New mathematical model for amyloid formation

Summary: Mathematical model describes the chemical reactions responsible for the formation of Alzheimer’s-linked protein.

Source: American Institute of Physics

Amyloids are aggregates consisting of stacks of thousands of proteins bound tightly together. Their formation is involved in several widespread disorders, including Alzheimer’s disease and Type II diabetes.

In this week’s Journal of Chemical Physics, by AIP Publishing, scientists report on a mathematical model for the formation of amyloid fibrils. The model sheds light on how the aggregation process can occur in a catalytic manner, something that has not been previously well understood.

The investigators applied their model to the aggregation of a specific protein associated with Alzheimer’s, Ab40. The results show the initiation of the aggregation process for Ab40 fibrils typically occurs at interfaces, such as near the surface of a liquid solution or the glass wall of a test tube. This has important implications for the interpretation of laboratory data used in the study of Alzheimer’s and other diseases.

The model consists of a set of mathematical equations, known as rate equations, that describe how protein aggregate concentrations change over time. Each reaction step in the model is shown to be analogous to those in reactions involving enzymes. The role of the enzyme is played by either the tip or side of a growing fiber or, possibly, a surface of the reaction vessel.

This is a diagram from the study
Comparison of Protein Aggregation to Enzyme Kinetics. Image is credited to Alexander J. Dear.

The investigators found the mathematical form of their model was related to the famous Michaelis-Menten equations, first published in 1913 to describe the rates of enzyme reactions. It has a far simpler mathematical form than all previous models used for amyloid formation and has the additional advantage that the equations can be solved by hand, without the need for computer simulations.

“We expect the methodology developed in this paper will underpin future efforts to model new amyloid formation phenomena,” co-author Alexander Dear said.

One of the key features of the mathematical solution found for Michaelis-Menten-type equations is a phenomenon known as saturation. When saturation occurs, the catalytic sites become fully occupied at high protein concentrations. In the case of the Ab40 studies, saturation also shows that the process initiating aggregation involves a surface, such as the wall of a test tube.

While the conclusions do not directly apply to the body itself, co-author Tuomas Knowles said, “This work will be central in facilitating the study of amyloid formation in the presence of other species found in body fluids.”

Co-author Sara Linse said, “This work takes the analysis of experimental data to a new level that will be essential for deriving potent inhibitors of amyloid formation.”

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Source:
American Institute of Physics
Media Contacts:
Larry Frum – American Institute of Physics
Image Source:
The image is credited to Alexander J. Dear.

Original Research: Open access
“The catalytic nature of protein aggregation”. Alexander Dear, Georg Meisl, Thomas C. T. Michaels, Manuela R. Zimmermann, Sara Linse and Tuomas P. J. Knowles.
Journal of Chemical Physics doi:10.1063/1.5133635.

Abstract

The catalytic nature of protein aggregation

The formation of amyloid fibrils from soluble peptide is a hallmark of many neurodegenerative diseases such as Alzheimer’s and Parkinson’s diseases. Characterization of the microscopic reaction processes that underlie these phenomena have yielded insights into the progression of such diseases and may inform rational approaches for the design of drugs to halt them. Experimental evidence suggests that most of these reaction processes are intrinsically catalytic in nature and may display enzymelike saturation effects under conditions typical of biological systems, yet a unified modeling framework accounting for these saturation effects is still lacking. In this paper, we therefore present a universal kinetic model for biofilament formation in which every fundamental process in the reaction network can be catalytic. The single closed-form expression derived is capable of describing with high accuracy a wide range of mechanisms of biofilament formation and providing the first integrated rate law of a system in which multiple reaction processes are saturated. Moreover, its unprecedented mathematical simplicity permits us to very clearly interpret the effects of increasing saturation on the overall kinetics. The effectiveness of the model is illustrated by fitting it to the data of in vitro Aβ40 aggregation. Remarkably, we find that primary nucleation becomes saturated, demonstrating that it must be heterogeneous, occurring at interfaces and not in solution.

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