Table 1.
Model symbols.
Table 2.
Estimated empirical parameters for different prion strains.
Figure 1.
Relationships between the empirical parameters and
.
The reproductive ratio is plotted against the rate of growth. The downward trend is not well described by the linear model with negative angular coefficient () and an intercept (
) (dotted blue line). In addition, the model prediction with
fixed (dashed-dot black line) fails to precisely represent the data, even if it provides a more reasonable relationship (notice that high stable prions, such as MK4985, would always be associated to positive
values). Introducing one more degree of freedom (exponent
) yields a higher
value (red line,
). This result corresponds to a prediction of
. In addition, we tested a further simplified model version (where
is considered to be much smaller than
) according to which
(i.e.
, shown in green). Similar conclusions could be drawn.
Table 3.
Figure 2.
In (A) and (B) the stability against denaturation is plotted against the reproductive ratio and the rate of growth. A direct proportionality links to
. As expected, an inverse proportionality emerges between
and
, reinforcing the previous results.
Table 4.
Estimated model parameter for different prion strains.
Figure 3.
Kinetic model and prion pathways.
The cartoon describes the pathways of kinetic replication of two prion strains with a different stability against denaturation: a stable one (high ) and an unstable one (low
) are drawn. These act as templates bringing the same cellular prion protein (triangle) to the two different strain conformations (
▴→
▪, ♦). The model assumes that the aggregation of monomers to polymers produces a very fast change of conformation and that this aggregation is unfavorable below a critical size (
), which is assumed to be independent of the prion strain in our model. The experimental data suggest that stable prions are characterized by a higher
and a corresponding lower
. In the model, this is translated into strain-specificity of the rates of breakage and of aggregation (which are both lower for stable prions). This implies that stable fibrils are longer and prefer to proliferate while maintaining themselves as fibrils larger than the nucleus size (pathway on the left). On the contrary, unstable prions are more frangible (i.e. more sensitive to breakage), implying a shorter mean length. This means that breakage events are more likely to be associated with the formation of very short fibrils, even under the critical size. The increase in the aggregation rate is not enough to avoid an increased growth in the number of fibrils. We can therefore hypothesize that an apoptotic pathway is most likely for these last strains (pathway on the right). These conclusions are in agreement with the working hypothesis of oligomer toxicity [44].