Diana J. Cole

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.

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 α-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 α-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.

Determining the Parametric Structure of Models

In this paper we develop a comprehensive approach to determining the parametric structure of models. This involves considering whether a model is parameter redundant or not and investigating model identifiability. The approach adopted makes use of exhaustive summaries, quantities that uniquely define the model. We review and generalise previous work on evaluating the symbolic rank of an appropriate derivative matrix to detect parameter redundancy, and then develop further tools for use within this framework, based on a matrix decomposition. Complex models, where the symbolic rank is difficult to calculate, may be simplified structurally using reparameterisation and by finding a reduced-form exhaustive summary. The approach of the paper is illustrated using examples from ecology, compartment modelling and Bayes networks. This work is topical as models in the biosciences and elsewhere are becoming increasingly complex.

A hybrid symbolic-numerical method for determining model structure

In this article, we present a method for determining whether a model is at least locally identifiable and in the case of non-identifiable models whether any of the parameters are individually at least locally identifiable. This method combines symbolic and numeric methods to create an algorithm that is extremely accurate compared to other numeric methods and computationally inexpensive. A series of generic computational steps are developed to create a method that is ideal for practitioners to use. The algorithm is compared to symbolic methods for two capture-recapture models and a compartment model.

Determining parameter redundancy of multi-state mark–recapture models for sea birds

Multi-state mark–recapture models are structurally complex models, and in particular the complexity increases when there are unobservable states. Until recently, determining whether or not such models were parameter redundant was only possible numerically. In this paper, we show how it now possible to examine parameter redundancy of such models symbolically. The advantage of this approach is that you can determine exactly how many parameters can be estimated in a model for any number of years of marking and recovery, as well as which combinations of parameters can be estimated. Here, we illustrate how the new methodology works for multi-state models. We further develop rules for determining the parameter redundancy status of a whole family of multi-state mark–recapture models.

Parameter redundancy in mark-recovery models

We provide a definitive guide to parameter redundancy in mark-recovery models, indicating, for a wide range of models, in which all the parameters are estimable, and in which models they are not. For these parameter-redundant models, we identify the parameter combinations that can be estimated. Simple, general results are obtained, which hold irrespective of the duration of the studies. We also examine the effect real data have on whether or not models are parameter redundant, and show that results can be robust even with very sparse data. Covariates, as well as time- or age-varying trends, can be added to models to overcome redundancy problems. We show how to determine, without further calculation, whether or not parameter-redundant models are still parameter redundant after the addition of covariates or trends.

Does your species have memory? Analyzing capture–recapture data with memory models

We examine memory models for multisite capture–recapture data. This is an important topic, as animals may exhibit behavior that is more complex than simple first-order Markov movement between sites, when it is necessary to devise and fit appropriate models to data. We consider the Arnason–Schwarz model for multisite capture–recapture data, which incorporates just first-order Markov movement, and also two alternative models that allow for memory, the Brownie model and the Pradel model. We use simulation to compare two alternative tests which may be undertaken to determine whether models for multisite capture–recapture data need to incorporate memory. Increasing the complexity of models runs the risk of introducing parameters that cannot be estimated, irrespective of how much data are collected, a feature which is known as parameter redundancy. Rouan et al. (JABES, 2009, pp 338–355) suggest a constraint that may be applied to overcome parameter redundancy when it is present in multisite memory models. For this case, we apply symbolic methods to derive a simpler constraint, which allows more parameters to be estimated, and give general results not limited to a particular configuration. We also consider the effect sparse data can have on parameter redundancy and recommend minimum sample sizes. Memory models for multisite capture–recapture data can be highly complex and difficult to fit to data. We emphasize the importance of a structured approach to modeling such data, by considering a priori which parameters can be estimated, which constraints are needed in order for estimation to take place, and how much data need to be collected. We also give guidance on the amount of data needed to use two alternative families of tests for whether models for multisite capture–recapture data need to incorporate memory.

Generalized linear mixed models for strawberry inflorescence data

Strawberry inflorescences have a variable branching structure. This paper demonstrates how the inflorescence structure can be modelled concisely using binomial logistic generalized linear mixed models. Many different procedures exist for estimating the parameters of generalized linear mixed models, including penalized likelihood, EM, Bayesian techniques, and simulated maximum likelihood. The main methods are reviewed and compared for fitting binomial logistic generalized linear mixed models to strawberry inflorescence data. Simulations matched to the original data are used to show that a modified EM method due to Steele (1996) is clearly the best, in terms of speed and mean-squared-error performance, for data of this kind.

Parameter redundancy in discrete state-space and integrated models.

Discrete state‐space models are used in ecology to describe the dynamics of wild animal populations, with parameters, such as the probability of survival, being of ecological interest. For a particular parametrization of a model it is not always clear which parameters can be estimated. This inability to estimate all parameters is known as parameter redundancy or a model is described as nonidentifiable. In this paper we develop methods that can be used to detect parameter redundancy in discrete state‐space models. An exhaustive summary is a combination of parameters that fully specify a model. To use general methods for detecting parameter redundancy a suitable exhaustive summary is required. This paper proposes two methods for the derivation of an exhaustive summary for discrete state‐space models using discrete analogues of methods for continuous state‐space models. We also demonstrate that combining multiple data sets, through the use of an integrated population model, may result in a model in which all parameters are estimable, even though models fitted to the separate data sets may be parameter redundant.

Reply to determining structural identifiability of parameter learning machines

The paper Ran and Hu (2014, Neurocomputing) examines identifiability and parameter redundancy in classes of models used in machine learning. This note discusses the results on global identifiability and also clarifies that the papers results on parameter redundancy already exist in the paper Cole et al. (2010, Mathematical Biosciences).