Publications

Recent experiments have shown that the congener Abeta(1-40)[D23-K28], in which the side chains of charged residues Asp23 and Lys28 are linked by a lactam bridge, forms amyloid fibrils that are structurally similar to the wild type (WT) Abeta peptide, but at a rate that is nearly 1000 times faster. We used all atom molecular dynamics simulations in explicit water, and two force fields, of the WT dimer, a monomer with the lactam bridge (Abeta(10-35)-lactam[D23-K28]), and the monomer and dimers with harmonically constrained D23-K28 salt bridge (Abeta(10-35)[D23-K28]) to understand the origin of the enhanced fibril rate formation. The simulations show that the assembly competent fibril-like monomer (N*) structure, which is present among the conformations sampled by the isolated monomer, with strand conformations in the residues spanning the N and C termini and a bend involving residues D(23) VGSNKG(29), are populated to a much greater extent in Abeta(10-35)[D23-K28] and Abeta(10-35)-lactam[D23-K28] than in the WT, which has negligible probability of forming N*. The salt bridge in N* of Abeta(10-35)[D23-K28], whose topology is similar to that found in the fibril, is hydrated. The reduction in the free energy barrier to fibril formation in Abeta(10-35)[D23-K28] and in Abeta(10-35)-lactam[D23-K28], compared to the WT, arises largely due to entropic restriction which enables the bend formation. A decrease in the entropy of the unfolded state and the lesser penalty for conformational rearrangement including the formation of the salt bridge in Abeta peptides with D23-K28 constraint results in a reduction in the kinetic barrier in the Abeta(1-40)-lactam[D23-K28] congener compared to the WT. The decrease in the barrier, which is related to the free energy cost of forming a bend, is estimated to be in the range (4-7)k(B)T. Although a number of factors determine the growth of fibrils, the decrease in the free energy barrier, relative to the WT, to N* formation is a major factor in the rate enhancement in the fibril formation of Abeta(1-40)[D23-K28] congener. Qualitatively similar results were obtained using simulations of Abeta(9-40) peptides and various constructs related to the Abeta(10-35) systems that were probed using OPLS and CHARMM force fields. We hypothesize that mutations or other constraints that preferentially enhance the population of the N* species would speed up aggregation rates. Conversely, ligands that lock it in the fibril-like N* structure would prevent amyloid formation.

Experiments show that for many two-state folders the free energy of the native state, DeltaG(ND)([C]), changes linearly as the denaturant concentration, [C], is varied. The slope {m = [dDeltaG(ND)([C])]/(d[C])}, is nearly constant. According to the transfer model, the m-value is associated with the difference in the surface area between the native (N) and denatured (D) state, which should be a function of DeltaR(g)(2), the difference in the square of the radius of gyration between the D and N states. Single-molecule experiments show that the R(g) of the structurally heterogeneous denatured state undergoes an equilibrium collapse transition as [C] decreases, which implies m also should be [C]-dependent. We resolve the conundrum between constant m-values and [C]-dependent changes in R(g) using molecular simulations of a coarse-grained representation of protein L, and the molecular transfer model, for which the equilibrium folding can be accurately calculated as a function of denaturant (urea) concentration. In agreement with experiment, we find that over a large range of denaturant concentration (>3 M) the m-value is a constant, whereas under strongly renaturing conditions (<3 M), it depends on [C]. The m-value is a constant above [C] > 3 M because the [C]-dependent changes in the surface area of the backbone groups, which make the largest contribution to m, are relatively narrow in the denatured state. The burial of the backbone and hydrophobic side chains gives rise to substantial surface area changes below [C] < 3 M, leading to collapse in the denatured state of protein L. Dissection of the contribution of various amino acids to the total surface area change with [C] shows that both the sequence context and residual structure are important. There are [C]-dependent variations in the surface area for chemically identical groups such as the backbone or Ala. Consequently, the midpoints of transition of individual residues vary significantly (which we call the Holtzer effect) even though global folding can be described as an all-or-none transition. The collapse is specific in nature, resulting in the formation of compact structures with appreciable populations of nativelike secondary structural elements. The collapse transition is driven by the loss of favorable residue-solvent interactions and a concomitant increase in the strength of intrapeptide interactions with a decreasing [C]. The strength of these interactions is nonuniformly distributed throughout the structure of protein L. Certain secondary structure elements have stronger [C]-dependent interactions than others in the denatured state.

We develop an analytical method for studying the properties of a noninteracting wormlike chain (WLC) in confined geometries. The mean-field-like theory replaces the rigid constraints of confinement with average constraints, thus allowing us to develop a tractable method for treating a WLC wrapped on the surface of a sphere, and fully encapsulated within it. The efficacy of the theory is established by reproducing the exact correlation functions for a WLC confined to the surface of a sphere. In addition, the coefficients in the free energy are exactly calculated. We also describe the behavior of a surface-confined chain under external tension that is relevant for single molecule experiments on histone-DNA complexes. The force-extension curves display spatial oscillations, and the extension of the chain, whose maximum value is bounded by the sphere diameter, scales as f(-1) at large forces, in contrast to the unconfined chain that approaches the contour length as f(-1/2). A WLC encapsulated in a sphere, that is relevant for the study of the viral encapsulation of DNA, can also be treated using the mean-field approach. The predictions of the theory for various correlation functions are in excellent agreement with Langevin simulations. We find that strongly confined chains are highly structured by examining the correlations using a local winding axis. The predicted pressure of the system is in excellent agreement with simulations but, as is known, is significantly lower than the pressures seen for DNA packaged in viral capsids.

Abeta peptide is an essential protein in the pathogenesis of Alzheimer's disease and is derived from amyloid precursor protein (APP) in the membrane by beta- and gamma-secretase cleavage. An experimental study has shown that a pairwise replacement of Gly with Leu in APP enhances homodimerization but leads to a drastic reduction of Abeta secretion. To resolve this apparent discrepancy, we predicted the wild-type (WT) and mutant APP dimer conformations by replica-exchange molecular dynamics simulations using the implicit membrane model IMM1. The simulations illustrate large conformational differences between the WT and mutant APP fragments in the membrane. Dimerization of the WT is due to two C(alpha)-H...O hydrogen bonds between two APP fragments, whereas dimerization of the mutant is due to hydrophobic interactions. In the mutant, each APP fragment is more tilted, and the gamma-cleavage site is shifted toward the center of the membrane. This position produces a mismatch between the active site of gamma-secretase and the gamma-cleavage site of APP that might prohibit Abeta production.

We use Metropolis Monte Carlo and umbrella sampling to calculate the free energies of interaction of two methane molecules and their charged derivatives in cylindrical water-filled pores. Confinement strongly alters the interactions between the nonpolar solutes and completely eliminates the solvent separated minimum (SSM) that is seen in bulk water. The free energy profiles show that the methane molecules are either in contact or at separations corresponding to the diameter and the length of the cylindrical pore. Analytic calculations that estimate the entropy of the solutes, which are solvated at the pore surface, qualitatively explain the shape of the free energy profiles. Adding charges of opposite sign and magnitude 0.4e or e (where e is the electronic charge) to the methane molecules decreases their tendency for surface solvation and restores the SSM. We show that confinement induced ion-pair formation occurs whenever l(B)/D approximately O(1), where l(B) is the Bjerrum length and D is the pore diameter. The extent of stabilization of the SSM increases with ion charge density as long as l(B)/D<1. In pores with D

It is suspected that correlated motions among a subset of spatially separated residues drive conformational dynamics not only in multidomain but also in single domain proteins. Sequence and structure-based methods have been proposed to determine covariation between two sites on a protein. The statistical coupling analysis (SCA) that compares the changes in probability at two sites in a multiple sequence alignment (MSA) and a subset of the MSA has been used to infer the network of residues that encodes allosteric signals in protein families. The structural perturbation method (SPM), that probes the response of a local perturbation at all other sites, has been used to probe the allostery wiring diagram in biological machines and enzymes. To assess the efficacy of the SCA, we used an exactly soluble two dimensional lattice model and performed double-mutant cycle (DMC) calculations to predict the extent of physical coupling between two sites. The predictions of the SCA and the DMC results show that only residues that are in contact in the native state are accurately identified. In addition, covariations among strongly interacting residues are most easily identified by the SCA. These conclusions are consistent with the DMC experiments on the PDZ family. Good correlation between the SCA and the DMC is only obtained by performing multiple experiments that vary the nature of amino acids at a given site. In contrast, the energetic coupling found in experiments for the PDZ domain are recovered using the SPM. We also predict, using the SPM, several residues that are coupled energetically.

Allostery forms the basis of intra-molecular communications in various enzymes, however the underlying conformational changes are largely elusive. Recently, we have proposed to employ an elastic model based normal mode analysis to investigate the allosteric transitions in several molecular nanomachines (including myosin II, DNA polymerase and chaperonin GroEL). After combining with bioinformatics analysis of the evolutionary sequence variations, we have been able to identify the highly conserved and robust modes of collective motions that are capable of transmitting molecular signals over long distances.

Understanding how RNA molecules navigate their rugged folding landscapes holds the key to describing their roles in a variety of cellular functions. To dissect RNA folding at the molecular level, we performed simulations of three pseudoknots (MMTV and SRV-1 from viral genomes and the hTR pseudoknot from human telomerase) using coarse-grained models. The melting temperatures from the specific heat profiles are in good agreement with the available experimental data for MMTV and hTR. The equilibrium free energy profiles, which predict the structural transitions that occur at each melting temperature, are used to propose that the relative stabilities of the isolated helices control their folding mechanisms. Kinetic simulations, which corroborate the inferences drawn from the free energy profiles, show that MMTV folds by a hierarchical mechanism with parallel paths, i.e., formation of one of the helices nucleates the assembly of the rest of the structure. The SRV-1 pseudoknot, which folds in a highly cooperative manner, assembles in a single step in which the preformed helices coalesce nearly simultaneously to form the tertiary structure. Folding occurs by multiple pathways in the hTR pseudoknot, the isolated structural elements of which have similar stabilities. In one of the paths, tertiary interactions are established before the formation of the secondary structures. Our work shows that there are significant sequence-dependent variations in the folding landscapes of RNA molecules with similar fold. We also establish that assembly mechanisms can be predicted using the stabilities of the isolated secondary structures.

We elucidate the structural transitions in a helical off-lattice homopolymer induced by crowding agents, as a function of the number of monomers (N) and volume fraction (varphi c) of crowding particles. At varphic=0, the homopolymer undergoes transitions from a random coil to a helix, helical hairpin HH, and helix bundle HB structures depending on N, and temperature. Crowding induces chain compaction that can promote HH or HB formation depending on varphic. Typically, the helical content decreases which is reflected in the decrease in the transition temperatures that depend on varphic, N, and the size of the crowding particles.

Sequence-dependent variations in the growth mechanism and stability of amyloid fibrils, which are implicated in a number of neurodegenerative diseases, are poorly understood. We have carried out extensive all-atom molecular dynamics simulations to monitor the structural changes that occur upon addition of random coil (RC) monomer fragments from the yeast prion Sup35 and Abeta-peptide onto a preformed fibril. Using the atomic resolution structures of the microcrystals as the starting points, we show that the RC --> beta-strand transition for the Sup35 fragment occurs abruptly over a very narrow time interval, whereas the acquisition of strand content is less dramatic for the hydrophobic-rich Abeta-peptide. Expulsion of water, resulting in the formation of a dry interface between 2 adjacent sheets of the Sup35 fibril, occurs in 2 stages. Ejection of a small number of discrete water molecules in the second stage follows a rapid decrease in the number of water molecules in the first stage. Stability of the Sup35 fibril is increased by a network of hydrogen bonds involving both backbone and side chains, whereas the marginal stability of the Abeta-fibrils is largely due to the formation of weak dispersion interaction between the hydrophobic side chains. The importance of the network of hydrogen bonds is further illustrated by mutational studies, which show that substitution of the Asn and Gln residues to Ala compromises the Sup35 fibril stability. Despite the similarity in the architecture of the amyloid fibrils, the growth mechanism and stability of the fibrils depend dramatically on the sequence.

The influence of hydration on the nanosecond timescale dynamics of tRNA is investigated using neutron scattering spectroscopy. Unlike protein dynamics, the dynamics of tRNA is not affected by methyl group rotation. This allows for a simpler analysis of the influence of hydration on the conformational motions in RNA. We find that hydration affects the dynamics of tRNA significantly more than that of lysozyme. Both the characteristic length scale and the timescale of the conformational motions in tRNA depend strongly on hydration. Even the characteristic temperature of the so-called "dynamical transition" appears to be hydration-dependent in tRNA. The amplitude of the conformational motions in fully hydrated tRNA is almost twice as large as in hydrated lysozyme. We ascribe these differences to a more open and flexible structure of hydrated RNA, and to a larger fraction and different nature of hydrophilic sites. The latter leads to a higher density of water that makes the biomolecule more flexible. All-atom molecular-dynamics simulations are used to show that the extent of hydration is greater in tRNA than in lysozyme. We propose that water acts as a "lubricant" in facilitating enhanced motion in solvated RNA molecules.

Positively charged counterions drive RNA molecules into compact configurations that lead to their biologically active structures. To understand how the valence and size of the cations influences the collapse transition in RNA, small-angle X-ray scattering was used to follow the decrease in the radius of gyration (R(g)) of the Azoarcus and Tetrahymena ribozymes in different cations. Small, multivalent cations induced the collapse of both ribozymes more efficiently than did monovalent ions. Thus, the cooperativity of the collapse transition depends on the counterion charge density. Singular value decomposition of the scattering curves showed that folding of the smaller and more thermostable Azoarcus ribozyme is well described by two components, whereas collapse of the larger Tetrahymena ribozyme involves at least one intermediate. The ion-dependent persistence length, extracted from the distance distribution of the scattering vectors, shows that the Azoarcus ribozyme is less flexible at the midpoint of transition in low-charge-density ions than in high-charge-density ions. We conclude that the formation of sequence-specific tertiary interactions in the Azoarcus ribozyme overlaps with neutralization of the phosphate charge, while tertiary folding of the Tetrahymena ribozyme requires additional counterions. Thus, the stability of the RNA structure determines its sensitivity to the valence and size of the counterions.

Fibril formation from amyloidogenic peptides is a hallmark of a wide range of diseases, including Alzheimer's disease and type II diabetes. Characterization of the aggregation process should include intrinsic factors, such as sequence variation, and extrinsic factors, such as crowding effects. To this end, we examined the interactions of dimers composed of residues 20-29 of human islet amyloid polypeptide (hIAPP), which form fibrils in vitro, and the nonamyloidogenic rat IAPP (rIAPP) using molecular dynamics simulations modeled at different peptide concentrations. There is a substantial free energy barrier to unbind the hIAPP dimer whereas no barrier exists for separating the rIAPP dimer. The profound differences in the free energy landscapes of the rIAPP and hIAPP dimers explains the lack of fibril formation in hIAPP upon substitution of the C-terminal residues by proline. Enhancing the extent of crowding has a substantial effect on both the barrier for separating a hIAPP beta-sheet dimer and the formation of potential beta-sheet nucleation sites. Our results show that the propensity for forming nucleation sites is dependent not only on the amino-acid sequence but also on the context in which it is found.

Single-molecule force spectroscopy methods can be used to generate folding trajectories of biopolymers from arbitrary regions of the folding landscape. We illustrate the complexity of the folding kinetics and generic aspects of the collapse of RNA and proteins upon force quench by using simulations of an RNA hairpin and theory based on the de Gennes model for homopolymer collapse. The folding time, tau(F), depends asymmetrically on deltaf(S) = f (S) - f (m) and deltaf (Q) = f (m) - f (Q) where f (S) (f (Q)) is the stretch (quench) force and f (m) is the transition midforce of the RNA hairpin. In accord with experiments, the relaxation kinetics of the molecular extension, R(t), occurs in three stages: A rapid initial decrease in the extension is followed by a plateau and finally, an abrupt reduction in R(t) occurs as the native state is approached. The duration of the plateau increases as lambda = tau (Q)/tau (F) decreases (where tau (Q) is the time in which the force is reduced from f (S) to f (Q)). Variations in the mechanisms of force-quench relaxation as lambda is altered are reflected in the experimentally measurable time-dependent entropy, which is computed directly from the folding trajectories. An analytical solution of the de Gennes model under tension reproduces the multistage stage kinetics in R(t). The prediction that the initial stages of collapse should also be a generic feature of polymers is validated by simulation of the kinetics of toroid (globule) formation in semiflexible (flexible) homopolymers in poor solvents upon quenching the force from a fully stretched state. Our findings give a unified explanation for multiple disparate experimental observations of protein folding.

Aggregation of Amyloid beta (Abeta) peptide has been linked to the neurodegenerative Alzheimer's Disease and implicated in other amyloid diseases including cerebral amyloid angiopathy. Abeta peptide is generated by cleavage of the amyloid precursor protein (APP) by transmembrane proteases. It is crucial to determine the structures of beta-amyloid peptides in a membrane to provide a molecular basis for the cleavage mechanism. We report the structures of amyloid beta peptide (Abeta(1-40) and Abeta(1-42)) as well as the 672-726 fragment of APP (referred to as Abeta(1-55)) in a membrane environment determined by replica-exchange molecular dynamics simulation. Abeta(1-40) is found to have two helical domains A (13-22) and B(30-35) and a type I beta-turn at 23-27. The peptide is localized at the interface between membrane and solvent. Substantial fluctuations in domain A are observed. The dominant simulated tertiary structure of Abeta(1-40) is observed to be similar to the simulated Abeta(1-42) structure. However, there are differences observed in the overall conformational ensemble, as characterized by the two-dimensional free energy surfaces. The fragment of APP (Abeta(1-55)) is observed to have a long transmembrane helix. The position of the transmembrane region and ensemble of membrane structures are elucidated. The conformational transition between the transmembrane Abeta(1-55) structure, prior to cleavage, and the Abeta(1-40) structure, following cleavage, is proposed.

The mechanism of addition of a soluble unstructured monomer to a preformed ordered amyloid fibril is a complex process. On the basis of the kinetics of monomer disassociation of Abeta(1-40) from the amyloid fibril, it has been suggested that deposition is a multistep process involving a rapid reversible association of the unstructured monomer to the fibril surface (docking) followed by a slower conformational rearrangement leading to the incorporation onto the underlying fibril lattice (locking). By exploiting the vast time scale separation between the dock and lock processes and using molecular dynamics simulation of deposition of the disordered peptide fragment (35)MVGGVV(40) from the Abeta peptide onto the fibril with known crystal structure, we provide a thermodynamic basis for the dock-lock mechanism of fibril growth. Free energy profiles, computed using implicit solvent model and enhanced sampling methods with the distance (delta(C)) between the center of mass of the peptide and the fibril surface as the order parameter, show three distinct basins of attraction. When delta(C) is large, the monomer is compact and unstructured and the favorable interactions with the fibril results in stretching of the peptide at delta(C) approximately 13 A. As delta(C) is further decreased, the peptide docks onto the fibril surface with a structure that is determined by a balance between intrapeptide and peptide fibril interactions. At delta(C) approximately 4 A, a value that is commensurate with the spacing between beta-strands in the fibril, the monomer expands and locks onto the fibril. Using simulations with implicit solvent model and all atom molecular dynamics in explicit water, we show that the locked monomer, which interacts with the underlying fibril, undergoes substantial conformational fluctuations and is not stable. The cosolutes urea and TMAO destabilize the unbound phase and stabilize the docked phase. Interestingly, small crowding particles enhance the stability of the fibril-bound monomer only marginally. We predict that the experimentally measurable critical monomer concentration, C(R), at which the soluble unbound monomer is in equilibrium with the ordered fibril, increases sharply as temperature is increased under all solution conditions.

Urea titration of RNA by urea is an effective approach to investigate the forces stabilizing this biologically important molecule. We used all atom molecular dynamics simulations using two urea force fields and two RNA constructs to elucidate in atomic detail the destabilization mechanism of folded RNA in aqueous urea solutions. Urea denatures RNA by forming multiple hydrogen bonds with the RNA bases and has little influence on the phosphodiester backbone. Most significantly we discovered that urea engages in stacking interactions with the bases. We also estimate, for the first time, the m-value for RNA, which is a measure of the strength of urea-RNA interactions. Our work provides a conceptual understanding of the mechanism by which urea enhances RNA folding rates.

Knowledge-based contact potentials are routinely used in fold recognition, binding of peptides to proteins, structure prediction, and coarse-grained models to probe protein folding kinetics. The dominant physical forces embodied in the contact potentials are revealed by eigenvalue analysis of the matrices, whose elements describe the strengths of interaction between amino acid side chains. We propose a general method to rank quantitatively the importance of various inter-residue interactions represented in the currently popular pair contact potentials. Eigenvalue analysis and correlation diagrams are used to rank the inter-residue pair interactions with respect to the magnitude of their relative contributions to the contact potentials. The amino acid ranking is shown to be consistent with a mean field approximation that is used to reconstruct the original contact potentials from the most relevant amino acids for several contact potentials. By providing a general, relative ranking score for amino acids, this method permits a detailed, quantitative comparison of various contact interaction schemes. For most contact potentials, between 7 and 9 amino acids of varying chemical character are needed to accurately reconstruct the full matrix. By correlating the identified important amino acid residues in contact potentials and analysis of about 7800 structural domains in the CATH database we predict that it is important to model accurately interactions between small hydrophobic residues. In addition, only potentials that take interactions involving the protein backbone into account can predict dense packing in protein structures.

We study the effect of the osmolyte, Trimethylamine N-Oxide (TMAO), which accumulates in cells in response to osmotic stress, on the stability of RNA hairpins. All atom molecular dynamics (MD) simulations of a nucleotide and the 22-nucleotide RNA hairpin P5GA in an aqueous TMAO solution show that TMAO preferentially interacts with the base through the formation of a single hydrogen bond. To circumvent the difficulties of adequately sampling the conformational space of polynucleotides, we used coarse-grained models (including one that is inspired by the results of all-atom MD simulations of a single nucleotide) to probe the effects of osmoyltes on the stability of P5GA. If, as revealed by our MD simulations, the cosolute specifically interacts with only one base at a time, then we find practically no change in hairpin stability as measured by Delta T m = T m(Phi) - T m, where T m(Phi) and T m are the melting temperatures at volume fraction Phi of the osmolyte and Phi = 0, respectively. This finding is in qualitative agreement with recent experiments. If the interactions between the RNA and osmolytes are repulsive, which is appropriate for mimicking the effects of crowding, Delta T m can vary from 5 to 15 K depending on the size of the osmolyte and the nature of RNA-osmolyte interactions. Cosolutes that interact favorably with multiple bases simultaneously can stabilize the hairpin more than a crowding agent of the same size. The implications of our predictions for experiments are briefly outlined.

The bacterial chaperonin GroEL and the co-chaperonin GroES assist in the folding of a number of structurally unrelated substrate proteins (SPs). In the absence of chaperonins, SP folds by the kinetic partitioning mechanism (KPM), according to which a fraction of unfolded molecules reaches the native state directly, while the remaining fraction gets trapped in a potentially aggregation-prone misfolded state. During the catalytic reaction cycle, GroEL undergoes a series of allosteric transitions (T<-->R-->R"-->T) triggered by SP capture, ATP binding and hydrolysis, and GroES binding. We developed a general kinetic model that takes into account the coupling between the rates of the allosteric transitions and the folding and aggregation of the SP. Our model, in which the GroEL allosteric rates and SP-dependent folding and aggregation rates are independently varied without prior assumption, quantitatively fits the GroEL concentration-dependent data on the yield of native ribulose bisphosphate carboxylase/oxygenase (Rubisco) as a function of time. The extracted kinetic parameters for the GroEL reaction cycle are consistent with the available values from independent experiments. In addition, we also obtained physically reasonable parameters for the kinetic steps in the reaction cycle that are difficult to measure. If experimental values for GroEL allosteric rates are used, the time-dependent changes in native-state yield at eight GroEL concentrations can be quantitatively fit using only three SP-dependent parameters. The model predicts that the differences in the efficiencies (as measured by yields of the native state) of GroEL, single-ring mutant (SR1), and variants of SR1, in the rescue of mitochondrial malate dehydrogenase, citrate synthase, and Rubisco, are related to the large variations in the allosteric transition rates. We also show that GroEL/S mutants that efficiently fold one SP at the expense of all others are due to a decrease in the rate of a key step in the reaction cycle, which implies that wild-type GroEL has evolved as a compromise between generality and specificity. We predict that, under maximum loading conditions and saturating ATP concentration, the efficiency of GroEL (using parameters for Rubisco) depends predominantly on the rate of R-->R" transition, while the equilibrium constant of the T<-->R has a small effect only. Both under sub- and superstoichiometric GroEL concentrations, enhanced efficiency is achieved by rapid turnover of the reaction cycle, which is in accord with the predictions of the iterative annealing mechanism. The effects are most dramatic at substoichiometric conditions (most relevant for in vivo situations) when SP aggregation can outcompete capture of SP by chaperonins.