Elsevier

Biophysical Chemistry

Volume 77, Issues 2–3, 29 March 1999, Pages 139-152
Biophysical Chemistry

Quantifying the kinetic parameters of prion replication

https://doi.org/10.1016/S0301-4622(99)00016-2Get rights and content

Abstract

The mechanism of protein-only prion replication is controversial. A detailed mathematical model of prion replication by nucleated polymerisation is developed, and its parameters are estimated from published data. PrP-res decay is around two orders of magnitude slower than PrP-sen decay, a plausible ratio of two parameters estimated from very different experiments. By varying the polymer breakage rate, we reveal that systems of short polymers grow the fastest. Drugs which break polymers could therefore accelerate disease progression. Growth in PrP-res seems slower than growth in infectious titre. This can be explained either by a novel hypothesis concerning inoculum clearance from a newly infected brain, or by the faster growth of compartments containing smaller polymers. The existence of compartments can also explain why prion growth sometimes reaches a plateau. Published kinetic data are all compatible with our mathematical model, so the nucleated polymerisation hypothesis cannot be ruled out on dynamic grounds.

Introduction

Transmissible spongiform encephalopathies (TSEs) are fatal neurodegenerative diseases. TSEs are found in a variety of mammals, including scrapie in sheep, bovine spongiform encephalopathy in cattle, and kuru and Creutzfeldt–Jakob disease (CJD) in humans. The nature of the infectious agent in TSEs has long been controversial, since its properties are unlike those of any previously known virus or viroid. It was proposed that the infectious agent, known as a prion, consists solely of a proteinaceous particle 1, 2. This was identified as a form of the PrP protein [1]. According to this hypothesis, an infectious proteinase K-resistant form of PrP (PrP-res or PrPSc) converts the normal proteinase K-sensitive form of PrP (PrP-sen or PrPC) into PrP-res. The protein-only hypothesis has not been directly proved, but an abundance of circumstantial evidence has won it widespread support. There are many recent reviews critically examining the evidence 3, 4, 5, 6.

Self-replication of a protein agent is a novel concept. The mechanism is not immediately clear, and several mechanisms have been proposed. In the heterodimer mechanism [7] (Fig. 1a), a single PrP-res molecule catalyses the conformational change of a single PrP-sen molecule into PrP-res. According to co-operative autocatalysis 8, 9 (Fig. 1b), a mixed aggregate of PrP-res and PrP-sen converts to an aggregate of PrP-res via allosteric interactions. According to nucleated polymerisation 8, 10, 11 (Fig. 1c), PrP-res is a polymeric form of PrP, while PrP-sen is monomeric. Polymerisation is very slow below a critical size. Above this size, the polymer is stabilised, and further polymerisation is comparatively rapid. The slow nucleation process can be circumvented by adding an infectious `seed'.

Prion diseases have some unusual kinetic features. Spontaneous disease is rare, but disease progresses inevitably after infection. Disease is characterised by an extremely long and precisely reproducible incubation period, followed by a brief and invariably fatal clinical disease. The length of the incubation period is dependent on the inoculum dosage, the prion strain, and the level of PrP expression in the animal.

Kinetic modelling is a tool which uses these unusual kinetic features to help determine whether a proposed mechanism is plausible. Any proposed mechanism will contain a number of assumptions. Formulating a mechanism as a mathematical model makes these assumptions more transparent. The model can then be critically examined for internal consistency and consistency with available data. To do this, it is helpful to determine the values of parameters specified in the model. For many purposes, it is enough to estimate a parameter to within an order of magnitude. If a model creates no direct contradictions and the calculated parameter values seem realistic, then the assumptions are reasonable and the model is plausible. If not, then a new model or mechanism should be considered. If a simple model fits the data, then a large number of modified, more complex models will fit the same data, and there is no way to choose between the many complex models. For this reason, if a simple model fits the data as well as a more complex model, then we use the simpler model.

Some hypothesised mechanisms include molecules other than the PrP protein, such as a virino [12] or protein X [13]. Here we examine the simplest possible models first, these being models without cofactors. If the simpler models are sufficient, this does not prove that no cofactors are present, since cofactors which are present in excess will not be apparent in kinetic data. It does show them to be unimportant for the dynamics of the system. If simpler models prove kinetically insufficient, we will then consider the kinetic contribution of cofactors.

A consistent model must explain why spontaneous prion disease is so rare, whereas disease progresses inevitably after inoculation. A single infectious particle will spread and cause disease unless it is rapidly degraded, so spontaneous prion production must be significantly slower than prion degradation. When this is considered in calculating the parameters of a heterodimer model, only an implausible parameter range is possible [8]. This makes the heterodimer mechanism seem highly unlikely.

Co-operative autocatalysis and nucleated polymerisation do not suffer from this difficulty [8]. These two mechanisms are also supported by the failure to dissociate infectivity from aggregated forms of PrP. The rich diversity of prion strains is also harder to explain by the multiple, non-inter-converting conformations of a single protein chain than it is by a mechanism involving the geometric interactions between protein subunits.

Scrapie-associated fibrils or `prion rods' colocalise with disease-specific PrP and may well be the pathogenic form of PrP. They appear as unbranched linear polymers when observed using electron microscopy [14]. For this reason, we assume that polymers are linear on a macroscopic level, although they may be helical on a microscopic level.

The nucleated polymerisation mechanism has two slight advantages over the co-operative autocatalysis mechanism, although there is no hard evidence to distinguish the two. Firstly, co-operative behaviour is usually associated with globular rather than linear aggregates, and pathogenic PrP appear to be linear. Secondly, nucleated polymerisation is much simpler.

The advantages and disadvantages of the three mechanisms are summarised in Table 1. Nucleated polymerisation seems the most likely candidate, and worthy of further development and testing. In this paper we develop and extend a formal model of nucleated polymerisation proposed by Nowak et al. [15]. We use published data on the unusual kinetic features of prion diseases to quantify the parameters.

Section snippets

Nucleated polymerisation model

Formal models of linear nucleated polymerisation have been developed for other biological systems 16, 17, 18. Our prion model, illustrated in Fig. 2, is more explicit in incorporating terms λ for production, d and a for degradation and b for polymer breakage. Polymer extension is treated as a one-way process with rate β. Nucleation is considered negligibly slow. The terms x, y and z count PrP-sen, PrP-res polymers and total PrP-res.

More formally, let x be the abundance of PrP-sen monomers and

Quantifying kinetic parameters

The model is specified by six independent parameters, given as a, b, β, d, n and λ. The model can be rearranged as shown in Eq. (12) in Appendix A, and can be given in terms of six different independent parameters: r, R0, d, n, s̄, and one of β, λ or X0. In Section 3we quantify r, R0 and d from published data and estimate n and s̄. Unfortunately, this gives us only five parameters, so the model is under-specified by current data. Nevertheless, r and R0 can be rearranged to give a, and this can

Clearance phenomena

When brain infectivity is assayed a few days after intracerebral inoculation, the majority of the infectious agent can no longer be found. The rate of this clearance is highly variable (Table 4). This variability can explain anomalies arising from growth rate measurements.

The factors influencing the extent of inoculum clearance have not been definitively determined, but the size and strain of the inoculum and the exact protocol of inoculation may be important. The organ of origin [53] and

Impact of the breakage parameter b

In our model, we assume that all polymers are subjected to the same breaking forces and that a polymer is equally likely to break at any position along its length. The cause of the breakage is not specified, but knowing the cause could help justify this assumption.

Shearing forces could well be significant, but these are dependent on the environment in which prion replication occurs. It is not yet known whether prion replication occurs in the extracellular space, at the plasma membrane or within

Acknowledgements

We thank R. May, B. Caughey, D. Krakauer, R. Payne and R. Arnaout for stimulating discussions and critical reading of the manuscript. VAAJ gratefully acknowledges the support of The Wellcome Trust and Linacre College and JM of the Rhodes Trust.

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