Make or Break—Equation of Everything Fibrillar?
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Partygoers familiar with conga line chaos probably know more about the formation of amyloid fibrils than they realize. There’s the instigator (nucleation), others gradually join in (extension), and when the line gets too unwieldy it is liable to break up into smaller segments that conga on their own (fragmentation). But for researchers, a deeper understanding of the dynamics of amyloid assembly could be the key to cracking Alzheimer’s and a host of other amyloidoses. In the December 11 Science, researchers led by Chris Dobson and Mark Welland at the University of Cambridge, U.K., report a series of mathematical equations that explain the selfassembly of filamentous structures. Interestingly, their equations suggest that secondary nucleation—which occurs when growing fibrils break apart to form new seeds—can be the dominant driving force in fibril formation. The findings could not only lead to a better understanding of what drives formation of fibrils, but might also have implications for drug developers, since compounds that are designed to break up protein aggregates could, in theory, increase nucleation.
Without getting into polynomials, integrals, and exponentials, the mathematical formulas derived by first author Tuomas Knowles and colleagues make up a “master equation” that explains a variety of experimental observations. The equation comprises terms that account for the principal fibrillization steps of nucleation, extension, fragmentation, and secondary nucleation. The researchers tested the equation by pitting it against experimental data. Key to the equation are two constants, k+ and k, which represent the elongation and fragmentation rates, respectively. The researchers found numbers for these rate constants that, when plugged into their equations, allowed them to accurately predict growth of fibrils of a variety of amyloidogenic proteins under a diverse set of conditions.
The equation explained the scientists’ experimental data on invitro aggregation of insulin, for example, which follows different kinetics depending on the starting concentration of protein monomer. Low and high starting monomer concentrations yielded convex and sigmoidal plots of aggregation over time, respectively, and the equations predicted this behavior. The math also explained the rates of aggregation of the WW domain of mouse FBP26 protein reported previously (see Ferguson et al., 2003), as well as the shorter lag phases and higher rates of polymerization seen when βlactoglobulin aggregates are experimentally sheared to induce fragmentation (see Hill et al., 2006). The latter case demonstrates how important secondary nucleation can be to the overall rate of aggregation, since higher shear rates result in much faster overall growth of filaments, write the authors.
Significantly, Knowles and colleagues found that, for fibrillization processes that have a secondary nucleation component, the maximal rate of aggregation seems to depend primarily on one single parameter in their equation, κ, which reflects the total protein concentration and the k+ and k rate constants. They found that both the lag phase and the maximal growth rate, parameters that together would predict the extent of fibrillization, depend primarily on κ.
Having filament growth depend primarily on one parameter has some interesting corollaries, the most important having to do with scaling, according to the authors. Natural phenomena often don’t scale as one might think, such that beefing up the concentration of components is no guarantee that things will go any faster. In the case of fibrillization, the dominance of the κ parameter predicts that the lag time should scale only weakly with monomer concentration. In fact, the authors found that this prediction tracked with experimental data for six different fibrillogenic proteins, including four yeast prions, insulin, and β2microglobulin: As monomer concentrations increased, the lag time for fibrillization changed little.
What does this mean for Alzheimer’s and other amyloidoses? Knowles and colleagues did not address Aβ fibrillogenesis directly, but they did offer some insight into polyglutamine disorders, such as Huntington disease. Traditional models relying on primary nucleation as the dominant parameter to explain lag phases poorly fit the kinetics of polyglutamine aggregation, according to the authors, because these models end up predicting negative numbers of growth nuclei for aggregation—something that clearly makes no sense. Negative nuclei numbers do not emerge using the “lessthanlinear” scaling of Knowles and colleagues’ κbased predictions. What does emerge is that “breakage is likely to be a key factor in the aggregation of polyglutamine peptides and hence potentially an important contributor for the development of disorders such as Huntington’s disease,” the authors write. Preventing this breakage, then, might slow the overall rate of fibrillization.—Tom Fagan.
Reference:
Knowles TPJ, Waudby CA, Devlin GL, Cohen SIA, Aguzzi A, Vendruscolo M, Terentjev EM, Welland ME, Dobson CM. An analytical solution to the kinetics of breakable filament assembly. Science 2009 October 23; 60:378389. Abstract
References
Paper Citations
 Hill EK, Krebs B, Goodall DG, Howlett GJ, Dunstan DE. Shear flow induces amyloid fibril formation. Biomacromolecules. 2006 Jan;7(1):103. PubMed.
 Knowles TP, Waudby CA, Devlin GL, Cohen SI, Aguzzi A, Vendruscolo M, Terentjev EM, Welland ME, Dobson CM. An analytical solution to the kinetics of breakable filament assembly. Science. 2009 Dec 11;326(5959):15337. PubMed.
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Papers
 Knowles TP, Waudby CA, Devlin GL, Cohen SI, Aguzzi A, Vendruscolo M, Terentjev EM, Welland ME, Dobson CM. An analytical solution to the kinetics of breakable filament assembly. Science. 2009 Dec 11;326(5959):15337. PubMed.
Primary Papers
 Knowles TP, Waudby CA, Devlin GL, Cohen SI, Aguzzi A, Vendruscolo M, Terentjev EM, Welland ME, Dobson CM. An analytical solution to the kinetics of breakable filament assembly. Science. 2009 Dec 11;326(5959):15337. PubMed.
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