Insight into the correlation between lag time and aggregation rate
in the kinetics of protein aggregation
Supplementary material
Stefan Auer1,* and Dimo Kashchiev2
1
Centre for Molecular Nanoscience, University of Leeds, Leeds, LS2 9JT, UK
Institite of Physical Chemistry, Bulgarian Academy of Sciences, ul. Acad. G. Bonchev 11,
Sofia 1113, Bulgaria
2
*Corresponding author. E-mail address: S.Auer@leeads.ac.uk
Kolmogorov-Johnson-Mehl-Avrami (KJMA) Theory and Protein Aggregation
Since the KJMA theory1-5 has the potential to provide a new insight into the kinetics of the
overall process of protein aggregation, applying this theory to the process is of considerable interest.
This was the idea behind the use of the KJMA-type Eq. (1) in the article for analysis of available
experimental (t) curves. However, the analysis showed that these curves are describable with values
of the kinetic index n predominantly between 6 and 22, rather than between 2 and 4, as predicted by
the original KJMA theory. It is this discrepancy for which Eq. (1) was used solely for a formal
description of the (t) data. This discrepancy indicates that the original KJMA theory needs an
appropriate modification in order to take into account the peculiarities in the evolution of the protein
aggregates, e.g., the aggregate fragmentation, and in this way to become applicable to the kinetics of
overall protein aggregation. As explained below, such a modification is by far not straightforward.
The KJMA theory1-5 describes the kinetics of new-phase formation at the mesoscopic level,
i.e. without using parameters characterizing processes at the molecular level, such as the frequencies
of attachment and detachment of molecules to and from the new-phase aggregates. Rather, this theory
operates solely with the aggregate nucleation and growth rates. The general KJMA equation for the
volume (or mass) fraction  of the new phase at time t is of the form (e.g., Ref. 6)
 (t )  1  exp[ V (t ) / V0 ]
(A)
where V is the total volume of the new phase at t in the absence of contacts between the new-phase
aggregates, and V0 is the volume of the system. Under the assumption that the aggregates nucleate in
the volume (and not on the surface) of the old phase and that they all grow isomorphically, V is given
by (Ref. 6)
t
V (t )  cV0  J (t)[ R(t, t )]d dt .
0
(B)
Here J (t ) is the nucleation rate at the moment t   t when a given aggregate was born, R(t , t ) is
the effective radius of this aggregate at the considered time t, c is the aggregate shape factor (e.g.,
c  4 / 3 for spheres), and d is the aggregate growth dimensionality ( d  1 , 2 or 3, e.g., for
cylinders with fixed diameter, disks with fixed thickness or spheres, respectively).
2
The classical example for the use of Eqs. (A) and (B) is new-phase formation by progressive
nucleation of aggregates at time-independent rate J0 (m3 s1) and radial growth of these aggregates at
time-independent rate G (m/s). Then J (t )  J 0 and R(t , t )  Gt  Gt  under the assumption that
R  0 at t  t . These simple time dependences of J and R allow the integration in Eq. (B) to be
readily performed analytically so that substitution of the resulting V(t) dependence in Eq. (A) yields
(Ref. 6)
(C)
 (t )  1  exp[ (t /  )n ]
where
(D)
n  d 1
d 1 /( d 1)
  [(1  d ) / cJ 0G )]
.
(E)
Equation (D) says that n is between 2 and 4, because d is between 1 and 3. As long as in protein
aggregation the kinetic index n is most often between 6 and 22, obviously, a serious reconsideration
of the V(t) dependence from Eq. (B) is necessary if the KLMA theory is to be applied to this process.
How should however this reconsideration be done? Regrettably, the answer to this question is
presently unknown. Yet, the strategy of reconsidering the KJMA theory is more or less clear. Namely,
J and R in Eq. (B) should be reformulated more generally and their time dependences should be
accounted in a proper way. This is particularly important with respect to J in the case of
fragmentation of the growing protein aggregates at a later stage of the overall aggregation process.
The fragmentation-mediated birth of secondary aggregates may be taken into account if J is
conceived more generally as the nucleation rate of both the primary and the secondary aggregates.
Then the time dependence of J in Eq. (B) could be determined from nucleation and fragmentation
theories. While however the general nucleation theory is rather well developed (e.g., Refs. 6, 7), that
is not the case with the theory of protein fragmentation. In addition, the time dependences of J and R
in Eq. (B) become mathematically quite complicated at the advanced stage of the overall aggregation
process because of the solution desupersaturation resulting from the consumption of dissolved protein
by the growing aggregates. Accounting for this desupersaturation requires formulation of a massbalance equation which has to be considered simultaneously with the KJMA Eqs. (A) and (B).
Analytically, the whole problem then becomes almost intractable except, perhaps, at the expense of
rather constraining approximations, and solutions ought to be obtained mostly by numerical methods.
References
1. Kolmogorov AN. On the statistical theory of crystallization of metals. Bull Acad Scis URSS
(Classe Scis Math Natur) 1937;3:355-359. (in Russian)
2. Johnson WA, Mehl RF. Reaction kinetics in processes of nucleation and growth. Trans AIME
1939;135:416-442.
3. Avrami M. Kinetics of phase change. I. General theory. J Chem Phys 1939;7:1103–1112.
4. Avrami M. Kinetics of phase change. II. Transformation-time relations for random distribution of
nuclei. J Chem Phys 1940;8:212–224.
5. Avrami M. Granulation, phase change, and microstructure. Kinetics of phase change. III.
1941;9:177-184.
6. Kashchiev D. Nucleation: Basic Theory with Applications. Oxford: Butterworth-Heinemann;
2000. 529 p.
7. Vehkamaki H. Classical Nucleation Theory in Multicomponent Systems. Berlin: Springer;
2006. 176 p.