MIL-CELL: a tool for multi-scale simulation of yeast replication and prion transmission

The single-celled baker's yeast, Saccharomyces cerevisiae, can sustain a number of amyloid-based prions, the three most prominent examples being [URE3], [PSI+], and [PIN+]. In the laboratory, haploid S. cerevisiae cells of a single mating type can acquire an amyloid prion in one of two ways (i) spontaneous nucleation of the prion within the yeast cell, and (ii) receipt via mother-to-daughter transmission during the cell division cycle. Similarly, prions can be lost due to (i) dissolution of the prion amyloid by its breakage into non-amyloid monomeric units, or (ii) preferential donation/retention of prions between the mother and daughter during cell division. Here we present a computational tool (Monitoring Induction and Loss of prions in Cells; MIL-CELL) for modelling these four general processes using a multiscale approach describing both spatial and kinetic aspects of the yeast life cycle and the amyloid-prion behavior. We describe the workings of the model, assumptions upon which it is based and some interesting simulation results pertaining to the wave-like spread of the epigenetic prion elements through the yeast population. MIL-CELL is provided as a stand-alone GUI executable program for free download with the paper. MIL-CELL is equipped with a relational database allowing all simulated properties to be searched, collated and graphed. Its ability to incorporate variation in heritable properties means MIL-CELL is also capable of simulating loss of the isogenic nature of a cell population over time. The capability to monitor both chronological and reproductive age also makes MIL-CELL potentially useful in studies of cell aging.

A structured model and likelihood approach to estimate yeast prion propagon replication rates and their asymmetric transmission

Prion proteins cause a variety of fatal neurodegenerative diseases in mammals but are generally harmless to Baker’s yeast (Saccharomyces cerevisiae). This makes yeast an ideal model organism for investigating the protein dynamics associated with these diseases. The rate of disease onset is related to both the replication and transmission kinetics of propagons, the transmissible agents of prion diseases. Determining the kinetic parameters of propagon replication in yeast is complicated because the number of propagons in an individual cell depends on the intracellular replication dynamics and the asymmetric division of yeast cells within a growing yeast cell colony. We present a structured population model describing the distribution and replication of prion propagons in an actively dividing population of yeast cells. We then develop a likelihood approach for estimating the propagon replication rate and their transmission bias during cell division. We first demonstrate our ability to correctly recover known kinetic parameters from simulated data, then we apply our likelihood approach to estimate the kinetic parameters for six yeast prion variants using propagon recovery data. We find that, under our modeling framework, all variants are best described by a model with an asymmetric transmission bias. This demonstrates the strength of our framework over previous formulations assuming equal partitioning of intracellular constituents during cell division.

In this work we investigate the transmissible [PSI⁺] phenotype in yeast. The agents responsible for this phenotype are propagons, misfolded protein aggregates of a naturally occurring protein. These propagons increase in number within a cell and are distributed between cells during division. We use mathematical modeling to infer the replication rate of propagons within cells and if propagons are transmitted equally or unequally during cell division. Prior models in this area assumed only symmetric transmission when fitting replication rates. We couple this model with a novel likelihood framework allowing us to exclude influential outliers from our datasets when inferring parameters. We find that for all six protein variants we study, propagons are transmitted asymmetrically with different biases. Our results can be reproduced with the code and data available at https://github.com/FS-CodeBase/propagon_replication_and_transmission/.

A Crump-Mode-Jagers Branching Process Model of Prion Loss in Yeast

The yeast Saccharomyces cerevisiae has emerged as an ideal model system to study the dynamics of prion proteins which are responsible for a number of fatal neurodegenerative diseases in humans. Within an infected cell, prion proteins aggregate in complexes which may increase in size or be fragmented and are transmitted upon cell division. Recent work in yeast suggests that only aggregates below a critical size are transmitted efficiently. We formulate a continuous-time branching process model of a yeast colony under conditions of prion curing. We generalize previous approaches by providing an explicit formula approximating prion loss as influenced by both aggregate growth and size-dependent transmission.

A Crump-Mode-Jagers Branching Process Model of Prion Loss in Yeast

The yeast Saccharomyces cerevisiae has emerged as an ideal model system to study the dynamics of prion proteins which are responsible for a number of fatal neurodegenerative diseases in humans. Within an infected cell, prion proteins aggregate in complexes which may increase in size or be fragmented and are transmitted upon cell division. Recent work in yeast suggests that only aggregates below a critical size are transmitted efficiently. We formulate a continuous-time branching process model of a yeast colony under conditions of prion curing. We generalize previous approaches by providing an explicit formula approximating prion loss as influenced by both aggregate growth and size-dependent transmission.

A Discrete-Time Branching Process Model of Yeast Prion Curing Curves

The infectious agent of many neurodegenerative disorders is thought to be aggregates of prion protein, which are transmitted between cells. Recent work in yeast supports this hypothesis, but suggests that only aggregates below a critical size are transmitted efficiently. The total number of transmissible aggregates in a typical cell is a key determinant of strain infectivity. In a discrete-time branching process model of a yeast colony with prions, prion aggregates increase in size according to a Poisson process and only aggregates below a threshold size are transmitted during cell division. The total number of cells with aggregates in a growing population of yeast is expressed.

Reproduction-time statistics and segregation patterns in growing populations

Pattern formation in microbial colonies of competing strains under purely space-limited population growth has recently attracted considerable research interest. We show that the reproduction time statistics of individuals has a significant impact on the sectoring patterns. Generalizing the standard Eden growth model, we introduce a simple one-parameter family of reproduction time distributions indexed by the variation coefficient δ∈[0,1], which includes deterministic (δ=0) and memory-less exponential distribution (δ=1) as extreme cases. We present convincing numerical evidence and heuristic arguments that the generalized model is still in the Kardar-Parisi-Zhang (KPZ) universality class, and the changes in patterns are due to changing prefactors in the scaling relations, which we are able to predict quantitatively. With the example of Saccharomyces cerevisiae, we show that our approach using the variation coefficient also works for more realistic reproduction time distributions.

Fungal prions

For both mammalian and fungal prion proteins, conformational templating drives the phenomenon of protein-only infectivity. The conformational conversion of a protein to its transmissible prion state is associated with changes to host cellular physiology. In mammals, this change is synonymous with disease, whereas in fungi no notable detrimental effect on the host is typically observed. Instead, fungal prions can serve as epigenetic regulators of inheritance in the form of partial loss-of-function phenotypes. In the presence of environmental challenges, the prion state [PRION(+)], with its resource for phenotypic plasticity, can be associated with a growth advantage. The growing number of yeast proteins that can switch to a heritable [PRION(+)] form represents diverse and metabolically penetrating cellular functions, suggesting that the [PRION(+)] state in yeast is a functional one, albeit rarely found in nature. In this chapter, we introduce the biochemical and genetic properties of fungal prions, many of which are shared by the mammalian prion protein PrP, and then outline the major contributions that studies on fungal prions have made to prion biology.

Kinetic models of guanidine hydrochloride-induced curing of the yeast [PSI+] prion

A population of [PSI(+)] Saccharomyces cerevisiae cells can be cured of the [PSI(+)] prion by the addition of guanidine hydrochloride (GdnHCl). In this paper we extend existing nucleated polymerisation simulation models to investigate the mechanisms that might underlie curing. Our results are consistent with the belief that prions are dispersed through the cells at division following GdnHCl addition. A key feature of the simulation model is that the probability that a polymer is transmitted from mother to daughter during cell division is dependent upon the length of the polymer. The model is able to reproduce the essential features of data from several different experimental protocols involving addition and removal of GdnHCl.

Prion dynamics and the quest for the genetic determinant in protein-only inheritance

According to the prion hypothesis, proteins may act in atypical roles as genetic elements of infectivity and inheritance by undergoing self-replicating changes in physical state. While the preponderance of evidence strongly supports this concept particularly in fungi, the detailed mechanisms by which distinct protein forms specify unique phenotypes are emerging concepts. A particularly active area of investigation is the molecular nature of the heritable species, which has been probed through genetic, biochemical, and cell biological experimentation as well as by mathematical modeling. Here, we suggest that these studies are converging to implicate small aggregates composed of prion-state conformers as the transmissible genetic determinants of protein-based phenotypes.

The number and transmission of [PSI] prion seeds (Propagons) in the yeast Saccharomyces cerevisiae

Background

Yeast (Saccharomyces cerevisiae) prions are efficiently propagated and the on-going generation and transmission of prion seeds (propagons) to daughter cells during cell division ensures a high degree of mitotic stability. The reversible inhibition of the molecular chaperone Hsp104p by guanidine hydrochloride (GdnHCl) results in cell division-dependent elimination of yeast prions due to a block in propagon generation and the subsequent dilution out of propagons by cell division.

Principal Findings

Analysing the kinetics of the GdnHCl-induced elimination of the yeast [PSI⁺] prion has allowed us to develop novel statistical models that aid our understanding of prion propagation in yeast cells. Here we describe the application of a new stochastic model that allows us to estimate more accurately the mean number of propagons in a [PSI⁺] cell. To achieve this accuracy we also experimentally determine key cell reproduction parameters and show that the presence of the [PSI⁺] prion has no impact on these key processes. Additionally, we experimentally determine the proportion of propagons transmitted to a daughter cell and show this reflects the relative cell volume of mother and daughter cells at cell division.

Conclusions

While propagon generation is an ATP-driven process, the partition of propagons to daughter cells occurs by passive transfer via the distribution of cytoplasm. Furthermore, our new estimates of n₀, the number of propagons per cell (500–1000), are some five times higher than our previous estimates and this has important implications for our understanding of the inheritance of the [PSI⁺] and the spontaneous formation of prion-free cells.

Computational methods for yeast prion curing curves

If the chemical guanidine hydrochloride is added to a dividing culture of yeast cells in which some of the protein Sup35p is in its prion form, the proportion of cells that carry replicating units of the prion, termed propagons, decreases gradually over time. Stochastic models to describe this process of 'curing' have been developed in earlier work. The present paper investigates the use of numerical methods of Laplace transform inversion to calculate curing curves and contrasts this with an alternative, more direct, approach that involves numerical integration. Transform inversion is found to provide a much more efficient computational approach that allows different models to be investigated with minimal programming effort. The method is used to investigate the robustness of the curing curve to changes in the assumed distribution of cell generation times. Matlab code is available for carrying out the calculations.

Cell division is essential for elimination of the yeast [PSI+] prion by guanidine hydrochloride

Guanidine hydrochloride (Gdn.HCl) blocks the propagation of yeast prions by inhibiting Hsp104, a molecular chaperone that is absolutely required for yeast prion propagation. We had previously proposed that ongoing cell division is required for Gdn.HCl-induced loss of the [PSI+] prion. Subsequently, Wu et al.[Wu Y, Greene LE, Masison DC, Eisenberg E (2005) Proc Natl Acad Sci USA 102:12789-12794] claimed to show that Gdn.HCl can eliminate the [PSI+] prion from alpha-factor-arrested cells leading them to propose that in Gdn.HCl-treated cells the prion aggregates are degraded by an Hsp104-independent mechanism. Here we demonstrate that the results of Wu et al. can be explained by an unusually high rate of alpha-factor-induced cell death in the [PSI+] strain (780-1D) used in their studies. What appeared to be no growth in their experiments was actually no increase in total cell number in a dividing culture through a counterbalancing level of cell death. Using media-exchange experiments, we provide further support for our original proposal that elimination of the [PSI+] prion by Gdn.HCl requires ongoing cell division and that prions are not destroyed during or after the evident curing phase.

Prion stability

The rate of spontaneous change from psi(-) to the psi(+) condition, determined in yeast by states of the Sup35p protein, is briefly discussed together with the conditions necessary for such change to occur. Conditions that promote and which affect the rate of induction of psi(+) in Sup35p and of other prion-forming proteins to their respective prion forms are also discussed. These include the influence of the amount of non-prion protein, the presence of other prions, the activity of chaperones, and brief descriptions of the role of native sequences in the proteins and how alteration of sequences in prion-forming proteins influences the rate of induction of [prion(+)] and amyloid forms. The second part of this article discusses the conditions which affect the reversion of psi(+) to psi-, including factors which affect the copy-number of prion "seeds" or propagons and their partition. The principal factor discussed is the activity of the chaperone Hsp104, but the existence of other factors, such protein sequence and of other, less well-studied agents is touched upon and comparisons are made, as appropriate, with studies with other yeast prions. We conclude with a discussion of models of maintenance, in particular that of Tanaka et al. published in Nature (2006), which provides much insight into the phenotypic and genetic parameters of the numerous "variants" of prions increasingly being described in the literature.

Approximations for expected generation number

A deterministic formula is commonly used to approximate the expected generation number of a population of growing cells. However, this can give misleading results because it does not allow for natural variation in the times that individual cells take to reproduce. Here we present more accurate approximations for both symmetric and asymmetric cell division. Based on the first two moments of the generation time distribution, these approximations are also robust. We illustrate the improved approximations using data that arise from monitoring individual yeast cells under a microscope and also demonstrate how the approximations can be used when such detailed data are not available.

The [PSI+] prion of yeast: A problem of inheritance

The [PSI(+)] prion of the yeast Saccharomyces cerevisiae was first identified by Brian Cox some 40 years ago as a non-Mendelian genetic element that modulated the efficiency of nonsense suppression. Following the suggestion by Reed Wickner in 1994 that such elements might be accounted for by invoking a prion-based model, it was subsequently established that the [PSI(+)] determinant was the prion form of the Sup35p protein. In this article, we review how a combination of classical genetic approaches and modern molecular and biochemical methods has provided conclusive evidence of the prion basis of the [PSI(+)] determinant. In so doing we have tried to provide a historical context, but also describe the results of more recent experiments aimed at elucidating the mechanism by which the [PSI(+)] (and other yeast prions) are efficiently propagated in dividing cells. While understanding of the [PSI(+)] prion and its mode of propagation has, and will continue to have, an impact on mammalian prion biology nevertheless the very existence of a protein-based mechanism that can have a beneficial impact on a cell's fitness provides equally sound justification to fully explore yeast prions.

New Approximations to the Malthusian Parameter

Approximations to the Malthusian parameter of an age-dependent branching process are obtained in terms of the moments of the lifetime distribution, by exploiting a link with renewal theory. In several examples, the new approximations are more accurate than those currently in use, even when based on only the first two moments. The new approximations are extended to include a form of asymmetric cell division that occurs in some species of yeast. When used for inference, the new approximations are shown to have high efficiency.