Editing Amyloid (section)

== Formation ==
[[File:Three phases of amyloid fibril formation.tif|thumb|upright=1.35|Three phases of amyloid fibril formation: [[Incubation period|lag phase]], [[exponential function|exponential phase]] and [[plateau effect|plateau phase]]]]
Amyloid is formed through the [[polymerization]] of hundreds to thousands of monomeric [[peptides]] or [[proteins]] into long fibers. Amyloid formation involves a ''[[Incubation period|lag]] phase'' (also called ''[[nucleation]] phase''), an ''[[Exponential growth|exponential]] phase'' (also called ''growth phase'') and a ''[[plateau effect|plateau]] phase'' (also called ''saturation phase''), as shown in the figure.<ref name="pmid8490014"> {{cite journal | vauthors = Jarrett JT, Berger EP, Lansbury PT | title = The carboxy terminus of the β amyloid protein is critical for the seeding of amyloid formation: implications for the pathogenesis of Alzheimer's disease | journal = Biochemistry | volume = 32 | issue = 18 | pages = 4693–7 | date = May 1993 | pmid = 8490014 | doi = 10.1021/bi00069a001 }}</ref><ref name="pmid10507029"> {{cite book | vauthors = Ferrone F | title = Amyloid, Prions, and Other Protein Aggregates | chapter = Analysis of protein aggregation kinetics | series = Methods in Enzymology | volume = 309 | pages = 256–74 | date = 1999 | pmid = 10507029 | doi = 10.1016/s0076-6879(99)09019-9 | isbn = 9780121822101 }}</ref><ref name="pmid19071235"> {{cite journal | vauthors = Morris AM, Watzky MA, Finke RG | title = Protein aggregation kinetics, mechanism, and curve-fitting: a review of the literature | journal = Biochimica et Biophysica Acta (BBA) - Proteins and Proteomics | volume = 1794 | issue = 3 | pages = 375–97 | date = March 2009 | pmid = 19071235 | doi = 10.1016/j.bbapap.2008.10.016 }}</ref><ref name="pmid20007899">{{cite journal | vauthors = Knowles TP, Waudby CA, Devlin GL, Cohen SI, Aguzzi A, Vendruscolo M, Terentjev EM, Welland ME, Dobson CM | s2cid = 6267152 | display-authors = 6 | title = An analytical solution to the kinetics of breakable filament assembly | journal = Science | volume = 326 | issue = 5959 | pages = 1533–7 | date = December 2009 | pmid = 20007899 | doi = 10.1126/science.1178250 | bibcode = 2009Sci...326.1533K }}</ref> Indeed, when the quantity of fibrils is plotted versus time, a [[sigmoid function|sigmoidal]] time course is observed reflecting the three distinct phases.

In the simplest model of 'nucleated polymerization' (marked by red arrows in the figure below), individual unfolded or partially unfolded [[polypeptide chains]] (monomers) convert into a [[Cell nucleus|nucleus]] ([[monomer]] or [[oligomer]]) via a [[thermodynamics|thermodynamically]] unfavourable process that occurs early in the lag phase.<ref name="pmid19071235"/> Fibrils grow subsequently from these [[Nucleation|nuclei]] through the addition of [[monomer]]s in the exponential phase.<ref name="pmid19071235"/>

A different model, called 'nucleated conformational conversion' and marked by blue arrows in the figure below, was introduced later on to fit some experimental observations: monomers have often been found to convert rapidly into misfolded and highly disorganized oligomers distinct from nuclei.<ref name="pmid10958771"> {{cite journal | vauthors = Serio TR, Cashikar AG, Kowal AS, Sawicki GJ, Moslehi JJ, Serpell L, Arnsdorf MF, Lindquist SL | display-authors = 6 | title = Nucleated conformational conversion and the replication of conformational information by a prion determinant | journal = Science | volume = 289 | issue = 5483 | pages = 1317–21 | date = August 2000 | pmid = 10958771 | doi = 10.1126/science.289.5483.1317 | bibcode = 2000Sci...289.1317S }}</ref> Only later on, will these aggregates reorganise structurally into nuclei, on which other disorganised oligomers will add and reorganise through a templating or induced-fit mechanism (this 'nucleated conformational conversion' model), eventually forming fibrils.<ref name="pmid10958771"/>

Normally [[folded proteins]] have to unfold partially before aggregation can take place through one of these mechanisms.<ref name="pmid19088715"> {{cite journal | vauthors = Chiti F, Dobson CM | title = Amyloid formation by globular proteins under native conditions | journal = Nature Chemical Biology | volume = 5 | issue = 1 | pages = 15–22 | date = January 2009 | pmid = 19088715 | doi = 10.1038/nchembio.131 }}</ref> In some cases, however, folded proteins can aggregate without crossing the major [[energy barrier]] for unfolding, by populating native-like conformations as a consequence of [[thermal fluctuations]], ligand release or local unfolding occurring in particular circumstances.<ref name="pmid19088715"/> In these native-like conformations, segments that are normally buried or structured in the fully folded and possessing a high propensity to aggregate become exposed to the solvent or flexible, allowing the formation of native-like aggregates, which convert subsequently into nuclei and fibrils. This process is called 'native-like aggregation' (green arrows in the figure) and is similar to the 'nucleated conformational conversion' model.

A more recent, modern and thorough model of amyloid fibril formation involves the intervention of secondary events, such as 'fragmentation', in which a fibril breaks into two or more shorter fibrils, and 'secondary nucleation', in which fibril surfaces (not fibril ends) catalyze the formation of new nuclei.<ref name="pmid20007899"/> Both secondary events increase the number of fibril ends able to recruit new monomers or oligomers, therefore accelerating fibril formation through a positive feedback mechanism. These events add to the well recognised steps of primary nucleation (formation of the nucleus from the monomers through one of models described above), fibril elongation (addition of monomers or oligomers to growing fibril ends) and dissociation (opposite process).

Such a new model is described in the figure on the right and involves the utilization of a [[master equation]] that includes all steps of amyloid fibril formation, i.e. primary nucleation, fibril elongation, secondary nucleation and fibril fragmentation.<ref name="pmid20007899"/><ref name=":0">{{cite journal | vauthors = Michaels TC, Šarić A, Habchi J, Chia S, Meisl G, Vendruscolo M, Dobson CM, Knowles TP | display-authors = 6 | title = Chemical Kinetics for Bridging Molecular Mechanisms and Macroscopic Measurements of Amyloid Fibril Formation | journal = Annual Review of Physical Chemistry | volume = 69 | issue = 1 | pages = 273–298 | date = April 2018 | pmid = 29490200 | doi = 10.1146/annurev-physchem-050317-021322 | bibcode = 2018ARPC...69..273M | doi-access = free }}</ref> The [[rate constant]]s of the various steps can be determined from a global fit of a number of time courses of aggregation (for example [[Thioflavin|ThT fluorescence]] emission versus time) recorded at different protein concentrations.<ref name="pmid20007899"/> The general master equation approach to amyloid fibril formation with secondary pathways has been developed by [[Tuomas Knowles|Knowles]], [[Michele Vendruscolo|Vendruscolo]], Cohen, Michaels and coworkers and considers the time evolution of the concentration <math>f(t,j)</math> of fibrils of length <math>j</math> (here <math>j</math> represents the number of monomers in an aggregate).<ref name=":0" /> <math display="block">\begin{align}
\frac{\partial f(t,j)}{\partial t}  & = 2k_+ m(t)f(t,j-1) - 2k_+ m(t)f(t,j) \\ & + 2k_{\rm{off}}f(t,j+1)-2k_{\rm{off}}f(t,j) \\ & + k_-\sum_{i=j+1}^\infty f(t,i)-k_-(j-1)f(t,j) \\ &  +k_1m(t)^{n_1}\delta_{j,n_1}+k_2m(t)^{n_2}M(t)\delta_{j,n_2}  \\
\\
\end{align}  </math>where <math>\delta_{i,j}  </math> denotes the [[Kronecker delta]]. The physical interpretation of the various terms in the above master equation is straight forward: the terms on the first line describe the growth of fibrils via monomer addition with rate constant <math>k_+  </math> (elongation). The terms on the second line describe monomer dissociation, i.e. the inverse process of elongation. <math>k_{\rm{off}}  </math> is the rate constant of monomer dissociation. The terms on the third line describe the effect of fragmentation, which is assumed to occur homogeneously along fibrils with rate constant <math>k_-  </math>. Finally, the terms on the last line describe primary and secondary nucleation respectively. Note that the rate of secondary nucleation is proportional to the mass of aggregates, defined as <math>M(t)=\sum_{j=n_1}^\infty jf(t,j)  </math>.

Following this analytical approach, it has become apparent that the lag phase does not correspond necessarily to only nucleus formation, but rather results from a combination of various steps. Similarly, the exponential phase is not only fibril elongation, but results from a combination of various steps, involving primary nucleation, fibril elongation, but also secondary events. A significant quantity of fibrils resulting from primary nucleation and fibril elongation may be formed during the lag phase and secondary steps, rather than only fibril elongation, can be the dominant processes contributing to fibril growth during the exponential phase. With this new model, any perturbing agents of amyloid fibril formation, such as putative [[drugs]], [[metabolites]], [[Amino acid replacement|mutations]], [[molecular chaperones|chaperones]], etc., can be assigned to a specific step of fibril formation.