Team:Waterloo/Aggregation

Lab and Design
Math Modelling

Aggregation Model

Overview

In order to understand the aggregation process of amyloids in yeast cells as well as to understand the curing process of these amyloids, a simulation was created. Additional objectives of the model were to give a approximate timescale for checkpoints during experiments for the lab team, predicting projected responses of the system after experimental runtimes, as well as to confirm necessary levels Hsp104 required for different system responses found in literature.[5],[6],[7]

Background Information

The life cycle of an prion protein can be summarized down into 4 stages: synthesis, conversion, fragmentation, and transmission[1]. Synthesis is when healthy Sup35 is created by the the yeast cell. It is then possible for it to be converted into the prion state either through spontaneous changes[2] or recruited to a growing aggregate, and then converted to the prion state as a part of the aggregate[3].The aggregate later fragments into smaller aggregates[1], and is eventually transmitted through cell division into the daughter cell[4].

This is shown in the figure below, where the squares are the [PSI+] Sup35 and the triangles are healthy Sup35.

Aggregate fragmentation is crucial to spreading the aggregates throughout the population, due to aggregates larger than 20 monomers being unable to be transmitted to daughter cell and thus propagated[1]. Fragmentation can be induced by several factors, such as the chaperone molecule, Hsp104[5].

It has been found that the concentration of Hsp104 induces three different responses in the [PSI+] of the yeast population. Higher concentrations of Hsp 104 will "cure" the [PSI+] state and revert the Sup35 to its [psi-] soluble form[6] due to a large increase in fragmentation induced by the Hsp104[7].Under-expression of Hsp104 will also decrease of the portion of the yeast population infected with prions, due to a lack of fragmentation, thus forming aggregates too large to pass to a daughter cell, allowing the aggregates to remain in the mother until cell death[5]. However, a more medial concentration will propagate the production of [PSI+] Sup35 through fragmentation of aggregates, producing smaller aggregates able to be inherited by the daughter cell.

These three responses are shown in the figure below, where the yellow pentagons represent the Hsp104 hexamers.

Modelling Aggregation

Algorithm

The agent-based model was written in Python with Cell objects each containing a list of amyloids with associated sizes, and a number of healthy Sup35 molecules.When the cell buds every 1.5 hours[8], the aggregates under the 20 monomer size threshold[1], are passed to the mother cell with a binomial distribution with a probability of 0.4[9], as well as a number of healthy Sup35.

Between budding times (a generation), at each timestep (a minute), a Gillespie algorithm with 6 different "reactions" simulates the different Hsp104, Sup35 and aggregate interactions in the cell, as follows in the table below.

In the reactions, s is Sup35, Yi is an amyloid of size i with addition or subtraction of monomers from the original amyloid size.

The propensities associated with the Gillespie's algorithm are based on rates found in literature [10] and adjusted with different factors, such as healthy Sup35 levels, number of amyloids present in the system, as well as the total number of bonds connecting aggregate monomers.

The spontaneous appearance of [PSI+] cells was considered to be included in the model, but due to its low probability of occurrence (in the order of 10-7[2]) in relation to the sample size of the model, it was excluded in favour of a faster simulation. The full cell population was not used for the simulation; instead, at each generation a set number of cells was sampled randomly from the population and kept for budding and amyloid interactions in the next generation. This was done to generate a faster simulation with minimal loss of accuracy.

Modelling Curing of [PSI+] State

The initial state of the cell population was that of 100% of the cells in the [PSI+]. Through adjustments of different curing rates of Hsp104, where a monomer was broken off during a single reaction, the effects of Hsp104 concentration on [PSI+] cells populations could be effectively simulated and in agreement with results found in literature.

100% propagation of the [PSI+] state throughout the population can be induced through concentrations of Hsp104 within a 'Goldilocks' zone[6]. It was found through running model simulations that the rate of curing and fragmentation increases with Hsp104 concentration, as expected. Assuming both of these rates have a linear relationship with Hsp104 concentrations, a state of propagation can be induced.

The following figure shows the different responses elicited by the system at the 20th generation of yeast cells as follows: varying fragmentation rates with h (top), varying both fragmentation and curing rates with h (middle), varying curing rates with h (bottom), where h is a variable representing levels of Hsp104 present in the system.

These graphs demonstrate that our assumption is correct; the bell curve of the middle graph represents partially cured cell populations at its ends and its peak is the 'Goldilocks' zone.

The over-expression condition of the system was simulated and produced graphs similar to the figure shown below:

As predicted, a curing curve of a characteristic 'S' shape was created under Hsp104 overexpression conditions. This shape is in agreement with curing curves found with experimental data[1], and can be viewed in the figure below. This result was also replicated by the lab team during investigations of Hsp104 overexpression and confirms the expected system reaction.

The curing curves produced during the simulated under-expression of Hsp104, such as in the figure below, confirms the effectiveness of this method in curing the prion states of a cell population. The under-expression was simulated with no Hsp104 in a system with non-Hsp104 related fragmentation rates of amyloids. In the future, tools to induce this Hsp104 under-expression, such as using CRISPR, could possibly be utilized to cure prion-based diseases.

References

[1] Sindi, S. S., & Olofsson, P. (2013). A Discrete-Time Branching Process Model of Yeast Prion Curing Curves*. Mathematical Population Studies, 20(1), 1–13. doi:10.1080/08898480.2013.748566

[2] Lancaster, A. K., Bardill, J. P., True, H. L., & Masel, J. (2010). The spontaneous appearance rate of the yeast prion [PSI+] and its implications for the evolution of the evolvability properties of the [PSI+] system. Genetics, 184(2), 393–400. doi:10.1534/genetics.109.110213

[3] Collins, S. R., Douglass, A., Vale, R. D., & Weissman, J. S. (2004). Mechanism of prion propagation: amyloid growth occurs by monomer addition. PLoS Biology, 2(10), e321. doi:10.1371/journal.pbio.0020321

[4] Liebman, S. W., & Chernoff, Y. O. (2012). Prions in yeast. Genetics, 191(4), 1041–72. doi:10.1534/genetics.111.137760

[5] Shorter, J., & Lindquist, S. (2008). Hsp104, Hsp70 and Hsp40 interplay regulates formation, growth and elimination of Sup35 prions. The EMBO Journal, 27(20), 2712–24. doi:10.1038/emboj.2008.194

[6] Shorter, J., & Lindquist, S. (2004). Hsp104 catalyzes formation and elimination of self-replicating Sup35 prion conformers. Science (New York, N.Y.), 304(5678), 1793–7. doi:10.1126/science.1098007

[7] Romanova, N. V, & Chernoff, Y. O. (2009). Hsp104 and prion propagation. Protein and Peptide Letters, 16(6), 598–605. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/19519517

[8] Brewer, B. J., Chlebowicz-Sledziewska, E., & Fangman, W. L. (1984). Cell cycle phases in the unequal mother/daughter cell cycles of Saccharomyces cerevisiae. Molecular and Cellular Biology, 4(11), 2529–31. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/6392855

[9] Olofsson, P., & Sindi, S. S. (2014). A Crump-Mode-Jagers branching process model of prion loss in yeast. Journal of Applied Probability, 51(2), 453–465. doi:10.1239/jap/1402578636

[10]Derdowski, A., Sindi, S. S., Klaips, C. L., DiSalvo, S., & Serio, T. R. (2010). A size threshold limits prion transmission and establishes phenotypic diversity. Science (New York, N.Y.), 330(6004), 680–3. doi:10.1126/science.1197785

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