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INTRODUCTION |
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.
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EXPERIMENTAL PROCEDURES |
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
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),
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(Eq. 1)
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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.
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(Eq. 2)
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(Eq. 3)
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(Eq. 4)
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Gcontact (Equation 5) was calculated
from Kcontact. By assuming a linear dependence
of
Gcontact on urea concentration (19), we
extrapolated to 0 M denaturant using Equation 6, where
m is determined empirically from plots of
Gcontact versus urea concentration (19, 20). Our estimated error for
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).
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(Eq. 5)
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(Eq. 6)
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For comparison of data sets it was convenient to fit the
dissociation curves to a unimolecular equilibrium,
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(Eq. 7)
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where mfc is mass fraction capsid, and
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).
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(Eq. 8)
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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.
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RESULTS |
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,
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.
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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.
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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
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.
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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
Gcontact of
3.7 kcal mol
1
(17).
The value of
Gcontact,H2O was
determined from urea titrations by measuring
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,
Gcontact is
3.8 kcal
mol
1 (see Equations 3-5).
The calculated
Gcontact,H2O was
roughly twice the value determined in assembly studies (Table
I) (17). The
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
Gcontact from dissociation and assembly experiments
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.
Gcontact,assembly values from Ceres and Zlotnick
(17) were concentration-independent. ND, not determined.
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Unlike
Gcontact from assembly studies (17),
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
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
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
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
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
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.
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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 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
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.
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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
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 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.
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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
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 |
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.