Modeling the growth of cytosolic TDP-43 inclusion bodies and accumulated neurotoxicity of misfolded oligomers in neurons

Abstract

This paper introduces a mathematical model for the growth of transactive response DNA binding protein of 43 kDa (TDP-43) inclusion bodies in neuron soma. The parameter representing the accumulated neurotoxicity caused by misfolded TDP-43 oligomers is also introduced. The models equations enable the numerical calculation of the concentrations of TDP-43 monomers, dimers, free oligomers, and oligomers deposited in inclusion bodies. By simulating the deposition of free oligomers into inclusion bodies, the model predicts the size of TDP-43 inclusion bodies. An approximate solution to the model equations is derived for the scenario where protein degradation machinery is dysfunctional, leading to infinite half-lives for TDP-43 dimers, monomers, and both free and deposited oligomers. This solution, valid at large times, predicts that the radius of the inclusion body increases proportionally to the cube root of time, whereas the accumulated neurotoxicity increases linearly with time. To the best of the authors knowledge, this study is the first to model the relationship between the size of TDP-43 inclusion bodies and time, and the first to introduce the concept of accumulated neurotoxicity caused by misfolded TDP-43 oligomers. Sensitivity analysis of the approximate solution indicates that the inclusion body radius and accumulated neurotoxicity become independent of the kinetic constants at large timescales. While the numerical solution of the full mathematical model continues to work with finite half-lives, the approximate solution becomes invalid for scenarios with physiologically relevant (finite) half-lives of TDP-43 dimers, monomers, and free misfolded oligomers. In contrast to the scenario with infinite half-lives, the numerical solution reveals that, for different values of the kinetic constants, the curves representing the inclusion body radius and accumulated neurotoxicity converge to distinct curves over time.