**2. Intracellular dynamics**

As we discussed in Section 1, prion dynamics is an inherently multi-scale process. However, the bulk of mathematical modeling on prion aggregation has considered only the microscopic scale, intracellular dynamics. Because the formation of the initial stable aggregate is a rare occurrence, most of these models have focused on the dynamics of the prion aggregates themselves in a well-mixed compartment leading to models that reflect what occurs microscopically within a single cell or are taken to be the average behavior of a population or tissue. In addition, these models assume either discrete or continuous aggregate sizes employing ordinary differential equations (ODEs) or partial differential equations (PDEs), respectively. This literature has emphasized establishing results on the global stability of prion aggregates as well as their asymptotic size distribution. More recently, mathematical approaches have been developed to consider the spread of prions between cells in a tissue or even organisms, but only under severely simplified molecular scales. At present, no single multi-scale framework has yet been formulated for prion dynamics leaving many unexplained phenotypes, such as yeast sectored colony phenotypes (see **Figure 1(C)**).

protein-based infectious agent of scrapie. Since then, a large number of mathematical models have emerged which track the size density of prion aggregates under different biochemical processes but typically include conversion of normal protein to prion aggregates and fragmentation of aggregates. Most are based whole or in

*Nucleated polymerization model. The NPM (see [39, 40]) is the canonical model for prion aggregate dynamics. In this formulation normally folded protein (circles) can be created with rate λ, decayed with rate d, converted to the misfolded form, and added onto the end of an existing aggregate (linear complex of squares) with rate β. Aggregates have their own degradation rate, a, that is independent of aggregate size, and number of aggregates increases through fragmentation which occurs at a rate b times the number of fragmentation sites in an aggregate. (Note that in this diagram, we allow normal protein to be recovered if an aggregate is fragmented below the minimal nucleus size n0* ¼ *2 depicting the aggregate dynamics evolving according to Eqs. (7)–(9).)*

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

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

Nowak and colleagues developed the NPM in a series of papers [39, 40] based on a model previously introduced by Eigen [41]. In this mathematical formulation, the state of the system at time *t* is the concentration of proteins in the normal confor-

normal conformation is assumed to be created at rate *λ* and decays at rate *d*, while aggregates of all sizes decay at rate *a* (see **Figure 2**) Conversion occurs through contact between aggregates and protein in the normal conformation at a rate depending on the size of the aggregate, *βi*. Finally, the total number of aggregates increases through fragmentation. In their most general formulation, Nowak et al. specify the rate that aggregates of size *j* fragment to create an aggregate of size *i* as *b <sup>j</sup>*,*<sup>i</sup>* and that during fragmentation no mass is lost, i.e., an aggregate of size *j* is always fragmented into two aggregates of size *i* and ð Þ *j* � *i* . Translating these biochemical kinetic assumptions into a set of differential equations results in the

ð Þ*t* . Protein in the

ð Þ*t* (1)

ðÞ�*<sup>t</sup>* <sup>X</sup> *i*�1

*j*¼1

*bi*, *<sup>j</sup> yi* ð Þ*t*

(2)

part on the notion of the nucleated polymerization model (NPM).

mation, *x t*ð Þ, and prion aggregates of every discrete size *i*, *yi*

following infinite system of ordinary differential equations:

*dt* <sup>¼</sup> *<sup>λ</sup>* � *dx t*ðÞ�X<sup>∞</sup>

ðÞ�*t ayi*

*i*¼1

ðÞþ*<sup>t</sup>* <sup>X</sup><sup>∞</sup>

*j*¼*i*þ1

*βix t*ð Þ *yi*

*b <sup>j</sup>*,*<sup>i</sup>* þ *b <sup>j</sup>*,*i*� *<sup>j</sup>* � � *<sup>y</sup> <sup>j</sup>*

*dx*

*dt* <sup>¼</sup> *<sup>β</sup><sup>i</sup>*�<sup>1</sup>*x t*ð Þ *yi*�<sup>1</sup>ðÞ�*<sup>t</sup> <sup>β</sup>ix t*ð Þ *yi*

*dyi*

**69**

**Figure 2.**

In this section we review the development of mathematical models in the study of prion dynamics. (Readers interested in a more detailed view on mathematical models of prion disease can consult [33].) Although many of these models share commonality with more general aggregation processes in neurodegenerative disease, we focus only on prion aggregation. (Readers interested in mathematical models of more general neurodegenerative phenomena can consult [34].)

#### **2.1 Intracellular models of prion phenotypes**

The prion hypothesis, the idea that proteins themselves could encode information about their structure without a DNA intermediary, was first suggested by experimental studies in 1966–1967 (see [35–37]). Remarkably, the first mathematical formulation of prion aggregation was published in 1967 when Griffiths [38] offered a simple mathematical autocatalytic process that was consistent with a

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

#### **Figure 2.**

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

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

As we discussed in Section 1, prion dynamics is an inherently multi-scale process. However, the bulk of mathematical modeling on prion aggregation has considered only the microscopic scale, intracellular dynamics. Because the formation of the initial stable aggregate is a rare occurrence, most of these models have focused on the dynamics of the prion aggregates themselves in a well-mixed compartment leading to models that reflect what occurs microscopically within a single cell or are taken to be the average behavior of a population or tissue. In addition, these models assume either discrete or continuous aggregate sizes employing ordinary differential equations (ODEs) or partial differential equations (PDEs), respectively. This literature has emphasized establishing results on the global stability of prion aggregates as well as their asymptotic size distribution. More recently, mathematical approaches have been developed to consider the spread of prions between cells in a tissue or even organisms, but only under severely simplified molecular scales. At present, no single multi-scale framework has yet been formulated for prion dynamics leaving many unexplained phenotypes, such as yeast sectored colony

In this section we review the development of mathematical models in the study of prion dynamics. (Readers interested in a more detailed view on mathematical models of prion disease can consult [33].) Although many of these models share commonality with more general aggregation processes in neurodegenerative disease, we focus only on prion aggregation. (Readers interested in mathematical models of more general neurodegenerative phenomena can consult [34].)

The prion hypothesis, the idea that proteins themselves could encode information about their structure without a DNA intermediary, was first suggested by experimental studies in 1966–1967 (see [35–37]). Remarkably, the first mathematical formulation of prion aggregation was published in 1967 when Griffiths [38] offered a simple mathematical autocatalytic process that was consistent with a

yeast growth and proliferation on the scale of an entire population.

heterogeneity.

**2. Intracellular dynamics**

*Apolipoproteins,Triglycerides and Cholesterol*

phenotypes (see **Figure 1(C)**).

**68**

**2.1 Intracellular models of prion phenotypes**

*Nucleated polymerization model. The NPM (see [39, 40]) is the canonical model for prion aggregate dynamics. In this formulation normally folded protein (circles) can be created with rate λ, decayed with rate d, converted to the misfolded form, and added onto the end of an existing aggregate (linear complex of squares) with rate β. Aggregates have their own degradation rate, a, that is independent of aggregate size, and number of aggregates increases through fragmentation which occurs at a rate b times the number of fragmentation sites in an aggregate. (Note that in this diagram, we allow normal protein to be recovered if an aggregate is fragmented below the minimal nucleus size n0* ¼ *2 depicting the aggregate dynamics evolving according to Eqs. (7)–(9).)*

protein-based infectious agent of scrapie. Since then, a large number of mathematical models have emerged which track the size density of prion aggregates under different biochemical processes but typically include conversion of normal protein to prion aggregates and fragmentation of aggregates. Most are based whole or in part on the notion of the nucleated polymerization model (NPM).

Nowak and colleagues developed the NPM in a series of papers [39, 40] based on a model previously introduced by Eigen [41]. In this mathematical formulation, the state of the system at time *t* is the concentration of proteins in the normal conformation, *x t*ð Þ, and prion aggregates of every discrete size *i*, *yi* ð Þ*t* . Protein in the normal conformation is assumed to be created at rate *λ* and decays at rate *d*, while aggregates of all sizes decay at rate *a* (see **Figure 2**) Conversion occurs through contact between aggregates and protein in the normal conformation at a rate depending on the size of the aggregate, *βi*. Finally, the total number of aggregates increases through fragmentation. In their most general formulation, Nowak et al. specify the rate that aggregates of size *j* fragment to create an aggregate of size *i* as *b <sup>j</sup>*,*<sup>i</sup>* and that during fragmentation no mass is lost, i.e., an aggregate of size *j* is always fragmented into two aggregates of size *i* and ð Þ *j* � *i* . Translating these biochemical kinetic assumptions into a set of differential equations results in the following infinite system of ordinary differential equations:

$$\frac{d\mathbf{x}}{dt} = \lambda - d\mathbf{x}(t) - \sum\_{i=1}^{\infty} \beta\_i \mathbf{x}(t) \,\boldsymbol{y}\_i(t) \tag{1}$$

$$\frac{d\boldsymbol{y}\_{i}}{dt} = \beta\_{i-1}\mathbf{x}(t)\,\boldsymbol{y}\_{i-1}(t) - \beta\_{i}\mathbf{x}(t)\,\boldsymbol{y}\_{i}(t) - a\mathbf{y}\_{i}(t) + \sum\_{j=i+1}^{\infty} \left(b\_{j,i} + b\_{j,i-j}\right)\mathbf{y}\_{j}(t) - \sum\_{j=1}^{i-1} b\_{i,j}\mathbf{y}\_{i}(t) \tag{2}$$

for *i* ¼ 1, 2, … , since there could possibly be infinitely many aggregate sizes. They then considered two simplifying assumptions that allowed a moment closure. First, the conversion rate is assumed to be independent of aggregate size. This occurs when assuming that the templated conversion of normal protein to the misfolded state can only occur when interacting with either end of a linear aggregate. Second, the fragmentation of an aggregate is assumed to be linearly proportional to its size, and the fragmentation kernel is assumed to be uniform (i.e., fragmentation is equally likely to occur at any monomer junction in a linear aggregate). Under these two simplifications, the infinite system of differential equations has the following three-dimensional moment closure:

$$\frac{d\mathbf{x}}{dt} = \lambda - d\mathbf{x}(t) - \beta \mathbf{x}(t)Y(t) \tag{3}$$

Eqs. (7)–(9) (or appropriately modified Eqs. (1) and (2)) are referred to as the nucleated polymerization model (NPM). The dynamics of the NPM are similar to those presented in Nowak's first model; however, the minimal nucleus size modifies the resulting equations slightly. First, the quantities *Y t*ð Þ and *Z t*ð Þ now represent the

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

The NPM, and its variants, has been extensively studied. In their paper, Masel,

Mathematicians continued to formalize the NPM through the twenty-first century. Prüss and colleagues [45] demonstrated that the prion phenotypes were glob-

ally asymptotically stable, and not merely locally stable, through deriving a Lyapunov function. Engler et al. [46] analyzed the well-posedness of a generalization of the NPM where aggregate sizes were continuous, instead of discrete. As such, rather than an infinite system of ordinary differential equations, the system consisted of a single ODE for protein in the normal configuration and a PDE specifying the distribution of aggregate sizes. While this formulation departs from the physically discrete nature of aggregates, in the limit of large aggregate sizes, these formalisms are provably equivalent [47], and the use of PDEs permits a wider array of mathematical techniques. Most notably, the continuous relaxation on aggregate sizes has permitted determination of the explicit asymptotic density [44, 46]. In comparison, the asymptotic density for the aggregate model with discrete aggregate sizes, while first approximated in 2003 by Pöschel et al. [48], was derived only recently by Davis and Sindi and required special functions [49]. Mathematical models of prion aggregate dynamics have been formulated under many more general kinetic assumptions such as nonlinear conversion rates, aggregate joining, and general fragmentation kernels (see [50–53]). More recently, models have been developed which consider prion aggregates along with other intracellular species (such as molecular chaperones) which impact their biochemical dynamics [54, 55], and others have considered interactions between prions and

Jensen, and Nowak conducted an detailed analysis of the NPM [39] including extensive linking of experimental observations on the time to appearance of prion disease symptoms with the kinetic parameters of the NPM and determining a viable range of minimal nucleus sizes *n*0. Overall, there was remarkable consistency between parameters predicted from different experimental data sets analyzed providing support at the time for this mathematical formulation. In addition, Masel et al. [39] (and then Greer and colleagues with a generalization [44]) demonstrated that the dynamics of aggregates under the NPM are consistent with the long incubation time observed for prion phenotypes. If prion disease begins with the introduction of a small amount of prion protein (in the form of aggregates), those aggregates will first have to increase in size until there are enough fragmentation

sites to permit aggregate amplification through fragmentation.

other protein aggregation diseases such as Alzheimer's [56].

While some of the models discussed in the previous section were not explicit about their biological context, in nearly all cases the mathematical models of prion aggregation processes neglected the multicellular scale of the establishment of prion phenotypes. However, there is an emerging literature on multicellular models of prion disease dynamics. In all cases, the intracellular scale is either simplified or ignored.

Several models have emerged to consider the special case of heterogeneity of prion aggregates in yeast colonies. Such models have benefitted from the extensive

**2.2 Multicellular models**

*2.2.1 Multicellular yeast colonies*

**71**

aggregates above this critical minimal size, *n*0.

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

$$\frac{dY}{dt} = bZ(t) - (a+b)Y(t) \tag{4}$$

$$\frac{dZ}{dt} = \beta \mathbf{x}(t)Y(t) - aZ(t) \tag{5}$$

where *Y t*ðÞ¼ <sup>P</sup><sup>∞</sup> *<sup>i</sup>*¼<sup>1</sup> *yi* ð Þ*t* represents the total number of aggregates and *Z t*ðÞ¼ P<sup>∞</sup> *<sup>i</sup>*¼<sup>1</sup>*iyi* ð Þ*t* represents the total amount of prion protein. We note that mathematically *Y t*ð Þ and *Z t*ð Þ correspond to the zeroth and first moments of the distribution of aggregate sizes and, as such, the time evolution under these kinetic simplifications is determined by the zeroth and first moments.

Nowak and colleagues demonstrated this model (Eqs. (3)–(5)) was mathematically equivalent to previously developed viral models studied in mathematical epidemiology and derived a basic reproductive number for their model. The basic reproductive number, or *R*0, is a common parameter in epidemiology and specifies the number of secondary infections (in this case infectious aggregate) created by an infectious aggregate in an initially susceptible population. In the case that *R*<sup>0</sup> . 1, we expect exponential growth of aggregates and therefore a stable nontrivial prion aggregate distribution. If *R*<sup>0</sup> , 1, we expect the prion aggregates to exponentially decay in number and the system to return to the aggregate-free state.

In their final formalization, Nowak and colleagues [39] considered the nucleated polymerization assumption of prion aggregates. Namely, it is believed that aggregates below a critical size are not thermodynamically stable. (This is thought to be part of the reason spontaneous appearance of prion disease is so rare because it is a nucleation-limited process [42, 43].) Since Nowak and colleagues did not consider spontaneous nucleation, the nucleus size enters only when aggregate fragmentation would create an aggregate of size below the minimum stable nucleation size *n*0. In this case, the monomers in that aggregate return to the pool of normally folded protein. Under the previous simplifications on kinetic rates, this changes the resulting moment closure of the infinite system of ODEs as follows:

$$Y(t) = \sum\_{i=n\_0}^{\infty} \mathcal{y}\_i(t) \quad \text{and} \quad Z(t) = \sum\_{i=n\_0}^{\infty} i \mathcal{y}\_i(t) \tag{6}$$

$$\frac{d\mathbf{x}}{dt} = \lambda - d\mathbf{x}(t) - \beta \mathbf{x}(t)Y(t) + b(n\_0)(n\_0 - \mathbf{1})Y(t) \tag{7}$$

$$\frac{dY}{dt} = bZ(t) - (a + b(2n\_0 - 1))Y(t) \tag{8}$$

$$\frac{dZ}{dt} = \beta \mathbf{x}(t)Y(t) - aZ(t) - b(n\_0)(n\_0 - 1)Y(t). \tag{9}$$

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

Eqs. (7)–(9) (or appropriately modified Eqs. (1) and (2)) are referred to as the nucleated polymerization model (NPM). The dynamics of the NPM are similar to those presented in Nowak's first model; however, the minimal nucleus size modifies the resulting equations slightly. First, the quantities *Y t*ð Þ and *Z t*ð Þ now represent the aggregates above this critical minimal size, *n*0.

The NPM, and its variants, has been extensively studied. In their paper, Masel, Jensen, and Nowak conducted an detailed analysis of the NPM [39] including extensive linking of experimental observations on the time to appearance of prion disease symptoms with the kinetic parameters of the NPM and determining a viable range of minimal nucleus sizes *n*0. Overall, there was remarkable consistency between parameters predicted from different experimental data sets analyzed providing support at the time for this mathematical formulation. In addition, Masel et al. [39] (and then Greer and colleagues with a generalization [44]) demonstrated that the dynamics of aggregates under the NPM are consistent with the long incubation time observed for prion phenotypes. If prion disease begins with the introduction of a small amount of prion protein (in the form of aggregates), those aggregates will first have to increase in size until there are enough fragmentation sites to permit aggregate amplification through fragmentation.

Mathematicians continued to formalize the NPM through the twenty-first century. Prüss and colleagues [45] demonstrated that the prion phenotypes were globally asymptotically stable, and not merely locally stable, through deriving a Lyapunov function. Engler et al. [46] analyzed the well-posedness of a generalization of the NPM where aggregate sizes were continuous, instead of discrete. As such, rather than an infinite system of ordinary differential equations, the system consisted of a single ODE for protein in the normal configuration and a PDE specifying the distribution of aggregate sizes. While this formulation departs from the physically discrete nature of aggregates, in the limit of large aggregate sizes, these formalisms are provably equivalent [47], and the use of PDEs permits a wider array of mathematical techniques. Most notably, the continuous relaxation on aggregate sizes has permitted determination of the explicit asymptotic density [44, 46]. In comparison, the asymptotic density for the aggregate model with discrete aggregate sizes, while first approximated in 2003 by Pöschel et al. [48], was derived only recently by Davis and Sindi and required special functions [49]. Mathematical models of prion aggregate dynamics have been formulated under many more general kinetic assumptions such as nonlinear conversion rates, aggregate joining, and general fragmentation kernels (see [50–53]). More recently, models have been developed which consider prion aggregates along with other intracellular species (such as molecular chaperones) which impact their biochemical dynamics [54, 55], and others have considered interactions between prions and other protein aggregation diseases such as Alzheimer's [56].

### **2.2 Multicellular models**

While some of the models discussed in the previous section were not explicit about their biological context, in nearly all cases the mathematical models of prion aggregation processes neglected the multicellular scale of the establishment of prion phenotypes. However, there is an emerging literature on multicellular models of prion disease dynamics. In all cases, the intracellular scale is either simplified or ignored.

#### *2.2.1 Multicellular yeast colonies*

Several models have emerged to consider the special case of heterogeneity of prion aggregates in yeast colonies. Such models have benefitted from the extensive

for *i* ¼ 1, 2, … , since there could possibly be infinitely many aggregate sizes. They then considered two simplifying assumptions that allowed a moment closure. First, the conversion rate is assumed to be independent of aggregate size. This occurs when assuming that the templated conversion of normal protein to the misfolded state can only occur when interacting with either end of a linear aggregate. Second, the fragmentation of an aggregate is assumed to be linearly proportional to its size, and the fragmentation kernel is assumed to be uniform (i.e., fragmentation is equally likely to occur at any monomer junction in a linear aggregate). Under these two simplifications, the infinite system of differential equations

*dt* <sup>¼</sup> *<sup>λ</sup>* � *dx t*ðÞ� *<sup>β</sup>x t*ð Þ*Y t*ð Þ (3)

*dt* <sup>¼</sup> *bZ t*ðÞ� ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup> Y t*ð Þ (4)

*dt* <sup>¼</sup> *<sup>β</sup>x t*ð Þ*Y t*ðÞ� *aZ t*ð Þ (5)

ð Þ*t* represents the total number of aggregates and *Z t*ðÞ¼

ð Þ*t* represents the total amount of prion protein. We note that mathematically *Y t*ð Þ and *Z t*ð Þ correspond to the zeroth and first moments of the distribution of aggregate sizes and, as such, the time evolution under these kinetic simplifications

Nowak and colleagues demonstrated this model (Eqs. (3)–(5)) was mathematically equivalent to previously developed viral models studied in mathematical epidemiology and derived a basic reproductive number for their model. The basic reproductive number, or *R*0, is a common parameter in epidemiology and specifies the number of secondary infections (in this case infectious aggregate) created by an infectious aggregate in an initially susceptible population. In the case that *R*<sup>0</sup> . 1, we expect exponential growth of aggregates and therefore a stable nontrivial prion aggregate distribution. If *R*<sup>0</sup> , 1, we expect the prion aggregates to exponentially

In their final formalization, Nowak and colleagues [39] considered the nucleated polymerization assumption of prion aggregates. Namely, it is believed that aggregates below a critical size are not thermodynamically stable. (This is thought to be part of the reason spontaneous appearance of prion disease is so rare because it is a nucleation-limited process [42, 43].) Since Nowak and colleagues did not consider spontaneous nucleation, the nucleus size enters only when aggregate fragmentation would create an aggregate of size below the minimum stable nucleation size *n*0. In this case, the monomers in that aggregate return to the pool of normally folded protein. Under the previous simplifications on kinetic rates, this changes the

ð Þ*<sup>t</sup>* and *Z t*ðÞ¼ <sup>X</sup><sup>∞</sup>

*i*¼*n*<sup>0</sup> *iyi*

*dt* <sup>¼</sup> *bZ t*ðÞ� ð Þ *<sup>a</sup>* <sup>þ</sup> *<sup>b</sup>*ð Þ <sup>2</sup>*n*<sup>0</sup> � <sup>1</sup> *Y t*ð Þ (8)

*dt* <sup>¼</sup> *<sup>λ</sup>* � *dx t*ð Þ� *<sup>β</sup>x t*ð Þ*Y t*ðÞþ *b n*ð Þ<sup>0</sup> ð Þ *<sup>n</sup>*<sup>0</sup> � <sup>1</sup> *Y t*ð Þ (7)

*dt* <sup>¼</sup> *<sup>β</sup>x t*ð Þ*Y t*ð Þ� *aZ t*ðÞ� *b n*ð Þ<sup>0</sup> ð Þ *<sup>n</sup>*<sup>0</sup> � <sup>1</sup> *Y t*ð Þ*:* (9)

ð Þ*t* (6)

has the following three-dimensional moment closure:

*Apolipoproteins,Triglycerides and Cholesterol*

where *Y t*ðÞ¼ <sup>P</sup><sup>∞</sup>

P<sup>∞</sup> *<sup>i</sup>*¼<sup>1</sup>*iyi*

**70**

*<sup>i</sup>*¼<sup>1</sup> *yi*

is determined by the zeroth and first moments.

*dx*

*dY*

*dZ*

decay in number and the system to return to the aggregate-free state.

resulting moment closure of the infinite system of ODEs as follows:

*Y t*ðÞ¼ <sup>X</sup><sup>∞</sup>

*dY*

*dx*

*dZ*

*i*¼*n*<sup>0</sup> *yi* work on yeast cell division which has facilitated the cellular scale of prion dynamics. We discuss two approaches taken to model this heterogeneity.

fragmentation occurs in a rate-limiting fashion and that aggregate transmission was biased such that larger aggregates are retained by mother cells. However, neither study considered the spatial growth of cells in a colony and, as such, cannot be

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

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

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 mecha-

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.

As described in Sections 1 and 2, prion diseases offer a particularly intriguing biological phenomenon for mathematical and computational analysis because such diseases cover many different systems and spatial scales. At the level of a population, prion diseases can be studied as a classical epidemic model where infections are spread among a susceptible population. Alternatively, prion diseases can be studied on a microscopic scale as a genetic disease whose phenotype is caused by prion aggregates that cause a gain of function mutation in certain genes. Current models of protein aggregation in yeast have successfully provided further insight into important mechanisms driving prion disease dynamics (i.e., conversion and fragmentation), but there is a need to develop models that consider the underlying microscopic processes of protein aggregation together with macroscopic properties of the environment in which they are taking place. This requires modeling frameworks that consider the impact of processes taking place on many different spatiotemporal scales. One such class of models that has been developed primarily for studying biological processes from many different scales is cell-based models.

adapted to the question of yeast colony sectoring.

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

*2.2.2 Mammalian tissue and organismal-level models*

with late stages of neurodegenerative disease.

**3. Cell-based modeling approaches**

**73**

nisms responsible for within the host aggregation dynamics.

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 observed biases in cell division.

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 strains.

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 fragmentation occurs in a rate-limiting fashion and that aggregate transmission was biased such that larger aggregates are retained by mother cells. However, neither study considered the spatial growth of cells in a colony and, as such, cannot be adapted to the question of yeast colony sectoring.
