![](bg4.png)
hence may serve as good approximations of the
nucleation mass (Fig 3). It was also observed in
course of the simulations that an initial concentration
of 10 µM made the simulation erratic for a wide
range of rate constants (because of increased
dynamism and stochasticity in the system with lower
molecular count of the species rendering the ODEs
unstable). So, we generated these semi-log plots for
the different regimes (data not shown) by removing
the data points for 10 µM. Indeed, these curves show
a more stable relationship between the lag times and
the initial concentrations, and we find close to linear
behavior for n=10,11 (in Regime 1), n=12,13 (in
Regime 2) and n=15,16,17 (in Regime 3).
The next question is whether 10,11,…,17 is the
right range for the nucleation mass, or can we
further reduce it? Fig 4 shows the concentration
curve for F against time and different initial
concentrations. One requirement for the rate
constants reported above is that these curves must
saturate to the same peak as expected
mathematically. So we considered this to be another
constraint that reduced the range of feasible
nucleation masses to n=10, 11…,14. Note that n=15,
16, 17 did not allow the concentration curves to
saturate (data not shown), and hence were ruled out
as possible candidates for the nucleation mass.
Figure 4: Simulated fluorescence change curves for
different initial concentrations with n=12.
3.2 Can we Compare Simulated and
Experimentally Observed Lag
Times?
The experimental ThT fluorescence plots show the
cumulative effect of all oligomers of a certain size
(and beyond). The results shown above plot the
concentration of F which model the cumulative
effect of all the nucleated oligomers in the pathway.
However, it is assumed that all nucleated oligomers
show up on the ThT curves (this is generally not the
case from actual experiments). Hence, the lag times
estimated from our model are lower than that seen
experimentally. Also, it is not yet known what size
of oligomers actually show up ThT positive and
hence the experimental estimates are at best the
maximum limits of the lag times for each initial Aβ
concentration. To get around this problem, we varied
the rate constants to estimate the maximum possible
lag times for each value of the nucleation mass. This
is still an approximation of the actual system and
needs further study. Ideally, we need to know what
sizes of oligomers are considered ThT positive such
that the experimental curves can be meaningfully
compared to the simulated plots. The present paper,
however, gives us a feasible range of nucleation
masses to work with in order to build a complete
simulation of the on-pathway. The rate constants
estimated in this exercise can serve as a guidance for
the complete simulation where we will need a more
detailed model (with separate parameters for each
post-nucleation oligomer) to properly model their
effects on the system.
4 CONCLUSIONS
In this paper, we have studied the lag times in the
sigmoidal Aβ fibril formation pathway. We also
reported that the nucleation mass can potentially be
in the range 10,11,…, 14 mers. In order to reduce
the complexity of the entire fibril formation
pathway, we used the rate constants that we have
earlier estimated for the post-nucleation stage into a
modified model that can approximately characterize
the complete pathway. These estimates will serve as
the basis for implementing a complete and accurate
simulation of the pathway wherein we have
approximately estimated all the 6 variables involved.
Such a simulation will pave the path to study the
complete system dynamics of Aβ aggregation
leading to a better understanding of AD in general.
ACKNOWLEDGEMENTS
This work was supported by NSF-1158608.
REFERENCES
Selkoe, D. J., Schenk, D. (2003). Alzheimer's disease:
molecular understanding predicts amyloid-based
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Harper, J. D., Lansbury, P. T., Jr. (1997). Models of
amyloid seeding in Alzheimer's disease and scrapie:
mechanistic truths and physiological consequences of
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PEPTIDE AGGREGATION
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