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D V A
Single-Molecule FRET
Spectroscopy and the Polymer
Physics of Unfolded and
Intrinsically Disordered
Proteins
Benjamin Schuler, Andrea Soranno, Hagen Hofmann,
and Daniel Nettels
Department of Biochemistry, University of Zurich, 8057 Zurich, Switzerland;
email: schuler@bioc.uzh.ch
Annu. Rev. Biophys. 2016. 45:207–31
Keywords
The Annual Review of Biophysics is online at
biophys.annualreviews.org
single-molecule fluorescence, correlation spectroscopy, protein dynamics,
protein folding, internal friction
This article’s doi:
10.1146/annurev-biophys-062215-010915
c 2016 by Annual Reviews.
Copyright All rights reserved
Abstract
The properties of unfolded proteins have long been of interest because of
their importance to the protein folding process. Recently, the surprising
prevalence of unstructured regions or entirely disordered proteins under
physiological conditions has led to the realization that such intrinsically disordered proteins can be functional even in the absence of a folded structure.
However, owing to their broad conformational distributions, many of the
properties of unstructured proteins are difficult to describe with the established concepts of structural biology. We have thus seen a reemergence
of polymer physics as a versatile framework for understanding their structure and dynamics. An important driving force for these developments has
been single-molecule spectroscopy, as it allows structural heterogeneity, intramolecular distance distributions, and dynamics to be quantified over a
wide range of timescales and solution conditions. Polymer concepts provide an important basis for relating the physical properties of unstructured
proteins to folding and function.
207
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Contents
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SINGLE-MOLECULE FRET OF UNFOLDED
AND DISORDERED PROTEINS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quantifying Distance Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quantifying Chain Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
POLYMER PROPERTIES OF UNFOLDED
AND DISORDERED PROTEINS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Denaturants and the Collapse Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Charge Interactions and Polyampholyte Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chain Dimensions and Scaling Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Temperature-Induced Compaction and Solvation Free Energies . . . . . . . . . . . . . . . . . .
Crowding and Excluded-Volume Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chain Dynamics of Unfolded and Intrinsically Disordered Proteins . . . . . . . . . . . . . . .
Internal Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CURRENT DEVELOPMENTS AND FUTURE CHALLENGES . . . . . . . . . . . . . . . .
208
208
209
213
214
214
215
217
218
219
220
221
223
INTRODUCTION
Intrinsically
disordered protein
(IDP): polypeptide
chain lacking stable
tertiary structure
under physiological
conditions
Förster resonance
energy transfer
(FRET): the
nonradiative transfer
of an energy quantum
from its site of
absorption (the donor
chromophore) to an
acceptor chromophore
by resonance
interaction between
the two
208
Polymer physics has a long history in biology and was applied to biopolymers long before the routine availability of high-resolution structural information from crystallography or nuclear magnetic
resonance (NMR). For proteins, however, the compelling atomistic detail provided by structural
biology resulted in a predominant interest in their precise three-dimensional structures. Only
in some areas (e.g., protein folding) did polymer ideas continue to be employed and developed
(20, 45, 73, 141). A recent change in trend has been stimulated by the increasing awareness that
unstructured regions or entirely disordered proteins are surprisingly widespread, especially in eukaryotes (164). For these intrinsically disordered proteins (IDPs), polymer properties—such as
structural heterogeneity, broad conformational distributions, and long-range fluctuations—are of
obvious relevance (5).
A broad range of powerful methods for investigating unfolded proteins and IDPs exist (162),
among them NMR (75) and small-angle X-ray scattering (SAXS) (9). Recently, single-molecule
spectroscopy has developed into a complementary approach for probing such systems (15, 22, 46)
for several reasons: It has the capability of resolving structural and dynamic heterogeneity and
avoiding many complications of ensemble averaging (46); it enables access to a wide spectrum
of timescales for investigating dynamics (137); and it allows investigations over a broad range of
solution conditions, from very low sample concentrations, where nonidealities such as aggregation
are negligible, to highly heterogeneous or crowded environments, including live cells (1, 83, 130).
The two predominant types of single-molecule techniques used for biomolecules and suitable
for probing their polymer properties are force (126, 182) and fluorescence spectroscopy. Here
we focus on work that uses fluorescence spectroscopy, particularly in combination with Förster
resonance energy transfer (FRET).
SINGLE-MOLECULE FRET OF UNFOLDED
AND DISORDERED PROTEINS
Single-molecule FRET is an attractive method for probing long-range distances and distance
dynamics in unfolded proteins because it is most sensitive in the range of ∼2 to 10 nm, an
Schuler et al.
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intramolecular distance range typical of polypeptide segments from several tens of to a few
hundred residues, depending on the compactness of the chain. The theoretical basis of the FRET
process was established by Theodor Förster (51), who showed that the rate coefficient of excitation
energy transfer, kT , between a suitable donor and acceptor chromophore is proportional to the
inverse sixth power of their distance,
6
R0
kT = kD
,
1.
r
where kD −1 = τ D is the fluorescence lifetime of the donor in the absence of the acceptor (Figure 1),
and r is the distance between donor and acceptor. R0 , the Förster radius, can be obtained from
readily measurable quantities (51): the overlap integral between the donor emission and the
acceptor absorption spectra, the donor fluorescence quantum yield, the refractive index of the
medium between the dyes, and the relative orientation of the chromophores. For cases in which
the rotational reorientation of the chromophores is fast compared to the donor excited-state
lifetime, the orientational contribution, κ 2 , averages to a value of 2/3, which simplifies the application of FRET considerably [the applicability of κ 2 = 2/3 can be tested by polarization-sensitive
detection (34, 146)]. Assuming a single fixed distance, the probability that a photon absorbed by
the donor will lead to energy transfer to the acceptor, called the FRET efficiency, E, is given by
kF /(kF + kD ), which (with Equation 1) leads to the relation (Figure 1)
E(r) =
R06
.
R06 + r 6
2.
The Förster radius is thus the characteristic distance that results in a transfer efficiency of 1/2. E is
determined experimentally either by counting the number of emitted donor and acceptor photons
[E = nA /(nA + nD ), including the necessary corrections (34)] or by measuring fluorescence
lifetimes, for example, of the donor in the presence (τ DA ) and absence (τ D ) of the acceptor (E =
1 − τ DA /τ D for a single fixed distance).
Because the molecules are observed individually (either from bursts of photons originating
from single molecules diffusing through the confocal observation volume, or in fluorescence time
traces of immobilized molecules), structural heterogeneity can be resolved if the underlying conformations differ in transfer efficiency and if their interconversion dynamics are sufficiently slow
compared to the time resolution of the measurement. A simple example is the equilibrium between
folded and unfolded molecules freely diffusing in solution, resulting in separate subpopulations in
the transfer efficiency histogram (Figures 1c and 3a) if they interconvert on a timescale slower
than approximately a millisecond, the typical diffusion time through the confocal volume. In this
case, the events corresponding to unfolded molecules can be singled out, and the properties of
the unfolded state can be investigated without overlap from the signal of folded molecules (and of
molecules lacking an active acceptor dye). The same idea can be extended to systems with more
than two subpopulations (8, 121). For a more detailed introduction to the technical aspects and
the versatility of multiparameter detection and analysis, we refer the reader to recent reviews (136
and 146).
Quantifying Distance Distributions
An important aspect that needs to be taken into account for unfolded proteins and IDPs is that
we are not dealing with a single fixed intramolecular distance, but with a distribution of distances,
P(r). Even if we have separated the fluorescence events of, for example, folded and unfolded
molecules, we still have to factor in the conformational heterogeneity within the unfolded-state
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209
210
0.8
1.0
0
250
500
Schuler et al.
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gij (τ)
1.8
2.0
2.2
0.0
0.2
0.4
h
0.0
d
c
0.4
0.6
–250
Lag time, τ (ns)
0
250
AA
DD
AD
0.8
1.0
Unfolded Folded
〈E〉
Transfer efficiency, E
0.2
D yn
amic
Sta
tic
a
D
I(t)
0
e
r
⎡
⎣D
дr
a
L
kexD
P(r)
A
10
дr
д
=
⎤
I + K0 (r) p(r,t)
⎦
P(r)
1
дt
дp(r,t)
Time, t (ns)
5
a
kD
D*A
⌠
I(t) = I0⌡P(r)e –(kT (r)+kD )tdr
D
д
L
⌠
〈E〉 = ⌡P(r)E(r)dr
A
b
15
DA
G(r)
P(r)
0
0
0
2
4
6
0.0
0.1
0.2
0.0
E(r) 0.5
1.0
i
R0
⎛ R0 ⎞6
⎝r⎠
5
=
E(r) =
k T (r)
DA distance, r (nm)
5
Diffusion in PMF
PMF: G(r) = –kBT ln P(r)
R 60 + r 6
R 60
k T (r) +k D
DA distance, r (nm)
k T (r ) = k D
g
f
kA
DA*
kexA
k T (r)
10
10
22 April 2016
τDA 0.6
τD
Number of molecules
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ARI
Excitation
pulse (t0 )
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ensemble. Dynamics in proteins above the microsecond timescale usually involve persistent tertiary
structure or cooperative interactions (41). In the absence of such folded or misfolded structures, the
dynamics within unfolded or intrinsically disordered states are thus expected to be fast compared
to the typical photon detection rates of ∼103 –105 s−1 (137); this case results in a narrow, shot
noise–dominated peak in the transfer efficiency histogram (56, 57). If the fluorophores reorient
rapidly compared to the donor fluorescence lifetime, which is often the case for unfolded proteins
and IDPs, as indicated by fluorescence anisotropy measurements (83, 113), κ 2 can be assumed to
average to 2/3, and the mean transfer efficiency of the unfolded state, E, results as
lc
E =
E(r)P(r) dr,
3.
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c
where c is the distance of closest approach of the dyes, lc is the contour length of the polypeptide
segment probed, and P(r) is a suitably normalized probability density function (Figure 1). [If these
requirements are not met, different types of averaging need to be used (139, 170).]
Based on these considerations, the shot noise–dominated peak corresponding to the unfolded
state in a transfer efficiency histogram can be analyzed only in terms of a single parameter, its mean
FRET efficiency, E, and Equation 3 is thus underdetermined unless additional information about
the functional form of P(r) is available. Different approaches have been chosen to obtain suitable
distance distributions. The simplest approximation is the use of polymer-physical models to reflect
the underlying chain statistics—for example, the Gaussian chain (67, 109, 138, 142), worm-like
chain (109, 114, 139), excluded volume chain (98, 116), or a weighted Flory-Fisk distribution (70,
180) (Figure 2). Equation 3 can then be solved numerically for the single adjustable parameter
in these models (e.g., for the mean-squared end-to-end distance in the case of a Gaussian chain).
Provided that the segment length between the dyes is much greater than the persistence length of
the chain, and that the mean of the distribution is near the Förster radius, these distributions usually
provide very similar results in terms of average chain dimensions (109). Alternatively, molecular
simulations (at various levels of coarse graining) have been employed to obtain information about
the shape of P(r) (11, 81, 100, 110, 116, 117, 139) and to assess deviations from simple polymer
models (116, 139) (Figure 2).
Additional experimental information about P(r) is available from the analysis of fluorescence
lifetimes (67, 90, 147) (Figure 1): Because the dynamics of disordered states (for the chain lengths
accessible in single-molecule FRET) are much slower than the fluorescence lifetimes of the
dyes (∼1 to 4 ns), the distribution of distances results in a distribution of fluorescence intensity decay rates (60). In confocal experiments employing time-correlated single-photon counting,
a subpopulation-specific analysis of fluorescence intensity decays can thus be used to test the validity of specific functional forms of P(r), ideally by a global analysis of both donor and acceptor
Shot noise: the
variation in count rates
about fixed means due
to the discrete nature
of the signal. In
single-molecule
Förster resonance
energy transfer
(FRET), its
contribution is often
important because of
the relatively small
number of photons
detected per unit time
Excluded volume:
describes the net twobody interaction
between monomers in
a chain, including
steric repulsion and
charge- or
solvent-mediated
interactions
Persistence length:
a measure of polymer
stiffness, defined as the
characteristic length
along the chain over
which orientational
correlations between
segments decay
←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
Figure 1
Experimental observables and data analysis in single-molecule Förster resonance energy transfer spectroscopy of unfolded-state
dimensions and dynamics. (a) The signal from individual molecules labeled with donor (D) and acceptor (A) fluorophores freely
diffusing through the confocal volume is detected and (b,f ) analyzed in terms of the distance-dependent transfer efficiency between
donor and acceptor. (c) Transfer efficiency histograms illustrate the separation of subpopulations (here, folded state at high E, unfolded
state at intermediate E, and donor-only population at E ≈ 0). (d ) The combination with fluorescence lifetime analysis (e) provides
additional information about distance distributions, especially via subpopulation-specific fluorescence intensity decays, I(t), and
( g) their analysis in terms of distance distributions based on polymer models or simulations (Figure 2). (h) Chain reconfiguration times
are accessible via nanosecond fluorescence correlation spectroscopy (i ) and a global analysis of donor-donor (DD), acceptor-acceptor
(AA), and donor-acceptor (AD) correlation functions in terms of diffusion in a potential of mean force (PMF) ( g) based on the distance
probability density function, P(r). K0 (r) is the rate matrix that describes the distance-dependent kinetics of interconversion between the
electronic states (b), and p(r,t) is the vector of their populations.
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211
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a
b
Gaussian chain
0.020
⎧ 3 ⎫
⎩ 2π〈r 2〉 ⎭
⎧
⎩
3/2
P(r) = 4πr 2
exp −
3 r
2
2
2 〈r 〉
⎫
⎭
Gaussian
WLC
SAW
Sanchez
0.015
P(r)
0.010
Worm-like chain (WLC)
P(r, lp , lc ) =
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f(lp , lc ) =
4π(r/lc )2 f (lp ,lc )
2 9/2
lc ( 1–(r/lc ) )
⎧
⎫
⎪
⎩ 4lp( 1–(r/lc )2 ) ⎭
0.005
–3lc
exp⎪
1
π 3/2 e –αα –3/2 (1+ 3/α + 15/(4α 2 ))
with α =
0.000
3 lc
4 lp
⎧ r ⎫2.3 ⎧ ⎧ r ⎫2.5⎫
exp⎪−b ⎪
1/2 ⎪
1/2 ⎪
1/2 ⎪ ⎪
〈r 2 〉 ⎭
〈r 2 〉 ⎩ 〈r 2 〉 ⎭
⎩
⎩
⎭
lc /2
100
120
140
Gaussian
WLC
SAW
Sanchez
0.010
0.005
⌠
P(r) = ⌡ P(r |Rg)P(Rg,ε,RgΘ)dRg
Rc
⎧ r ⎫ 9⎧ r ⎫ 3 ⎧ r ⎫
1
3⎪
P(r |Rg) =
⎪
⎪+ ⎪
⎪− ⎪
δ · Rg ⎩ δ · Rg⎭ 4 ⎩δ · Rg ⎭ 16 ⎩δ · Rg ⎭
⎩
80
0.015
P(r)
⎧
60
0.020
Sanchez
P(Rg,ε,RgΘ) = Z –1R 6g exp⎪ –
40
c
a
2
20
r(Å)
Self-avoiding walk (SAW)
P(r) =
0
3
5
0.000
0
20
40
60
80
100
120
140
r(Å)
2
⎫
7 Rg 1
1−ϕ
ln (1− ϕ) ⎪
+ Nεϕ − N
2 R 2gΘ 2
ϕ
⎭
Figure 2
(a) End-to-end distance probability density functions, P(r), of polymer models commonly used in data analysis for unfolded proteins.
Note that in all cases, a single model parameter is adjusted to fit the experimental data (r2 is the mean-squared end-to-end distance, lp
the persistence length, ε the mean interaction energy between amino acids); all other parameters are known independently. Rg is the
root-mean-squared radius of gyration of the state (see the section Chain Dimensions and Scaling Laws), and ϕ is the volume fraction
of the chain. lc is the contour length of the chain segment probed; N is the number of segments; δ is a correction factor (180); and Z, a,
and b are normalization factors (116). (b,c) Comparison of P(r) from these models (lines) with simulation results (symbols) of the
denatured-state end-to-end distance distributions of protein L at (b) 2.4 M and (c) 6 M guanidinium chloride (GdmCl) using a
Cα -side-chain model and the molecular transfer model (116, 117). Panels b and c modified from Reference 116.
intensity decays (67, 90, 147). A helpful diagnostic is a two-dimensional representation of the
donor fluorescence lifetime versus transfer efficiency from count rates (Figure 1), in which the
existence of a distance distribution resulting from rapidly interconverting conformations shows
up as a deviation from the linear dependence of τ DA on E expected for a single fixed distance (59,
78, 147).
Residual secondary structure or the formation of long-range interactions can lead to deviations
from the distance distributions based on simple models or simulations, even if global aspects such as
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length scaling are preserved (48, 116). Experimentally, these effects can be quantified by mapping
different segments of the protein (by varying the positions of the fluorophores) and testing the
consistency with the models employed (67, 70, 110, 116, 147) for example, in terms of the scaling
of the measured average distance with the length of the segment probed (67, 147). Alternatively,
Stokes radii can be obtained from fluorescence correlation spectroscopy (FCS) measurements, and
although hydrodynamic properties can be difficult to relate to polymer models quantitatively, they
provide complementary information about changes in chain dimensions and length scaling (70, 71,
97, 144). Finally, combining single-molecule FRET with complementary methods sensitive to the
presence of structure, such as circular dichroism spectroscopy (68) or NMR (under conditions in
which interference with the signal from folded or structured states can be excluded), will become
increasingly important.
Quantifying Chain Dynamics
Single-molecule fluorescence can be used to probe conformational dynamics over a wide range of
timescales (137). A benefit of single-molecule experiments is the possibility of extracting information on dynamics, even from equilibrium measurements, without requiring a synchronization of
the system by perturbation methods, as frequently employed in ensemble experiments. For proteins, equilibrium dynamics have been investigated based on correlation functions (21, 39, 108,
112, 113, 115, 143, 147, 152), the analysis of broadening and exchange between subpopulations
in FRET efficiency histograms (25, 58, 68, 138), and from fluorescence trajectories of immobilized molecules (25, 26, 87, 88, 121, 124, 125). In this way, both the equilibrium distributions
and the kinetics can often be obtained from the same measurement. Alternatively, nonequilibrium single-molecule dynamics can be investigated by microfluidic or manual mixing experiments
(137).
The reconfiguration dynamics of unfolded proteins and IDPs can be probed with nanosecond
fluorescence correlation spectroscopy (nsFCS) combined with FRET (55, 112, 113). Distance
fluctuations between the donor and the acceptor fluorophores then result in fluctuations in fluorescence intensity on the timescale of interconversion between the different configurations of the
polypeptide chain. For relating the measured decay time of the fluorescence correlation curves
to the reconfiguration time of the chain, the relative motion of the two labeled positions can be
represented as a diffusive process in a potential of mean force that corresponds to the equilibrium
dye-to-dye distance distribution of the unfolded protein (Figure 1). In this simplified description, the dynamics of the system are determined (a) by the shape of the potential [i.e., the free
energy as a function of the distance between the dyes, G(r), which is obtained directly from the
distance distribution function, P(r), as described above] and (b) by an effective diffusion coefficient,
D, that determines the relaxation time in the potential (55, 63, 112, 150, 167) (Figure 1). This
relaxation time corresponds to the reconfiguration time, τ r , of the chain segment between the
dyes.
In summary, the single-molecule approach described here can provide two essential quantities that characterize the global properties of unfolded or intrinsically disordered proteins: intramolecular distance distributions (from transfer efficiency and fluorescence lifetime measurements) and reconfiguration dynamics (from fluorescence correlation analysis). These are key
quantities in the statistical mechanics of polymers (20, 30, 38, 50) and thus make the results
amenable to the application (and even the testing) of concepts in polymer physics, with the goal
of providing a coherent and quantitative framework for the behavior of unfolded proteins and
IDPs.
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Stokes radius: the
radius of a sphere that,
according to the
Stokes-Einstein
relation, has the same
diffusion coefficient as
the molecule of
interest
Fluorescence
correlation
spectroscopy (FCS):
a method used to
quantify the dynamics
of molecular systems,
such as their
translational diffusion
or conformational
changes, via the
resulting fluctuations
in fluorescence
emission rates
Reconfiguration
time: the relaxation
time of the
autocorrelation
function of the
distance between two
monomers in a chain
213
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POLYMER PROPERTIES OF UNFOLDED
AND DISORDERED PROTEINS
Denaturants and the Collapse Transition
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Polymers respond to changes in solvent quality by a change in chain dimensions: In poor solvent, where intrachain interactions dominate over chain-solvent interactions, they collapse into
compact globules; in good solvent, where chain-solvent interactions are favorable, they populate
rather open, expanded conformations (20, 49). In water, many proteins fold into well-defined,
compact structures and/or tend to aggregate; in solutions of denaturants such as urea or guanidinium chloride (GdmCl), however, proteins and peptides are highly soluble and (in the absence
of disulfides) form expanded unfolded states (80, 127). Correspondingly, the chain dimensions of
folded globular proteins, quantified, e.g., in terms of the radius of gyration, Rg , exhibit a dependence on the number of peptide segments in the chain, N, that is in good agreement with the
scaling behavior expected for compact globules, Rg ∝ N1/3 (36, 159, 168). For unfolded proteins
in high concentrations of denaturant, measurements of intrinsic viscosity, NMR, and small-angle
scattering have shown good agreement with the length scaling of chain dimensions with N3/5 (84,
154, 155, 168), as expected for a polymer in good solvent.
From a polymer physics perspective, even in the absence of a transition to a folded structure
or populated folding intermediates, denatured states are expected to exhibit a compaction akin
to heteropolymer collapse when the solvent quality (here modulated by the denaturant concentration) is reduced (2, 20, 27, 71, 117). This behavior has been demonstrated particularly clearly
by the molecular transfer model (117) (Figure 3b), which combines simulations in the absence
of denaturants with experimentally determined transfer free energies of side chain and backbone
moieties from water to denaturant solution (153). However, detecting this behavior in ensemble
experiments had been challenging, as the corresponding signal change is hard to separate from
the folding transition. The separation of folded and unfolded subpopulations in single-molecule
FRET experiments (33) led to the identification of this unfolded-state compaction in terms of
a continuous shift in transfer efficiency (138, 142) (Figure 3a); it has since been observed for
many proteins (65, 137) and is in agreement with a range of other experiments, theory, and simulations (71, 94, 107, 117, 155, 156, 158, 174, 181). However, the detectability of unfolded-state
compaction in SAXS experiments has remained controversial (175).
Particularly interesting is the analysis of denaturant-dependent unfolded-state compaction in
terms of the theory of coil-globule transitions (35, 61, 131), which allows chain compaction or expansion to be related to changes in effective intrachain interaction energies between the residues.
Haran and coworkers (142, 180) used the theory of Sanchez (131), which provides an expression
for the probability density function of Rg in the form of a Boltzmann-weighted Flory-Fisk distribution (Figure 2). Ziv & Haran (180) analyzed a large set of results in which the mean transfer
efficiency of unfolded proteins had been obtained over a broad range of denaturant concentrations
(Figure 3c,d ). By determining the mean interaction energy between amino acids from E and
computing the free energy of the chain based on the Sanchez theory, they found that the difference between the free energy of unfolded proteins and their most collapsed states depends
approximately linearly on the concentration of GdmCl and that the change of this free energy
with denaturant concentration is close to the equilibrium m value, the derivative of the free energy
difference between the folded and unfolded states with respect to the denaturant concentration.
Ziv & Haran concluded that a major part of the free energy change for protein folding originates
from the free energy change of collapse, thus providing a link between the polymeric properties
of the unfolded state and protein folding mechanisms (180).
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Number of events (×100)
b
5
0.0 M
0
5
0.8 M
1
0.9
0.8
0
5
CspTm
1.5 M
〈E 〉
0
7
2.0 M
0
15
3.5 M
0
10
5.0 M
0.7
0.6
0.5
0.4
0.3
0
0
2
20
0
0.0
0.5
4
6
GdmCl concentration (M)
8.0 M
1.0
E
c
d
×10 –2
1.5
20
ε (kBT )
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a
ε
P(r)
1.0
10
0.5
0
0
4
8
12
0
r (nm)
2
4
6
8
GdmCl or urea (M)
Figure 3
Denaturant-dependent unfolded-state compaction. (a) Changes in unfolded-state dimensions can be
detected in single-molecule Förster resonance energy transfer (FRET) experiments owing to the separation
of folded and unfolded populations, here shown for cold shock protein (Csp) as a function of guanidinium
chloride (GdmCl) concentration (67, 138). (b) Denaturant dependence of the experimental transfer
efficiencies for Csp (67, 100) ( filled symbols) compared to predictions from the molecular transfer model (117)
(open symbols); folded state (triangles), unfolded state (squares), average signal (circles). (c) Sanchez theory
enables the analysis of single-molecule FRET experiments and the underlying changes in chain dimensions
in terms of their mean-field interaction energy ε as a function of denaturant concentration (180) and (d ) was
used by Ziv & Haran (180) to relate ε for the unfolded states of many small proteins to their equilibrium
unfolding m values. Panel b modified from Reference 117 with permission, copyright 2008, National
Academy of Sciences, USA; panel d modified from Reference 180 with permission, copyright 2009,
American Chemical Society.
Charge Interactions and Polyampholyte Theory
Proteins are heteropolymers, and their effective intrachain interaction energy, as reflected by
the mean interaction energy between amino acids as expressed in the Sanchez theory (Figure 2),
comprises different contributions, including hydrogen bonding and hydrophobic and electrostatic
interactions. For IDPs, charge interactions are expected to be particularly pronounced, as their
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a
c
b
3.2
70
2.8
Hydropat
Rg (nm)
2.0
50
nsic
ally
disordered
Swollen
coil Disordered Swollen
coil
globule
f+
〈rij 〉 (Å)
hy
Intri
2.8
2.4
4.0
3.6
EV
FRC
LJ
40
30
20
3.2
10
2.8
4.8
0
0
4.4
10
20
30
40
50
60
| j−i|
4.0
3.6
d
3.2
10
2.8
8
7
Denaturant concentration (M)
5
0
2
4
6
Rg (nm)
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60
1.6
Folded
f–
2.4
ν=
8
0. 5
3
4
ν=
2
0. 3
1
hCyp
10
100
1,000
N
Figure 4
Effect of sequence composition on chain dimensions. (a) Diagram of states to classify predicted conformational properties of
intrinsically disordered proteins (IDPs) as a function of the fraction of positively ( f+ ) and negatively ( f− ) charged residues, and the
hydropathy of the sequence (29, 97, 164). Green dots correspond to disordered sequences from the DisProt database (145).
(b) Dependence of the apparent radii of gyration (Rg ) on the concentration of guanidinium chloride (GdmCl) ( filled circles) and urea (open
circles) for unfolded CspTm, IN, ProTαN, and ProTαC (from top to bottom) (109). Fits to a binding model for the urea dependence
(colored dashed lines) and to polyampholyte theory for the GdmCl dependence (black solid lines) are shown. The contributions of GdmCl
binding and electrostatic repulsion are indicated as continuous and dashed gray lines, respectively. The colored squares indicate the
changes in Rg upon addition of 1 M KCl. (c) Averaged interresidue distances, Rij , versus sequence separation, | j − i |, based on
simulations for ten naturally occurring IDPs (symbols), illustrate the dependence of chain dimensions on the content of charged residues
and their distribution in the sequence (29). The solid lines are reference curves corresponding to an excluded volume chain (EV, red ), a
Flory random coil (FRC, black), and a compact globule (Lennard–Jones, blue). (d ) Rg for unfolded cyclophilin as a function of the
number of bonds, N, at different GdmCl concentrations from 0 M (blue) to 6 M (red ) (70). Colored dashed lines are fits according to
Rg = Rg0 Nν . Gray circles are RG values determined for unfolded proteins by small-angle X-ray scattering (SAXS) (84). Open blue
circles are RG values of denatured proteins under native conditions determined with SAXS (163). Black solid lines are fits of the data
taken from Reference 84 and of monomeric folded proteins from the Protein Data Bank with fits of the form Rg ∝ Nν ; the resulting
scaling exponents, ν, are indicated. Panel a modified from Reference 164, panel b modified from Reference 109, panel c modified from
Reference 29, and panel d modified from Reference 70.
amino acid composition is strongly biased toward low hydrophobicity and high charge content (97,
161). The conformational properties of proteins, especially their degree of expansion, as a function
of these contributions can be classified in terms of a state diagram (29, 97, 164) (Figure 4a). One
way of treating the underlying interactions of charges within the chain quantitatively is through
polyampholyte theory (37), which encompasses both repulsive interactions between charges of the
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same sign and attractive interactions between charges of opposite sign. The approach of Higgs
& Joanny (66), for example, considers a chain in which monomers carry a positive charge with
probability f and a negative charge with probability g. The monomers have an excluded volume vb3
owing to steric repulsion (with a segment length b = 0.38 nm, the Cα -Cα distance in a polypeptide)
and interact via screened Coulomb interactions between each pair of monomers, with distances
assumed to follow Gaussian chain statistics. The effect of the electrostatic interactions on chain
dimensions can then be treated as a contribution to an effective excluded volume, v ∗ b 3 . Introducing
the expansion factor α as the ratio of an effective segment length b1 and the real segment length
b, one can then describe the chain dimensions using the expressions
4 3 3/2 1/2 ∗
b1
4.
N ν , where α = ,
α5 − α3 =
3 2π
b
4π l B ( f − g)2
π l B2 ( f + g)2
.
5.
−
κ2
κ
The second term of the right-hand side in Equation 5 accounts for repulsive interactions due to the
net charge of the polypeptide, resulting in an increase of v ∗ b 3 , as in related forms of polyelectrolyte
theory (62). The third term leads to a reduction of v ∗ b 3 through attractive interactions between
charges of opposite sign. lB is the Bjerrum length and 1/κ the Debye length.
Müller-Späth et al. (109) applied this theory to a series of unfolded and intrinsically disordered
proteins for which the importance of the net charge was qualitatively obvious from the different
response of chain dimensions with urea, which is uncharged, and with GdmCl, which is a salt
(69) (Figure 4b): With increasing urea concentration, a monotonic expansion was observed for all
chains, but the screening of charged residues in the IDPs by GdmCl caused a pronounced collapse
of charged IDPs between 0 and 0.5 M (Figure 4b), as expected for a polyelectrolyte (62, 118).
Remarkably, however, unfolded cold shock protein (Csp), which contains approximately equal
numbers of acidic and basic amino acids, exhibited signs of polyampholyte behavior: The addition
of salt led to an expansion rather than a collapse, indicating that attraction between charge residues
of opposite sign causes compaction. The response of the entire set of proteins was described well
by the theory of Higgs and Joanny, combined with a simple binding model (96) accounting for
denaturant-induced chain expansion (109). The result demonstrated that charge interactions can
dominate the dimensions of unfolded proteins and IDPs, in accord with molecular simulations
(97), and that a mean-field theory can account for these observations. However, charge patterning
within the sequence (29) or the formation of persistent local or long-range structure (44, 93) can
lead to deviations from this relatively simple behavior (Figure 4c). For instance, segregation of
charges along the sequence can lead to structure formation, and simulations or more advanced
theoretical approaches are required for a quantitative treatment of these effects (29, 132).
and ν ∗ b 3 = νb 3 +
Polyampholyte
theory: describes the
effect of charge
interactions on the
conformations of a
polymer, including
both repulsive
interactions between
charges of the same
sign and attractive
interactions between
charges of opposite
sign
Polyelectrolyte:
a macromolecule in
which a substantial
fraction of the
monomers contain
ionizable or ionic
groups
Polyampholyte:
a polyelectrolyte that
contains both
positively and
negatively charged
groups
Chain Dimensions and Scaling Laws
Denaturants and charge interactions are only two of the factors that have a pronounced influence
on the dimensions of denatured proteins and IDPs. Given the quasi-continuous spectrum of their
amino acid compositions (161) and the broad range of relevant solution conditions, a quantitative
way of classifying the degree of expansion or collapse of unfolded and intrinsically disordered
proteins is essential (Figure 4a). That some denatured proteins are neither as expanded as fully
unfolded states nor as compact as folded proteins was already realized in the 1980s. Both equilibrium and kinetic intermediates in protein folding with such “molten globule” properties were
observed (123), and the scaling of chain dimensions with the length of the polypeptide was found
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Scaling laws: describe
the scale invariance
observed for many
natural phenomena
and are frequently
used in polymer
physics, for example,
to quantify the scaling
of chain dimensions
with chain length
conditions:
conditions under
which attractive
interactions within the
chain counterbalance
the steric repulsion
between monomers,
and the polymer
follows the statistics of
an ideal chain to good
approximation
12:11
to be between the N1/3 and the N3/5 behavior of folded and fully unfolded proteins, respectively
(155).
One possibility is to relate the denaturant-dependent collapse and other effects on chain compaction quantitatively to the effective interaction energies between the residues via the Sanchez
theory of coil-globule transitions described above (131, 142, 180). Doing so requires knowledge
of the dimensions of the chain at its point, where chain-chain and chain-solvent interactions of
a real chain balance in a way that its length scaling approaches that of an ideal chain (i.e., Rg ∝
N1/2 ). However, systematic variation in the number of monomers over a large range—the classic
way of determining length scaling in polymers—is difficult for proteins. Hofmann et al. (70) thus
made use of the finding that previously determined values of the prefactor, Rg0 , in scaling laws of
the type Rg = Rg0 Nν vary only from ∼0.2 nm to ∼0.3 nm, even when comparing polypeptides
in high concentrations of denaturants and folded globular proteins (48, 84, 168, 176), and can
be interpolated in this range (70). With this constraint, the resulting scaling exponents, ν, can be
quantified for a broad range of proteins (Figure 4d ) and depend sensitively on their sequence composition, net charge, and denaturant concentration. This approach also allows the conditions
to be identified based on the chain dimensions. Interestingly, protein sequences with a moderate
net charge and average hydrophobicity were found to be close to conditions in water (70). It
has been suggested that folding from the state provides an ideal evolutionary compromise by
favoring intrachain interactions while allowing for rapid sampling of different conformations (17).
Length scaling exponents of unstructured peptides close to 1/3, as found for polyglutamine (27),
are in line with the high aggregation tendency of these pathogenic sequences. Based on singlemolecule FRET experiments of an FG-repeat nucleoporin, a connection between chain collapse
and the capacity to form hydrogels has been proposed (103). In summary, length scaling exponents
are thus an appealing way of quantifying the degree of expansion of unfolded proteins and IDPs
because they are independent of the length of the individual polypeptide (160) and can be used
to reflect differences in effective solvent quality brought about both by differences in amino acid
composition (28, 70, 71) and by changes in solution conditions such as denaturant concentration
(65, 70, 137).
Temperature-Induced Compaction and Solvation Free Energies
The dimensions of unfolded proteins and IDPs show a response to temperature that may be somewhat counterintuitive. Typical homopolymers in organic solvent, for example, expand monotonically with increasing temperature (149), as expected if thermal energy increases compared with attractive intrachain interactions (131). For proteins, however, laser temperature-jump experiments
had indicated a compaction of the acid-denatured protein BBL with increasing temperature (129),
and NMR measurements of the C-terminal domain of protein L9 in the cold-denatured state had
shown increasing radii of hydration as temperature was reduced (92). The separation of folded
and unfolded subpopulations in single-molecule FRET experiments enabled an investigation of
this issue over a wider range of temperatures. For Csp, a continuous collapse of the unfolded state
between ∼280 K and 340 K was observed (114). Similarly, denatured frataxin collapsed over a
broad range of temperatures, spanning both the cold- and the heat-denatured state (4).
Although this temperature-induced compaction is suggestive of hydrophobic interactions
as a driving force (92, 129), highly charged IDPs exhibit the most pronounced collapse (114,
172), indicating the importance of additional contributions related to changes in solvation free
energy as a function of temperature. A critical role of the solvent contribution is supported by
molecular simulations of unfolded proteins with different water models (12, 114). Interestingly,
the compaction observed for five different unfolded or intrinsically disordered proteins could be
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a
b
3.6
0.9
λ = 6.9
3.8
1.9
0.9
0.8
0.7
0
0.1
gcSPT
Short chains
0.3
0.4
renFH
Long chains
gcSPT
3.4
SPT
3.2
FH
3.0
0.2
FH
1
ϕ
5 10
Renormalized FH
50 100
c
Cytosol
50
0
50
Nucleus
0
500
0
Degree of polymerization
0.5
1
Transfer efficiency
gDD (τ)
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Rg (nm)
Rg(ϕ)/Rg(0)
1
SPT
Number of bursts
BB45CH10-Schuler
4.2
4.0
3.8
–1
–0.5
0
0.5
1
τ (μs)
Figure 5
Effects of crowding and cellular conditions on intrinsically disordered proteins. (a) Crowding effect on the conformation of a
self-avoiding-walk chain as a function of crowder volume fraction, φ, and the ratio of chain dimensions and the crowder size, λ, for solid
spherical crowders based on simulations (79). An illustration of the chain and the crowding particles for λ = 1.9 and φ = 0.3 is shown
on top. (b, top) Graphical representation of scaled-particle theory (SPT), Gaussian cloud model (gcSPT), Flory-Huggins theory (FH) in
the short-chain regime, and renormalized Flory-Huggins theory (renormalized FH) in the long-chain regime. (b, bottom) Radius of
gyration of ProTαC as a function of the degree of polymerization of PEG at φ = 0.15. Fits according to the different theories are
shown as black (SPT), gray (gcSPT), cyan (FH), and blue (renormalized FH) lines. Solid lines indicate the regime for which the
respective theories were derived; otherwise, dashed lines are used. (c) Single-molecule Förster resonance energy transfer (FRET)
measurements in live cells after microinjection of labeled unfolded molecules: (top) Schematic illustration, (middle) single-cell FRET
efficiency histograms of prothymosin α (ProTα) in the cytosol and nucleus of HeLa cells, and (bottom) nanosecond fluorescence
correlation spectroscopy (nsFCS measurements of chain dynamics in live cells (83). Panel a modified from Reference 79, panel b
modified from Reference 148, and panel c modified from Reference 83.
reproduced semiquantitatively in implicit solvent simulations by taking into account the solvation
free energies of the constituent amino acids, with the solvation properties of the most hydrophilic
residues playing a large part in determining the collapse (172). Temperature-dependent studies
can thus provide an important link between unfolded-state dimensions and the solvation
properties of the constituent amino acids, similar in spirit to the molecular transfer model (117).
Crowding and Excluded-Volume Screening
Ultimately, we want to understand the behavior of unfolded proteins and IDPs in a cellular
environment. An important difference compared to typical experiments in vitro is the high concentration of cellular components, resulting in molecular crowding effects (177). In view of their
lack of persistent structure, IDPs and unfolded proteins are expected to be particularly sensitive
to the effects of crowding (Figure 5a). Indeed, experiments indicate that some unfolded proteins
or IDPs gain structure upon crowding (32), while others do not (99, 151) but may change their dimensions (72, 77, 101, 148). The theoretical concept most widely used for rationalizing crowding
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Molecular crowding:
describes the effects of
high concentrations of
inert cosolutes on
molecular interactions
and conformations
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Semidilute regime:
regime in which chains
are present at
sufficiently high
concentration for
them to overlap so that
the interactions
between polymer coils
can dominate their
behavior
Excluded volume
screening: the
screening of repulsive
interactions within a
chain by the
interpenetration with
other polymers at
concentrations in the
semidilute regime and
above
12:11
effects is scaled-particle theory, which provides an estimate of the change in free energy required
for creating a cavity equivalent to the size of the IDP (or the crowded molecule in general) in
a solution of hard spheres with a radius corresponding to the size of the crowding agent (104).
Correspondingly, creating a cavity sufficiently large for accommodating an expanded chain becomes entropically more unfavorable with increasing crowder concentration, thus promoting the
formation of more compact configurations (79, 106, 177).
However, this treatment can fail in the presence of polymeric crowders, most obviously when
the degree of polymerization of the crowder is varied, as in the experiments of Soranno et al.
(148), who studied several IDPs with single-molecule FRET in the presence of polyethylene glycol
(Figure 5b). To explain the results, the authors had to include the polymeric properties of both
IDPs and crowders in the model explicitly to account both for the interpenetration between IDPs
and crowders and for the mutual overlap between crowding polymers in the semidilute regime
and above, which for long chains is reached already for volume fractions of a few percent (43, 133).
Advanced Flory-Huggins-type theories (76, 134) can account for the resulting excluded volume
screening effects quantitatively (Figure 5b) (148). Surprisingly, both recent single-molecule FRET
measurements of the highly charged IDP prothymosin α (83) and ensemble FRET experiments
of polyethylene glycol (54) in live eukaryotic cells have not indicated a pronounced effect of
crowding on chain dimensions (Figure 5c). One possible reason may be that the macromolecular
concentrations in cultured eukaryotic cells are lower (119, 169) than is commonly assumed based
on estimates for microbial cells (179). Another aspect that will require future study and that
may be difficult to treat with simple models is how crowding effects are modulated by attractive
intermolecular interactions (105).
Chain Dynamics of Unfolded and Intrinsically Disordered Proteins
Conformational dynamics in proteins can occur over an enormous span of timescales owing to the
broad range of enthalpic and entropic contributions involved. The slowest conformational change
for an individual structured protein will usually be its global folding and unfolding, which can take
place on timescales from microseconds to hours and beyond. Within unfolded or intrinsically
disordered states, however, cooperative interactions linked to the formation of specific structures
are much less prevalent; correspondingly, conformational dynamics are expected to be much faster.
Unexpectedly slow dynamics up to the seconds timescale have been reported both for chemically
unfolded ribonuclease H (87) and for some proteins previously classified as intrinsically disordered
(24), but the structural origins (including the possible role of peptidyl-prolyl cis-trans isomerization)
have remained unresolved. For α-synuclein, slow dynamics are induced by the interaction with
membranes (47).
In the absence of persistent intra- or intermolecular interactions, an unfolded polypeptide chain
is expected to approach dynamics that are limited by monomer diffusion through the solvent and
described by simple polymer models, such as the Rouse or Zimm theories (38). In the Rouse
theory, chain dynamics are represented by harmonic modes, arising from the motion of beads
representing individual chain segments connected by ideal springs, and whose dynamics are thus
controlled only by the neighboring beads and the viscous drag from the solvent. For several
proteins unfolded in high concentrations of denaturant, good agreement with these models has
been observed, corresponding to reconfiguration times, τ r , of several tens of nanoseconds for 50to 70-residue segments (147). Note that according to Onsager’s regression hypothesis, this time
also corresponds to the collapse time of the chain (112), in agreement with laser temperaturejump experiments (129), and is thus closely linked to the pre-exponential factor in protein folding
220
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12:11
kinetics (40, 112, 129, 173). In terms of an effective end-to-end diffusion coefficient (150), these
dynamics are in a similar range as those of unstructured peptides (39, 63, 89, 107).
However, with decreasing denaturant concentration, τ r (corrected for solution viscosity) typically increases concomitant with chain compaction (14, 112, 147), in contrast to the behavior
expected from the Rouse or Zimm model, where chain compaction leads to a decrease in τ r (38).
This discrepancy is a clear indication for the presence of internal friction (112). Defined operationally as a deviation from proportionality between relaxation time and solvent viscosity, internal
friction had previously been identified in folded proteins (3) and in the context of protein folding
reactions (18, 64). Its molecular origin was mainly assigned to the large degree of desolvation in
the very compact folded or transition states and the resulting decoupling from the solvent. For
unfolded proteins, however, where the expansion of the chain relative to the folded state entails
pronounced solvation even in the absence of denaturant (67, 70), the origin of internal friction is
less obvious.
Internal friction: a
dissipative force within
a medium (in contrast
to hydrodynamic
friction with the
solvent), here within
an unfolded protein,
that slows down its
dynamics
Internal Friction
There is a long history of theoretical concepts in polymer dynamics that address the question of
internal friction (or internal viscosity). Early ideas go back to Kuhn & Kuhn (85), Cerf (19), and de
Gennes (30). A particularly simple class of models that allow internal friction to be incorporated
rigorously (23) are those of Rouse (128) and Zimm (38, 178). Internal friction can be included
in terms of an internal friction coefficient that impedes the relative motion of two beads (7, 82)
(Figure 6a). Notably, this additional term does not change the eigenmodes of the system, but it
increases all relaxation times by the same internal friction time, τ i , yielding a spectrum of relaxation
times,
τ (n) = τRouse /n2 + τi ,
n = 1, 2, . . . ,
6.
where τ Rouse is the Rouse time, corresponding to the slowest relaxation mode of the Rouse chain
in the absence of internal friction, and n is the mode number. A simple common assumption is
that only the dynamics corresponding to τ Rouse /n2 depend on solvent viscosity, η, whereas τ i does
not (30). Because the overall solvent-dependent relaxation time, τ s , with contributions from all
relaxation modes, is the same as τ Rouse to within a numerical factor (95), we arrive at the relation
τr (η) = τs (η0 )η/η0 + τi ,
7.
where η0 is the viscosity of water. In other words, if the chain reconfiguration time is measured
as a function of solution viscosity and extrapolated linearly to zero viscosity, the internal friction
time would result as the intercept. The same behavior is found for the Zimm model when internal friction is included analogously, only with a mode dependence of n−2/3 compared to n−2 in
Equation 6 (23).
The solvent viscosity dependence predicted by Equation 7 is in good agreement with experimental observations for unfolded Csp (147), with increasing values of τ i as the denaturant
concentration is decreased (Figure 6b). This result was confirmed by an orthogonal approach not
requiring changes in viscosity: Because the chain exhibits a spectrum of relaxation modes (Equation 6), the relative contribution of internal friction to the reconfiguration dynamics will depend
on the length of the segments probed (82, 95). By placing the FRET dyes in different positions
within the chain, one can determine τ i independently. Very similar results are obtained (147) using
the Zimm model with internal friction (23). For chemically unfolded spectrin domains, significant
internal friction was present even at the highest GdmCl concentrations (14), indicating a possible
role for residual local interactions and secondary structure formation. Notably, spectrin domains
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a
12:11
150
b
c
2.0 M
50
0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 4
η (cP)
η (cP)
d
η (cP)
70
60
50
40
30
20
10
0
0M
2.0 M
4.0 M
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5
η/η0
η/η0
η (cP)
e
0.02
η/η0
f
4
100
50
3
τ (ns)
GS4
τ–fp
(ns)
150
τi (ns)
6.0 M
100
0
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4.0 M
τr (ns)
τr (ns)
1.3 M
2
0.01
1
0
0.4
0.6
0.8
1.0
1.2
1.4
lp (nm)
1.6
0
0
1
2
3
0
0
1
η/η0
2
3
η/η0
Figure 6
Internal friction in unfolded proteins and intrinsically disordered proteins (IDPs). (a) Schematic of a Rouse chain (represented as beads
coupled by harmonic springs) with internal friction (indicated as dashpots) and terminal dyes. (b) Solvent viscosity (η) dependences of
chain reconfiguration times, τ r , of terminally labeled cold shock protein at different guanidinium chloride concentrations (indicated in
each panel) and fits with the Rouse model with internal friction. The values of the internal friction time, τi , correspond to the intercepts
(red arrows). (c) Results analogous to panel b from atomistic simulations (42). (d ) The dependence of τi on chain expansion of different
unfolded proteins and IDPs, as indicated by their apparent persistence length, lp , shows a trend of more internal friction for more
compact chains [prothymosin α (blue), integrase ( yellow), Csp (red ), spectrin R17 (dark green), spectrin R15 (light green)] (147).
(e) End-to-end contact formation time, τ GS4
f p , of a (GlySer)4 peptide from atomistic simulations where viscosity is varied by rescaling
the solvent mass (140), with linear (solid line) and power-law (dashed line) fits, and a fit based on a Rouse model for contact formation
including internal friction (dotted line). ( f ) The low viscosity dependence of the isomerization rate for a four-bead butane-like molecule
(inset) indicates that solvent memory effects can contribute to the signature of internal friction observed experimentally (31). Panel a
modified from Reference 82; panel b modified from Reference 147; panel c modified from Reference 42, copyright 2014, American
Chemical Society; panel e modified from Reference 140, copyright 2012, American Chemical Society; and panel f modified with
permission from Macmillan Publishers Ltd: Nature Communications, copyright 2014, Reference 31.
are α-helical proteins, whereas Csp is an all-β protein, supporting suggestions based on simulations that interactions such as hydrogen bonds (140) local in sequence contribute more strongly
to internal friction in unfolded proteins than the highly nonlocal interactions characteristic of β
structure (31).
What is the role of internal friction in IDPs? Both the N-terminal domain of HIV integrase
and the highly negatively charged prothymosin α (ProTα) exhibit much less internal friction than
unfolded Csp or spectrin in the absence of denaturant (147). A plot of τ i versus the dimensions of
the chains in terms of the radius of gyration shows an overall trend corresponding to an increase
in τ i with chain compaction (Figure 6d ), suggestive of the general importance of intrachain
interactions for internal friction. Simulations have started providing important insights as to the
molecular origin of internal friction. Echeverria et al. (42) simulated unfolded Csp at different
GdmCl concentrations, found remarkable agreement with the experimentally determined dynamics (Figure 6c), and concluded from their analysis that dihedral angle transitions provide the
dominant mechanism of internal friction. It will be interesting to further investigate the possible
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connection to intrachain interactions such as hydrogen bonding, which has been suggested to
cause internal friction based on atomistic simulations of peptides (140) (Figure 6e). Another
potentially very important contribution is the low sensitivity of dihedral angle isomerization to
solvent viscosity, as predicted theoretically by Portman et al. (122) and recently identified to
affect protein folding dynamics by de Sancho et al. (31) in simulations (Figure 6f ).
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CURRENT DEVELOPMENTS AND FUTURE CHALLENGES
Polymer concepts have proven very valuable for a physical understanding and classification of
the structural and dynamic behavior of unfolded and intrinsically disordered proteins. This is
the case especially in the context of the global properties typically probed with single-molecule
FRET. Obviously, the applicability of simple models needs to be evaluated for each case, and
in the presence of persistent secondary or tertiary structure, the role of specific interactions and
protein folding may have to be included in a physical description. For many questions, a closer
combination with experimental methods complementary to single-molecule spectroscopy will
thus be important, especially circular dichroism, scattering techniques (9, 53), and NMR (75).
Although often limited to conditions in which the unfolded or disordered state is populated
exclusively, these methods can provide essential additional information, such as the local structural
and dynamic properties available from NMR. This type of integrated approach will lead to a more
comprehensive view of the behavior of unfolded proteins and IDPs (162). In regard to singlemolecule spectroscopy, challenges include developing a more complete use of the broad span
of timescales accessible from correlation analysis (136, 146), taking full advantage of three-color
FRET experiments (52, 102), and the extension to intracellular measurements (1, 83).
At the same time, simulations will become increasingly important for linking experimental
observations and chain statistics to molecular detail. The traditional focus on folded proteins for
the parameterization of the energy functions for molecular dynamics simulations has typically
resulted in unfolded states that were much more compact than observed experimentally (13, 120).
The increasing interest in IDPs, however, has triggered developments to remedy this deficiency,
in particular the optimization of suitable implicit (166, 172) and explicit water models (12, 13, 111,
120), and also the explicit representation of the fluorophores (10, 11, 100, 135). The more routine
availability of such simulations ideally complements the further development of analytical theory
(6, 86, 132) and will enable us to obtain increasingly accurate representations of unfolded proteins
and IDPs even though their heterogeneous conformational ensembles are underdetermined by
experimental observables alone (75, 110, 165).
This improved quantitative understanding paves the way for addressing questions of increasing
complexity, such as the interactions of IDPs with their biological targets, be they other proteins,
nucleic acids, lipids, small molecules, or metal ions (171). For instance, many IDPs are known to
remain partially, largely, or possibly even completely unstructured in the bound state (157) and
thus retain much of their polymer dynamics. Closely connected are the processes of liquid-liquid
phase separation and coacervation (16), which have recently seen a revival in the context of nonmembrane bound compartments in cells (74). A key challenge will be to bridge the gap between
the mesoscopic picture emerging from the concepts of phase separation (74) and the underlying
microscopic/molecular mechanisms (91), both of them deeply rooted in polymer physics. Methods
such as single-molecule spectroscopy and NMR are ideally suited for characterizing the resulting
heterogeneous conformational ensembles and understanding the mechanisms of complex formation. In summary, a large array of problems related to the conformations and dynamics of unfolded
and intrinsically disordered proteins is waiting to be addressed, and concepts from polymer and
soft matter physics will be essential for progress.
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DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdings that
might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS
We thank Alessandro Borgia for discussions and for providing raw data for figures, and all members
of our research group for discussions over the years. Our work has been supported by the Swiss
National Science Foundation, the European Research Council, and the Human Frontier Science
Program.
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