Amyloid fibrils are associated with a number of pathologies including Alzheimer's, Parkinson's, Huntington's, prion disease, and type II diabetes (Horwich and Weissman 1997; Kelly 1998; Dobson 1999; Rochet and Lansbury 2000). Therefore it is of fundamental medical interest to understand the mechanisms of fibrillogenesis, with the ultimate goal of designing inhibitors. One important and still unanswered question regarding amyloid fibril formation is the specificity with which the amino acid sequence determines β-aggregation propensity and the atomic details of the fibril structure. Because of the difficulties in obtaining detailed structural information by X-ray crystallography or solution phase NMR spectroscopy, computational approaches are needed to guide experiments, e.g., to determine short segments of amyloid-like proteins that share the same biophysical properties of the full-length proteins (Balbirnie et al. 2001) and identify those elements which are essential for the formation of protein fibrils (Tenidis et al. 2000; von Bergen et al. 2000). As aggregation conditions vary sensibly with the composition and especially the sequence of the polypeptide, single amino acid substitutions have been used to investigate the fibril formation (Chiti et al. 1999), and complementary theoretical studies proposed relative rate equations to predict the change of aggregation rate upon mutation (Chiti et al. 2003; Tartaglia et al. 2004). Although the application of relative rate equations shows high correlation with experimental data, these models require the a priori knowledge of wild-type aggregation rates.
We report here an absolute rate equation derived from both first principles and analysis of aggregating sequences designed by a computational approach. The latter is based on a genetic algorithm optimization in sequence space and molecular dynamics sampling of conformation space. The equation does not need any information except the amino acid sequence and two environmental factors (i.e., temperature and concentration). Our model gives both the aggregation rate and the “amyloid spectrum” of a protein, identifying those segments involved in β-aggregation. In addition, the model distinguishes between the parallel and anti-parallel β-sheet organization within the fibrils and shows that mammalian and nonmammalian prion proteins have different amyloid spectra.
Results and Discussion
Absolute rate prediction
Predicted and experimentally measured rates are shown in logarithmic scale in Figure 1. The correlation is 95% and extends over 90 data points and about 15 natural logarithmic units. This is a remarkable result considering that the rate is calculated solely from the primary structure with the addition of two external factors, i.e., temperature and concentration. Interestingly, the correlation is good for different proteins and also within mutants of the same protein. For single-point mutants of long sequences (Acylphosphatase and Titin), the error is rather large because of the poor signal-to-noise ratio due to the average over the entire sequence. The model was subjected to statistical tests to assess the chance correlation. In Figure 2A, the experimentally measured rates were randomly permutated to generate about 107 “scrambled” data sets. The calculated rates were fitted to each scrambled set, giving an extremely small likelihood for high correlations. In Figure 2B, ∼107 data sets were randomly generated within the range of experimental rates. The predictive ability andcorrelation of the modelare much higher than the corresponding values obtained upon randomization of the experimental rates. These statistical tests show that chance correlation is not present.
Prediction of β-aggregating segments
There is in vivo evidence that amyloid fibrils originate from misfunctions of the degradation machinery and cleavage of fragments that have high propensity for β-aggregation (Stefani and Dobson 2003). Moreover, even proteins not implicated in amyloid diseases were recently found to form amyloid fibrils in vitro under denaturing conditions, indicating that fibrillogenesis is a common feature of proteins (Chiti et al. 1999; Dobson 1999; Stefani and Dobson 2003). Our approach to estimate aggregation rates can be also used to identify segments with high aggregation propensity. The method is tested on the following proteins: α-synuclein, apolipoprotein, amyloid precursor protein (APP), gelsolin, islet amyloid precursor protein (IAPP), lactadherin, prion, serum amyloid A, transthyretin, ABri, ADan, fibrinogen, β2-microglobulin, insulin, Sup35, and tau. The former nine proteins represent all hits of a combined search for “amyloid” and “human” at http://www.expasy.org (Gasteiger et al. 2003) in September 2004; the latter seven proteins result from a literature search (references are reported in Table 1). As indicated in Figure 3, the data set contains
regions known to promote aggregation
segments found to aggregate in vivo (often after degradation)
stretches extracted from the precursors and shown to aggregate in vitro
Each sequence in the data set is scanned by shifting a window of fixed size one residue at a time starting from the N terminus. The extracted stretches are ranked using the aggregation propensity π (see Materials and Methods). The procedure is repeated for different window sizes (3–25 amino acids), each time storing the positions of the three stretches having the highest π. These positions are then used to build the histogram of Figure 3. Peaks of the histogram represent positions of stretches with the highest β-aggregation propensity (“windows' consensus”). All the sequences except fibrinogen and prion show main peaks in segments known to promote aggregation. For prion, amyloidogenic areas are—up to now—not known and few experiments have been performed and on limited portions of the protein (Vanik et al. 2004). Following the protein-only hypothesis (Prusiner 1988; Soto and Castilla 2004), we suggest that the peak found at position 150 may be determinant for prion transmissions (in the subsection Prions, the same peak is numbered with 175 because of the alignment with other prion sequences). For transthyretin, only one of the two experimentally known β-aggregating fragments has been found with our analysis. We speculate that the corresponding area promotes the aggregation of the entire protein, which is consistent with NMR data (Jaroniec et al. 2002).
To furthertest the sensitivity of our model, we focused on the segments that are experimentally known to aggregate. For this purpose, we used a window size of five consecutive residues, as in a previous work (Fernandez Escamilla et al. 2004) (Table 1). Interestingly, several five-residue stretches are found in segments that were shown to aggregate, e.g., FGAIL contained in IAPP NFGAILSS, FILDL in gelsolin's SFNNGDCFILD, SVQFV in lacthaderin's NFGSVQFV, and YQQYN in Sup35's PQGGYQQYN (Azriel and Gazit 2001). For APP, three stretches are found in correspondence of the segment LVFFA, which is known to be involved in the aggregation of Aβ40 (Williams et al. 2004) (see subsection Amyloid Protein Precursor). Importantly, all the stretches are ranked among those having the highest π in the respective precursor proteins (see Table 1), which suggests that a small window size is sufficient for the identification of amyloidogenic regions. In Table 1, we also list β-aggregating segments that have not yet been investigated with experiments in vitro (e.g., YVDVL in apolipo-protein A–I and ENYCN in insulin) and indicate the predicted parallel or anti-parallel arrangement of the individual segments in the fibril.
Amyloid protein precursor
Using a window size of five residues, the amyloid spectrum of the 750-residue APP (Fig. 4) shows a predominant peak at position 671 for the stretch LVFFA. Furthermore, the predicted β-aggregating stretches AIIGL and IGLMV are consistent with solid-state NMR (Antzutkin et al. 2002; Bond et al. 2003) and scanning proline mutagenesis (Williams et al. 2004). The stretches with the highest rate for each window size in the range 3–25 are shown in Table 2 for Aβ42. Most of the high-aggregation stretches contain the segment LVFFA and are parallel. As in experiments (Gordon et al. 2004), the segment KLVFFAE has a preferential anti-parallel arrangement, while Aβ42 is parallel(Antzutkin et al. 2000; Torok et al. 2002). As shown in clinical reports and oligomerization experiments performed with photo-induced cross-linking of unmodified proteins (Bitan et al. 2003), we found that Aβ42 has a higher aggregation propensity than Aβ40 (ln πAβ42 = −7, ln πAβ40 = −). Interestingly, the experimental evidence indicates that the Ile41–Ala42 extension of the 1–40 segment affects the rate of amyloid formation rather than the fibril stability (Jarrett et al. 1993).
To further investigate the usefulness of our model, the amyloidogenic propensities of the prion protein from different organisms were evaluated using a moving window of five residues along the entire sequence. To compare the amyloid spectra, prion sequences have been aligned using ClustalW (Thompson et al. 1994). It is remarkable that prion sequences in mammals show a peak at position 175 corresponding to the segment SNQNN in human prion (Fig. 5; Table 3; all the notations used to number stretches refer to the major prion proteins, i.e., signal- and/or propeptides are omitted). Such a peak is absent in the chicken and the turtle. Interestingly, the peak is located in a glutamine/asparagine-rich region, which shows high propensity to self-propagate in amyloid fibrils (Michelitsch and Weissman 2000). Other peaks correspond to β-strand 2 (segment NQVYY, conserved in mammals and nonmammals and mutated in NRVYY in chicken) and helix 1 of human prion (segment YEDRY in mammals, WNENS in turtle, and WSENS in chicken), which are known to form ordered aggregates in vitro (Nguyen et al. 1995; Kozin et al. 2001). Furthermore, the amyloid profiles are similar within mammals (e.g., 97% correlation between man and cow) and different between mammals and nonmammals (e.g., 55% correlation between man and turtle).
To compare with experiments in vitro (Vanik et al. 2004), we analyzed the unstructured region of the prion protein (residues 1–122) in human, mouse, and hamster prion peptides. We found thathuman and mouse prions share similar amyloid spectra (i.e., 98% correlation), while the hamster prion diverges from them at position 143 (position 116 in the nonaligned human sequence). More specifically, the stretch 143–148 of hamster prion (position 116–121 in the nonaligned human sequence) is found to be less amyloidogenic than the corresponding segment in mouse and human (ln πhamster = −16, ln πmouse = −12, and ln πhuman = −12), which is consistent with the prion 1–122 species barrier observed in vitro (Vanik et al. 2004).
The gene for Huntington's disease consists of 67 hexons and contains an open reading frame for a polypeptide of > 3140 residues. Using a window size of five residues, our model identifies the N-terminal poly(Gln) repeat and the stretch IFFFL in the middle of the sequence as the two most prone to induce ordered aggregates. With window sizes larger than 20, the N-terminal poly(Gln) repeat dominates and the peak in the middle of the sequence disappears.
Our model is not sensitive enough to discriminate repeats of fewer than 38 glutamine residues from those with > 41 glutamine residues; the former are harmless, whereas the latter are responsible for toxic aggregates (Perutz et al. 1994; Perutz 1999). Alternatively, the dramatic difference in toxicity observed at a repeat length of ∼40 might require the context of a much longer polypeptide sequence.
The model presented here was motivated by the challenging tasks of predicting aggregation propensity and identifying β-aggregating stretches in polypeptide sequences. An essential element in the derivation of the equation was the analysis of a large pool of β-aggregating peptide sequences designed by a computational approach based on molecular dynamics and genetic algorithm optimization in sequence space (G.G. Tartaglia and A. Caflisch, in prep.). The very good correlation between calculated and experimental rates for a large and heterogeneous set of polypeptide chains has allowed us to use the model to successfully identify β-aggregating segments and predict the parallel or anti-parallel arrangement. Fibrils formed by short segments of a protein might have a different molecular structure than the fibril of the full-length protein. Yet our results, as well as previous experimental (Chiti et al. 1999, 2003; Balbirnie et al. 2001) and computational (Fernandez Escamilla et al. 2004) works by others, indicate that the amyloid-forming part of a protein could be only a short segment of the entire chain. That a function based on simple physicochemical principles is able to predict aggregation ratesand identify β-aggregating fragments in proteins might be a consequence of the essential role of side-chain interactions in β-sheet aggregates (Gazit 2002; Gsponer et al. 2003; Linding et al. 2004).
Although some of the physicochemical properties in our model are similar to those used in previous works by others, it is important to distinguish approaches based on parameter optimization for a multiterm equation (Chiti et al. 2003; DuBay et al. 2004) from first-principle models like the one of this work and that of Tartaglia et al. (2004). On a very similar test set of peptides and proteins, the multi-parameter approach gives results comparable to those obtained with our model, but it is likely to have a lower predictive ability. As an example, positional effects are taken into account in our model, whereas they are neglected in the multiparameter approach (DuBay et al. 2004), which is mainly based on amino acid composition and alternation of hydrophobic–hydrophilic residues (Broome and Hecht 2000). Recent scanning proline mutagenesis, combined with critical concentration analysis and NMR hydrogen–deuterium exchange, indicate a strong positional effect on both the aggregation kinetics and structural properties of the Aβ40 fibril (Williams et al. 2004). Most importantly, the multiparameter approach cannot be used to identify β-aggregating segments as explicitly mentioned by the investigators (DuBay et al. 2004).
Recently, an approach based on secondary structure propensity and estimation of desolvation penalty (TANGO) has been shown to accurately predict the sequence-dependent and mutational effects on the aggregation of a large data set of peptides and proteins (Fernandez Escamilla et al. 2004). TANGO is based on the assumption that the probability of finding > 2 ordered segments in the same polypeptide is negligible. The investigators report that TANGO allows quantitative comparison within the same polypeptide chain or mutants. On the other hand, only qualitative comparison between different polypeptide chains is possible with TANGO (Fernandez Escamilla et al. 2004), whereas our model allows for the prediction of absolute rates (Fig. 1).
In conclusion, we have identified the physicochemical properties of amino acids that are essential for ordered aggregation and proposed a model that takes into account sequence effects for aromatic and charged residues, as well as composition. Compared with the models previously published by others, our equation is the only one that takes explicitly into account π-stacking. Very recent high-resolution structural data (electron and X-ray diffraction) have provided strong evidence for the importance of aromatic side chains for amyloid formation (Makin et al. 2005).
Our model derived from first principles and analysis of in silico designed sequences is able to predict aggregation rates and identify β-aggregating segments with high accuracy, suggesting possible biological implications as in the prion protein case. For nonmammalian prions, the absence of the peak at position 175 observed in mammals decreases the overall aggregation propensity, indicating a species-specific behavior consistent with experiments (Marcotte and Eisenberg 1999; Matthews and Cooke 2003) and supporting the hypothesis of a species barrier in the transmission of the prion disease (Hill et al. 2000).
In the accompanying article we present a bioinformatics application of our model that reveals an anti-correlation between organism complexity and proteomic β-aggregation propensity (Tartaglia et al. 2005).
Materials and methods
Absolute rate equation
An equation based on physicochemical properties of natural amino acids is introduced to estimate the aggregation rate of proteins and identify β-aggregating segments. Aromaticity, β-propensity, and formal charges play a major role in our model, as they are known in the literature to be determinant for fibrillization (Gazit 2002; Tjernberg et al. 2002; Chiti et al. 2003). Polar and nonpolar surfaces, as well as solubility, are also taken into account following an analysis of sequences designed to aggregate into β-sheets. The design of β-aggregating sequences was performed by structural sampling using molecular dynamics and peptide sequence optimization by a genetic algorithm (Tartaglia et al. 2004; G.G. Tartaglia and A. Caflisch, in prep.) (see subsection Derivation of the Equation). The aggregation propensity πil of an l-residue segment starting at position i in the sequence is evaluated as:
The factor Φil contains exponential functions and is position-dependent
where Ail, Bil, and Cil are functionals related to the aromaticity, β-propensity, and charge, respectively. The factor ϕil depends almost exclusively on the amino acid composition
where Sai, Spi, Sti, and σi—weighted by their average over the 20 standard amino acids (hatted values)—are the side-chain apolar, polar, total water-accessible surface area, and solubility, respectively (see subsection Parallel and Anti-Parallel Configuration). The functionals θ↑↑ and θ↑↓ include positional effects and reflect the parallel or anti-parallel tendency to aggregate if the majority of residues is apolar or polar, respectively. Considering the high correlation between measured and predicted changes in aggregation rate upon single point mutations (Chiti et al. 2003; DuBay et al. 2004; Tartaglia et al. 2004), it is possible to utilize the propensity πil to predict the absolute rate νil
where α(c,T) is introduced to take into account concentration and temperature (see subsection Concentration and Temperature).
Parallel and anti-parallel configuration
The functional for the parallel or anti-parallel configuration was introduced following the analysis of sequences designed by genetic algorithm optimization (see subsection Derivation of the Equation; Fig. 6):
The parallel in-register β-sheet organization within fibrils is favored by the number of side chains involved in π-stacking (Tyr, Phe, and Trp) and apolar interactions (Ala, Gly, Ile, Leu, Met, Pro, and Val) (McGaughey et al. 1998; Azriel and Gazit 2001; Jenkins and Pickersgill 2001; Makin et al. 2005). The number of aromatic and apolar residues is indicated with naromatic and napolar, respectively. Hydrogen bonds between polar residues are not considered for the parallel aggregation because the number of polar residues decreases significantly during the optimization of parallel aggregated sequences (Fig. 6).
The anti-parallel configuration is mainly determined by the electric dipole moment of the polypeptide (Hwang et al. 2004). Sequences abounding in polar residues show a small tendency for the parallel in-register aggregation because of unfavorable dipole–dipole interactions between side chains. Hence, the anti-parallel organization is promoted by the number of polar residues (Arg, Asn, Asp, Cys, Gln, Glu, His, Lys, Ser, and Thr), which is indicated with npolar. In some specific positions, charged (Arg, Lys, Asp, and Glu) and aromatic amino acids contribute to the anti-parallel aggregation. “Specific positions” means that one or more couples of opposite charged residues or one or more aromatic residues are symmetrically placed with respect to the center of the sequence (Balbach et al. 2000; Hwang et al. 2004; Makin et al. 2005). In this specific case, the number of charged and aromatic residues is labeled as nscharge and nsaromatic, respectively.
In Equation 3, a parallel configuration is preferred if napolar + naromatic > npolar + nscharge + nsaromatic. Since the number of aromatic residues in symmetric position is always smaller than the total amount of aromatic residues, naromatic ≥ nsaromatic (e.g., in the APP stretch: LVFFA naromatic = 2, nsaromatic = 1), we used a stricter condition for the parallel arrangement napolar > npolar + nscharge. The stricter condition allows the factorization of aromatic contributions in Equation 1. In the ϕil factor of Equation 3, θ↑↑ and θ↑↓ are
It is useful to explain the effect of the θ↑↑ and θ↑↓ functional by some examples. The segment LVFFA at position 671–676 of the APP is predicted to be parallel because it satisfies the parallel condition napolar > npolar + nscharge with napolar = 3 and npolar = nscharge = 0 (θ↑↑ = 1). The segment KLVFFAE (at position 670–677 of the APP), with two opposite charged residues, has anti-parallel propensity because it satisfies the anti-parallel condition napolar < npolar + nscharge with napolar = 3 and npolar = nscharge = 2 (θ↑↓ = 1).
The IAPP stretch FGAIL at position 22–26 is predicted to be parallel (napolar = 4 and npolar = nscharge = 0, i.e., θ↑↑ = 1), in agreement with experimental results (Kayed et al. 1999a; Azriel and Gazit 2001; Gazit 2002). As in Azriel and Gazit (2001), the following stretches are predicted to be parallel: SVQFV at position 289–292 of lactadherin; DCFIL, CFILD, and FILDL at position 187–191, 188–192, and 189–193 of gelsolin, respectively; FFSFL, FSFLG, and SFLGE at position 3–7, 4–8, and 5–9 of serum amyloid, respectively.
Poly(Gln), poly(Asn), and poly(Lys) homopolymers are predicted to be in an anti-parallel arrangement, as proposed in Perutz et al. (1994), Scherzinger et al. (1997), and Michelitsch and Weissman (2000) and observed by Tanaka et al. (2001) and Dzwolak et al. (2004). Moreover, it is likely that completely aliphatic sequences result in amorphous aggregates if N and C termini are capped, while a tendency to the anti-parallel arrangement is expected for short stretches with charged termini (e.g., transthyretin's stretch IAALL). Capping groups are neglected in the present version of the model.
The fragment GNNQQNY from the Sup35 yeast prion is predicted to be anti-parallel (napolar = 1, npolar = 5, and nscharge = 0, i.e., θ↑↓ = 1), in contrast with the parallel packing suggested on the basis of X-ray diffraction and Fourier transform infrared (FTIR) data (Balbirnie et al. 2001). On one hand, it is important to note that the experimental data supporting a parallel arrangement are not conclusive, and, in particular, FTIR can be misleading on this point. In fact, in the unit cell of the microcrystals, the parallel β-sheets are proposed to be in anti-parallel contact along the fibril axis. On the other hand, a possible reason for the parallel configuration is that the π-interactions between the Tyr side chains are much less favorable in the anti-parallel configuration.
Aromatic side chains contribute to the parallel aggregation with π-interactions (McGaughey et al. 1998; Azriel and Gazit 2001; Makin et al. 2005). The density of aromatic residues naromatic/l is used to distinguish two regimes for the aromatic contribution Ail of Equation 2:
where 3/20 is the aromatic density averaged over the 20 standard amino acids and naromatic was defined in the previous subsection. In the case of low aromatic density (naromatic/l≤3/20), Aillow takes into account the polar/apolar environment. Following the results obtained by the genetic algorithm optimization of β-aggregation-prone sequences (see Fig. 6), Aillow has a positive effect for mainly apolar sequences and a negative contribution for mainly polar sequences:
The variables napolar, npolar, and nscharge are defined in the previous subsection.
As an example, the APP stretch LVFFAEDVGSNKGAIIGLMVGGVVI shows low aromatic density (naromatic/l = 2/25 < 3/20). Since i = 671, l = 25, napolar = 17, npolar = 6, and nscharge = 0, the aromatic contribution for LVFFAEDVGSNKGAIIGLMVGGVVI is Alow671 25 = 2 [17 − 6] 25−1 = 0.88.
In the case of a high aromatic density (naromatic/l > 3/20), the model takes into account the number of aromatic residues:
As an example, the APP stretch LVFFA shows high aromatic density (naromatic/l = 2/5 > 3/20). Since i = 671 and l = 5, the aromatic contribution for LVFFA is Ahigh6715 = 2.
Besides the total amount of aromatic residues and the position dependence, which enters Equation 2 through Aillow, the different polar and apolar side-chain surfaces, solubility, and β-propensity of Phe, Tyr, and Trp are taken into account in the factor ϕil. Hence, the mutation F22Y for the IAPP (islet β-amyloid protein precursor) stretch NFGAILSS produces a sensible change of rate (ln πwt = −6,ln πF22Y = −7), compatible with experiments in vitro (Porat et al. 2003).
The β-propensity is evaluated as the fraction of residues that stabilize the β-sheet more than the α-helix:
The function βil is defined as:
The variables αj and βj correspond to the α-helix and β-sheet stabilizing effects of the amino acid at position j (Fersht 1999). Values of αj and βj are normalized from 0 (low stabilization) to 1 (high stabilization) to have the same range of variability. In the function Bil, the offset value of 1/2 is introduced so that Bil > 0 if at least one-half of the residues in the sequence is more stable in a β-sheet rather than in an β-helix conformation (i.e., βil > l−1/2).
In the case of the APP stretch LVFFA, values are i = 671, l = 5, β672 = β673 = β674 = 1, and β671 = β675 = 0. The predicted β-propensity for LVFFA is β671 5 = 3/5 − 1/2 = 0.1.
As in other models, we consider that the electrostatic repulsion of charged sequences penalizes the aggregation (Chiti et al. 2003; Tartaglia et al. 2004). In addition, our model takes into account the fact that side-chain pairs with opposite charges and positioned symmetrically with respect to the center of the segment contribute to the anti-parallel aggregation, as found in experiments (Gordon et al. 2004). In Equation 2, the charge contribution Cil is
where Cj is the charge of the side chain and ncharge is the number of charged residues. The first term of the functional Cil takes into account the electrostatic repulsion between polypeptides with net charge different from zero. The second term counts the number of pairs of opposite charged side chains that are symmetrically placed with respect to the central residue of the sequence:
In the case of the APP stretch KLVFFAE, the residues K670 and E676 have opposite charges and are symmetrically placed with respect to the central amino acid F673. Since i = 670, l = 7, C670 = +1, and C1340 + 7 − 670 − 1 = C676 = −1, the net charge for KLVFFAE is |∑i+l−1j=iCj| = 0 and the oppositely charged K670 and E676 give C670 7 = δcharge670 + δcharge676 = 2.
Surfaces and solubility
For sequences that are predominantly apolar (θ↑↑ = 1; see subsection Parallel and Anti-Parallel Configuration), the apolar water-accessible surface Saj measures the contribution of hydrophobic side chains to aggregation. For mostly polar sequences (θ↑↓ = 1), the polar water-accessible surface Spj takes into account the propensity to form hydrogen bonds between polar residues. The total surface Sjt = Saj + Spj is used to weight polar and apolar surfaces by the total area. Values of apolar and polar side-chain surfaces are given in our previous work (Tartaglia et al. 2004) and span the intervals 44–195 Å2 and 27–107 Å2, respectively. Averaged values are Ŝa = 108 Å2 and Ŝp = 54 Å2. In the case of poly(Gln), values of surfaces are Sa = 53 Å2 and Sp = 91 Å2. Since Gln is polar and θ↑↓ = 1, the surface contribution is Sp/Ŝp•Ŝt/St = (91/54)(162/144) = 1.9.
The variable σj takes into account the water solubility of the side chain at position j. In our model, aggregation propensity and solubility are inversely proportional to introduce a penalty for highly soluble polypeptides. Most of the solubility values are available at http://acrux.igh.cnrs.fr/proteomics/densities_pi.html (Nahway 1989). The missing values (Cys, Lys, and Thr) were taken from http://www.formedium.com/Europe/amino_acids_and_vitamins.htm. The variable σj spans the interval 0.04–162 g/100 g, with average = 3.95 g/100 g. In the case of poly(Gln), /σ = 3.95/2.5 = 1.5, which indicates low solubility in agreement with experiments of β-aggregation (Perutz et al. 1994; Perutz 1999).
Concentration and temperature
The function α(c,T) captures the effects of concentration (c) and temperature (T) in Equation 4:
The aggregation rate ν is approximated to be proportional to the temperature because the probability of collision and elongation of peptides increases with temperature (Kusumoto et al. 1998). Although aggregation rate and temperature are not expected to correlate above physiological values (Massi and Straub 2001), we used a simple linear dependence, which is preferable for the small extent of experimentally accessible values of the temperature. In fact, the temperature ranges from 298 K to 310 K in the data set of Figure 1.
In agreement with quasielastic light-scattering experiments of fibrillogenesis of Aβ40, the aggregation rate ν is assumed to be proportional to the concentration for c < c* mM (c* = 0.1 mM) and to be independent of concentration above the critical value c = c* (Lomakin et al. 1996, 1997) (see also subsection-Derivation of the Equation). The hyperbolic function 1/c was introduced to decrease the aggregation rate ν for c > 1 mM, as there is experimental evidence that a very high concentration opposes formation of ordered aggregates (Munishkina et al. 2004). The concentration ranges from 0.01 mM to 20 mM in the data set of Figure 1.
Derivation of the equation
Functionals for aromaticity, β-propensity, and charge were taken from our relative rate equation (Tartaglia et al. 2004). The aromatic term was modified according to the results obtained by the genetic algorithm optimization of aggregating sequences (Fig. 6) (G.G. Tartaglia and A. Caflisch, in prep.). The functional for β-propensity, previously based on a single scale (Tartaglia et al. 2004), now takes into account β-versus α-propensity. Scales for β- and α-propensity are taken from Fersht (1999) and normalized in the range 0–1. The term used for the β-propensity was tested on 100 globular proteins: 82% of the β-sheet content is successfully recognized (data not shown). The functional for charged residues was modified with the addition of a term for symmetrically placed charges of opposite signs, which is consistent with experimental data (Gordon et al. 2004). The function ncharge/l replaces the constant factor in the relative rate (Tartaglia et al. 2004) and is introduced to weight the overall charge by the charge density. The three functionals for aromaticity, charge, and β-propensity can be zero. Exponential functions were introduced so that their product is different from zero.
The product of the three functionals was plotted versus available experimental rates (see next subsection), obtaining a correlation of 80%, while the individual correlations for aromaticity, charge, and β-propensity are 76%, 81%, and 70%, respectively.
The dependence on concentration and temperature was introduced to derive aggregationrates from propensities (Lomakin et al. 1997; Kusumoto et al. 1998; Massi and Straub 2001; Munishkina et al. 2004). With the concentration alone, the correlation improves to 85%. The correlation is 82% without the hyperbolic function for high concentrations (c > 1 mM). With the temperature function, the correlation improves to 88%.
The factor for polar/apolar contributions ϕil in Equation 1 was added upon the analysis of sequences produced by computational design (Fig. 6). The term is a linear combination of normalized surfaces and has a nonzero minimum. The correlation improves to 92%. The solubility dependence was added at the very end and introduces a penalty for highly soluble sequences. The correlation improves to 95%.
Most of the experimental rates were kindly provided by Dr. F. Chiti and Dr. M. Vendruscolo (Chiti et al. 2003; DuBay et al. 2004). The remainder data set was taken from previous experimental studies (Litvinovich et al. 1998; Konno et al. 1999; Ferguson et al. 2003). The absolute aggregation rates were determined from in vitro experiments of denaturated polypeptide chains without taking into account the presence of cellular components as chaperones and proteases. Aggregation rates were obtained from kinetic traces in different ways: thioflavin T fluorescence, turbidity, CD, sedimentation, size exclusion chromatography, and filtration. Lag phases were not considered in the analysis, as they were not reported or difficult to extract from published data (DuBay et al. 2004). Since a comprehensive understanding of lag phases in protein aggregation is lacking (Kayed et al. 1999b; Padrick and Miranker 2002) (e.g., it is not known whether fibrils form by addition of monomers or oligomers and how growth conditions influence the amyloid formation), the aggregation kinetics was analyzed after the lag phase. The elongation phase showing an exponential behavior is fitted to the function z = α (1 − e−νt) where ν is the rate measured in sec−1.
We thank Prof. C. Dobson, Prof. F. Chiti, Dr. M. Vendruscolo, and Dr. J. Zurdo for providing rates of several proteins. The molecular dynamics simulations were performed on the Matterhorn Beowulf cluster at the Informatikdienste at the University of Zurich. We thank C. Bolliger, Dr. T. Steenbock, and Dr. A. Godknecht for setting up and maintaining the cluster. This work was supported by the Swiss National Science Foundation and the NCCR “Neural Plasticity and Repair.”
The program for the calculation of aggregation rates is available from the corresponding author upon request.