Grupo de Reciclado e Valorización de Residuos (REVAL), Instituto de Investigacións Mariñas (IIMCSIC), C/ Eduardo Cabello 6, CP36208 Vigo, Spain
Abstract
Backgrounds
The process of amyloid proteins aggregation causes several human neuropathologies. In some cases,
Results
In this manuscript, experimental data of insulin, Aβ42 amyloid protein and apomyoglobin fibrillation from recent bibliography were selected to evaluate the capability of a bivariate sigmoid equation to model them. The mathematical functions (logistic combined with Weibull equation) were used in reparameterized form and the effect of inhibitor concentrations on kinetic parameters from logistic equation were perfectly defined and explained. The surfaces of data were accurately described by proposed model and the presented analysis characterized the inhibitory influence on the protein aggregation by several chemicals. Discrimination between true and apparent inhibitors was also confirmed by the bivariate equation. EGCG for insulin (working at pH = 7.4/T = 37°C) and taiwaniaflavone for Aβ42 were the compounds studied that shown the greatest inhibition capacity.
Conclusions
An accurate, simple and effective model to investigate the inhibition of chemicals on amyloid protein aggregation has been developed. The equation could be useful for the clear quantification of inhibitor potential of chemicals and rigorous comparison among them.
Background
The aggregation and fibrillation of proteins has been commonly associated with numerous degenerative disorders in humans including Alzheimer’s, Parkinson’s, prion’s, diabetes type II and Huntington’s diseases
In general, much of the success in the future application of chemicals to inhibit the
Although the use of empirical sigmoid equations, mainly the logistic equation, does not provide a direct explanation of the molecular steps that underlie in the generation of fibrils, it is a robust tool to examine protein aggregation kinetic data and to address all the phases of the process
In the present work, the capability of fit and experimental data predictability of a sigmoid bivariate model that simulates the growth of aggregation process on different proteins along with the effects of inhibitory chemicals on the kinetic parameters is explored in selected cases obtained from the literature. The results reveal its efficacy and validity to analyze the most relevant parameters that describe geometrically and macroscopically the mentioned process.
Methods
Experimental data
Amyloid protein aggregation data were collected from results previously reported in the bibliography and digitized from the published curves using GetData Graph Digitizer 2.24. The kinetics of insulin inhibition induced by (−)epigallocatechin3gallate (EGCG) were selected from Wang et al.
On the other hand, the aggregation kinetics of Aβ42 amyloid protein inhibited by apigenin and taiwaniaflavone were selected from Thapa et al.
Mathematical modelling
The model developed to simulate the process of aggregation and hence insulin fibrillation was defined by a bivariate equation. Such model is based on the combination of Weibull function as chemicalconcentration model
where,
Insulin aggregation kinetics measured by absorbance or fluorescence
Amyloid aggregation growth measured as absorbance at 600 nm, relative ThT fluorescence intensity (%) and ThT fluorescence intensity at 482 nm or 490 nm. Units: absorbance units (AU) or (%).
Time. Units: h or d
Maximum aggregation growth. Units: AU or %
Maximum aggregation rate. Units: AU h^{−1}, AU d^{−1} or % h^{−1}
Lag phase. Units: h or d
Maximum insulin aggregation affected by chemical agent. Units: AU or %
Maximum insulin aggregation rate affected by chemical agent. Units: AU h^{−1}, AU d^{−1} or % h^{−1}
Lag phase affected by chemical agent. Units: h or d
Concentration effects on insulin aggregation kinetics
Concentration of chemical agent. Units: mM or μM
Maximum response affecting on
Concentration corresponding to the semimaximum response affecting on
Shape parameter affecting on
Maximum response affecting on
Concentration corresponding to the semimaximum response affecting on
Shape parameter affecting on
Maximum response affecting on
Concentration corresponding to the semimaximum response affecting on
Shape parameter affecting on
Additionally, a global parameter (
Numerical methods and statistical analysis
The fitting procedures and parametric estimates from the experimental results were performed by minimizing the sum of quadratic differences between the observed and modelpredicted values using the nonlinear leastsquares (quasiNewton) method provided by the ‘
Excel spreadsheet used for modeling the Ab42amyloid apigenin case.
Click here for file
Results and discussion
Characteristics and simulations of bivariate model
In the description of amyloidogenic fibrillation growths, the logistic equation used to formalize their kinetic profiles is always formulated through an explicit expression based on the time required to reach 50% of the maximal aggregation (
Left, Graphical description of the kinetic parameters (
Left, Graphical description of the kinetic parameters (
In contrast, the fit using the reparameterized functions can be used to easily calculate the confidence intervals of the parameters. The algebraic steps required to obtain the corresponding reparameterization of
The absolute correlation between the lag phase and the aggregation growth was recently demonstrated in several sets of protein data
Figure
Profiles obtained by simulation under the numerical conditions specified in Table
Profiles obtained by simulation under the numerical conditions specified in Table
Simulation conditions
Parameters
A
B
C
D
E
F
G
Aggregation growth
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.25
0.25
0.25
0.25
0.50
0.20
0.50
3.00
3.00
3.00
3.00
4.00
5.00
4.00
Effect on
1.00
1.00
1.00
1.00



5.00
5.00
5.00
5.00



2.00
2.00
2.00
2.00



Effect on
1.00
1.00


0.60
0.60

4.00
4.00


3.00
8.00

2.00
2.00


2.00
2.00

Effect on
1.00


1.00
4.00

4.00
2.00


2.00
10.00

10.00
2.00


2.00
2.00

2.00
Inhibitory effect of EGCG on insulin aggregation
The representation of the experimental data from selected cases and surfaces predicted by equation (1) are depicted in Figure
Insulin fibrillation kinetics at different concentrations of EGCG, diC7PC and methylglyoxal (points) and fittings to equation (1) (surface)
Insulin fibrillation kinetics at different concentrations of EGCG, diC7PC and methylglyoxal (points) and fittings to equation (1) (surface).
Parameters
EGCG_1
EGCG_2
diC7PC
Methylglyoxal
Statistical values of adjusted coefficient of multiple determination (
Aggregation model
1.12±0.03
1.28±0.06
99.85±4.91
541.20±11.38
0.18±0.01
0.25±0.04
1.55±0.27
219.68±20.98
4.78±0.28
53.34±0.28
124.62±5.86
2.92±0.12
Effect on
0.43±0.03
0.94±0.02
0.75±0.05
1.00±0.44
0.68±0.08
0.01±0.00
0.65±0.12
1.80±1.25
1.65±0.45
0.96±0.20
4.04±2.37
0.99±0.23
Effect on
NS
NS
0.86±0.04
0.99±0.01
NS
NS
0.47±0.06
0.42±0.07
NS
NS
2.51±2.35
0.74±0.08
Effect on
0.81±0.10
NS
NS
NS
0.50±0.08
NS
NS
NS
0.98±0.24
NS
NS
NS
0.44
0.006
0.47
0.33
7.65
55.02
150.68
3.92
<0.001
<0.001
<0.001
<0.001
0.88
0.80
1.00
1.04
1.23
1.40
1.18
1.15
0.992
0.977
0.984
0.974
At an acidic pH and a high temperature, the effect of EGCG on the kinetic parameters was not significant for the maximum aggregation rate but statistically significant for
In contrast, different surface and parametric responses were observed for EGCG_2 up to a concentration of 0.2 mM polyphenol with only significant modifications in the maximum aggregation growth parameter (Figure
The global description of chemical inhibition was calculated by means of a single index:
Inhibitory effect of diC7PC and methylglyoxal on insulin aggregation
Figure
Regarding the effects, the two kinetic parameters
Similar inhibitory responses were also observed with methylglyoxal; thus,
Inhibitory effect of mono and biflavonoids on Aβ42 amyloid protein
The dependence of the kinetic parameters of fibrillation on the apigenin and taiwaniaflavone concentrations is represented in Figures
Amyloid protein aggregation kinetics at different concentrations of apigenin, ectoine, taiwaniaflavone and trehalose (points) and fittings to equation (1) (surface)
Amyloid protein aggregation kinetics at different concentrations of apigenin, ectoine, taiwaniaflavone and trehalose (points) and fittings to equation (1) (surface).
Parameters
Apigenin
Ectoine
Hidroxyectoine
Taiwaniaflavone
Trehalose
The case of apomyoglobin affected by trehalose is also shown. Statistical values of adjusted coefficient of multiple determination (
Aggregation model
17.43±0.50
(5.07±0.23) × 10^{−3}
(4.96±0.24) × 10^{−3}
16.55±0.55
10.49±0.28
16.10±1.49
(0.24±0.05) × 10^{−3}
(0.30±0.08) × 10^{−3}
17.01±2.09
3.31±0.33
0.25±0.03
11.60±2.26
13.50±2.31
0.27±0.06
5.76±0.14
Effect on
0.41±0.06
1.00±0.99
1.00±0.90
0.78±0.05
0.44±0.03
6.52±1.93
11.82±11.03
66.70±65.78
2.32±0.38
51.58±7.27
0.91±0.25
0.19±0.13
0.43±0.43
0.98±0.16
1.36±0.31
Effect on
0.38±0.12
0.83±0.06
0.86±0.16
1.14±0.60
0.53±0.08
9.29±5.35
0.33±0.32
1.35 (NS)
1.88±1.73
33.22±17.26
1.35±1.13
0.36±0.14
0.34±0.20
0.40±0.21
0.66±0.33
Effect on
1.76±0.70
NS
NS
1.27±1.13
NS
2.16±0.29
NS
NS
2.53±1.45
NS
2.39±0.99
NS
NS
2.33±2.31
NS
2.45
0.96
2.20
0.91
21.43
0.75
17.31
20.52
0.65
6.54
<0.001
<0.001
<0.001
<0.001
<0.001
0.99
1.04
1.00
1.02
1.03
1.06
1.09
1.18
1.11
1.08
0.995
0.988
0.984
0.990
0.995
The numerical values of
Inhibitory effect of osmolytes on Aβ42 amyloid protein
Different types of molecules have been recently suggested as drug candidates for the treatment of neurodegenerative disorders caused by the antiaggregation properties of amyloid proteins
Inhibition of apomyoglobin fibrils formation by trehalose
The effect of trehalose on the apomyoglobin aggregation kinetics was also studied using model (1), and excellent statistical results were obtained from the modeling and the description of the experimental data (Figure
Conclusions
In summary, a general bivariate model that combines the logistic equation for the description of kinetics and the Weibull equation for the chemicalconcentration response has been proposed for the characterization of the inhibitory effects produced by several chemicals on the growth of the aggregation of amyloid proteins. In all cases, the inhibitory effects on the kinetic parameters were established, and the theoretical response surfaces were in perfect agreement with the selected data. In addition, the recent definition of true and apparent inhibitors reported by Martins
Appendix
Reparameterization of logistic equation
It is wellknown the autocatalytic origin of the logistic equation based on the following differential equation:
which, integrated between
The parameter τ is defined as the time required to obtain the semimaximum fibrillation growth (when
The inflection point (
The value of aggregation when
The slope in the inflection point (
The lag phase (
with
Reorganizing terms, two reparameterized functions can be defined:
When parameters from both equations are influenced by chemical agent concentration, they can be rewritten as follows:
On the other hand, the calculation of the time τ for
The most interesting form of Weibull equation for doseresponse modelling is expressed as follows
where,
This equation can be modified according with the graphical tendencies of chemicals effects (Figure
Thus, decrease of
When the equations (A.15) are inserting directly on equation (A.10) or (A.11), the bivariate model (1) is obtained.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The authors also thank the Unit of Information Resources for Research (URICICSIC) for the cofinancing of this publication in Open Access.
Prepublication history
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