From reaction kinetics to dementia: A simple dimer model of Alzheimer's disease etiology

Abstract

Oligomers of the amyloid β-protein (Aβ) have been implicated in the pathogenesis of Alzheimer's disease (AD) through their toxicity towards neurons. Understanding the process of oligomerization may contribute to the development of therapeutic agents, but this has been difficult due to the complexity of oligomerization and the metastability of the oligomers thus formed. To understand the kinetics of oligomer formation, and how that relates to the progression of AD, we developed models of the oligomerization process. Here, we use experimental data from cell viability assays and proxies for rate constants involved in monomer-dimer-trimer kinetics to develop a simple mathematical model linking Aβ assembly to oligomer-induced neuronal degeneration. This model recapitulates the rapid growth of disease incidence with age. It does so through incorporation of age-dependent changes in rates of Aβ monomer production and elimination. The model also describes clinical progression in genetic forms of AD (e.g., Down's syndrome), changes in hippocampal volume, AD risk after traumatic brain injury, and spatial spreading of the disease due to foci in which Aβ production is elevated. Continued incorporation of clinical and basic science data into the current model will make it an increasingly relevant model system for doing theoretical calculations that are not feasible in biological systems. In addition, terms in the model that have particularly large effects are likely to be especially useful therapeutic targets.

Fig 1. Model scheme.

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 ${\mathcal{D}}_M$ and ${\mathcal{D}}_D$, 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.

Fig 3. Incidence and prevalence.

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 $\widehat{R}$ vs number of TBIs, n. The error bars represent one standard error. Model fit: $\widehat{R}\left(n\right)={(1+an)}^2$ with $a=A/\overline{S}$ to be estimated. The fitted value is a = 0.231.

Fig 5. Spatial model.

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 $\overline{M}$, $\overline{D}$, and $\overline{S}$, 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.