Simulating tubulin-associated unit transport in an axon: using bootstrapping for estimating confidence intervals of best-fit parameter values obtained from indirect experimental data

Abstract

In this paper, we first develop a model of axonal transport of tubulin-associated unit (tau) protein. We determine the minimum number of parameters necessary to reproduce published experimental results, reducing the number of parameters from 18 in the full model to eight in the simplified model. We then address the following questions: Is it possible to estimate parameter values for this model using the very limited amount of published experimental data? Furthermore, is it possible to estimate confidence intervals for the determined parameters? The idea that is explored in this paper is based on using bootstrapping. Model parameters were estimated by minimizing the objective function that simulates the discrepancy between the model predictions and experimental data. Residuals were then identified by calculating the differences between the experimental data and model predictions. New, surrogate ‘experimental’ data were generated by randomly resampling residuals. By finding sets of best-fit parameters for a large number of surrogate data the histograms for the model parameters were produced. These histograms were then used to estimate confidence intervals for the model parameters, by using the percentile bootstrap. Once the model was calibrated, we applied it to analysing some features of tau transport that are not accessible to current experimental techniques.

Simulating the effect of formation of amyloid plaques on aggregation of tau protein

“…(2.32)At the axon terminal, we imposed the following boundary conditions: At x* = L*: equation (2.33b), j * tot,tau,x=L is the flux of tau into the terminal. Following[52], equation(2.33b) can be restated as − D * free ∂n * free ∂x * − D * mt ∂n * dif ∂x * + v * a n * a − v * r n * r = A 1 − exp − for the concentrations of misfolded tau protein in the axon and in the soma and for the number of tau protein molecules in the soma are, respectively, At t * = 0 : n * mis,ax (0) = 0, N * mis,s (0) = 0, N * s (0) = N * s,i . (2.35) (d) Estimating values ofq * tau and h * tau To estimate values ofq * tau and h * tau , we stated the conservation of monomeric tau in the soma, free tau − j tot,tau,x=0 v * a n * tot,ax,x=0 = 0, (2.36) where j tot,tau,x=0 = j * tot,tau,x=0 n * tot,ax,x=0 v * a = h * tau n * tot,ax,x=0 v * a N * s,i = h * tau V * s N * s,i A * ax v /journal/rspa Proc.…”

How the formation of amyloid plaques and neurofibrillary tangles may be related: a mathematical modelling study

“…In ref. [30], we estimated the remaining 18 parameters by minimizing the discrepancy between the model predictions and experimental results for the total tau concentration, reported in ref. [31], and for the tau average velocity, reported in ref.…”

“…The best-fit values are given in electronic supplementary material, table S3. For details of how the discrepancy between the model predictions and experimental results is minimized see [30]. It is likely that in the beginning of AD, tau aggregation into NFTs plays a neuroprotective role by sequestering toxic tau oligomers [53,54].…”

“…In ref. [30], we suggested a model of tau transport in the axon, which simulates slow axonal transport of tau protein due to motor-driven transport of tau (by kinesin and dynein motors), diffusion of free cytosolic tau, and transport of a sub-population of MT-bound tau due to its diffusion along the MT lattice. The model depends on 24 parameters, six of which were estimated from the literature.…”

Utilization of the bootstrap method for determining confidence intervals of parameters for a model of MAP1B protein transport in axons

“…We applied the developed method for estimating best fit parameters, and their confidence intervals, to a model describing MAP1B transport in an axon. We believe that the suggested approach can be generalized to many situations where model parameters (and their uncertainly) are hard to estimate directly and there are few experimental data to fit the model predictions (for example, we recently applied this method to a model of tau protein transport in Kuznetsov and Kuznetsov, 2016b).…”