Conceived and designed the experiments: ARM. Analyzed the data: HRF. Wrote the paper: HRF ARM. Performed the mathematical modelling and analysis: HRF.
The authors have declared that no competing interests exist.
Understanding the circumstances under which exposure to transmissible spongiform encephalopathies (TSEs) leads to infection is important for managing risks to public health. Based upon ideas in toxicology and radiology, it is plausible that exposure to harmful agents, including TSEs, is completely safe if the dose is low enough. However, the existence of a threshold, below which infection probability is zero has never been demonstrated experimentally. Here we explore this question by combining data and mathematical models that describe scrapie infections in mice following experimental challenge over a broad range of doses. We analyse data from 4338 mice inoculated at doses ranging over ten orders of magnitude. These data are compared to results from a withinhost model in which prions accumulate according to a stochastic birthdeath process. Crucially, this model assumes no threshold on the dose required for infection. Our data reveal that infection is possible at the very low dose of a 1000 fold dilution of the dose that infects half the challenged animals (ID50). Furthermore, the dose response curve closely matches that predicted by the model. These findings imply that there is no safe dose of prions and that assessments of the risk from low dose exposure are right to assume a linear relationship between dose and probability of infection. We also refine two common perceptions about TSE incubation periods: that their mean values decrease linearly with logarithmic decreases in dose and that they are highly reproducible between hosts. The model and data both show that the linear decrease in incubation period holds only for doses above the ID50. Furthermore, variability in incubation periods is greater than predicted by the model, not smaller. This result poses new questions about the sources of variability in prion incubation periods. It also provides insight into the limitations of the incubation period assay.
During the 1980s and 1990s, millions of Britons were orally exposed to bovine spongiform encephalopathy (BSE), yet fewer than 200 individuals have been diagnosed with variant Creutzfeldt–Jakob disease (vCJD). Understanding the circumstances under which exposure to transmissible spongiform encephalopathies (TSEs) leads to infection is important for managing risks to humans and animals from TSEs. With regard to the vCJD epidemic, it is clear that the species barrier was important in curtailing cases, but other mechanisms may also have played a role. For example, it is unclear how the probability of infection changes with the level of TSE exposure, especially at very low doses. Based upon ideas in toxicology and radiology
In the case of TSEs, there is some evidence to suggest such a threshold may exist. Mathematical models have been used to investigate the replication kinetics of prions – a general term for the proteinaceous etiological infectious agents of TSEs. It has been proposed that the smallest such infectious agent of TSEs is a polymer, made up of several monomers of PrP^{Sc}
We investigate the existence of a threshold infectious dose of prions using data from a very large collection of experiments (N = 127) in which mice were inoculated with varying doses of tissue from mice infected with mousepassaged scrapie isolates. We also use these data to understand more about TSE incubation periods. The data includes records of whether each mouse became ill, and the incubation periods for those mice that developed symptoms. We compare these data to a simple stochastic model of prion replication. In the model the onset of disease is defined as the moment at which the number of prions reaches a predetermined limit
Is there evidence of a threshold dose of prions below which the probability of infection is zero?
How does mean incubation period change with dose; is it true that mean incubation period decreases linearly with logarithmic increase in dose?
How does the variation in incubation period change with dose; is it true that TSE incubation periods are highly reproducible?
To address the first of these questions we measure how the probability of infection in mice changes with dose. We show that infection is still possible at very low doses indeed, namely three orders of magnitude lower than the ID50 – the dose at which 50% of challenged hosts become infected. Furthermore the shape of this doseresponse curve is consistent with the model of prion replication in which there exists no threshold dose of prions. Taken together these findings imply that there is no safe dose of prions. We find evidence to support the assumption of a linear relationship between dose and probability of infection in assessing the risk from low dose exposure.
In the remaining two questions we challenge two common assertions. The first is that mean incubation periods decrease linearly with logarithmic decrease in the dose of infectious material. The model predicts that the linear decrease in incubation period holds only for doses above the ID50; for doses below the ID50, incubation period is relatively invariant to dose. We observe precisely this pattern in the data. The second assertion is that incubation periods of prion diseases are highly reproducible. We measure the variability in incubation periods and compare it to model predictions. We find that although murine scrapie incubation periods appear very reproducible they are actually markedly more variable than predicted by the model. These findings have implications for prion studies that use incubation periods as a method to quantify the dose of an inoculum, emphasising that this method is only reliable at doses above the ID50. Furthermore they reveal a new perspective on prion incubation periods asking ‘why are they so variable?’ rather than ‘why are they so constant?’
To investigate evidence for the existence of a threshold dose of prions below which the probability of infection is zero we used a mathematical model and data collated from 127 different experiments in the murine scrapie model
First we analysed how the probability of infection changed with the dose of the inoculum. To enable us to meaningfully collate the data from different experiments we made use of the concept of the relative dose. This required us to calculate the ID50 for each experiment. However, for eight experiments it was not possible to estimate the ID50 with reasonable accuracy, thus data from these experiments (195 mice) were excluded from further analysis leaving 4143 mice to analyse. The rules used for the ID50 calculations are described in
A) The proportion of mice infected increases as the relative dose increases. The data reveals a sigmoidal pattern that fits well with the model that predicts that infection probability approaches zero at low doses and approaches one at high doses. In the data, infection probability was non zero as low as relative dose –3, but at relative doses –4 (the lowest dose tested) no mice were infected. However, the data at relative dose 3 and the model both indicate that a much larger sample size than the one used (N = 11) would be needed to find at least one infected mice at this dose. These results are consistent with the hypothesis that there exists no safe dose of prions. The modelled infection probability has no free parameters (
Relative dose  Number of mice challenged  Number of mice infected  Proportion of mice infected  Mean incubation period  Variance of incubation period  Variance of difference from group mean incubation period (≥2 mice per group)  
In total  With an incubation period  In doseexperiment groups with ≥2 mice with an incubation period  
−4  11  0  0  0  0·000       
−3  92  1  0  0  0·011       
−2  294  6  5  0  0·020  333  3316   
−1  591  64  59  38  0·108  357  25491  2503 
0  712  375  352  330  0·527  337  17041  2935 
1  644  604  589  584  0·938  290  10651  1354 
2  461  452  439  439  0·980  256  9092  936 
3  337  333  329  329  0·988  240  11041  270 
4  207  205  203  203  0·990  200  7060  970 
5  90  90  90  89  1·000  184  1329  136 








In the model the host is inoculated with a certain number,
A) A mathematical model of prion replication in which the number of prions (
To compare the model to the doseresponse data (
To compare the model and observations further, we transformed the infection probability data according to the function
In risk assessments of exposure to TSE infectivity it is important to understand the relationship between dose and the probability of infection. The relationship can be used to translate exposure levels into predicted number of infections. Many such studies assume, either implicitly or explicitly, that the relationship between dose and infection probability is linear, i.e. doubling the dose leads to double the infection probability
There has been a long understanding that TSE incubation periods decrease approximately linearly with every tenfold increase in dose and are highly reproducible between hosts
We have used the model and murine data to investigate how mean incubation periods change with dose. The deterministic form of the model (
A) and B) show factors affecting the expectation of the incubation period. The expected incubation period is approximately invariant to the inoculating dose for relative doses less than approximately 0 (the ID50). Beyond approximately 0, the incubation period decreases linearly with relative dose. A) shows that the gradient of this slope is steeper if the net growth rate of prions (
For those mice for which incubation period data were available (N = 2086) we grouped the mice according to relative dose (
Since TSE incubation periods are regarded as being highly reproducible
We used the model and our estimate of the net growth rate (0.07 days^{−1}) to predict the variance of the murine incubation periods at different relative doses. This prediction is shown in
To understand whether the high level of observed variability was generated by variability between experiments, our second calculation aimed to eliminate such interexperiment variability. For each dose within each experiment we first calculated the mean incubation period. For doseexperiment ‘groups’ with two or more infected mice with incubation period data (N = 2032) for each mouse we then calculated the difference between their incubation period and the mean for their group. The ‘differences’ from each experiment were then grouped according to relative dose and the variance of these differences at each dose was calculated (
Since end point titrations take a long time and require large numbers of mice, a technique called the ‘incubation period assay’ is now often used as an alternative method for determining the TSE infectious doses
Our results imply that both the mean (
In this study we first asked whether there exists a threshold dose of prions below which the probability of infection is zero. By comparing the scrapie doseresponse curve observed in mice to model predictions we found no evidence that such a threshold exists. As the stochasticbirth death model predicts, the probability of infection simply becomes smaller as the dose decreases. Furthermore, we find evidence to support the assumption of a linear relationship between dose and probability of infection in assessing the risk from low dose exposure. Use of a linear relation for doses above the ID50 will lead to overestimation of the risk.
Although we find no evidence of a threshold, it must be emphasized that we cannot rule out this possibility. Though it was to be expected because of the limited sample size, no mice were infected at the lowest relative dose tested (–4), therefore a threshold may exist at this dose or lower. However, acquiring data to investigate this question further would require an unfeasibly large number of test animals. Furthermore, the observation of infection at a dose 1000 times more dilute than the ID50 shows that infection is still possible at very low doses. In practical terms this is low enough to regard there to be no safe dose.
Previous modelling work has focussed on understanding the molecular form of a prion and its mechanism of replication. We are not proposing a specific form or replication mechanism, rather we ask whether data on infection probabilities are consistent with the simplest model of replication that assumes no threshold dose. If evidence did emerge of a threshold dose, one could tie it to the hypothesis that the smallest infectious agent involved in TSEs (a prion) is a polymer consisting of multiple PrP^{Sc} monomers above a critical polymer length
In this study we also asked how TSE incubation periods change according to the inoculating dose. First, we asked whether it is true that mean incubation period decreases linearly with logarithmic decrease in dose. The model predicts that this relationship holds only at doses higher than the ID50. For doses below the ID50, mean incubation period is predicted to be invariant to dose. The murine data are consistent with these predictions. This finding delineates the situation in which there is a linear relationship between dose and provides insight into the limitations of the incubation period assay. Specifically, it suggests that the incubation period assay should be unable to distinguish between different doses below the ID50.
Second, we investigated variability in incubation periods and asked whether TSE incubation periods are highly reproducible. We revealed that they are markedly more variable than predicted by the model. Our findings lead us to question why this is so, especially given that the dose response curve and the mean incubation periods are in close agreement with the model. Could the mechanism of prion growth be incorrect or does the inconsistency lie with data collection or the relationship between prion numbers and clinical symptoms? The most popular model for prion growth, based upon polymer breakage and expansion, is underpinned by exponential growth dynamics and would predict the same variability as seen here. That model therefore also cannot explain the effect that we see. In regards to data collection, some variability is likely to arise from the difficulty in spotting symptoms and from small experimental variability in the dose of the inoculum. It must also be noted that some inaccuracies in the data could arise because the duration of the experiments was finite. At the end of each experiment all surviving mice were culled and examined for the presence of pathological lesion without symptoms might have progressed to disease if the experiments had run for longer. However this does not explain the high variability in incubations periods as if the data were not censored in this way, such unusually long incubation periods would only increase variability, not reduce it.
It is not difficult to propose ways in which prion infection is more complex than a stochastic birthdeath process in a homogeneous environment. All the heterogeneities of the in vivo situation: spatial, tissue, temporal and genetic
The data used in this study were collated from 127 murine scrapie titration experiments, conducted over 30 years (started between 1965 and 1993). In these experiments each mouse was inoculated with tissue from another mouse infected with a mousepassaged scrapie isolate. Within each experiment, the mice were controlled for mouse breed, scrapie strain, route of inoculation and the tissue from which the inoculum was derived. Between experiments these variables differed such that ten different mouse breeds and seven different scrapie strains were used. The maximum duration of observation also differed between experiments. In the majority of experiments, inoculation was intracranial and the source of infectious material was usually brain; however spleen tissue was occasionally used. In total, 4338 mice were inoculated at varying doses (10 fold dilutions). For each mouse the following information was recorded: 1) whether the animal was killed or died, 2) the number of days between inoculation and death and 3) whether there were clinical signs of scrapie at death. For most mice pathological signs of scrapie were also tested postmortem. Depending upon the combination of the these factors, each mouse was classified into one of the following categories: 1) uninfected at the end of the experiment (N = 1379), 2) infected with incubation period data (N = 2162), 3) infected without incubation period data (i.e. they died or were killed before clinical symptoms arose but were pathologically positive; N = 57), or 4) died prior to the end of the experiment not of scrapie (N = 740). The precise rules defining these categories, and more details of these experiments are provided in McLean and Bostock
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Analytic expressions derived from the stochastic model describing the probability of infection and the expectation and variance of incubation period.
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Derivations for analytic expressions describing the probability of infection and the expectation and variance of the incubation period.
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We are grateful to Moira Bruce for providing the data.