Fig 1.
A: Monomers are produced at rate S (zeroth order) and cleared at a rate κ (first order). Two monomers combine to form a dimer with rate constant ν (second order) and a dimer can dissociate at rate μ (first order) into two monomers. Monomers and dimers can combine to form trimers at rate ν (second order), with negligible backwards reactions. Neurons are killed at a rate σ times the dimer concentration. Thus, as the dimer concentration rises, so does the speed of neuronal death. Monomers and dimers diffuse with diffusivities and
, respectively. B: Representative production/loss rates of individual components when concentrations are at their baselines values in Table 1, without the rates changing with age. Incoming arrows represent gain/production; outgoing arrows represent loss/clearance. For example, in each second, 40% of the dimer concentration is lost due to dissociation (dimers have a very short lifespan) and gained from dimerization illustrating that the monomer-dimer equilibrium is fast relative to other equilibria, whereas cell viability is lost very slowly.
Fig 2.
Means of obtaining model parameters.
Some parameters (blue) were taken from published values in the literature; others (yellow-orange) were fitted based on experimental data; the value γ (green) is fitted from our overall model with reference to clinical data.
Table 1.
Values of parameters within the model.
See Fig 2 for an explanation of how these values were determined and their sources; Bars (e.g., ), indicate a quantity representative of that in a healthy brain; Lit. denotes values listed in literature; Inf. denotes values inferred or fit from published data; Mod. indicates a value fit from our model with reference to literature; Def. denotes the definition of a value used in our study in various calculations, derived from other quantities in the table.
Fig 3.
Comparison of static and dynamic models with clinical data for AD. The dotted green lines represent the line of best fit to clinical data [19, 34] on log-scale; The black solid lines are the lines of best fit to the dynamic model on log-scale. A: for prevalence, the clinical doubling time is 4.9 y and our dynamic model predicts 12 y. B: for incidence, the clinical doubling time is 4.9 y and our dynamic model predicts 11 y. The value γ is chosen so that clinical and dynamic model incidence agree at age 60.
Fig 4.
A: Time dependence of HV for the dynamic model with or without additional AD pathology. An HV of 1 is maximal. At age 75, the annual changes in hippocampal volume are −0.015% (static model, not shown), −0.29% (dynamic model), and −1.1% (AD pathology model—rates have been scaled to match this value). The HV ratio between those at age 71.6 (AD pathology) to age 63.4 (CN) is 0.859. We can also compare within models. The hippocampal volume ratios between age 71.6 to age 63.4 years are as follows: 0.999 (static), 0.984 (dynamic), and 0.944 (AD pathology). B: Traumatic Brain Injury. Our fit to clinical data [56] for the relative hazard rate vs number of TBIs, n. The error bars represent one standard error. Model fit:
with
to be estimated. The fitted value is a = 0.231.
Fig 5.
A: the excess monomer production is taken to be spherically symmetric. The distance (x−axis) denotes the distance from the center of the source. The dashed circle/lines represent the boundary where excess monomer production ceases. B: monomer and dimer concentrations, and monomer production rate, versus distance from center. These values have been nondimensionalized by ,
, and
, respectively. C: viability at various ages plotted against position.
Fig 6.
Significant variations of rate constants.
A: the prevalence for the time-dependent model as ω0 is scaled. B: clinically observed prevalence of AD in males (M) and females (F) [60] with inflection point marked by arrow.
Table 2.
The probability distribution here has that the mean U0 value is and the standard-deviation to mean ratio is consistent with (19).
Table 3.
Each result is presented in this manuscript. Three significant figures are used as the model results come from formulas and the parameters were stored with 3 significant figures.