- Open Access
A modified Stokes-Einstein equation for Aβ aggregation
© Achuthan et al; licensee BioMed Central Ltd. 2011
Published: 18 October 2011
In all amyloid diseases, protein aggregates have been implicated fully or partly, in the etiology of the disease. Due to their significance in human pathologies, there have been unprecedented efforts towards physiochemical understanding of aggregation and amyloid formation over the last two decades. An important relation from which hydrodynamic radii of the aggregate is routinely measured is the classic Stokes-Einstein equation. Here, we report a modification in the classical Stokes-Einstein equation using a mixture theory approach, in order to accommodate the changes in viscosity of the solvent due to the changes in solute size and shape, to implement a more realistic model for Aβ aggregation involved in Alzheimer’s disease. Specifically, we have focused on validating this model in protofibrill lateral association reactions along the aggregation pathway, which has been experimentally well characterized.
The modified Stokes-Einstein equation incorporates an effective viscosity for the mixture consisting of the macromolecules and solvent where the lateral association reaction occurs. This effective viscosity is modeled as a function of the volume fractions of the different species of molecules. The novelty of our model is that in addition to the volume fractions, it incorporates previously published reports on the dimensions of the protofibrils and their aggregates to formulate a more appropriate shape rather than mere spheres. The net result is that the diffusion coefficient which is inversely proportional to the viscosity of the system is now dependent on the concentration of the different molecules as well as their proper shapes. Comparison with experiments for variations in diffusion coefficients over time reveals very similar trends.
We argue that the standard Stokes-Einstein’s equation is insufficient to understand the temporal variations in diffusion when trying to understand the aggregation behavior of Aβ 42 proteins. Our modifications also involve inclusion of improved shape factors of molecules and more appropriate viscosities. The modification we are reporting is not only useful in Aβ aggregation but also will be important for accurate measurements in all protein aggregation systems.
Aberrant misfolding and aggregation of proteins have been implicated in over 40 different human pathologies including Alzheimer’s disease (AD), Parkinson’s disease, type 2 diabetes, transmissible spongiform encephalopathies (TSE) or Prion diseases, Huntington’s disease and Cruedfelt-Jacob disease (CJD). A biochemical commonality in these diseases is that the protein involved forms pathogenic aggregates, irrespective of whether the monomeric protein is in the misfolded or intrinsically disordered forms, that have a consensus structural moiety, commonly known as “amyloids”. In all amyloid diseases, such protein aggregates have been implicated fully or partly, in the etiology of the disease. Due to their significance in human pathologies, there have been unprecedented efforts towards physiochemical understanding of aggregation and amyloid formation over the last two decades. It is well understood that the process of aggregation towards amyloid fibrils from a monomeric state is a nucleation-dependant mechanism, analogous to crystal growth. In such a process, it is believed that there is a conformational change during the nucleation event followed by rapid aggregation to a state that forms large insoluble or soluble aggregates, which in turn is accompanied by a structural convergence to cross β-sheet conformation .
In Alzheimer’s disease (AD), a protein called amyloid-β (Aβ) peptide forms aggregates that deposit as senile plaques in brains. The nucleation-dependent process of Aβ aggregation is inferred by the occurrence of a ’lag-phase’ prior to fibril growth that shows a sigmoidal pattern . In the aggregation pathway, one important intermediate called “protofibrils” has been isolated and characterized by several groups . Protofibrils mainly differ from fibrils in their size and solubility; while fibrils can be sedimented with relatively smaller forces (19000g, 10min)-where g is the acceleration due to gravity- protofibrils require substantially high sedimentation forces and have smaller diameters than the fibrils . Protofibrils have propensities to both elongate (by monomer addition) as well as to laterally associate (protofibril-protofibril association) to grow into mature fibrils and have been well characterized . These two mechanisms of fibril growth depend on the structure and stability of protofibrils, which in turn depend on the factors affecting nucleation.
An important aspect of the Stokes-Einstein’s equation and its utility in protein aggregation systems in particular is that the viscosity of the bulk solvent in a given buffer solution is considered to be a constant.
firstly, molecules of concern to us are not necessarily spheres  although this is the simplest assumption one can make. In general for our problem, where we consider the aggregation of molecules, several classes of molecules (or n-mers) are assumed to populate the solvent. While it is not clear what the specific shape of each class (n-mer) is, it is most certainly not a sphere  for the most part.
The second point we make is that the SE relation takes the viscosity to be a constant, based upon the solvent viscosity. However we argue that it should depend upon the concentration of the various species of molecules present in the system. Any single molecule (individual or aggregated), during diffusion will experience the viscous forces of the liquid as well as due to the remaining molecules around it. As a result, the diffusion coefficient, k, is also a function of the concentration of the molecules. In fact, more generally, we will consider an effective diffusion coefficient k e where k e = k e (ϕ i ,η0) where i = 1, 2, …,n.
In this paper, we have considered these two important conditions and incorporated modifications to the Stokes-Einstein equation that may faithfully represent the changes occurring in amyloid aggregation systems, especially Aβ. The significance of non-sphericity in computing the diffusion coefficient has been realized in various contexts and the biophysics, chemistry, geological and chemical engineering literature is replete with discussions of the role of non-spherical particles, specifically spheroidal particles [19–23]. While the biophysics community has been well aware of the importance of non-sphericity, the importance of concentration upon the viscosity and hence diffusivity, has been largely neglected. The chemical engineering literature is however a very rich source of discussion on this matter; see  and references cited therein for more information on this subject.
While we are following the vast majority of physics and engineering literature in following this approach of modeling a mixture through its overall material parameters, we are also conscious that it not a rigorous mathematical approach. In his paper on the “viscosities of mixtures” Massoudi  provides an overview of the historical approach to the modeling of effective viscosities but more importantly, he points to the mathematically rigorous approach of modeling a mixture by examining the stress tensors. However, this is recognized to be a rather formidable challenge to verify experimentally and very tedious to implement in computations such as these. Even within this approach of modeling viscosities, it has been argued  that the effective viscosity of say two liquids would be not just a linear superposition of the two individual viscosities, but also include an interaction term, referred to “mutual viscosity” which again is very difficult to verify experimentally. All these point to the complexity of rigorously modeling even a seemingly simple parameter as viscosity. In the rest of the paper, we consider two specific and successful models applied in the engineering literature to our problem of Aβ aggregation. With regards to the shape of the molecules, there has been much discussed about the validity of spherical-shape assumption of biomolecules( [17, 18] and references therein) and while it is recognized that molecular shapes can be fairly complex and random, it still remains worthwhile to represent these arbitrarily composed molecules using some standard shapes that can be imagined to envelope the molecules in a reasonable manner. Our attempt in particular is to go beyond the assumption of the sphere. To do so we need at our disposal, the drag coefficients corresponding to arbitrarily shaped bodies. This is a very daunting challenge in fluid mechanics and at best relatively simpler shapes such as spheres, oblate and prolate spheroids (of any eccentricity) have been analyzed mathematically [26, 27]. However there is some literature in terms of empirical or computational relations which provide the drag coefficients for other shapes.
Case 1: a sphere-rod empirical model
We begin with an empirical model. Based upon the work of [28–31] we will consider the modification of the SE law by taking a suspension composing of spheres and rods with the former particles being considerably smaller than the latter such that the ratio a r /a s > 20, where a r is the major axis of the rod and a s is the radius of the sphere. This particular assumption can also be very meaningful in our case where we consider our sample primarily to be composed of smaller monomers and much larger protofibrils. In particular we follow the specific approach of  in obtaining an effective viscosity for such a suspension. In the words of  the primary assumption in the computation of viscosity is that:
... for each fraction of a given particle size, the smaller particles in suspension have the same effect as a homogeneous fluid with Newtonian viscosity similar to the effective viscosity of a suspension made up of the fraction of smaller spheres. In other words, the smaller suspended particles do not interact with the larger particles and are ’sensed’ by the large particles as part of the continuous suspending fluid.
If the net volume fraction of all the solute molecules is held fixed (i.e. Φ), then the diffusion coefficient can be written purely in terms of ϕ r or ϕ s .
Case 2: a mixture theory approach
where as in the sphere-rod model treated above, η i (i ≠ 0) where i ranges over all cyclic permutations of (1, …, j, j + 1, …,n), represents the relative viscosity of the suspension containing the j th solute in a “solvent” composed of the liquid of viscosity η0and all other solute particles. Note that the equation (16) is not unique. Other formulations of the effective viscosity are possible. The one provided here seems mathematically and physically reasonable and consistent with some models provided in the literature. The term represents the relative volume fraction of the j’th solute relative to the “solvent” that contains it and is independent for different j’s. Therefore, for an n-species suspension, the viscosity relation for the j solute in a background containing all other species and the pure solvent can be inferred from the algorithm: η j → η j +1→ …η n → η1→ …η j –1 where η a → η b indicates that η a is measured relative to η b in the sense of equation (8). In particular, the cyclic permutations of i are taken to ensure evenness in the definition of the volume fractions, for all j’s. This is clearly illustrated in the cases of 2 and 3-species which are examined in detail. We illustrate this by means of a few examples.
(i) The case of two species
(ii) The case of three species
where η e is defined by the equation (16) and where i, j ≠ 0 that is, the fluid is discounted. Since the diffusion that we are interested in only pertains to the solute species embedded in the surrounding solvent, we need only consider the volume fraction of each solute with respect to the others in the evaluation of the effective diffusion. In particular let us assume, for instance, that two classes (or species) or molecules populate the solvent; the shape are so selected since the shape factors are only known for a small class of shapes. The first are the molecules at the protofibril stage or beyond which are treated as being spheroidal in shape(diffusion of spheroidal bodies had been considered as far back as 1936 by Perrin ) and the second class are those smaller than the protofibril stage which will be treated as spheres, even though Zwanzig and Harrison  rightly argue that the shapes of molecules are not correctly described by either of these shapes, the latter being far more realistic than the former. We however feel that this is still an improvement over the current approach of treating all the molecules as hard spheres.
where ϕ(1) = ϕ. As ∈ → 0, the diffusion coefficient approaches the classical Stokes Einstein equation since then ϕ0→ 1.
Protofibril-protofibril lateral association experiment
Molecular level simulation of protofibril-protofibril lateral association
Next, we used some simple molecular level reaction models to predict the concentration of protofibrils of different size in the lateral association stage that can correspond to the diffusion coefficient estimates from DLS reported above. First, we assumed that the protofibrils are composed of 1600 monomers based on the previous reports [39, 40]. Here, F1600 denote the protofibrils (comprising 1600-mers and average length 64 nm) [39, 40]. In general, we will let F i denote a protofibril with i number of Aβ molecules binding to it during the lateral association phase.
We present separate reaction models considering different number of species formed during the lateral association reaction. The reactions involving these species were modeled under the assumption of a homogeneous mixture of protofibrils (with different initial concentrations) as used in the in vitro experiments. In particular, we considered a maximum of five different molecular species involved in this phase as follows: F1600, F3200, F4800 , F6400 and F8000. We have assumed that lateral association will result in at most a 5-fold increase in size of the protofibrils within the first 1.5 hrs of observation such that the system of reactions will not involve the formation of protofibrils beyond F8000. This assumption is purely theoretical although it is experimentally well known that three protofibrils associate together to form mature fibrils. We wanted to consider upto 5 species for the interest of modeling and calculations. The reaction fluxes in this are denoted by R(i, j) corresponding to the reaction between F i *1600 and F j *1600 forming F( i + j )*1600 for i = 1, …, 5 and j = i, i + 1,…, 5 – i. We have also assumed that the initial concentration of protofibrils comprise of only F1600’s and that of the other species are zero in the reaction models as only minute levels of associated protofibrils are expected to be present initially in the in vitro system.
In the following, we have considered three case studies to estimate the concentration change of the different molecular species involved in lateral association: 2-species (F1600, F3200), 3-species (F1600, F3200, F4800) and 5 species (F1600, F3200, F4800, F6400, F8000). Each reaction was considered reversible with k+ and k– being the forward and backward rate constants respectively. Following our previously validated lateral association stage model , we have considered k+ = 9.0 × 10–1h–1mM–1 and k– = 6.0 × 10–3h–1 respectively. While the 2-species and 3-species models are less realistic, we have seen that protofibrils do not grow beyond F8000 in the first 1.5 hrs of lateral association, and hence the 5-species model should be the best approximation of the in vitro system under study.
Case 1: 2-species reaction model
Case 2: 3-species reaction model
Case 3: 5-speciesreaction model
After the kinetic schemes are established in the reaction models above, we computed the corresponding reaction fluxes, and hence the differential equations that govern the temporal change of these species can be derived from material balances and reaction kinetics. The reaction flux and differential equations for each of the models reported here have also been shown alongside. The initial concentration of protofibrils ([F1600]) is set to 1µ M, 3µ M and 5µ M respectively following the in vitro experiments while the concentrations of the other species are assumed to be zero at the start (i.e., [F i *1600] = 0, i = 2,…, 5). Thus, the system of differential equations for each model is properly defined and were solved using Matlab’s ODE toolbox to estimate the concentration of the different species at various time points between 0 – 90 mins as shown in the previous section.
In this section, we consider some special cases to test our proposed models. We consider the cases of two, three and five species embedded in a solvent, introduced earlier. In these examples, we consider each species to be an ellipsoid since the length and width of the molecules considered here are significantly different so they cannot be correctly represented as spheres. As is indicated in Figure 3(c), associations of these spheroids occur laterally; therefore according to our model, the result of n-associations of spheroids of major axis c and minor axis b results in a new spheroid with major axis c and minor axis n × b. Our objective is to employ our model to estimate the effective diffusion coefficient of various systems and compare them to the experimentally measured values of diffusion.
Certain features of the computations merit immediate attention. In particular, the profile of the 1600, 3200 and 8000mers is striking. These include the declining values of the apparent intersection point with increasing number of species and also the slope of the red curves whose declining trend in the graph as plotted, gets more rapid with the larger molecule.
Discussion and future work
where i = 1,2,…,n. The new k e renders the equation highly nonlinear. The modified SE equation provided in this paper now allows for a more appropriate coupling of the solute-solvent dynamics. Under the realistic conditions of flow we have a two way coupling; the dynamics of the solute is described by the concentration equation (35), while the dynamics of the solvent would be given by the Navier-Stokes equation. Our next goal is to take on this multiscale approach coupled with a reaction term to understand the temporal and spatial variations in the solute (protofibrils and aggregates) distribution. Due to the contributions of this paper, this would be a very new and rigorous addition to the biophysics literature. The model proposed here, therefore merits serious attention since it provides a systemic treatment of the biophysics involved. The current paper is therefore a first step in this long term approach of understanding the problem of protein aggregation thoroughly and rigorously.
In this report, we have shown that the use of classical Stokes-Einstein equation may not be accurate for the measurement of hydrodynamic radii and consequently molecular weights of amyloid aggregates. Although it is known that appropriate assumption of the shape of solute molecules is essential for accurate size determination while using Stokes-Einstein equation, this is the first time we have shown that it is incorrect to assume solvent viscosity be a constant during the reaction. This is particularly true for reactions involving Aβ protofibrils, as large size of the solute molecules indicate significant effect on bulk solvent viscosity. We have attempted to rectify the problem by incorporating modifications to the Stokes-Einstein equation (Equation 16)that seem to be valid for the protofibril lateral association reaction. It is noteworthy that the modifications reflecting the viscosity changes can lead to realistic models not only for Aβ aggregation but also for a majority of amyloid aggregation systems.
Aβ protofibril preparation
Synthetic Aβ 42 peptide was obtained from synthesis facility at Mayo Clinic, Rochester, MN as a lyophilized powder. Aβ 42 protofibrils were generated and isolated as previously reported (9). Freshly purified Aβ 42 monomers (100 µ M) from size exclusion chromatography buffered in 10 mM Tris, 50 mM NaCl, pH 8.0 was agitated at room temperature for 48 h. The aggregation was monitored using ThT fluorescence. The sample was then centrifuged at 19000g for 12 min to spin out any fibril that may have formed. The supernatant was then fractionated by Superdex-75 size exclusion column to isolate protofibrils from unreacted monomers and smaller oligomers. The concentration of protofibrils was measured by UV-Vis with a molar extinction coefficient of 1450 cm–1M–1 corresponding to Aβ 42.
Dynamic light scattering (DLS)
The DLS experiments were performed using Zetasizer nanoseries instrument ( Malvern Instruments, Inc.). The samples were equilibrated for 120s and a total of 90 measurements of 10s each were made with a time interval of 1min after each measurement. The data was exported as diffusion coefficient using Zetasizer software version 6.20 and processed using origin 6.0 software.
Reaction flux and differential equations for the reaction pathway
The initial concentration of F1600s is equal to the amount of protofibrils added initially and is the main driving force for the downstream reactions. The concentrations of the other species are assumed to be zero at the start (i.e., [F i ×1600] = 0, i = 1, …, 5). This set of differential equations is properly defined and can be solved using Matlab’s ode15s.
The study was conceptualized by PG,VR and AV. PG designed the computational model for the reaction pathway and VG designed the ThT fluorescence and DLS experiments. AV,SA and BJC performed the mathematical modeling and the related calculations. All authors read and approved the final manuscript.
The authors thank the following agencies for their financial support: NSF EAGER(1049962) and Mississippi INBRE (P20RR016476) (for PG and VR); American Heart Association (10GRNT4190124) (for VR). The authors thank Amit Kumar for his help with biophysical experiments and Dr. Cannon for letting us use the DLS instrument.
This article has been published as part of BMC Bioinformatics Volume 12 Supplement 10, 2011: Proceedings of the Eighth Annual MCBIOS Conference. Computational Biology and Bioinformatics for a New Decade. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2105/12?issue=S10.
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