1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Nucleated polymerization with secondary pathways. I. Time evolution of the principal moments
Rent:
Rent this article for
USD
10.1063/1.3608916
/content/aip/journal/jcp/135/6/10.1063/1.3608916
http://aip.metastore.ingenta.com/content/aip/journal/jcp/135/6/10.1063/1.3608916

Figures

Image of FIG. 1.
FIG. 1.

Schema illustrating the microscopic processes of polymerisation with secondary pathways treated in this paper. Primary nucleation (a) leads to the creation of a polymer of length n c from soluble monomer. Filaments grow linearly (b) from both ends in a reversible manner with monomers also able to dissociate from the ends (c). The secondary pathways (d) and (e) lead to the creation of new fibril ends from pre-existing polymers; fragmentation (d) is discussed in the first part of this paper, and monomer-dependent secondary nucleation (e) is discussed in the second part.

Image of FIG. 2.
FIG. 2.

Growth kinetics in the early time limit with different initial seed concentrations (from left to right): M(0) = 6.25 × 10−7, M(0) = 1.25 × 10−7 M, M(0) = 2.5 × 10−8 M, M(0) = 5 × 10−9. The other parameters are: k off = 0, P(0) = M(0)/5000, k + = 5 × 104 M−1 s−1, k off = 0, k = 10−9 s−1 k n = 0, m tot = 5 × 10−5 M. (a) shows the zeroth moment, P(t), of the distribution f(t, j) given by Eq. (22) and (b) shows the first moment, M(t), from Eq. (23). The dashed lines in (b) show the initial rate dM/dt| t = 0 = 2k + m tot P| t = 0.

Image of FIG. 3.
FIG. 3.

Time evolution of the second moment Q(t) of the length distribution in the early time limit for differing primary nucleation rates (from left to right): k n = 2 × 10−3 M−1 s−1, k n = 2 × 10−4 M−1 s−1, k n = 2 × 10−5 M−1 s−1. The other parameters are: k + = 5 × 104 M−1 s−1, k off = 0, k = 2 × 10−8 s−1, m tot = 5 × 10−6 M, n c = 2, M(0) = P(0) = 0. The dashed lines show the analytical result Eq. (28) derived by neglecting the third central moment. The solid lines show the numerical result from the master equation, Eq. (2), which accounts for all central moments.

Image of FIG. 4.
FIG. 4.

Kinetics of fibrillar growth. Growth through nucleation, elongation, and fragmentation leads to sigmoidal kinetic curves for the mass concentration of fibrils as a function of time. Solid line: first moment M(t) computed from the numerical solution of the master equation Eq. (2). Dashed curve: analytical solution given in Eq. (33). Dotted curve: early time limit from Eq. (23). The parameters are: k + = 5 × 104 M−1 s−1, k off = 0, k = 2 × 10−8 s−1, m tot = 5 × 10−6 M, k n = 2 × 10−5 M−1 s−1, n c = 2, M(0) = P(0) = 0. (b) shows the maximal growth rate r max, polymer concentration corresponding to the maximal growth rate M max, time of maximal growth rate t max, and lag time τlag.

Image of FIG. 5.
FIG. 5.

Effect of the depolymerisation rate. Sigmoidal reaction profiles for increasing depolymerisation rates are shown. Depolymerisation rates are given as a percentage of k + m tot. In most cases of practical interest (Refs. 36 and 58), k offk + m tot. Solid lines: first moment M(t) computed from the numerical solution of the master equation Eq. (2). Dashed lines: analytical solution given in Eq. (33). The parameters are: k + = 5 × 104 M−1 s−1, k = 2 × 10−8 s−1, m tot = 1 × 10−6 M, k n = 5 × 10−5 M−1 s−1, n c = 2, M(0) = 1 × 10−8 M, P(0) = M(0)/5000.

Image of FIG. 6.
FIG. 6.

Low breakage rate limit, k → 0. (a) shows Eq. (33) evaluated for successively smaller breakage rates k = 10−7 s−1, k = 10−8 s−1, k = 10−9 s−1, k = 10−10 s−1. The other parameters are k + = 5 × 104 M−1 s−1, k off = 0, m tot = 50 × 10−6 M, k n = 10−6 M−1 s−1, M(0) = P(0) = 0. (b) shows an expanded portion of the t → 0 limit showing the progressive transition from exponential to polynomial growth (red line, shows Eq. (45)) as a function of time when nucleation takes over from breakage as the most important source of new fibril ends.

Image of FIG. 7.
FIG. 7.

Effect of elongation rate and breakage rate on the growth kinetics in the absence of nucleation. In (a) the elongation rate k + is varied, from left to right: k + = 7.3 × 105 M−1 s−1, k + = 2.9 × 105 M−1 s−1, k + = 1.2 × 105 M−1 s−1, k + = 4.7 × 104 M−1 s−1, k + = 1.9 × 104 M−1 s−1, and k + = 7.5 × 103 M−1 s−1 and the other parameters are k off = 0, k n = 0, k = 10−8 s−1, m tot = 50 μM, M(0) = 1 nM and P(0) = M(0)/1000. In (b) the breakage rate is varied from left to right: k = 6.4 × 10−8 s−1, k = 3.2 × 10−8 s−1, k = 1.6 × 10−8 s−1, k = 8.0 × 10−9 s−1, k = 4.0 × 10−9 s−1, and k = 2.0 × 10−9 s−1 and the other parameters are: k off = 0, k n = 0, k + = 1 × 104 M−1 s−1, m tot = 50 μM, M(0) = 100 nM and P(0) = M(0)/1000.

Image of FIG. 8.
FIG. 8.

Effect of monomer concentration on the growth kinetics in the absence of nucleation. (a) Polymer mass concentration M(t) Eq. (33) as a function of time for different total monomer concentrations, from left to right: m tot = 20 μM, m tot = 40 μM, m tot = 80 μM, m tot = 160 μM, m tot = 320 μM, and m tot = 640 μM. The values used for the other parameters are: k off = 0, k n = 0, k + = 2 × 104 s−1 M−1, k = 10−9 s−1, M(0) = 100 nM, P(0) = M(0)/1000. (b) shows the normalised polymer mass fractions M(t)/m tot as a function of time, for the same monomer concentrations as in (a).

Image of FIG. 9.
FIG. 9.

Polymer number concentration. The time evolution of the polymer number concentration P (red) from Eq. (47) and mass concentration M (blue) from Fig. 4 are shown for the same parameter values as in Fig. 4. The solid lines show the numerical results, the dashed lines are the analytical solutions and the dotted lines are early time limits from Eqs. (22) and (23).

Image of FIG. 10.
FIG. 10.

Time evolution of the central moments of the filament length distribution. (a) shows the mean filament length computed numerically (solid black line) and a comparison to the early time linear solution (dotted blue line), given by M 0(t)/P 0(t) from Eqs. (23) and Eq. (22), and the full closed-form solution (dashed blue line), given by M(t)/P(t) from Eqs. (33) and (47). The standard deviation of the length distribution is shown in (b). The black line is the result from evaluating the standard deviation from a numerical solution to the master equation Eq. (2), the blue dotted line shows the linear solution valid for early times Eq. (50) and the dashed line shows the approximation σ(t) ∼ μ(t) valid for late times Eq. (51). The kinetic parameters are the same as for Fig. 4.

Image of FIG. 11.
FIG. 11.

Effect of initial seed concentration. (a) Polymer mass concentration M(t), Eq. (33), as a function of time for different initial seed concentrations, from left to right: M(0) = 729 nM. M(0) = 243 nM, M(0) = 81 nM, M(0) = 27 nM, M(0) = 9 nM and M(0) = 3 nM. The values used for the other parameters are: k off = 0, k n = 0, k + = 5 × 104 s−1 M−1, k = 10−9 s−1, m tot = 50 μM, and P(0) = M(0)/1000. (b) shows the growth rate dM(t)/dt. Interestingly the maximal growth rate is independent of the seed concentration, except for seed concentrations that are higher than the critical concentration (M c = 260 nM for the parameters used) as described in the text. The maximal growth rate Eq. (62) (1.3 × 10−9 M s−1 for the parameters used) is shown as a dashed red line.

Image of FIG. 12.
FIG. 12.

Nucleated growth with varying monomer concentrations. (a) Polymer mass concentration M(t), Eq. (33), as a function of time for different total monomer concentrations, from left to right: m tot = 1.3 × 10−4 M, m tot = 6.4 × 10−5 M, m tot = 3.2 × 10−5 M, m tot = 1.6 × 10−5 M, m tot = 8.0 × 10−6 M, m tot = 4.0 × 10−6 M. The values used for the other parameters are: k off = 0, n c = 2, k n = 10−8 M−1 s−1, k + = 5 × 104 s−1 M−1, k = 5 × 10−8 s−1, M(0) = P(0) = 0. (b) shows the normalised polymer mass fractions M(t)/m tot as a function of time, for the same monomer concentrations as in (a).

Tables

Generic image for table
Table I.

Comparison of the first order self-consistent solutions for irreversible filament growth with secondary nucleation or fragmentation with the analogous results from the theory of nucleated polymerisation.

Loading

Article metrics loading...

/content/aip/journal/jcp/135/6/10.1063/1.3608916
2011-08-12
2014-04-20
Loading

Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Nucleated polymerization with secondary pathways. I. Time evolution of the principal moments
http://aip.metastore.ingenta.com/content/aip/journal/jcp/135/6/10.1063/1.3608916
10.1063/1.3608916
SEARCH_EXPAND_ITEM