Kinetics of prion growth

Abstract

We study the kinetics of prion fibril growth, described by the nucleated polymerization model analytically and by means of numerical experiments. The elementary processes of prion fibril formation lead us to a set of differential equations for the number of fibrils, their total mass, and the number of prion monomers. In difference to previous studies we analyze this set by explicitly taking into account the time-dependence of the prion monomer concentration. The theoretical results agree with experimental data, whereas the generally accepted hypothesis of constant monomer concentration leads to a fibril growth behavior which is not in agreement with experiments. The obtained size distribution of the prion fibril aggregates is shifted significantly toward shorter lengths as compared to earlier results, which leads to a enhanced infectivity of the prion material. Finally, we study the effect of filtering of the inoculated material on the incubation time of the disease.

FIGURE 1

Sketch of the prion aggregate growth model after Masel et al. (1999) but modified. For explanation see text.

FIGURE 2

Distribution of polymers w_i ≡ y_i/y over the polymer length i for fixed number of monomers x₀ = 500. We obtain excellent agreement between the analytical result Eq. 26 (line) and the results of a numerical simulation (circles). The parameters are n = 6, a = 0.05, b = 9 × 10⁻⁴, and β = 0.015.

FIGURE 3

Evolution of the average size s = z(t)/y(t) of PrP^sc polymers due to Eq. 30 (line) and results of a simulation (circles). The parameters are n = 6, a = 0.05, b = 9 × 10⁻⁴, and β = 0.015. The initial size is s(0) = 10.

FIGURE 4

The average growth velocity over the average size s due to Eq. 29. The parameters are n = 6, a = 0.015, b = 0.0009, β = 0.025, and x₀ = 500.

FIGURE 5

PrP^sc polymer growth scenarios for the case s^* < s₁. The parameters are n = 6, a = 0.05, b = 9 × 10⁻⁴, and β = 0.015, which corresponds to s^* = 66.44 and s₁ = 96.78. (Top) s(0) < s^* (y₁₀(0) = 5 × 10⁵); (middle) s(0) > s^* (y₂₅(0) = 8 × 10³); and (bottom) s(0) > s₁ (y₂₀₀(0) = 25,000). In all cases the initial inoculation is z(0) = 5 × 10⁶.

FIGURE 6

PrP^sc polymer growth scenarios for the case s^* > s₁. The parameters are the same as in Fig. 5 except for a = 0.1, which corresponds to s^* = 122.11 and s₁ = 96.76. (Top) s(0) < s₁ (y₁₀(0) = 5 × 10⁵); (middle) s^* > s(0) > s₁ (y₁₀₀(0) = 5 × 10⁴); and (bottom) s(0) > s^* (y₅₀₀(0) = 10⁴). In all cases the total initial inoculation is z(0) = 5 × 10⁶.

FIGURE 7

Evolution of the number of monomers x = x(t). The simulation starts with x(0) = 100 monomers and relaxes within a very short time (which is not visible on the timescale of the figure) to x(t) ≈ 500. This corresponds to the number of monomers x₀ = λ/d which provides the initial exponential growth. According to a complicated dynamics it eventually approaches its steady-state value x_st = 222.211 given by Eq. 38 (dashed line). The parameters are n = 6, a = 0.05, b = 9 × 10⁻⁴, β = 0.015, λ = 200,000, and d = 400.

FIGURE 8

Evolution of the average size of PrP^sc polymers s = z(t)/y(t). The simulation starts with s(0) = 200. The dashed line shows the analytical result, Eq. 37. The parameters are given in the caption of Fig. 7.

FIGURE 9

Evolution of the number of PrP^sc polymers y(t) (left) and the total mass of polymers z(t). For comparison, the dashed lines show the solution of the simplified set of equations, Eq. 14, with the exponents and coefficients given by Eqs. 16 and 17, respectively, with x₀ = λ/d. The parameters of the simulation are given in the caption of Fig. 7.

FIGURE 10

Size distributions of PrP^sc polymers for the full model, including x = x(t). The points display numerical simulations and the line shows the analytical result, Eq. 39. The parameters of the simulation are given in the caption of Fig. 7. For comparison, the dashed line shows the result for the simplified model with x(t) = λ/d = const. due to Eq. 26.

FIGURE 11

Evolution of the size distribution for the initial condition y₅₀(0) = 435, y_i(0) = 0 for i ≠ 50. The parameters of the simulation are given in the caption of Fig. 7.

FIGURE 12

Influence of the size distribution of the inoculation to the evolution of the distribution. The parameters are n = 6, d = 100, a = 0.027, b = 0.0048, λ = 10⁶, and β = 0.025. For explanation, see text.

FIGURE 13

Total mass of PrP^sc molecules, z(t) over time for different initial size distributions as shown in Fig. 12. The initial total mass, z(0) = 16,000, is identical in all cases. The full line corresponds to the left plot in Fig. 12, the dashed line to the middle plot, and the dotted line to the filtered inoculation drawn in the right plot.

FIGURE 14

The incubation time is sensitive to the size distribution of the inoculation. The figure shows the model incubation time over the level of filtering as defined by Eq. 41 for an identical number of PrP^sc units in the inoculation, i.e., for the same value of z(0).

FIGURE 15

Time-dependent number of PrP^sc polymers as it follows from the numerical simulation of the set of equations in Eq. 11, including the time-dependence of the number of monomers (full line) together with the experimental data (points) (Rubenstein et al., 1991). The abundance of the fibrils (given in this reference as a number of PrP^sc per a square element of the substrate) was obtained by negative-stain electron microscopy at various times after intracerebral inoculation. The measurements were performed for the spleens of Compton white mice and C57BL/6j mice. The dashed line show the prediction of the simplified model with the same rate constants but with a constant number of monomers, x₀ = λ/d; see Eq. 13. The parameters are n = 6, a = 0.027, b = 4.8 × 10⁻⁴, β = 0.8, λ = 1080, and d = 215.

FIGURE 16

The same as Fig. 15, but for the intraperitoneal inoculation. The parameters are n = 6, a = 0.018, b = 3.2 × 10⁻⁴, β = 0.32, λ = 1170, and d = 140.

FIGURE 17

To choose randomly the transition μ by which the system escapes from the present state we set up an array V containing the cumulative rates, i.e., V[0] = 0, V[1] = V[0] + A₁ = V[0] + λ, V[2] = V[1] + A₂ = V[1] + dx, V[3] = V[2] + A₃ = V[3] + ay_n, etc. Then we draw an equidistributed random number RND from the interval . The process i for which V[i − 1] < RND ≤ V[i] is chosen to be the next process.