Mathematical aspects of the kinetics of formation and degradation of linear peptide or protein aggregates

• In neurons, peptides form plaques related to Alzheimer’s disease.
• In vivo, the corresponding kinetics depend on peptide degradation.
• The model proposed takes this process into account.
• Steady-state size distributions of aggregates are illustrated.
• Typical transient kinetics are shown.
• Global feedbacks in the peptide production are also discussed.

In cells, peptides and proteins are sometimes prone to aggregation. In neurons, for example, amyloid β peptides form plaques related to Alzheimer’s disease (AD). The corresponding kinetic models either ignore or do not pay attention to degradation of these species. Here, the author proposes a generic kinetic model describing formation and degradation of linear aggregates. The process is assumed to occur via reversible association of monomers and attachment of monomers to or detachment from terminal parts of aggregates. Degradation of monomers is described as a first-order process. Degradation of aggregates is considered to occur at their terminal and internal parts with different rates and these steps are described by first-order equations as well. Irrespective of the choice of the values of the rate constants, the model predicts that eventually the system reaches a stable steady state with the aggregate populations rapidly decreasing with increasing size at large sizes. The corresponding steady-state size distributions of aggregates are illustrated in detail. The transient kinetics are also shown. The observation of AD appears, however, to indicate that the peptide production becomes eventually unstable, i.e., the growth of the peptide population is not properly limited. This is expected to be related to the specifics of the genetic networks controlling the peptide production. Following this line, two likely general networks with, respectively, global negative and positive feedbacks in the peptide production are briefly discussed.

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