A Three-Stage Kinetic Model of Amyloid Fibrillation
Amyloid fibrillation has been intensively studied because of its association with various neurological disorders. While extensive time-dependent fibrillation experimental data are available and appear similar, few mechanistic models have been developed to unify those results. The aim of this work was to interpret these experimental results via a rigorous mathematical model that incorporates the physical chemistry of nucleation and fibril growth dynamics. A three-stage mechanism consisting of protein misfolding, nucleation, and fibril elongation is proposed and supported by the features of homogeneous fibrillation responses. Estimated by nonlinear least-squares algorithms, the rate constants for nucleation were ~10,000,000 times smaller than those for fibril growth. These results, coupled with the positive feedback characteristics of the elongation process, account for the typical sigmoidal behavior during fibrillation. In addition, experiments with different proteins, various initial concentrations, seeding versus nonseeding, and several agitation rates were analyzed with respect to fibrillation using our new model. The wide applicability of the model confirms that fibrillation kinetics may be fairly similar among amyloid proteins and for different environmental factors. Recommendations on further experiments and on the possible use of molecular simulations to determine the desired properties of potential fibrillation inhibitors are offered.
Amyloid fibrillation is the process of native soluble proteins misfolding into insoluble fibrils comprising cross-?-sheets. More than 20 amyloidogenic diseases such as Alzheimer’s disease, Parkinson’s disease, and prion-associated encephalopathies have been found to share fibril formation as the common symptom (1). The presence of amyloid plaques correlates with disease, but whether fibrils themselves, misfolded oligomers, or other factors are the causal agents of diseases remains unclear . Although the proteins associated with each disease do not share sequence homology, they exhibit similar insoluble filaments and fibrillation responses . This suggests that the underlying fibril formation mechanisms may be common (7).
The typical fibril formation process starts with a lag phase in which the amount of amyloid proteins turned into of fibrils is not significant enough to be detected. Afterwards, a drastic elongation phase follows and fibril concentration increases rapidly (8). Eventually, the process reaches equilibrium when most soluble proteins are converted into fibrils. The length of lag times and fibril growth rates depend upon factors like the initial concentration and pH, both of which affect the degree of supersaturation in solution. The presence of seeded molecules and foreign surfaces can influence the kinetics of fibrillation, because of the ability to catalyze the reactions at these interfaces (9). Other factors include the ionic strength of the solution and the intensity of agitation (10). Although experimental data covering these many different conditions have been reported in the literature, there is a noticeable lack of quantitative mechanistic models to provide insight into the process and directions for further research.
Because of the commonly observed sigmoidal-shaped fibrillation response reported in the literature, fibrillation processes have been modeled as a number of reactions in series covering the assembly of oligomers, the formation of nuciei as well as the growth and the breakage of fibrils. Moreover, the two-stage mechanism of yeast prion fibrillation, in which fibrils act as enzymes to trigger nucleated conformational conversion by Michaelis-Menten kinetics, provides another valuable perspective (14). Empirical or semi-empirical exponential functions are popular choices to fit the data since they are computationally simple and match the observed data well . While suggestive, some of these models only depicted the sigmoidal trend without rigorous quantitative arguments; others have not provided details on how the nuclei form or explained the shortened lag-time resulting from seeding and an increase in the initial protein concentration.
The lag-time before fibril growth has been noted in numerous publications and resembles an incubation period (10,11). Explaining its existence is one of the key scientific challenges. The problem was approached by Shoghi-Jadid et al. (16) with introduction of the Heaviside function to force the separation of nucleation and fibrillation processes, while Uversky et al. (17) used an empirical exponential model with adjustable parameters. We suggest that nucleation theory and growth models could be valuable in describing the fibrillation process. Furthermore, the drastic rate increase in the fibrillar growth phase after the lag phase indicates that cooperativity or positive feedback mechanisms are involved.
Another critical but missing piece of information is the relationship between the observable response and the degree of fibrillation. Even though histological dyes like thioflavin T (ThT) and Congo Red have been the commonly used as indicators of the presence of amyloid fibrils, the relationship between fluorescence intensity and amount of amyloid fibril remain unclear. There are also physical property methods for measuring fibril formation like turbidity, absorbance, and sedimentation . Here, we assumed linearity between ThT fluorescence and fibril concentrations based on Beer-Lambert law as a measure of fibril content, and use ultraviol et-visible (UV-vis) absorbance at 280 nm as a quantitative measure of dissolved total protein.
Insulin was chosen as the model protein for the measurements in this study because it 1), is a well-studied fibril-forming protein and has recently been studied in our laboratory, is known to develop structurally similar cross-?-sheet plaques to those formed by other amyloids and is deposited in arterial walls of type II diabetes patients (23); and 4), is available in large quantities at reasonable price. Native insulin is well folded and in stable hexamer state associated with Zn^sup 2+^ molecule under physiological conditions. Yet it can be readily unfolded to form fibrils in solution by both increasing the temperature to 65°C and by reducing the pH to 1.6. Jiménez et al. (28) proposed that the ?-helical structure (58%) of native insulin becomes unfolded to expose the ?-sheet region (6%), which is the major component of the amyloid cross-? ribbon.
In the next section, we describe the proposed kinetic model for insulin fibrillation including the parameter estimation procedure. Since experimental protocols and responses of fibrillation are similar among amyloid proteins, the modeling approach presented here is also applicable to the fibrillation of other proteins. Afterwards, our model is compared with an empirical fitting function. A general description of the Experimental Materials and Methods follows. Then, in Results and Discussion, the new model is fitted to our insulin fibrillation data, to fibrillation of A?-40 and prionlike NM fragment of Sup-35 , and to data conducted under various conditions .
Three standard analytical steps were chosen to model insulin fibrillation; formulation of the appropriate kinetic reactions based on the polymerization and nucleation theories, conversions of the reaction set into a system of differential equations, and parameter estimation by nonlinear least-square algorithms to optimize the fit between simulation results and the experimental measurements.
Initially four species of insulin were considered during fibrillation: original hexamer, monomer, cluster, and fibril . While the original hexamer is composed of six monomers stabilized by Zn^sup 2+^ an insulin monomer refers to two chains of polypeptides connected with disulfide bonds . For systems other than insulin, different morphologies may be involved such as those for ?-microglobulin (26). By incorporating the four insulin species into the reaction scheme, the proposed kinetic mechanism for this study consists of three distinct stages: decomposition of hexamers, nucleation process, and fibrillation stage as summarized in Fig. 1 and Table 1. All the reactions listed are elementary reactions so the fluxes can be easily expressed as the products of reactant concentrations and the rate constant. Regarding notations, A^sub hex^ and A^sub i^ denote the concentration of original insulin hexamers and oligomers containing i monomers, respectively. All fibrils are abbreviated as F, regardless of their length. Even though physical reactions contributing to larger-size cluster formation and the entanglement between strands of fibrils have been reported , the actual active chemical reaction sites are assumed to be restricted to the fibril ends . Therefore, fibrils of different sizes can be considered as the same species.