Section on Physical Biochemistry, Laboratory of Biochemistry and
Genetics, National Institute of Diabetes and Digestive and Kidney
Diseases, National Institutes of Health, Bethesda, Maryland 20892-0830 USA
An equilibrium statistical-thermodynamic model for the
effect of volume exclusion arising from high concentrations of stable macromolecules upon the stability of a trace globular protein with
respect to denaturation by heat and by chaotropes is presented. The
stable cosolute and the native form of the trace protein are modeled by
effective hard spherical particles. The denatured state of the trace
protein is represented as an ensemble of substates modeled by random
coils having the same contour length but different rms end-to-end
distances (i.e., different degrees of compaction). The excess or
nonideal chemical potential of the native state and of each denatured
substate is calculated as a function of the concentration of stable
cosolute, leading to an estimate of the relative abundance of each
state and substate, and the ensemble average free energy of the
transition between native and denatured protein. The effect of the
addition of stable cosolute upon the temperature of half-denaturation
and upon the concentration of chaotrope required to half-denature the
tracer at constant temperature is then estimated. At high cosolute
concentration (>100 g/l) these effects are predicted to be large and
readily measurable experimentally, provided that an experimental system
exhibiting a fully reversible unfolding equilibrium at high total
macromolecular concentration can be developed.
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INTRODUCTION |
The equilibria and rates of a variety of
macromolecular reactions, including self- and heteroassociation,
condensation, and surface site binding, have been shown to be
significantly affected by volume exclusion in solutions of high total
macromolecular content, commonly referred to as "crowded" solutions
(Minton, 1997
; Zimmerman and Minton, 1993). On theoretical grounds,
volume exclusion would also be expected to affect macromolecular
isomerization in general (Minton, 1981, 1983) and protein denaturation
in particular, as an important example of isomerization. Zhou and Hall
(1996) (henceforth referred to as ZH) recently presented a
statistical-thermodynamic model for the effect of cosolute excluded
volume on the denaturation of proteins. According to their model, high
concentrations of larger volume-excluding cosolutes would stabilize
proteins against denaturation, while high concentrations of smaller
cosolutes would destabilize them. The purpose of the present
communication is to present an alternative model for the effect of
cosolute excluded volume on protein stability, to compare and contrast
the results of the new model with that of ZH, and to estimate the
magnitude of the excluded volume effect upon experimentally measured
parameters of thermal and isothermal denaturation. In the following
section we present a simple thermodynamic model for denaturation in
nonideal solutions. Next, a molecular structure-based
statistical-thermodynamic model is introduced to enable quantitative
estimation of the energetic consequences of volume exclusion.
Calculations are carried out for a model trace protein undergoing
reversible denaturation in the presence of arbitrary concentrations of
a second, stable model protein. The effect of volume exclusion on
experimentally measurable properties of the model protein solution is
estimated, and it is shown that they should be readily measurable.
Finally, results of the present model are compared with those obtained
from the model of ZH, and the applicability of the present model to the analysis of unimolecular DNA condensation is considered.
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THERMODYNAMIC MODEL FOR THE EFFECT OF A COSOLUTE ON THE
DENATURATION OF A TEST PROTEIN1 |
Consider a solution containing a test protein at low concentration,
together with a single inert cosolute, denoted S, at arbitrary concentration cs. We model denaturation of the
test protein as a reversible transition between a single globular
folded or native state and an ensemble of unfolded or denatured
states2
with chemical potentials given by
|
(1)
|
and
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(2)
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The chemical potential of all denatured states is defined relative
to that of a single standard state described below.3
Because at equilibrium µN = µDi
for all i, we may write the thermodynamic constant for
equilibrium between the native and any given denatured species,
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(3)
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and define an apparent equilibrium constant,
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(4)
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Defining the total concentration of denatured protein as the sum
over the denatured ensemble, Eq. 4 yields
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(5)
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Thus the denaturation model may be regarded as an effective
two-state model with
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(6)
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where fD denotes the fraction of tracer
protein that is denatured, as monitored via experiment, and
D is an apparent activity coefficient for the denatured
"state," given by
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(7)
|
We may therefore refer to the ensemble of D states as "the D
state," keeping in mind that the (ensemble average) properties of the
D state so defined are variable. The explicit inclusion of a functional
dependence of K on {c} serves to remind us
that the apparent equilibrium constant, unlike the thermodynamic
equilibrium constant, may vary with the concentrations of all solute
species present through the composition dependence of N
and Di.
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STATISTICAL-THERMODYNAMIC MODEL FOR THE DEPENDENCE OF ACTIVITY
COEFFICIENTS ON THE CONCENTRATION OF INERT COSOLUTE |
According to the McMillan-Mayer theory of solutions (McMillan and
Mayer, 1945), the activity coefficient of macrosolute species X may be
calculated as functions of the potential of mean force (POMF) acting
between a molecule of X and one, two, and higher numbers of other
macrosolute molecules of the same and other species in the solution. It
has been found that under conditions such that long-range electrostatic
interactions between protein molecules are damped out, the POMF acting
between native, globular protein molecules is well approximated by a
hard particle potential acting between rigid convex bodies, the size
and shape of which resemble those of the actual molecule viewed at low
resolution (Minton, 1983; Minton and Edelhoch, 1983; Ross et al., 1978;
Ross and Minton, 1977). Hard particle models have been found to account
quantitatively for the dependence of the thermodynamic activity of
compact globular proteins upon solute concentrations up to several
hundred grams per liter (Minton, 1983, 1995; Minton and Edelhoch, 1983;
Ross et al., 1978). Therefore, for the purpose of the model
calculations to follow, we shall represent both the inert cosolute
molecule and the native state of the test protein as rigid spheres of
fixed radii rs and rN,
respectively.4 Denatured states of the test protein will be
represented as random coils of varying extent of compaction, either
without internal excluded volume (a Brownian walk) or with internal
excluded volume (a self-avoiding walk). This model is depicted
schematically in Fig. 1.
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FIGURE 1
Schematic depiction of the equilibrium between the
native state N of the tracer (shaded sphere) and the
unfolded state Di (coil) in the presence of
stable cosolute S (open spheres). The dashed border around
the coil indicates the time-averaged convex hull of the Brownian walk
representing the coil, assumed to define an effective hard particle
excluding volume to molecules of S.
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Activity coefficient of N
The excess chemical potential of a rigid macrosolute species 1 at
limiting low concentration, in a solution containing an arbitrary
concentration of macrosolute species 2, may be calculated in the hard
particle approximation, using the following relation obtained from the
scaled particle theory of hard particle fluid mixtures (Boublík,
1974):
![<UP>ln</UP> &ggr;<SUB>1</SUB>=<FR><NU>&Dgr;A<SUB>1</SUB></NU><DE>kT</DE></FR>=<UP>−ln</UP>(1−&rgr;<SUB>2</SUB>V<SUB>2</SUB>)+[H<SUB>1</SUB>S<SUB>2</SUB>+S<SUB>1</SUB>H<SUB>2</SUB>+V<SUB>1</SUB>]](/content/vol78/issue1/fulltext/101/img009.gif)
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(8)
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where 
Ai is the change in the
(Helmholtz) free energy of the system associated with the introduction
of a molecule of species 1, k is Boltzmann's constant,
2 denotes the number density of species 2 (proportional
to w/v concentration w2), and
Hi, Si, and
Vi denote, respectively, the Kihara supporting
function (Kihara, 1953), surface area, and volume of the equivalent
convex particle representing species i. Because both N and S
are represented by equivalent hard spheres, ln N is
given by Eq. 8 with H1 = rN, S1 = 4
rN2, V1 = 4rN3/3,
H2 = rs,
S2 = 4rs2, and
V2 = 4rs3/3
(Minton, 1998).
Activity coefficient of Di
The ith denatured "state" is treated as a Brownian
or self-avoiding walk with a rms end-to-end distance
hi. We define the standard state of all
Di to be the isolated walk (i.e., solvated, but without
intersolute interaction) with a rms end-to-end distance h0, at unit concentration and standard
temperature and pressure. To a first approximation, the nonideal
contribution to the chemical potential (or excess chemical potential)
of tracer may be partitioned into a term deriving from intermolecular
interaction between Di and S, and a term deriving from the
configurational free energy of the isolated molecule of Di
with fixed center of mass:
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(9)
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The magnitude of µDiNI,intermol, the
equilibrium average excess free energy arising from repulsive
interactions between a molecule of Di and all molecules of
S, may be approximated by Eq. 8, where the effective particle
representing species 1 is the convex hull of the Brownian walk, with
H1 = (2/3)1/2hi, S1 = (2/3)hi2, V1 = 4(2/3)1/2hi3/27
(Jansons, 1991; Jansons and Phillips, 1990), and H2,
S2, and V2 are as given
above.5
Because the standard states of all Di are defined as the
unperturbed random coil with rms end-to-end distance
h0, µDiNI,conf
represents the difference between the equilibrium average free energies
of isolated random walks of identical contour length, constrained to
rms end-to-end distances hi and
h0, respectively. It arises from differences in
configurational entropy and, if present, intramolecular excluded
volume. The magnitude of µDiNI,conf is
estimated as follows. Jaeckel and Dayantis (1994a,b) have carried out
Monte Carlo calculations of the effect of confinement in a spherical
volume of radius R upon the rms end-to-end distances of
Brownian and self-avoiding walks and upon the entropy per unit volume
of the walks. Their results may be expressed in units that are scaled
to the rms end-to-end distance of the unrestricted walk
h0. Letting 
= R/h0 and 
= h/h0, the
effect of compression on the rms end-to-end distance of the walk is
well described by the empirical relation
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(10)
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over the range 0.4 
4, where
P1 ... P4 are equal to 0.072713, 0.045978, 0.25152, and 
0.062739, respectively, for a Brownian walk,
and 0.091936, 0.17012, 0.33811, and 0.076614, respectively, for a
self-avoiding walk. The entropy of the walk, relative to that for the
unconstrained walk (corrected for change in volume of the confining
sphere, hence appropriate for a fixed center of mass), is well
represented by the empirical function
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(11)
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over the range 0.4 4, where
Q1 ... Q4 = 0.88087, 1.6725, 1.1652, and 0.3211 for a Brownian walk, and approximately the same
for a self-avoiding walk. Assuming that the compression is essentially
athermal, one may calculate the dependence of
µNI,conf/kT = S/k on from Eqs.
10 and 11; this relationship is plotted in Fig.
2. As expected, the self-avoiding walk is
less compressible than the Brownian walk because of the presence of
intramolecular excluded volume.
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FIGURE 2
The excess chemical potential of an isolated molecule
of tracer in state Di, plotted as a function of
i. The solid line is calculated for a Brownian walk
representation of Di, the dashed line for a self-avoiding
walk representation.
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Equations 4 and 9 may be combined to yield the probability that a
molecule of denatured protein will be present in state Di at any particular concentration of cosolute S:
![p<SUB><UP>i</UP></SUB>(&eegr;<SUB><UP>i</UP></SUB>, c<SUB><UP>S</UP></SUB>)=<FR><NU><UP>exp</UP>[<UP>−</UP>&mgr;<SUP><UP>NI</UP></SUP><SUB><UP>D<SUB>i</SUB></UP></SUB>(c<SUB><UP>S</UP></SUB>, &eegr;<SUB><UP>i</UP></SUB>)]</NU><DE><LIM><OP>∑</OP><LL><UP>j</UP></LL></LIM> <UP>exp</UP>[<UP>−</UP>&mgr;<SUP><UP>NI</UP></SUP><SUB><UP>D<SUB>j</SUB></UP></SUB>(c<SUB><UP>S</UP></SUB>, &eegr;<SUB><UP>j</UP></SUB>)]</DE></FR>.](/content/vol78/issue1/fulltext/101/img014.gif)
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(12)
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A "REALISTIC" SIMULATION OF THE EQUILIBRIUM DENATURATION OF A
LABILE PROTEIN IN THE PRESENCE OF AN INERT STABLE PROTEIN |
As an illustration of the magnitude of excluded volume effects one
may expect to observe experimentally, we calculate the effect of the
addition of large concentrations of a model protein with the
approximate size and shape of ribonuclease A (M = 13,000) upon the native-denatured equilibrium of a second, dilute
model protein with the approximate size and shape of 
-lactalbumin
(M = 17,000). The native states of both RNAse A and
-lactalbumin are compact and quasispherical, the maximum dimension
of each being no greater than ~1.5 times the minimum dimension
(Acharya et al., 1991; Tilton et al., 1992). For purposes of
calculating excluded volume, such bodies may be well represented by an
equivalent hard sphere (Minton, 1998). The volume of an equivalent
convex particle representing a protein is assumed to be equal to the volume of solvent displaced by the protein, which is given by the
partial specific or partial molar volume. Thus equivalent hard sphere
radii rS and rN are
evaluated using
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(13)
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where Mx and
x denote the molar mass and partial
specific volume of X, and NA denotes Avogadro's
number. If it is assumed that
N = S 
0.72 cm3/g, a value
typical of most compact globular proteins (see Appendix 2 of Attri and
Minton, 1983), then rS = 1.54 nm and
rN = 1.72 nm. The rms end-to-end distance
of the unperturbed random coil representing the standard state for all
Di, h0, is taken to be 7.3 nm = 
rg, where rg
is the experimentally measured radius of gyration of fully denatured
-lactalbumin, 3 nm (Kataoka et al., 1997).
The value of ln Di = µDiNI/kT, calculated for both
Brownian and self-avoiding walks, using Eqs. 8-11, is plotted in Fig.
3 as a function of i = hi/h0 for values of
ws, the w/v concentration of S, ranging between
0 and 400 g/l. The value of
pi(hi,
cs) was calculated for a high density of discrete
states with 0.35 1.0, using Eq. 12 and the results
shown in Fig. 2. In the limit of high state density, the value of
pi(i,
cs) approaches that of the continuous probability
distribution p(, cs)d, which is plotted as a function of and ws for both Brownian and
self-avoiding walks in Fig.
4.6 Calculated distributions
for Brownian and self-avoiding walks do not differ qualitatively, and
the quantitative differences between them reflect the greater ease of
compression of the Brownian walk noted earlier. As the concentration of
S increases above ~200 g/l, the most probable value of begins to
decrease from 1 toward a value of 0.55-0.6 at
cs = 400 g/l.
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FIGURE 3
The excess chemical potential of Di plotted
as a function of i for varying concentrations of stable
cosolute S. Solid lines are calculated for a Brownian walk
representation of Di, dashed lines for a self-avoiding walk
representation of Di. Each family of curves represents the
calculated dependence for ws = 0, 10, 20, ... , 400 g/l from bottom to top.
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FIGURE 4
Normalized probability distribution of states of D with
normalized rms end-to-end distance at fixed
ws, plotted as a function of and
wS. Calculations are performed for D represented
as a Brownian walk (A) and as a self-avoiding walk
(B).
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D was calculated for both Brownian and self-avoiding
walks according to Eq. 7. The dependence of log N and
log D upon ws is plotted in Fig.
5. While the addition of S destabilizes
both N and D in an absolute sense (as would be expected in the case of
pure volume exclusion interactions), the more compact N is stabilized
relative to the less compact ensemble of D states. The dependence of
log(K/K°) upon ws, calculated
according to Eq. 6, is plotted in Fig.
6.7
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FIGURE 5
Common logarithm of the activity coefficient of tracer
in the native state N (solid line) and the average denatured
state D (Brownian walk representation, dotted line;
self-avoiding walk representation, dashed line) plotted as a
function of the concentration of stable cosolute S.
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FIGURE 6
Dependence of K, the equilibrium constant
for unfolding, upon the concentration of stable cosolute S, calculated
for two representations of the average denatured state (Brownian walk,
solid line; self-avoiding walk, dotted line).
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The basic qualitative conclusion derived from this calculation is that
the introduction of sufficiently large concentrations of a stable
globular protein of molar mass comparable to that of a labile trace
protein can reduce the equilibrium constant for denaturation of the
labile protein by between one and two orders of magnitude. Although the
specific model presented here is simplified and heuristic rather than
rigorous, particularly with respect to treatment of the denatured
ensemble, the major excluded volume contributions seem to be taken into
account in a reasonable fashion. Thus we believe that the predicted
order of magnitude of the excluded volume effect is likely to be correct.
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ESTIMATE OF THE MAGNITUDE OF THE EFFECT OF VOLUME EXCLUSION ON
THERMAL DENATURATION |
Within the context of the effective two-state model for
denaturation (Eq. 6), the dependence of the equilibrium constant for unfolding upon temperature may be described by the van't Hoff equation,
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(14)
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where H° represents the standard
state enthalpy change for the transition N 
D. Consider an interval
of temperature within which H° is approximately
independent of temperature. It follows from Eq. 14 that an isothermal
change in the equilibrium constant log K at any
temperature within this region will result in a change in
T50, the temperature at which the protein is
half-denatured (fD = 0.5, K = 1), given by
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(15a)
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Equation 15a may be generalized to the case of a
temperature-dependent H°:
![<FR><NU>1</NU><DE>T<SUP>(2)</SUP><SUB>50</SUB></DE></FR>−<FR><NU>1</NU><DE>T<SUP>(1)</SUP><SUB>50</SUB></DE></FR>=2.303R <LIM><OP>∫</OP><LL><UP>log</UP> K<SUP>(1)</SUP></LL><UL><UP>log</UP> K<SUP>(2)</SUP></UL></LIM> <FR><NU>d <UP>log</UP> K</NU><DE>&Dgr;H°[T<SUB>50</SUB>(<UP>log</UP> K)]</DE></FR>,](/content/vol78/issue1/fulltext/101/img020.gif)
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(15b)
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where H° is calculated as a function of
temperature and heat capacity change Cp via
the thermodynamic relation
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(15c)
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The dependence of T50 on log
K, calculated using Eqs. 15b, c for various values of
H° and Cp over ranges
spanning the majority of measured enthalpy and heat capacity changes
for protein denaturation (Pfeil, 1986), is plotted in Fig.
7. It may
be seen that when K is reduced by one to two orders of
magnitude, T50 is predicted to increase by
5--20°C.8
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FIGURE 7
Dependence of the temperature of half-denaturation upon
the change in K at constant temperature induced by excluded
volume, calculated for three values of H° (solid
line, 200 kJ/mol; dashed line, 300 kJ/mol; dotted
line, 400 kJ/mol). The upper curve of each line type was
calculated using Eqs. 15a-c with Cp = 0, and the lower curve was calculated with
Cp = 10 kJ/mol-°C.
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FIGURE 8
(A) Dependence of K for the
unfolding of the -subunit of tryptophan synthase upon urea
concentration at constant temperature, obtained by transformation of
the data of Ogasahara et al. (1993) as described in text.
(B) Change in the urea concentration of half-denaturation
compensating for change in K resulting from volume
exclusion, calculated as described in text.
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ESTIMATE OF THE MAGNITUDE OF THE EFFECT OF VOLUME EXCLUSION ON
ISOTHERMAL DENATURATION BY CHAOTROPES |
The dependence of protein denaturation on chaotrope concentration
may best be illustrated by an example taken from the literature. Ogasahara et al. (1993) have reported the fraction of tryptophan synthase -subunit denatured (fD) as a
function of urea concentration at constant temperature. By means of Eq. 6, these data may be converted into the relation between log
K and urea concentration plotted in Fig. 8
A.9
We define c50 to be that concentration of
urea at which the protein is half-denatured
(fD = 0.5, K = 1). Consider
the following process. To a solution of labile tracer protein
containing stable inert protein S at an initial concentration
csinitial, urea is added to a concentration
c50initial. Now a quantity of stable inert
protein is added, increasing the concentration to
cSfinal. The increase in
cs may either stabilize or destabilize the
tracer protein; if it stabilizes the tracer, fD
will decrease to a value less than 0.5, and if it destabilizes the
tracer protein, fD will increase to a value
greater than 0.5. Now the concentration of urea is adjusted, either
upward or downward, to regain the initial state of half-denaturation.
Let the new concentration of urea be denoted
c50final. Because the protein is once again
half-denatured, the total free energy change associated with the
addition of stable protein and adjustment of urea concentration must be
zero. If we assume that the influences of urea and inert protein are
energetically independent, then it follows that
(G°N
D)s, the change in
standard state free energy of denaturation arising from addition of S,
must be equal and opposite in sign to
(G°ND)urea, and
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(16)
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Thus by simple inversion of sign, the results shown in Fig. 8
A may be converted to those shown in Fig. 8 B,
which reveal that the decrease in K of between one and two
orders of magnitude expected to result from the excluded volume of
large concentrations of S would result in an increase of
c50 of at least 0.5 M and possibly as much as
several M.
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COMPARISON WITH THE MODEL OF ZHOU AND HALL |
The model presented differs from that of ZH (Zhou and Hall, 1996)
primarily in the treatment of the denatured state of the tracer
protein. In the present model, the denatured state is assumed to
exclude volume to the hard-sphere cosolute as would the convex hull of
a Brownian walk, while in the model of Zhou and Hall, the denatured
state is assumed to exclude volume to a hard-sphere cosolute as would a
chain of tangent hard spheres. The present model is limited to
consideration of the effect of hard sphere cosolutes with a mass
comparable to or larger than that of the tracer protein, so that the
denatured state and the hard sphere cosolute cannot significantly
interpenetrate. In this regime, the effect of the hard sphere cosolute
is to stabilize the native state relative to the denatured state.
Zhou and Hall's claim that small cosolutes preferentially stabilize
the denatured state derives from a severely unphysical assumption
regarding the size (volume) of the denatured state. It should be clear
from the definition of the potential of mean force (McMillan and Mayer,
1945) that to the extent that the actual potential of average force
between solute molecules resembles a hard particle interaction (i.e.,
consisting of a short-range steric repulsion, without significant
attraction or long-range repulsion), the size and shape of an
equivalent hard particle representation of a particular macrosolute
species should resemble those of the actual molecule at low resolution.
Significant attraction or long-range repulsion would result in apparent
dimensions of the effective hard particle best reproducing the behavior
of the actual macrosolute that are significantly smaller or larger,
respectively, than actual molecular dimensions (Minton, 1983, 1995;
Minton and Edelhoch, 1983). Because ZH state that the purpose of their
model is to investigate the role of excluded volume interactions only, and not other sources of intermolecular interaction, it follows that in
the context of such a model, the volume of a hard particle representation of a macrosolute is not a freely variable parameter and
must agree with available experimental information regarding the volume
of the macrosolute being modeled.
According to the dimensions published in table II of ZH, the model for
the denatured state of lysozyme has a volume that is 0.22 times that of
the corresponding native state model, and the model for the denatured
state of bovine pancreatic trypsin inhibitor has a volume equal to 0.30 times that of the corresponding native state model. Actual
solvent-excluded volume changes accompanying protein unfolding may be
measured directly (Katz et al., 1973a,b) or calculated from the
pressure dependence of the unfolding process (Brandts et al., 1970;
Zipp and Kauzmann, 1973). Volume changes for typical globular proteins
undergoing unfolding range between ~0 and 50 cm3/mol
for ribonuclease A (Brandts, et al., 1970) and between 50 and 115
cm3/mol for metmyoglobin (Katz et al., 1973; Zipp and
Kauzmann, 1973), depending upon conditions (temperature, pH, etc.).
These volume changes represent a fractional change in solvent-excluded
volume upon unfolding of less than 12%. If the volume of the
denatured state is constrained to be no smaller than 12% less than
that of the corresponding native state, then the model of Zhou and Hall
predicts only stabilization of the native state, as was recognized by
the authors.10
Even in the large cosolute regime, excluded volume effects predicted by
the ZH model differ qualitatively from those predicted by the present
model. According to figure 3 of ZH, addition of a stable solute
comparable in size to the native state of bovine pancreatic trypsin
inhibitor may decrease K for unfolding of bovine pancreatic
trypsin inhibitor (BPTI) by a factor as large as
109.11 In contrast, the present model predicts
a maximum reduction of K by a factor of 10-100. We believe
that this large discrepancy results, at least in part, from a
fundamental defect in the theory utilized by ZH to calculate the
chemical potential of the denatured state in the presence of hard
sphere cosolutes.
According to thermodynamic perturbation theory (Ghonasgi and Chapman,
1994) as implemented by ZH, the excess free energy of a molecule
modeled as a chain ("necklace") of n tangent hard
spheres of radius r1 ("beads") in a fluid of
uniform hard spheres of radius rs is calculated
as the sum of two contributions. The first is the free energy change
associated with the placement of n individual beads in the
fluid at a great enough distance from each other that they do not
interact with each other, which is just n times the excess
free energy of placing a single bead in the fluid. The second
contribution is the free energy change associated with forming the
necklace from the n isolated beads within the hard sphere
fluid. This is calculated as n 1 times the free
energy of bringing two beads from an infinite distance apart to the
contact distance, i.e., a center-to-center distance of
2r1, within the fluid. It is the calculation of
the second contribution that is flawed in principle. Upon closer
inspection one finds that at no point is the entire necklace treated.
The only species for which calculations are carried out are the
individual beads and bead doublets, or "dumbbells." At no point are
correlations between more than two beads considered. Hence the theory
used by ZH contains no specification of, or information regarding,
chain conformation. It is incapable in principle of treating
the relationship between chain conformation, configurational entropy,
and either intra- or intermolecular excluded volume, factors that are
explicitly treated in the present model and shown to be essential
determinants of the chemical potential of the denatured state.
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COMPARISON WITH EXPERIMENT |
The model presented here predicts that excluded volume in
concentrated solutions of stable "inert" macrosolutes may serve to
stabilize the globular native state of proteins against unfolding by
preferentially destabilizing the unfolded state (or ensemble of
unfolded states). The model further predicts that the average dimension
of the unfolded state will decrease with increasing concentration of
inert macrosolute. The magnitude of the changes in experimental
observables is predicted to be easily measurable. Why then have such
effects not yet been reported in the literature?
The investigation of excluded volume effects in concentrated protein
solution is complicated by the likely presence of direct intermolecular
interactions in addition to excluded volume interactions, the more so
as volume exclusion in concentrated solution is predicted to magnify
the consequences of weak interactions that would be negligible in
dilute solution (Minton, 1981, 1983; Zimmerman and Minton, 1993).
Moreover, the unfolding of a protein exposes hydrophobic residues
normally sequestered in the interior of the native protein; such
exposure increases the likelihood of interaction of the denatured state
of the unstable protein with nonpolar residues on the surface of the
stable protein cosolute. In this laboratory we have attempted to
characterize the thermal denaturation of individual proteins in
mixtures but have repeatedly observed that an unfolding transition that
is reversible in dilute solution becomes increasingly irreversible as
the concentration of "stable" cosolute protein increases. Such complications make it difficult to test a theory, such as the one
presented here, that is based upon the assumption of reversible equilibrium and treats only one of several possible types of
interactions that may be present in any real protein solution.
However, there is a well-studied model system that is in several
respects comparable to the native-denatured transition in proteins and
may be free of some of the complications noted above. This is the
unimolecular condensation of large double-helical DNA, which is known
to undergo a collapse to a compact (probably toroidal) form when a
sufficient fraction of the negative charge of the phosphate groups is
neutralized via interaction with multivalent cations (Bloomfield, 1996,
1997; Minagawa et al., 1991). The effect of an inert nonionic polymer,
PEG-10K, on this transition has been studied by Kidoaki and Yoshikawa
(1999), who report two interesting phenomena: 1) With increasing
concentrations of polyethylene glycol (PEG), the degree of charge
neutralization required to attain a condition of half-condensation is
lessened. 2) With increasing concentrations of PEG, the average size of
uncondensed DNA decreases. Observation 1 indicates that both increasing
charge neutralization and increasing PEG concentration stabilize the
condensed form of the DNA relative to the open, random-coil form. Hence
a condition of 50% condensation (K = 1 in the context
of the effective two-state model) may be achieved either with greater
charge neutralization and a lower concentration of PEG, or by less
charge neutralization and a higher concentration of PEG. This
observation qualitatively accords with the prediction of the model
presented here with respect to isothermal denaturation of proteins by
chaotropes, where the addition of chaotrope is energetically equivalent
to the removal of a charge-neutralizing ligand. Observation 2 indicates
that increasing concentration of the inert cosolute weights the
ensemble of uncondensed conformations in favor of more compactly coiled states, as predicted by the model presented here. The analogy between
protein folding and unimolecular DNA condensation is of course limited,
and a more detailed analysis of the effect of excluded volume on
unimolecular DNA condensation is in progress.
| |
Endnotes |
1. The theory presented here is formally applicable to
the constant volume system. Because the actual volume change associated with the unfolding of trace concentrations of a labile protein is
miniscule (see below), the numerical results derived therefrom should
be semiquantitatively applicable to the constant pressure (laboratory)
system. 2. The "linear" diagram presented here has
no mechanistic or kinetic implications and is only meant to indicate
the presence of equilibria between all species. 3. The
use of a single standard state in the definition of the chemical potential of all denatured states does not reflect an assumption of
energetic equality of the various denatured states. Instead, it
indicates an arbitrary assignment of energetic differences between these states to the nonideal part of the chemical potential, manifested in the uniquely defined activity coefficient of each denatured state. The utility of this particular assignment will become
evident with the introduction of a specific model for the denatured
states in the following section. 4. It is assumed that
the volume-excluding cosolute is more stable than the test protein and
does not undergo a conformational change under conditions promoting
denaturation of the test protein. 5. Because of this approximation, the present treatment is limited to globular cosolutes with molar mass comparable to that of the tracer species; i.e., the
molecules are sufficiently large that they do not penetrate into the
spatial domain of the random walk. 6. The shape of the plotted probability distribution in the vicinity of = 1 is
an artifact resulting from truncation of the calculated distribution at
= 1. In reality, denatured states with > 1 do exist,
but the calculations of Jaeckel and Dyantis (1994a,b) provide no
information on their relative entropies (free energies). It may be
readily shown that these states become rapidly depopulated upon the
addition of stable solute S because of the large increase in the value of
dµDiNI,intermol/dws
with increasing i. Thus neglect of denatured states with > 1 will not qualitatively affect our estimates of free
energy changes accompanying the addition of
S. 7. Although the calculated values of the activity
coefficients of the individual N and D species are quite sensitive to
the assumed radius of the hard sphere model cosolute, the ratio of
calculated activity coefficients is far less so. Thus, for a fixed
value of ws, the absolute value of the slope of
the dependence of log(K/K°) on wS
increases by 10% or less per Å increase in the radius of the hard
sphere model cosolute. Hence order-of-magnitude estimates of the effect
of excluded volume on the stability of the native relative to the unfolded state will not be affected by a slight uncertainty in the
assignment of an effective hard sphere radius to a globular macromolecular cosolute. 8. An additional consequence of
Eq. 15 is that in the case of cold denaturation (Privalov, 1990), where
the sign of the enthalpy change is negative rather than positive,
volume exclusion would be expected to decrease rather than increase
T50. 9. For the purpose of this
illustrative calculation, it is assumed that tryptophan synthase
-subunit undergoes an effective two-state N D transition, i.e.,
that intermediate states do not constitute a substantial fraction of
total protein at equilibrium. Deviations from strict two-state behavior
are not expected to alter the qualitative
conclusions. 10. The same result holds for any excluded
volume model that models the denatured state as a markedly aspherical
hard particle of volume approximately equal to that of the
quasispherical native state, as was originally shown by Minton
(1981). 11. This result was obtained using a severe
underestimate of the actual volume of the denatured state of BPTI, as
mentioned above. If the volume of the denatured state had been assumed
to be roughly equal to that of the native state, the extent of
stabilization calculated by ZH would have been substantially larger.
The author thanks Dr. Peter McPhie, NIDDK, for a critical reading
of the initial draft of this report. Thanks are also due to Prof. K. Yutani and the staff and students of the Laboratory of Solution
Chemistry, Institute for Protein Research, Osaka University, for their
gracious hospitality and support during the tenure of a Visiting
Professorship, June-August 1997, during which the work reported here
was initiated.
Address reprint requests to Dr. Allen P. Minton, National Institutes of
Health, Bldg. 8, Rm. 226, Bethesda, MD 20892. Tel.: 301-496-3604; Fax:
301-402-0240; E-mail: minton{at}helix.nih.gov.
TOP
ABSTRACT
INTRODUCTION
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