure 1.**

*Multi-scale dynamics of yeast prion Phenotypes. (A) Prion aggregates are thought to be ordered linear structures (squares) capable of* converting *normally folded protein (circles) to the prion form and* fragmenting *into multiple aggregates. Prion aggregates are thought to have a minimal stable size* n0 *below which the aggregate structure will not be stable. (B) Aggregate dynamics occur within an individual yeast cell (large circle) but are spread to daughter cells through transmission through the bud neck. (C) Phenotypes in yeast are observed at the level of a colony which consists of thousands of individual cells. PSI*<sup>þ</sup> ½ � *phenotypes are associated with a color red ( psi* ½ � � *or no prion), pink ( PSI*<sup>þ</sup> ½ �*), or sectored (mixture of phenotypes).*

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

Then, these protein aggregates must be transmitted to other cells. Note that in yeast, this transmission occurs between mother and daughter cells during division [26, 27], while in mammals, this transmission typically occurs between cells in a tissue [28, 29]. Finally, prion phenotypes are determined by examining populations of cells. In yeast, prion phenotypes are assayed at the level of a yeast colony, and mammalian prion phenotypes are typically distinguished by their patterns of neurological decay [14].

Although there is growing appreciation for the multi-scale nature of mammalian and yeast prion phenotypes, the vast majority of mathematical models developed for studying such phenotypes typically focus on only one scale. In particular, the majority of mathematical model have focused on the isolated prion aggregate dynamics. While these models have been particularly useful in comparisons to in vitro prion aggregation, they are limited in their ability to insight to the in vivo systems. In a population of living cells, different cell behaviors such as growth, diffusion, and division are known to impact the abundances and concentrations of reactants and could have a large impact on protein aggregation or more specifically propagation of prion aggregates. Moreover, in the case of yeast, since the proteins are only transmitted during cell division [6] and yeast have an asymmetric cell cycle [30], a colony of yeast with a prion phenotype will exhibit considerable heterogeneity in prion aggregates and potentially other relevant cellular constitutes [27].

One particular yeast prion phenotype that is intimately linked to population heterogeneity and yet to be mathematically characterized is that of colony sectoring for the *PSI*<sup>þ</sup> ½ � yeast prion phenotype (**Figure 1(C)**). The *PSI*<sup>þ</sup> ½ � phenotype is associated with the color of a colony: a red colony indicates a ½ � *psi*� (prion-free) phenotype and *PSI*<sup>þ</sup> ½ � colony is in a shade of white or pink. The actual color displayed by the colony is associated with a loss of function of the normal Sup35 protein as in when in a prion aggregate the Sup35 monomers have greatly reduced normal function [31]. A sectored colony represents a case when cells in the colony display both ½ � *psi*� and *PSI*<sup>þ</sup> ½ � phenotypes. Such a colony is thought to appear when the founding cell contains prion aggregates, but at some point through division of cells and growth of the colony, some cells lose the ability to propagate the prion phenotype. Because spontaneous appearance of Sup35 prion aggregates is a rare event, any subsequent daughter cells will also lack prion aggregates. These sub-colonies associated with the loss of the prion phenotypes are thought to correspond to the red sectors. Since the number and pattern of red sectors is fairly similar under the same condition, this suggests an intimate connection between the lineage of ancestry of the cells which were the first to lose the ability to propagate the prion phenotype. Indeed a previous study that did not consider the spatial distribution of individual cells suggested that loss of prion aggregates for a particular variant of *PSI*<sup>þ</sup> ½ � occurred in a lineage-dependent fashion [27]. What is the mechanistic basis for this partial loss? Can these mechanisms be exploited in the context of mammalian prion diseases? Answering these questions requires developing mathematical frameworks that accurately represent all scales of the yeast prion system.

Beyond mammalian prion disease, yeast prions offer a particularly unique opportunity to investigate and model the emergence and role of heterogeneity in cellular populations. In the past, our knowledge of individual cell behaviors was based on ensemble measurements that do not account for effects of cellular heterogeneity within a population. However, emerging technologies such as single-cell gene expression analysis, single-cell genome sequencing, and single-cell imaging technologies [32] provide an unprecedented opportunity to quantitatively measure heterogeneity at the single-cell level. Molecular and live imaging experiments investigating growth and development of multicellular systems provide rich data

another infected host from the same (as in scrapie [9], chronic wasting disease [10], or Kuru [11]) or different species, as in when Bovine spongiform encephalopathy ("mad cow" disease) is acquired by humans [12] or scrapie is spread from sheep to goats [13]. Further, some prion diseases can occur through multiple modes, such as Creutzfeldt-Jakob which has been associated with each of these modes of transmission [14]. More generally, the amyloid structure of prion aggregates is common to

Alzheimer's, Parkinson's, and Huntington's disease, and evidence continues to suggest commonalities between such disorders and prions [15, 16], including their transmissibility [17]. At present, prion disease, and other amyloid disease such as Alzheimer's [18], has no effective methods for treatment or early detection. However, one potential source to gain insight into these pathogenic phenotypes is to study instances in biology where amyloids confer beneficial consequences, such as growth advantages in yeast and long-term memory storage in mammals [19], and therefore the amyloid dynamics can be studied absent of the disease state [16, 20]. One promising candidate for probing the mechanisms behind prion-associated phenotypes is the yeast *S. cerevisiae* [21–23]. In yeast, prion proteins were first identified in the 1980s when interrogating the genetic basis for non-Mendelian phenotypes [21, 23]. Today, nearly a dozen harmless phenotypes in yeast are known to propagate through misfolded protein aggregates [6]. Unlike their mammalian counterparts, yeast prion phenotypes have been associated with many different proteins including Sup35 ( *PSI*<sup>þ</sup> ½ � phenotype), Rnq1 (½ � *PIN* phenotype), and Ure2 (½ � *URE*3 phenotype). As for mammalian prions, each of these phenotypes has multiple variants (or strains) which themselves are associated with different properties such as infectivity and phenotype presentation. However, unlike mammalian prion disease, which can take months of years to present, yeast prion phenotypes manifest quickly and, most remarkably, can be made to appear or disappear through mild experimental manipulations which do not impact the host cell [6, 16, 24, 25]. The underlying aggregate dynamics are the same regardless of organism. As shown in **Figure 1**, the establishment of prion phenotypes in yeast (and of course mammals) is an inherent multi-scale process. First, there is the biochemistry of the interactions between normal and misfolded protein. Then, there is the level of an individual cell where these protein aggregates interact not only with the cellular environment including molecular chaperones and protein degradation factors.

*Multi-scale dynamics of yeast prion Phenotypes. (A) Prion aggregates are thought to be ordered linear structures (squares) capable of* converting *normally folded protein (circles) to the prion form and* fragmenting *into multiple aggregates. Prion aggregates are thought to have a minimal stable size* n0 *below which the aggregate structure will not be stable. (B) Aggregate dynamics occur within an individual yeast cell (large circle) but are spread to daughter cells through transmission through the bud neck. (C) Phenotypes in yeast are observed at the level of a colony which consists of thousands of individual cells. PSI*<sup>þ</sup> ½ � *phenotypes are associated with a color red*

*( psi* ½ � � *or no prion), pink ( PSI*<sup>þ</sup> ½ �*), or sectored (mixture of phenotypes).*

non-prion proteins associated with neurodegenerative disorders such as

*Apolipoproteins,Triglycerides and Cholesterol*

**Figure 1.**

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sets that can be used for the first time to understand the different signaling cascades and components that are necessary for cell behaviors to arise. This new level of insight makes it possible to reveal a more accurate picture of cellular behavior and highlights the importance of understanding cellular variation in a wide range of biological contexts. Developing models that describe heterogeneity throughout an entire population provides a quantitative way to understand the dynamical behavior of heterogeneous cell population characteristics/biological importance of heterogeneity.

In this chapter, we consider the different tools available for developing a mathematical framework for distinct scales of prion dynamics. We first review mathematical models for prion disease dynamics and discuss the few models which have considered multiple scales of prion aggregation. We then discuss cell-based models as a promising approach for coupling intracellular and intercellular scales. We discuss several common classes of cell-based models and emphasize the contributions they have made to the understanding of biological systems. We conclude with a discussion how one could build a multi-scale model for prion disease dynamics in yeast that couples current models for aggregation of proteins with a spatial model of yeast growth and proliferation on the scale of an entire population.
