Observed Hysteresis of Virus Capsid Disassembly Is Implicit in Kinetic Models of Assembly*

Sushmita Singh and Adam ZlotnickDagger

From the Department of Biochemistry and Molecular Biology, University of Oklahoma Health Sciences Center, Oklahoma City, Oklahoma 73190

Received for publication, November 8, 2002, and in revised form, March 10, 2003

    ABSTRACT
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ABSTRACT
INTRODUCTION
EXPERIMENTAL PROCEDURES
RESULTS
DISCUSSION
REFERENCES

For many protein multimers, association and dissociation reactions fail to reach the same end point; there is hysteresis preventing one and/or the other reaction from equilibrating. We have studied in vitro assembly of dimeric hepatitis B virus (HBV) capsid protein and dissociation of the resulting T = 4 icosahedral capsids. Empty HBV capsids composed of 120 capsid protein dimers were more resistant to dissociation by dilution or denaturants than anticipated from assembly experiments. Using intrinsic fluorescence, circular dichroism, and size exclusion chromatography, we showed that denaturants dissociate the HBV capsids without unfolding the capsid protein; unfolding of dimer only occurred at higher denaturant concentrations. The apparent energy of interaction between dimers measured in dissociation experiments was much stronger than when measured in assembly studies. Unlike assembly, capsid dissociation did not have the concentration dependence expected for a 120-subunit complex; consequently the apparent association energy systematically varied with reactant concentration. These data are evidence of hysteresis for HBV capsid dissociation. Simulations of capsid assembly and dissociation reactions recapitulate and provide an explanation for the observed behavior; these results are also applicable to oligomeric and multidomain proteins. In our calculations, we find that dissociation is impeded by temporally elevated concentrations of intermediates; this has the paradoxical effect of favoring re-assembly of those intermediates despite the global trend toward dissociation. Hysteresis masks all but the most dramatic decreases in contact energy. In contrast, assembly reactions rapidly approach equilibrium. These results provide the first rigorous explanation of how virus capsids can remain intact under extreme conditions but are still capable of "breathing." A biological implication of enhanced stability is that a triggering event may be required to initiate virus uncoating.

    INTRODUCTION
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ABSTRACT
INTRODUCTION
EXPERIMENTAL PROCEDURES
RESULTS
DISCUSSION
REFERENCES

Hysteresis, the lagging of effect behind cause (1), operationally defined as a failure of opposing reactions to equilibrate, can be an impediment to understanding the stability of macromolecular complexes. Examples of hysteresis include DNA melting and annealing as well as association-dissociation reactions for trimeric collagen fibrils (2), SNARE (soluble N-ethylmaleimide factor attachment protein receptor) complexes (3), and viruses (4, 5). In the course of investigating assembly of hepatitis B virus (HBV),1 we have observed a marked hysteresis between association and dissociation and identified a mechanism for hysteresis that is internally consistent with our understanding of the assembly process.

HBV is an enveloped DNA virus with an icosahedral core. Although it is found in two sizes (6), most HBV capsids (the protein shell of the core) are complexes of 120 homodimeric capsid proteins arranged with T = 4 quasi-symmetry (7, 8). A relatively rare smaller capsid is composed of 90 dimers. Image reconstruction of cores from HBV and homologous hepatitis viruses are essentially identical to capsids from a bacterial expression system (9) and to capsids assembled in vitro from the assembly domain of the capsid protein (10). The full-length capsid protein has 183 residues including the C-terminal RNA binding domain, which has 34 residues. We refer to dimers of the first 149 residues, which include the linker sequence that connects assembly and RNA binding domains (11), as Cp1492. The dimers are tetravalent; in the T = 4 capsid they are arranged as a network of 5-fold and quasi-6-fold vertices held together by 240 very similar quasi-equivalent contacts (12).

We are interested in the mechanisms of capsid assembly and dissociation. To interpret experimental observations, we have developed testable models of assembly reactions (13-15); capsid assembly is not well described using mathematical models developed for crystal or filament formation (cf. Ref. 16). Our models describe assembly as a cascade of low order reactions leading to formation of a closed polymer of specified size.

In vitro HBV capsid assembly is consistent with the predictions of the simplest case of the assembly model. Cp1492 assembles into capsids spontaneously in response to protein concentration and ionic strength (8, 10, 14). Assembly can be observed by light-scattering, fluorescence, and size exclusion chromatography (SEC). As predicted from the model, intermediates are rare and kinetics are sigmoidal; the sigmoidal shape occurs because of the time required to accumulate intermediates necessary to support assembly of subsequent intermediates and eventually capsid. A model-based analysis of assembly kinetics indicates that assembly is nucleated by a trimer of Cp1492 (14), which is particularly compatible with the HBV capsid geometry. Model studies predicted that weak interaction energies between multivalent subunits would be sufficient to form a stable particle and that assembly reactions can closely approach equilibrium (13, 15). We found that the interaction energy between adjacent dimers is on the order of -3 to -4 kcal mol-1 (17), a value that is in qualitative agreement with the prediction from the model and the energy calculated from the HBV crystal structure (12).

In the course of our studies on HBV, we became interested in developing alternative approaches to measuring capsid stability. An obvious approach is to use chaotropes to induce dissociation. Such an approach is desirable because not all viruses can be assembled in vitro and dissociation studies can be performed on virus capsids that have been isolated from any source. However, we found that there is a marked hysteresis between capsid association and dissociation reactions. Remarkably, the mathematical model, which was designed to emulate assembly reactions, predicted the hysteresis of dissociation. Examination of dissociation simulations allows identification of a plausible mechanism for hysteresis and suggests biological features of viruses that will accentuate it.

    EXPERIMENTAL PROCEDURES
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ABSTRACT
INTRODUCTION
EXPERIMENTAL PROCEDURES
RESULTS
DISCUSSION
REFERENCES

Capsid Preparation-- The dimeric HBV capsid protein assembly domain, Cp1492, was expressed in Escherichia coli and purified as described (18). Capsids for dissociation experiments were assembled by adding NaCl to Cp1492 to final concentrations of 25 µM Cp1492, 25 mM HEPES, pH 7.5, 2 mM dithiothreitol, 0.5 M NaCl (assembly buffer) then incubating at 4 °C for 24 h. Capsids were isolated by SEC using a Superose-6 column equilibrated with assembly buffer. Fractions were pooled and concentrated using Centricon YM-30 filters (Amicon Bioseparations, Millipore Corp. Bedford, MA) and dialyzed against assembly buffer. Protein concentration was determined by absorbance using epsilon 280 of 60,900 M-1 cm-1.

Dilution Experiments-- Purified capsids were diluted to final protein concentrations between 0.5 and 30 µM in 25 mM HEPES, pH 7.5, 0.15 M NaCl, 5 mM dithiothreitol and incubated at 21 °C for 2-5 days. The capsid concentration was quantified using SEC using a Superose-6 column equilibrated with the same buffer on a SMART chromatography system (Amersham Biosciences).

Capsid Dissociation by Chaotropes-- Guanidine HCl (GuHCl) or urea was added to purified capsids in the presence of specified NaCl concentrations and incubated for up to 48 h. In practice, most experimental measurements were made after 24 h; no significant change was observed at longer times. Experiments with urea were conducted in 50 mM Tris-HCl, pH 7.5, rather than HEPES to scavenge reactive cyanate from decomposing urea.

Tryptophan fluorescence at 21 °C was observed with a SPEX Fluoromax fluorometer (Edison, NJ) using a 3-mm path length cuvette (Hellma, Forest Hills, NY). Excitation and emission wavelengths were 280 and 324 nm, respectively. Light-scattering (LS) data were recorded with excitation and emission wavelengths at 320 nm. A 1.0 neutral optical density filter (Melles Griot, Irvine, CA) was in the excitation path for all experiments.

Circular dichroism spectra were measured using a JASCO J-715 spectropolarimeter. All spectra were obtained with 10 µM Cp1492 in 25 mM HEPES at 21 °C using a 1-mm cuvette (Hellma).

Data Analysis-- Raw data from tryptophan fluorescence, LS, and SEC could not be directly compared because fluorescence and LS were differentially sensitive to chaotrope concentration. Scaling data sets was facilitated by treating dissociation as a two-state mixture of capsid and dimer. This simplification is supported by data described under "Results." The normalized and base line-corrected data are expressed in terms of mass fraction capsid (mfc),


<UP>mfc = </UP>(F<SUB><UP>observed</UP></SUB><UP> − </UP>f<SUB><UP>dimer</UP></SUB>[<UP>denaturant</UP>])<UP>/</UP>([<UP>denaturant</UP>](f<SUB><UP>capsid</UP></SUB><UP> − </UP>f<SUB>dimer</SUB>)) (Eq. 1)
where Fobserved is the fluorescence intensity measured at a given concentration of urea or guanidine HCl. The coefficients fcapsid and fdimer were the fluorescence of capsid and dimer per concentration of denaturant; these were obtained by linear extrapolation of pre- and post-transition base lines, respectively. Because of the inherent noise of the raw data, the coefficients for the background subtractions were averaged from seven experiments. An identical treatment was used for LS data.

The capsid and dimer concentrations calculated from the mass fraction capsid were used to determine the values for the capsid association constant, Kcapsid, and the per contact association constant, Kcontact. Kcapsid was calculated based on the equilibrium expression for assembly of capsid from 120 dimers (Equation 2). This was more conveniently handled in logarithmic form (Equation 3). Calculation of Kcontact (Equation 4) is based on the geometry of the capsid (13) and the assumption that quasi-equivalent contacts form with about the same association constant (17). The 120 tetravalent dimers in the HBV capsid share 240 contacts; the degeneracy inherent in capsid assembly is (2119/120) (13, 15). In the equations below, R is the gas constant of 1.987 cal/(mol degree), and T is 294° K for these experiments.
K<SUB><UP>capsid</UP></SUB>=[<UP>capsid</UP>]<UP>/</UP>[<UP>Cp149<SUB>2</SUB></UP>]<SUP><UP>120</UP></SUP> (Eq. 2)

<UP>ln </UP>K<SUB><UP>capsid</UP></SUB><UP> = ln</UP>([<UP>capsid</UP>])<UP> − 120 × </UP>(<UP>ln</UP>([<UP>Cp149<SUB>2</SUB></UP>]) (Eq. 3)

K<SUB><UP>contact</UP></SUB><UP> = exp </UP>((<UP>ln</UP>K<SUB><UP>capsid</UP></SUB><UP> − ln</UP>(<UP>2<SUP>119</SUP>/120</UP>))<UP>/240</UP>) (Eq. 4)
Delta Gcontact (Equation 5) was calculated from Kcontact. By assuming a linear dependence of Delta Gcontact on urea concentration (19), we extrapolated to 0 M denaturant using Equation 6, where m is determined empirically from plots of Delta Gcontact versus urea concentration (19, 20). Our estimated error for Delta Gcontact,H2O (Equation 6) was extremely low for two reasons. First, we used an average m value for all extrapolations, which meant that the analytical problem of identifying the y intercept was reduced to fitting the sparse data from a titration with a line of known slope (21). Secondly, because we were reporting energy per contact, the error in the measurement of Kcapsid was decreased by moving into logarithmic values and by dividing this value by 240 for each intersubunit contact (Equation 4) (15).
&Dgr;G<SUB><UP>contact</UP></SUB><UP> =  −</UP>RT<UP>ln </UP>(K<SUB><UP>contact</UP></SUB>) (Eq. 5)

&Dgr;G<SUB><UP>contact</UP></SUB><UP> = &Dgr;</UP>G<SUB><UP>contact,H<SUB>2</SUB>O</UP></SUB><UP> + m</UP>[<UP>urea</UP>] (Eq. 6)

For comparison of data sets it was convenient to fit the dissociation curves to a unimolecular equilibrium,
<UP>mfc = 1/1 + </UP>e<SUP>((<UP>&Dgr;G + </UP>m[<UP>urea</UP>])<UP>/</UP>RT)</SUP> (Eq. 7)
where mfc is mass fraction capsid, and Delta G is an empirically fit pseudo-energy term. Equation 7 was only used for visual comparison of curves.

Simulations of Dissociation Reactions-- Simulations were based on the mathematical model developed for assembly of a 30-subunit icosahedron developed in previous studies, where the subunits topologically resemble those of HBV (15). The system of equations included a trimeric nucleation step, as does HBV (14). The microscopic forward rate constant for nucleation was 103 (M s)-1 compared with 105 for elongation. The rates used in simulations were based on a crude fit to HBV assembly kinetics (not shown). Numerical integrations were calculated using BERKELEY MADONNA (Berkeley Software, Berkeley, CA) with the Rosenbrock numerical integration methods. For simulations, we simultaneously calculated the concentrations of all intermediates, capsid, and free subunit. The concentration of a given intermediate n, with n subunits, was calculated by numerical integration of rate equations (Equation 8).


[n]={k<SUB>nf</SUB>[n−<UP>1</UP>][<UP>1</UP>]<UP> − </UP>k<SUB>(n<UP> + 1</UP>)f</SUB>[n][1]}−k<SUB>nb</SUB>[n]+k<SUB>(n<UP> + 1</UP>)b</SUB>[n + 1] (Eq. 8)
In Equation 8, knf is the forward rate constant for production of n from an intermediate with (n - 1) subunits and monomer, 1. This rate constant includes statistical coefficients as previously detailed (13). Dissociation rates (knb) were calculated from the forward rate and the association constant KnA(KnA = knf/knb), where the association constant is a function of the number of intersubunit contacts, the per contact microscopic association constant (see Equation 4), and a statistical factor reflection reaction degeneracy (13). Kcontact observed for HBV is 200-2000 M-1, depending on conditions. The Kcontact for the simulations described in this paper was varied from 20 to 10,000 M-1. Equation 8 emphasizes that association reactions are dependent on significant concentrations of intermediates and monomers, whereas the dissociation rate is strongly affected by the number of bonds to be broken.

For initial simulations, we assumed that all contacts made by a given subunit were formed or broken simultaneously. To mimic independent breakage of contacts on a single subunit, we inserted a probability factor; for an intact capsid with 30 tetravalent subunits forming 60 contacts, the odds of a second contact breaking on a subunit with one already broken contact were 3/59. For computational simplicity in these simulations, we adopted a simplest case rule requiring that two contacts be broken to release a subunit. Light scattering for simulations was calculated by assuming that the contribution of each intermediate to the signal was proportional to its mass and concentration, without any size- or shape-dependent form factor.

    RESULTS
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ABSTRACT
INTRODUCTION
EXPERIMENTAL PROCEDURES
RESULTS
DISCUSSION
REFERENCES

Capsid Dissociation Displays Hysteresis-- In agreement with model studies, we have observed that capsid assembly is very concentration-dependent (14, 17). This is expected for any system that incorporates 120 independent subunits. Simulations show that assembly reactions approach equilibrium rapidly. In practice, we found that HBV assembly reactions equilibrated within 24 h (17), which allowed us to calculate the per contact association energy, Delta Gcontact.

Based on assembly studies in 0.15 M NaCl (17), we expected that dilution of Cp1492 capsids would lead to a broad dissociation transition between about 15 and 40 µM total Cp1492. The midpoint of the transition, with 50% capsid, was expected at 20 µM total protein and nearly complete dissociation was expected below 14 µM. However, we observed little dissociation of capsids even after incubation in 0.15 M NaCl, 21 °C for up to 5 days (Fig. 1). A small amount of free Cp1492 was present in all samples, but the mass fraction of capsid remained approximately constant at >90% to at least 5 µM total protein. What little dimer was observed after 5 days was essentially identical to concentrations observed after 2 days (data not shown). Even at 0.5 µM, the lowest protein concentration examined in this study, less than a third of the protein was free dimer. In contrast, assembly studies led to the prediction that dissociation would be nearly quantitative after dilution to less than 15 µM. Our data give the appearance that the concentration-dependent dissociation transition will occur at lower initial [capsid], but we could not reliably measure these very low concentrations. These results are a first demonstration of hysteresis with HBV; that is, capsid dissociation and assembly reactions are not in equilibrium.


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Fig. 1.   Perturbing equilibrium by dilution reveals hysteresis of dissociation. Purified capsids were diluted into 25 mM HEPES, pH 7.5, 150 mM NaCl, and 2 mM dithiothreitol. After 5 days at 21 °C, were assayed for dissociation by SEC (open circles). The assembly isotherm predicted for these conditions from assembly studies (17) is shown as a solid line.

Denaturants Induce Two Transitions-- An alternative approach to measuring stability of a protein complex is with chaotropes such as GuHCl and urea. The effect of GuHCl on Cp1492 capsids was measured by intrinsic tryptophan fluorescence, LS, SEC, and circular dichroism (CD). After normalization for denaturant-dependent changes in LS and fluorescence signal (see "Experimental Procedures"), titrations observed by fluorescence, LS, and SEC measurements were coincident between 0 and 4 M GuHCl (Fig. 2A). SEC measurements showed a dissociation transition between 1 and 2 M GuHCl during which the concentration of Cp1492 increased at the expense of capsid. Only two major species, capsid and Cp1492, were observed throughout the dissociation transition. In light of the SEC observations, the concurrent decreases in LS, which is proportional to the average molecular weight, and fluorescence can be interpreted in terms of fraction of capsid (see "Experimental Procedures," Equation 1). Although fluorescence intensity decreased by a factor of 2 during dissociation, the emission maximum remained at 324 nm. This suggests that the environment of the tryptophan residues did not change dramatically through the transition and that Cp1492 remained folded (Fig. 2B). CD spectra support this last assertion; no significant change in secondary structure was observed between 0 and 2.5 M GuHCl (Fig. 2C); there is a small difference between CD spectra of capsid and dimer (8).


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Fig. 2.   Fluorescence, light scattering, and SEC allow observation of HBV capsid dissociation by denaturants. A, GuHCl-induced dissociation curves of 1.25 µM capsid protein in 0.3 M NaCl monitored by tryptophan fluorescence (circles), light scattering (triangles), and SEC (squares). The line through the data represents the curve fit for a unimolecular reaction. All three techniques are in excellent agreement in describing capsid dissociation. Tryptophan emission (B) and circular dichroism (C) spectra of Cp1492 in the presence of 0, 2.5, 3, and 5 M GuHCl and 0.3 M NaCl indicate that there is a second transition characterized by a "red shift" in fluorescence and loss of secondary structure.

Based on SEC and LS, there was no capsid remaining beyond 2.5 M GuHCl. At higher GuHCl, Cp1492 undergoes a second transition observable by fluorescence (Fig. 2B) and CD (Fig. 2C). From 3 to 5 M GuHCl, the fluorescence intensity increased, and the emission maximum shifted from 324 to 340 nm (Fig. 2B). The red shift in fluorescence suggests a progressive solvent exposure of tryptophans. Over this same interval, the CD spectra indicated a progressive loss of alpha  helical secondary structure (Fig. 2C). These data indicate that two transitions, dissociation and unfolding, take place at different ranges of GuHCl concentrations. They also demonstrate the utility of fluorescence, LS and SEC for observation of dissociation, and the value of fluorescence for discerning dissociation from denaturation.

We found a similar separation of dissociation and denaturation curves using urea as the denaturant. Urea allows us to examine dissociation at constant ionic strength, which is significant because NaCl concentration is known to influence capsid stability (8, 17). As with GuHCl, the LS and fluorescence data from urea titrations were superimposable (Fig. 3, A and B). The correspondence of dissociation by urea with the transition observed by fluorescence and LS was again corroborated by SEC analysis (data not shown).


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Fig. 3.   Capsid dissociation displays much weaker concentration dependence than predicted from assembly reactions. The mass fraction capsid in 0.3 M NaCl is shown for 2 µM (circles), 4 µM (triangles), 8 µM (squares), and 16 µM (diamonds) protein, as calculated from fluorescence (A) and light-scattering data (B). For ease of comparison, the data are fit to a unimolecular transition, where the same family of curves is used for both fluorescence and light-scattering data. C, observed concentration dependence of dissociation (solid lines with symbols) is compared with the concentration dependence predicted for the 120th order reaction (dashed lines) for the lowest (2 µM, circles) and the highest (16 µM, diamonds) protein concentrations used. The experimental data in panel C are from panel A of this figure.

Chaotrope Studies Also Indicate Hysteresis of Dissociation-- Chaotrope titrations allowed investigation of the concentration dependence of dissociation as well as capsid stability. Capsid stability is more easily understood in terms of the average bimolecular association constant between pairs of subunits, Kcontact, than the overall association constant based on the Kcapsid, which is in units of M-119 (see "Experimental Procedures," Equations 2 and 4). From assembly studies, we know that Kcontact is very weak at 21 °C and 0.3 M NaCl, corresponding to a Delta Gcontact of -3.7 kcal mol-1 (17).

The value of Delta Gcontact,H2O was determined from urea titrations by measuring Delta Gcontact at different urea concentrations and extrapolating to 0 M urea (19, 20). For these extrapolations, a single m value of 1.04 ± 0.19 (kcal mol-1)/[urea] was used. This m value is the average from 15 titrations at 4 protein concentrations measured by fluorescence and LS (Fig. 3, A and B). The m value has been related to the change in the amount of hydrophobic surface exposed in the unfolding (or dissociation) transition (22), which should be independent of concentration or ionic strength. Note that the data presented (Fig. 3) are in terms of mass fraction of capsid; at the midpoint of the transition, when capsid and dimer mass fraction are equal for 2 µM total Cp1492, Delta Gcontact is -3.8 kcal mol-1 (see Equations 3-5).

The calculated Delta Gcontact,H2O was roughly twice the value determined in assembly studies (Table I) (17). The Delta Gcontact,H2O determined from urea dissociation of 1.25 µM Cp1492 in 0.15 M NaCl was -5.8 kcal mol-1. This result corresponds well with the data in Fig. 1, where capsid was dissociated by dilution. This substantial difference between association energy measured in assembly reactions and by dissociation further demonstrates hysteresis.


                              
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Table I
Delta Gcontact from dissociation and assembly experiments
Delta Gcontact,H2O is extrapolated from the midpoint of dissociation curves using an m-value of 1.04 for all experiments; the error shown in this table reflects only uncertainty in determining the midpoint. Delta Gcontact,assembly values from Ceres and Zlotnick (17) were concentration-independent. ND, not determined.

Unlike Delta Gcontact from assembly studies (17), Delta Gcontact,H2O was not constant at different protein concentrations. Purified Cp1492 capsids at 2, 4, 8, and 16 µM concentrations in 0.3 M NaCl were incubated with urea (Fig. 3, A and B). The systematic variation of Delta Gcontact indicates that the dissociation reaction was not well represented by an equilibrium between capsid and 120 dimers. This is demonstrated by comparing calculated and observed dissociation curves. Given a Delta Gcontact,H2O and an m value, it is straightforward to back-calculate the mass fraction of capsid at any urea concentration (Fig. 3C, dashed lines). The dissociation curve calculated for 2 µM Cp1492 fit that data. However, the curve calculated for 16 µM Cp1492 using the Delta Gcontact,H2O derived from the 2 µM data does not fit. In summary, HBV capsid dissociation can be induced by chaotropes, but it does not show the predicted concentration dependence expected for assembly of a 120-mer (Equation 2). Therefore, although the dissociation energies determined in urea titrations are much greater than those determined from assembly, they are at least consistent with an effort to evaluate dissociation by dilution (Fig. 1).

Examination of HBV dissociation kinetics was not straightforward. Dissociation by urea or GuHCl was largely complete within a few hours; there was no measurable difference between the degree of dissociation measured at 24 and 48 h (data not shown). Early times in a dissociation reaction could be observed by fluorescence and LS; however, these results could not be interpreted because of the undeterminable relative contributions of capsids and intermediates to the signal (see results of simulations, this section). In contrast, assembly reactions have very low concentrations of intermediates (13).

Capsid Dissociation Is Reversible-- Equilibrium between HBV capsid and dimer implies a flux between the two states. This leads to the predictions that capsids will assemble in urea and that dissociation, like assembly, will be ionic strength-dependent (8, 14, 17). Conversely, if capsids assemble irreversibly, one would expect dissociated dimers to be inert and that there would be no ionic strength dependence of dissociation.

Urea-induced capsid dissociation was observed at 0.15, 0.325, and 0.5 M NaCl (Table I). The midpoint of dissociation in these reactions was influenced by the ionic strength, varying from 1.75 to 3.25 M urea, although curve shape was not changed. The m value of 1.04 (kcal mol-1)/[urea] fit data at all three salt concentrations. The effect of [NaCl] on dissociation and assembly was qualitatively the same. This suggests that dissociation was inhibited by high salt in the same manner that assembly is induced by it.

Cp1492 derived from urea-dissociated capsid readily reassembles. In fact, urea dissociation and reassociation are part of the purification protocol for assembly active Cp1492 (8, 10, 14). We determined the Delta Gcontact in the presence of urea from reassembly experiments for comparison to values determined from dissociation. Assembly of 10 µM Cp1492 in varying concentrations of urea was driven by the addition of NaCl to 0.3 M (Fig. 4). We found that 0.3 M NaCl was sufficient to drive assembly of an appreciable amount of capsids in 0 to 0.5 M urea. The observed concentrations of capsid and dimer in 0 M urea, 2.7 and 2.3 µM, respectively, were in good agreement with the concentrations expected for these assembly conditions (17). At urea concentrations up to 1 M, some capsid was detectable but could not be reliably quantified. The effect of [urea] on Delta Gcontact observed in these assembly experiments corresponds to an m value of ~0.4 (kcal mol-1)/[urea], much less than the 1.04 (kcal mol-1)/[urea] value from the dissociation data. These data demonstrate that capsids are stable at urea concentrations that are too high to support de novo assembly, another example of a hysteresis cycle.


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Fig. 4.   Capsid assembly in increasing concentrations of urea. Capsid assembly reactions were performed with 5 µM Cp1492 in 50 mM Tris-HCl, pH 7.5, 0.3 M NaCl in the presence of varying concentrations of urea (open circles, solid line). Mass fraction capsid was calculated from SEC analysis of samples. For comparison, we include a dissociation curve (dashed line) for 4 µM Cp1492 in the same solution conditions.

Simulations of Dissociation Kinetics Predict Hysteresis-- By themselves, the preceding results only show that association and dissociation reactions are distinctly different and that dissociation does not conform to most expectations for a reaction approaching equilibrium. Assembly does conform to those expectations (17). The difference between association and dissociation can be reconciled by examining dissociation of a 30-subunit model. We found that simulations of dissociation reactions allowed us to identify reaction features that resulted in hysteresis, suggesting that the source of the hysteresis is kinetic.

We initially expected that capsids would collapse as soon as they began the process of dissociation. A simple analogy is the effect of removing of one brick from an arch. Contrary to this expectation, the estimated LS calculated from dissociation simulations did not generate first order kinetics (Fig. 5A). A closer inspection of the kinetic trajectories for [capsid] shows that the decrease in concentration was not monotonic. Initially, [capsid] decreased rapidly; then [capsid] increased slightly; then there was a gradual decline (Fig. 5B). An advantage to model reactions is that all populations can be examined. What has happened in these dissociation simulations? A population of 30-mer capsids dissociated to generate a variety of intermediates and free subunit. For some intermediates, in particular the 29-mer, the resulting concentration of intermediate and free subunit was sufficiently high that reassociation was favored over dissociation. The dissociation of the 29-mer was delayed and partially prevented by competition with reassembly.


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Fig. 5.   Simulations predict trapped dissociation kinetics. The kinetics of capsid dissociation (in terms of assembled subunit) for different Delta Gcontact were calculated for reactions beginning with complete 30-mer capsid. A, simulations of kinetics (solid lines) show that kinetics were not first order (dashed line, fit to Delta Gcontact = -2.3 kcal mol-1 simulation). B, a close-up of capsid concentration from dissociation simulations is shown. All simulations began with 2 µM subunit, all in capsid form.

As a result of this "local" barrier to dissociation, these simulations were very slow to reach equilibrium. Simulations of assembly reactions rapidly reached equilibrium; there was no divergence between calculated equilibrium concentrations of capsid and subunit for a given Delta Gcontact and the concentrations observed after a 24-h simulation (see Fig. 6). Shorter simulations of assembly (~2 h) resulted in a close approximation of the equilibrium value (not shown); the presence of a stochastic equilibrium between intermediates and capsid has been demonstrated by the effect of altering the dissociation rate (14, 15). However, simulations of dissociation reactions using the same model, beginning with "pure" capsid, showed an obvious hysteresis. Very long simulations marginally decreased the hysteresis.


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Fig. 6.   Simulations of denaturant titrations demonstrate hysteresis. A, the concentration of capsid (for 2 µM total subunit) is reported as a function of input Delta Gcontact. Assembly simulations result in a near equilibrium concentration of capsid (open symbols). Dissociation simulations, beginning with pure capsid, show much less dissociation (solid symbols). Reactions are shown for subunits where all contacts break simultaneously (circles) and where at least one contact must be broken previously in order for the subunit to dissociate (diamonds). B, simulations of association reactions for the 30 subunit model capsid show strong concentration dependence. C, simulations of dissociation show that the concentration dependence is attenuated by hysteresis. For both sets of calculations, simulations were performed for 2 µM (circles), 4 µM (squares), 8 µM (triangles), and 16 µM (diamonds) subunit.

The initial simulations of association and dissociation incorporated an assumption that all contacts made by a given subunit were made and broken simultaneously. There was a dramatic enhancement of hysteresis when we relaxed this assumption by allowing contacts to break independently of one another and requiring that a subunit have at least two broken contacts to dissociate (Fig. 6). This was modeled by incorporating a probability coefficient into the model (see "Experimental Procedures"). Because it did not alter contact energy, this two-contact rule had no significant effect on the rate or extent of association. In the simulations, the disparity between association and dissociation is attributable to hysteresis; it is not a result of insufficient equilibration time.

Dissociation of HBV (Fig. 3) and other viruses (4, 5) shows a much weaker than expected concentration dependence for dissociation. The hysteresis observed in simulations mirrored this effect (Fig. 6C). Simulations for assembly and dissociation were conducted for total subunit concentrations of 2, 4, 8 and 16 µM to parallel the experimental work. Instead of urea concentration, we varied the input Delta Gcontact to mimic the effect of urea. As expected, the assembly reactions showed a high degree of concentration dependence, whereas dissociation occurred at lower energies (corresponding to higher concentrations of urea) and reduced concentration dependence. The greater the hysteresis of dissociation, the weaker was its concentration dependence.

    DISCUSSION
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ABSTRACT
INTRODUCTION
EXPERIMENTAL PROCEDURES
RESULTS
DISCUSSION
REFERENCES

HBV capsids assemble from 120 copies of the dimeric capsid protein. Kinetic simulations predict that the assembly reaction will approach the concentrations predicted from reactant concentration and association energy, whereas dissociation will demonstrate marked hysteresis. Observations of assembly and dissociation support this prediction. Assembly reactions show the expected steep concentration dependence (17). Dissociation reactions do not reach the same concentration of products as assembly reactions nor do they show the expected concentration dependence. However, the free subunits collected from dissociation reactions are not denatured and are competent to reassemble. The per contact association energy estimated from denaturation studies yields a result that is consistent with dissociation by dilution. We conclude that hysteresis is isolated to the dissociation reaction.

We demonstrate the most likely source of hysteresis by taking advantage of kinetic simulations that were originally developed to describe assembly (13, 14, 17). Hysteresis is observed in simulations of dissociation but not association, although they are constrained to the same path. Consequently, hysteresis does not violate the principle of microscopic reversibility (23, 24). Kinetic simulations reveal dissociation to be a complex reaction where accumulation of temporally stable intermediates results in kinetic traps that impede the reaction, creating an energy barrier to dissociation. During dissociation, intermediates may reach concentrations that are orders of magnitude greater than at equilibrium; however, they may still be difficult to quantify experimentally. This last effect leads to the two-state appearance of assembly and dissociation reactions observed experimentally and in simulations.

An important concern with simple models is whether they can accurately reflect the behavior of a complex system. In dissociation simulations, we observed that the rate-controlling intermediates were nearly complete capsids (especially 29-mers). This is largely because they are slow to dissociate further but readily re-associate with excess monomer. Because all dissociation paths necessarily begin with intermediates of N-1 subunits, we suggest that our simplest case model of dissociation, with its single path, results in a reasonable representation more complex dissociation paths.

Association and dissociation reactions with their many possible paths can be considered in terms of an energy landscape (25-29). As a general rule, each successive intermediate in an association reaction is progressively more stable because a greater number of intersubunit contacts are formed (13, 14). The resulting landscape can be characterized as a steep downhill slope with capsid as the only significant minimum. The landscape for dissociation of the 30-mer model (and presumably HBV capsids) is very different. In order for the first few tetravalent subunits to dissociate from the capsid, three or four contacts must be broken; in contrast, the capsid stability is roughly proportional to the energy of two contacts (15). This last effect is analogous to the hysteresis observed in molecular dynamics simulations of a dimer (30). In our model studies, metastable intermediates contribute to a kinetic and energetic barrier for capsid dissociation. Early in dissociation, these species are present at concentrations that are high enough to favor reassembly, although these same conditions will not support de novo assembly. Dissociation proceeds because a fraction of the metastable intermediates dissociates irretrievably to smaller unstable forms, but is much slower than would be predicted for a simple two-state reaction.

Hysteresis is inherent in capsid dissociation. It arises from the closed geometry of an icosahedral particle, or any nominally symmetrical oligomer. It can be traced to intermediate reactions where association is favored over the globally favored dissociation. It is clear that hysteresis would be accentuated if subunits were constrained by a membrane or tethered by nucleic acid. Simulations described in this study also suggest that the independent non-cooperative breaking of intersubunit contacts may make a significant contribution to the hysteresis of dissociation. We deduce that hysteresis can also be increased by any mechanism that introduces differentially stabilized intermediates that would form additional minima in the energy landscape of dissociation.

Weber et al. (31) suggest an alternative explanation for hysteresis by postulating that populations of capsids (and other multimers) dissociate irreversibly. Irreversible dissociation would lead to a relatively narrow transition that could be broadened if the oligomer were heterogeneous and each subspecies had a slightly different stability. In the case of HBV, we have shown that dissociation is reversible. Any heterogeneity of the capsids would broaden the dissociation transition and actually decrease the magnitude of the calculated association energy.

We suggest that hysteresis has a biological role. Capsid proteins may associate with modest per contact energy to minimize kinetic trap formation during assembly (13, 14) yet still yield a stable particle. The hysteresis effect would then preserve capsids under non-ideal conditions, for example, between hosts or at low concentrations within a new host. This scenario is appealing because the combination of weak inter-subunit contact energy combined with independent disruption of contacts can lead to the breathing observed in capsids of several viruses (32-34), where intersubunit contacts break and re-anneal to transiently expose buried protein segments to antibodies and proteases. Thus, capsids remained stable even where individual contacts were not. Hysteresis can mask the underlying fragility of a virus. A particularly revealing observation that supports this argument is that poliovirus receptor acts as transition state catalyst to lower the activation energy for conversion of 160 S poliovirus to an infectious intermediate (35). This final point also suggests that where there is hysteresis, a trigger may be required for uncoating, giving the virus an important regulatory control.

    ACKNOWLEDGEMENTS

We thank Pablo Ceres and Robert Turner for contributions to this work, Stephen Stray and Jennifer Johnson for critical reading of the manuscript, and Paul Wingfield and Stephen Stahl for helping us get started on this project.

    FOOTNOTES

* This research was supported by American Cancer Society Grant RSG-99-339-04-MBC.The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Dagger To whom correspondence should be addressed. Tel.: 405-271-9030; Fax: 405-271-3910; E-mail: adam-zlotnick@ouhsc.edu.

Published, JBC Papers in Press, March 13, 2003, DOI 10.1074/jbc.M211408200

    ABBREVIATIONS

The abbreviations used are: HBV, hepatitis B virus; SEC, size exclusion chromatography; GuHCl, guanidine HCl; LS, light scattering.

    REFERENCES
TOP
ABSTRACT
INTRODUCTION
EXPERIMENTAL PROCEDURES
RESULTS
DISCUSSION
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