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Thermodynamics and stability of a ß-sheet complex: Molecular dynamics simulations on simplified off-lattice protein models

¹ Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27695-7905, USA
² Department of Physiology and Biophysics, State University of New York at Buffalo, Buffalo, New York 14214, USA

Reprint requests to: Carol K. Hall, Department of Chemical Engineering, North Carolina State University, Raleigh, NC 27695-7905, USA; e-mail: hall{at}turbo.che.ncsu.edu; fax: (919) 515-3465.

(RECEIVED April 25, 2003; FINAL REVISION September 3, 2003; ACCEPTED September 17, 2003)

Article and publication are at http://www.proteinscience.org/cgi/doi/10.1110/ps.03162804.

	Abstract

TOP Abstract Introduction Models and simulation method Results Discussion References

 
We have performed discontinuous molecular dynamics simulations of the thermodynamics and stability of a tetrameric ß-sheet complex that contains four identical four-stranded antiparallel ß-sheet peptides. The potential used in the simulation is a hybrid Go-type potential characterized by the bias gap parameter g, an artificial measure of the preference of a model protein for its native state, and the intermolecular contact parameter , which measures the ratio of intermolecular to intramolecular native attractions. Despite the simplicity of the model, a complex set of thermodynamic transitions for the ß-sheet complex is revealed that shows there are three distinct oligomer (partially ordered, ordered, and highly ordered ß-sheet complex) states and four noninteracting monomers phases. The thermodynamic properties of the three oligomer states strongly depend on both the size of the intermolecular contact parameter and the temperature. The partially ordered ß-sheet complex is made up of four ordered globules and is observed at intermediate to large at high temperatures. The ordered ß-sheet complex contains four native ß-sheets and is located at small to intermediate at low temperatures in the phase diagram. The highly ordered ß-sheet complex has fully-stiff ß-sheet strands, the same as the global energy minimum structure, and is observed for all at low temperatures.

Keywords: fibril; amyloid; discontinuous molecular dynamics; Go-type potential; ß-sheet complex; bias gap; intermolecular contact parameter

	Introduction

TOP Abstract Introduction Models and simulation method Results Discussion References

 
Ordered protein aggregates are implicated in a number of so-called protein deposition diseases (Massry and Glasscock 1983; Eaton and Hofrichter 1990; Clark and Steele 1992; Gallo et al. 1996; Moore and Melton 1997). The best known of these diseases is Alzheimer’s disease (Selkoe 1991; Simmons et al. 1994), which is characterized, on a molecular level, by the assembly of normally soluble proteins into insoluble fibril plaques in the extracellular space of brain tissue. The fibrils found in Alzheimer’s disease victims are composed of ß-sheets of the ß-amyloid peptide (Aß), whose monomeric form is a random coil or -helix (Esler et al. 1996, 2000; Sunde et al. 1997; Benzinger et al. 1998, 2000; Burkoth et al. 1998; Lazo and Cowning 1998; Lynn and Meredith 2000; Zhang et al. 2000). Although amyloid fibrils commonly exhibit a cross-ß structure, the exact nature of the ß-sheet conformations in these aggregates is still ambiguous (Serpell 2000). Furthermore, the causes of amyloid fibril formation and the fundamental mechanisms underlying this type of aggregation are largely unknown.

The mechanism whereby isolated proteins change their conformations and assemble into an ordered oligomer is the subject of a number of recent investigations. The most popular view of protein aggregation is that partially unfolded states serve as precursors to fibril formation (Kelly 1996), because hydrophobic residues that are exposed to solution tend to cluster together. This view is supported by several recent experimental (Booth et al. 1997; Fink 1998; Guijarro et al. 1998; Chiti et al. 1999; Quintas et al. 1999; Khurana et al. 2001) and simulation (Harrison et al. 1999, 2001) studies of protein aggregation but is, by no means, universal. An alternative mechanism is indicated by the work of Silow et al. (1999), who have shown that ordered protein aggregates are formed directly from unfolded proteins in the two-state folders U1A and CI2; such behavior has also been observed in a recent simulations study of protein aggregation (Dima and Thirumalai 2002). Whether or not either mechanism explains amyloid formation in general is unclear; there may be a number of different mechanisms behind protein aggregation just as isolated proteins exhibit a number of distinct folding mechanisms on the molecular level.

The long-term goal of our work is to reveal the molecular-level mechanisms underlying protein aggregation and fibril formation, with particular focus on the role played by ß-strands. Our approach is to use low-resolution or simplified protein models; such models are, in our opinion, best suited for simulating multiprotein systems because they allow us to study folding behavior over relatively long time-scales. Investigations based on these coarse-grained protein models by other investigators have already provided some insights into the thermodynamic and kinetic properties of protein folding for isolated chains (Lau and Dill 1989; Skolnick and Kolinski 1991; Miller et al. 1992; Chan and Dill 1994; Guo and Thirumalai 1995, 1996; Kolinski et al. 1995, 1999; Guo and Brooks III 1997; Gupta and Hall 1997; Zhou and Karplus, 1997a,b, 1999; Dokholyan et al. 1998, 2000; Nymeyer et al. 1998; Pande and Rokhsar 1998; Gupta et al. 1999; Shea et al. 2000; Jang et al. 2002a, b) and of protein aggregation for multichain systems (Dill and Stigter 1995; Gupta and Hall 1998; Harrison et al. 1999, 2001; Bratko and Blanch 2001; Dima and Thirumalai 2002; Jang et al. 2003). We recently introduced three minimalist models of four-strand antiparallel ß-strand peptides: the ß-sheet, the ß-clip, and the ß-twist (Jang et al. 2002a, b). In our study of the folding thermodynamics of these three models (Jang et al. 2002a), discontinuous molecular dynamics (DMD) simulations (Alder and Wainwright 1959; Rapaport 1978; Smith et al. 1996) were performed in order to determine how the thermodynamic properties of an isolated peptide vary with temperature. Despite the simplicity of these models, they undergo a complex set of protein transitions similar to those observed in experimental studies on real proteins (Ptitsyn 1995). Starting from high temperature, these transitions include a collapse transition, a disordered-to-ordered globule transition, a folding transition, and a liquid-to-solid transition. In our study of the folding kinetics for the same models (Jang et al. 2002b), the ß-sheet exhibits a fast-track folding pathway without becoming trapped in any intermediate. In contrast, the ß-clip and ß-twist exhibit multiple folding pathways that include trapping in intermediates and direct folding to the native state. The folding speed of the model proteins with different native state topologies strongly depends on the contact order in the native state (Plaxco et al. 1998).

In this article, we investigate the thermodynamics and stability of a tetrameric ß-sheet complex, consisting of four identical four-stranded antiparallel ß-sheet peptides (the ß-sheet peptides studied previously), with each peptide containing 39 connected residues (beads). We have already investigated the kinetics and assembly of the ß-sheet complex (Jang et al. 2003). In that study, four separate random coils were quenched from a high temperature to a low temperature, the temperature at which the peptides are in their ß-sheet native state. After quenching, the four monomer chains experienced a conformational change toward the highly ordered ß-sheet complex (folded state) or to a partially folded or disordered (misfolded) state. The formation kinetics and resulting structure at fixed temperature strongly depended upon the size of the intermolecular contact parameter , which measures the ratio of intermolecular to intramolecular native attractions. For small , most folding trajectories follow the following path: four monomers dimer and two monomers trimer and monomer or two dimers tetramer. The folding yield is low, and secondary structure begins to form before an ordered ß-sheet complex assembles. For intermediate , most folding trajectories follow the following path: four monomers dimer and two monomers trimer and monomer tetramer. The folding yield is very high, and four partially folded chains assemble into a highly ordered ß-sheet complex. For large , most folding trajectories follow the same path as seen in the intermediate model. The folding yield is very low, and the four random coil monomers directly assemble into a highly ordered ß-sheet complex. The results of the kinetics study for the ß-sheet complex presented in our previous article set the stage for this work in which we examine the thermodynamics and stability of the ß-sheet complex.

Our model of the ß-sheet complex was designed in part to mimic the small Aß oligomers that are observed (albeit indirectly) in the early stages of fibril formation (Harper et al. 1997). These oligomers are now widely believed to serve as the nuclei that seed the growth of the fibrils that characterize Alzheimer’s and other amyloid diseases (Pallitto and Murphy 2001). In fact, recent evidence indicates that it is these oligomers, rather than the fully formed fibrils, that are the toxic species in Alzheimer’s disease (Kirkitadze et al. 2002). Our model was designed in part to mimic real Aß oligomer complex formation in several respects. First, our peptides are 39 residues long, and Aß is between 39 and 42 residues long. Second, Aß has several extended stretches of hydrophobic residues (Aß[17–21], Aß[32–42]); our peptide has essentially four stretches of residues that act with an intermolecular hydrophobic interaction. Third, the Aß peptides in Aß fibrils (and, indeed, the peptides in all fibrils) experience intramolecular hydrogen bonding within the ß-sheets and intermolecular hydrophobic interactions between the ß-sheets. The model peptides in our ß-sheet complex experience intramolecular interactions within the sheets that are reminiscent of hydrogen bonding; these interactions are characterized by the Go-model bias gap parameter, g. They also experience intermolecular interactions between the sheets that are reminiscent of hydrophobic interactions; these interactions are characterized by the intermolecular contact parameter .

The Go-model bias gap parameter is set to an intermediate value (g = 0.9). Go models with intermediate values of the bias gap parameter are considered to be the most realistic compared with other values of the bias gap parameter, because all nonbonded pairs of beads are attracted to each other. The problems associated with large values of g are avoided (unfavorable repulsive forces between nonnative pairs of beads for g > 1.0 or no forces for g = 1.0), as are the problems associated with low values of g (near indistinguishability between native and nonnative interactions hinders the formation of the target native structure).

In our model, the intermolecular contact parameter , which essentially measures the ratio of the intermolecular (hydrophobic) interaction and the intramolecular (hydrogen bonding) interaction, is selected in the range between 0.2 1. The reasons for considering a range of intermolecular contact parameter values are as follows: (1) the best values for the relative strengths of the hydrophobic and hydrogen bonding interactions in simplified models are a matter of debate (Pace et al. 1998), and (2) intermolecular hydrophobic interaction strengths depend upon the intrinsic conditions of the protein (sequence and number of hydrophobic residues) and on external conditions (pH and temperature; Fraser et al. 1991a, b; Snyder et al. 1994; Kowalewski and Holtzman 1999). Furthermore, by exploring how the aggregation mechanism varies with the intermolecular contact parameter, we provide insights to other research workers interested in modeling aggregation phenomena.

In this article, we investigate the thermodynamic properties and stability of the tetrameric ß-sheet complex, the same model as the one we investigated in the study of kinetics and assembly of the ß-sheet complex (Jang et al. 2003). DMD simulations (Alder and Wainwright 1959; Rapaport 1978; Smith et al. 1996) were performed on the model systems at an intermediate value of the bias gap g = 0.9 for different intermolecular contact parameters in the range 0.2 10 at different temperatures. All simulations were started from folded states of the complex; these configurations were obtained from our previous kinetic simulations. For each value of and temperature, at least five independent simulations were performed to obtain equilibrium averages. The phase transitions of the ß-sheet complex were determined by calculating thermodynamic averages for the fraction of native contacts Q, the squared radius of gyration R_g², the specific heat C_v, the internal energy E, the root-mean-squared pair separation fluctuation _B, and the Lindemann disorder parameter _L. The results are summarized in a phase diagram for the ß-sheet complex in the temperature/intermolecular contact parameter plane.

In the following section, a description of the model and the simulation method are presented. Next is the Results section, which is divided into four subsections: The first presents the results for the fraction of native contacts and the radius of gyration, the second presents the results for the specific heat and the internal energy, the third presents the results for the bead fluctuations, and the fourth presents the phase diagram for the ß-sheet complex. The article concludes with a discussion of the key findings.

	Models and simulation method

TOP Abstract Introduction Models and simulation method Results Discussion References

 
We consider an off-lattice protein model of a ß-sheet complex that consists of four identical four-stranded antiparallel ß-sheet peptides. The global energy minimum structure for the model ß-sheet complex is shown in Figure 1. Each chain in the ß-sheet complex is shaded differently: chain 1 (white), chain 2 (light gray), chain 3 (dark gray), and chain 4 (black). A topology diagram representing a top view of the complex is shown at the bottom of the figure. In the topology diagram, thick arrows indicate chain connectivity at the top of the strands, and thin arrows indicate chain connectivity at the bottom of the strands. It can be seen from the topology diagram that the ß-sheet is not perfectly planar because it has an inflection in the middle. This is because ß-sheet complex with kinked ß-sheets has more intermolecular native contacts than that with planar ß-sheets. In the global energy minimum state of the ß-sheet complex, the total number of intramolecular native contacts is , the total number of intermolecular native contacts is , and the reduced squared radius of gyration for the entire system is .

View larger version (38K): [in this window] [in a new window]   Figure 1. Bead (A) and topology (B) diagrams of the global energy minimum structures for the model ß-sheet complex.

(1)

where , B_O, and are measures of the relative strengths of the energies associated with the intramolecular native, nonnative, and intermolecular native pair interactions in this Go-type potential (Taketomi et al. 1975; Go and Taketomi 1978, 1979; Ueda et al. 1978). Within a chain, nonbonded pairs of beads that are in contact in the global energy minimum structure experience an attractive interaction, that is, , when their square-wells overlap. On the other hand, within a chain and in different chains, nonbonded pairs of beads that are not in contact in the global energy minimum structure experience either an attractive interaction (B_O < 0) or a repulsive interaction (B_O > 0). The sign of the parameter B_O depends on the size of the bias gap parameter g,

(2)

where g is the bias gap (Zhou and Karplus 1997a,b, 1999; Jang et al. 2002a, b). The bias gap measures the ratio of the interaction strength between the intramolecular native contacts and nonnative contacts. Note that for g < 1, B_O > 0; in this case, nonnative contacts are repulsive so that the native state structure is strongly favored over any nonnative state structure. For 0 < g < 1, B_O < 0, all nonbonded contact pairs are attractive, but the intramolecular native contacts are always more favorable than are the nonnative contacts. For g = 0, the intramolecular native and nonnative contacts are equally favorable, , and the model reduces to a homopolymer. The bias gap is an artificial measure of the preference of a model protein for its native state; in a real protein, this preference for the native state might be measured, for example, by the energy difference between the native and nonnative state.

In different chains, bead pairs that are in contact in the global energy minimum structure experience an attractive interaction, that is, , when their square-wells overlap. The parameter depends on the size of the intermolecular contact parameter ,

(3)

where is the intermolecular contact parameter (Jang et al. 2003). The quantity measures the ratio of the interaction strength between the intermolecular and intramolecular native contacts. For = 1, the intramolecular and intermolecular native contacts are equally favorable, that is, . For < 1, the intramolecular native contacts are more favorable than the intermolecular native contacts, that is, . However, nonnative contacts are always less favorable than are the intramolecular and intermolecular native contacts, so that for g > 0. In this model, the intramolecular native interaction, intermolecular native interaction, and nonnative interaction might be regarded as mimicking hydrogen bonding, hydrophobic interactions, and van der Waals interactions, respectively, because in real fibrils, the molecular interactions along the sheets (generally hydrogen bonds) are different from the molecular interactions between the sheets (generally hydrophobic interactions).

The thermodynamic properties and stability of the ß-sheet complex are investigated by using the DMD algorithm (Alder and Wainwright 1959; Rapaport 1978; Smith et al. 1996). All simulations were started from the folded state of the complex, which was obtained from our previous kinetic simulations on the same model (Jang et al. 2003). The initial configurations of the folded ß-sheet complex were selected from different folding trajectories in the kinetic simulations to ensure that they are independent. All initial configurations of the folded ß-sheet complexes were pre-equilibrated for 100,000 collisions at the temperature of interest, and then equilibrium simulations were performed. For simulations at low temperatures, the initial configurations were pre-equilibrated at a relatively high temperature, above the solid-to-liquid transition. This procedure was used to prevent the system from becoming trapped in a metastable or frozen state. Equilibrium simulations were performed for up to 4 x 10⁸ collisions to ensure equilibration. For each value of , at least five independent simulations at the temperature of interest were performed to obtain the equilibrium averages. Equilibrium averages were typically taken after discarding data from the first half of the simulation.

The progression of a conformational change was monitored by introducing the fraction of total native contacts formed (Sali et al. 1994; Lazaridis and Karplus 1997), Q_total, defined by

(4)

where _eq denotes an average over the collisions after discarding the first half of the simulation, N_total is the total number of native contacts with N_total N_intra + N_inter, and is the total number of native contacts in the global energy minimum state with . The fraction of total native contacts is a weighted sum of the fraction of intramolecular native contacts, Q_intra, and the fraction of intermolecular native contacts, Q_inter,

(5)

where , and is a constant, . The degree of nativeness of the chains in the ß-sheet complex was determined by measuring the fraction of intramolecular native contacts, Q_intra, whereas the dissociation of the ß-sheet complex was monitored by examining the fraction of intermolecular native contacts, Q_inter, when Q_inter = 0.

To determine the locations of the thermodynamic transitions, the reduced specific heat and the reduced internal energy were determined. In calculating the specific heat and energy, the weighted histogram method (Ferrenberg and Swendsen 1989; Zhou et al. 1997) was used. Details of the method are reported elsewhere (Zhou et al. 1997). The squared radius of gyration for the entire system, R_g², which monitors the dissociation of the system at high temperatures and the chain stiffness at low temperatures; the root-mean-squared (RMS) pair separation fluctuation, _B, for the degree of bead mobility; and the Lindemann disorder parameter, _L, for the low temperature transition were calculated. The details of the equations are described in earlier publications (Jang et al. 2002a, 2003).

	Results

TOP Abstract Introduction Models and simulation method Results Discussion References

 
We performed DMD simulations to investigate the phase behavior of the ß-sheet complex as a function of the size of the intermolecular contact parameter and the reduced temperature, T^* k_BT / . Thermodynamic averages, including the fraction of native contacts Q, the squared radius of gyration R_g², the specific heat C_v, the internal energy E, the root-mean-squared pair separation fluctuation _B, and the Lindemann disorder parameter _L, were calculated as a function of temperature at intermolecular contact parameters, = 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. The major results presented in this paper are for three selected values of the intermolecular contact parameters, = 0.2, 0.5, and 0.8.

Fraction of native contacts and radius of gyration
The fraction of total native contacts, Q_total, the fraction of intramolecular native contacts, Q_intra, and the fraction of intermolecular native contacts, Q_inter, are shown in Figure 2 as a function of the reduced temperature, T*, for the three selected intermolecular contact parameters: (1) = 0.2, (2) = 0.5, and (3) = 0.8. Here, denotes the average over at least five independent simulations. The error bars are the standard deviation in the measured values and are only shown for values larger than the size of the symbol. The nativeness of the individual chains in the ß-sheet complex can be measured by the fraction of intramolecular native contacts, Q_intra. The dissociation of the complex can be identified by examining the fraction of intermolecular native contacts, Q_inter.

View larger version (71K): [in this window] [in a new window]   Figure 2. Average values of the fraction of total native contacts, Qtotal, the fraction of intramolecular native contacts, Qintra, and the fraction of intermolecular native contacts, Qinter, as a function of the reduced temperature, T*, for three selected values of the intermolecular contact parameter : = 0.2 (A), = 0.5 (B), and = 0.8 (C).

For = 0.5, the solid-to-liquid transition is observed at T* = 0.27 and the system dissociation transition is located at T* = 0.60, as indicated by the discontinuities in the Q_inter curve of Figure 2B. Both of the transition temperatures at = 0.5 are higher than those observed at = 0.2. At T* < 0.27, the oligomer is the highly ordered ß-sheet complex, which, as indicated earlier, is a solid-like phase. At much lower temperatures, T* 0.15, for the highly ordered ß-sheet complex, Q_total 1, indicating that the highly ordered ß-sheet complex is the same as the global energy minimum structure shown in Figure 1. At 0.27 < T* < 0.52, the oligomer is the ordered ß-sheet complex, which has the same physical properties as those observed at = 0.2. The ordered ß-sheet complex consists of four native ß-sheets with large Q_intra values and is loosely packed with smaller Q_inter values than the highly ordered ß-sheet complex. The ordered ß-sheet complex remains intact with the ß-sheets remaining in their native state at temperatures above the folding transition for the isolated ß-sheets, T* = 0.40 (Jang et al. 2002a). This indicates that the presence of the strong intermolecular native interactions stabilizes the ordered ß-sheet complex structures at high temperatures. At 0.52 < T* < 0.58, the ß-sheet complex is in a partially ordered state; we will refer to this as the partially ordered ß-sheet complex. In fact, the transition at T* = 0.52, between the ordered and the partially ordered ß-sheet complexes is determined by a peak in the specific heat (see below). The partially ordered ß-sheet complex is characterized by Q_intra values with 0.6 < Q_intra < 0.85; these are significantly smaller than the Q_intra values in the ordered ß-sheet complex where Q_intra > 0.85. This indicates that each chain in the partially ordered ß-sheet complex is not in its native state but is instead in an ordered globule state, a thermodynamic intermediate that is known as a molten globule (Ptitsyn 1995). The ordered globule state for g = 0.9 has Q values typically in the range 0.5 < Q < 0.9 (Jang et al. 2002a). The partially ordered ß-sheet complex, which consists of four ordered globules, is more loosely assembled than the ordered ß-sheet complex, and consists of four native ß-sheets, because it has smaller Q_inter values than the ordered ß-sheet complex. The partially ordered ß-sheet complex was not observed at = 0.2. At T* = 0.58, the partially ordered ß-sheet complex starts to dissociate and is completely separated into four chains at T* = 0.63 with Q_inter = 0. A number of thermodynamic intermediates are observed during this dissociation process. The four separated chains at T* = 0.63 are each in the ordered globule state. As T* increases, each chain undergoes the ordered-to-disordered globule transition at T* = 0.80 and the collapse transition at T* = 0.95 as seen in our thermodynamic study of the isolated ß-sheet chain systems (Jang et al. 2002a).

For = 0.8 in Figure 2C, the solid-to-liquid transition is observed at T* = 0.44, and the system dissociation transition is found at T* = 0.90 as indicated by the discontinuities observed in the Q_inter curve. These transition temperatures are higher than those at = 0.5, indicating that as increases, both transitions are shifted to higher temperatures. At T* < 0.44, the system is in the highly ordered ß-sheet complex, a solid-like phase. At even lower temperature, T* 0.25, the highly ordered ß-sheet complex with Q_total 1 is the same as the global energy minimum structure. At 0.44 < T* < 0.87, the system is the partially ordered ß-sheet complex because it has Q_intra values in the range 0.5 < Q_intra < 0.8. The physical properties of the partially ordered ß-sheet complex for = 0.8 are the same as those observed at = 0.5; that is, the partially ordered ß-sheet complex consists of four loosely aligned ordered globules. No ordered ß-sheet complex is observed for = 0.8. The partially ordered ß-sheet complex remains intact with the four ordered globules at temperatures above the ordered-to-disordered globule transition for the isolated ß-sheets, T* = 0.80. This indicates that the chains are more stable in the complex conformation than when they are isolated. At T* = 0.87, the system starts to dissociate and is completely separated into four chains with Q_inter = 0 by T* = 0.92. Various types of thermodynamic intermediates are observed during the system dissociation process. The four separated disordered globules at T* = 0.92 undergo a collapse transition toward the random coil state at T* = 0.95 as seen in our thermodynamic study of the isolated ß-sheet chain systems (Jang et al. 2002a).

The system size changes dramatically as the system separates into four monomers. This is reflected in the change in the reduced squared radius of gyration for the entire system, R_g²/²M, as a function of the reduced temperature, T*, as shown in Figure 3. Here, denotes the average over at least five independent simulations. In Figure 3A, as T* increases the R_g²/²M values increase rapidly as a function of temperature at T* = 0.27, 0.58, and 0.87 for = 0.2, 0.5, and 0.8, respectively, signaling the chain separation. These temperatures are consistent with the starting points of the system dissociation process at each shown in Figure 2. In Figure 3, B is the same as A, but shows only the behavior at low values of R_g²/²M. It is immediately apparent that as T* decreases, the R_g²/²M values of the complex increase slightly at T* = 0.09, 0.27, and 0.45 for = 0.2, 0.5, and 0.8, respectively, and then remain steady at even lower temperatures. The increase in the R_g²/²M values at low temperature is related to the liquid-to-solid transition and is consistent with the low temperature discontinuity in the Q_inter values in Figure 2. At this transition, the ordered ß-sheet complex at = 0.2 and 0.5 and the partially ordered ß-sheet complex at = 0.8 turn into the highly ordered ß-sheet complex by elongating their ß-strands. The size of the system increases at this transition as the R_g²/²M values increase at low temperature. The sizes of the ordered and the partially ordered ß-sheet complexes are smaller than that of the highly ordered ß-sheet complex as indicated by the depth of the "well" in the R_g²/²M curves. This indicates that the highly ordered ß-sheet complex has highly stiff ß-sheet strands, recovering the long molecular shape which increases the R_g²/²M values. The size of the highly ordered ß-sheet complex at very low temperatures is relatively independent of and is very close to that of the global energy minimum structure.

View larger version (28K): [in this window] [in a new window]   Figure 3. (A) Average values of the reduced squared radius of gyration per bead, Rg2/2M, on a logarithmic scale for three selected values of the intermolecular contact parameter, = 0.2, 0.5, and 0.8, as a function of the reduced temperature, T*. (B) The same plot, but only showing behavior at low values of Rg2/2M.

View larger version (25K): [in this window] [in a new window]   Figure 4. Average values of the reduced specific heat per bead, Cv*/M (A), and the reduced internal energy per bead, E*/M (B), for = 0.2 as a function of the reduced temperature, T*.

View larger version (26K): [in this window] [in a new window]   Figure 5. Average values of the reduced specific heat per bead, Cv*/M (A), and the reduced internal energy per bead, E*/M (B), for = 0.5 as a function of the reduced temperature, T*.

View larger version (26K): [in this window] [in a new window]   Figure 6. Average values of the reduced specific heat per bead, Cv*/M (A), and the reduced internal energy per bead, E*/M (B), for = 0.8 as a function of the reduced temperature, T*.

View larger version (27K): [in this window] [in a new window]   Figure 7. Average values of the root-mean-squared pair separation fluctuation, B (A), and the Lindemann disorder parameter, L (B), for = 0.2 as a function of the reduced temperature, T*.

View larger version (27K): [in this window] [in a new window]   Figure 8. Average values of the root-mean-squared pair separation fluctuation, B (A), and the Lindemann disorder parameter, L (B), for = 0.5 as a function of the reduced temperature, T*.

View larger version (27K): [in this window] [in a new window]   Figure 9. Average values of the root-mean-squared (rms) pair separation fluctuation, B (A), and the Lindemann disorder parameter, L (B), for = 0.8 as a function of the reduced temperature, T*.

View larger version (35K): [in this window] [in a new window]   Figure 10. Snapshots showing typical structures observed for the ß-sheet complex in the final state. (A) The partially ordered ß-sheet complex for = 0.5 at T* = 0.55 and for = 0.8 at T* = 0.85. (B) The ordered ß-sheet complex for = 0.2 at T* = 0.25 and for = 0.5 at T* = 0.33. (C) The highly ordered ß-sheet complex for = 0.2 at T* = 0.07 and for = 0.8 at T* = 0.30.

View larger version (36K): [in this window] [in a new window]   Figure 11. Phase diagram for the ß-sheet complex as a function of the reduced temperature, T*, and the intermolecular contact parameter, .

	Discussion

TOP Abstract Introduction Models and simulation method Results