single-molecule fret spectroscopy and the polymer ......forster resonance¨ energy transfer (fret),...

BB45CH10-Schuler ARI 8 June 2016 18:12

Single-Molecule FRETSpectroscopy and the PolymerPhysics of Unfolded andIntrinsically DisorderedProteinsBenjamin Schuler, Andrea Soranno, Hagen Hofmann,and Daniel NettelsDepartment of Biochemistry, University of Zurich, 8057 Zurich, Switzerland;email: [email protected]

Annu. Rev. Biophys. 2016. 45:207–31

First published online as a Review in Advance onMay 2, 2016

The Annual Review of Biophysics is online atbiophys.annualreviews.org

This article’s doi:10.1146/annurev-biophys-062215-010915

Copyright c© 2016 by Annual Reviews.All rights reserved

Keywords

single-molecule fluorescence, correlation spectroscopy, protein dynamics,protein folding, internal friction, Forster resonance energy transfer, FRET

Abstract

The properties of unfolded proteins have long been of interest because oftheir importance to the protein folding process. Recently, the surprisingprevalence of unstructured regions or entirely disordered proteins underphysiological conditions has led to the realization that such intrinsically dis-ordered proteins can be functional even in the absence of a folded structure.However, owing to their broad conformational distributions, many of theproperties of unstructured proteins are difficult to describe with the estab-lished concepts of structural biology. We have thus seen a reemergenceof polymer physics as a versatile framework for understanding their struc-ture and dynamics. An important driving force for these developments hasbeen single-molecule spectroscopy, as it allows structural heterogeneity, in-tramolecular distance distributions, and dynamics to be quantified over awide range of timescales and solution conditions. Polymer concepts pro-vide an important basis for relating the physical properties of unstructuredproteins to folding and function.

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http://www.annualreviews.org/doi/full/10.1146/annurev-biophys-062215-010915


Intrinsicallydisordered protein(IDP): polypeptidechain lacking stabletertiary structureunder physiologicalconditions

Forster resonanceenergy transfer(FRET): thenonradiative transferof an energy quantumfrom its site ofabsorption (the donorchromophore) to anacceptor chromophoreby resonanceinteraction betweenthe two

Contents

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208SINGLE-MOLECULE FRET OF UNFOLDED

AND DISORDERED PROTEINS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Quantifying Distance Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209Quantifying Chain Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

POLYMER PROPERTIES OF UNFOLDEDAND DISORDERED PROTEINS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Denaturants and the Collapse Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Charge Interactions and Polyampholyte Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Chain Dimensions and Scaling Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Temperature-Induced Compaction and Solvation Free Energies . . . . . . . . . . . . . . . . . . 218Crowding and Excluded-Volume Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219Chain Dynamics of Unfolded and Intrinsically Disordered Proteins . . . . . . . . . . . . . . . 220Internal Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

CURRENT DEVELOPMENTS AND FUTURE CHALLENGES . . . . . . . . . . . . . . . . 223

INTRODUCTION

Polymer physics has a long history in biology and was applied to biopolymers long before the rou-tine availability of high-resolution structural information from crystallography or nuclear magneticresonance (NMR). For proteins, however, the compelling atomistic detail provided by structuralbiology resulted in a predominant interest in their precise three-dimensional structures. Onlyin 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 thatunstructured regions or entirely disordered proteins are surprisingly widespread, especially in eu-karyotes (164). For these intrinsically disordered proteins (IDPs), polymer properties—such asstructural heterogeneity, broad conformational distributions, and long-range fluctuations—are ofobvious relevance (5).

A broad range of powerful methods for investigating unfolded proteins and IDPs exist (161),among them NMR (75) and small-angle X-ray scattering (SAXS) (9). Recently, single-moleculespectroscopy 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 andavoiding many complications of ensemble averaging (46); it enables access to a wide spectrumof timescales for investigating dynamics (137); and it allows investigations over a broad range ofsolution conditions, from very low sample concentrations, where nonidealities such as aggregationare 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 suitablefor probing their polymer properties are force (126, 182) and fluorescence spectroscopy. Herewe focus on work that uses fluorescence spectroscopy, particularly in combination with Forsterresonance energy transfer (FRET).

SINGLE-MOLECULE FRET OF UNFOLDEDAND DISORDERED PROTEINS

Single-molecule FRET is an attractive method for probing long-range distances and distancedynamics in unfolded proteins because it is most sensitive in the range of ∼2 to 10 nm, an

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intramolecular distance range typical of polypeptide segments from several tens of to a fewhundred residues, depending on the compactness of the chain. The theoretical basis of the FRETprocess was established by Theodor Forster (51), who showed that the rate coefficient of excitationenergy transfer, kT , between a suitable donor and acceptor chromophore is proportional to theinverse sixth power of their distance,

kT = kD

(R0

r

)6

, 1.

where kD−1 = τD 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 Forster radius, can be obtained fromreadily measurable quantities (51): the overlap integral between the donor emission and theacceptor absorption spectra, the donor fluorescence quantum yield, the refractive index of themedium between the dyes, and the relative orientation of the chromophores. For cases in whichthe rotational reorientation of the chromophores is fast compared to the donor excited-statelifetime, the orientational contribution, κ2, averages to a value of 2/3, which simplifies the appli-cation of FRET considerably [the applicability of κ2= 2/3 can be tested by polarization-sensitivedetection (34, 146)]. Assuming a single fixed distance, the probability that a photon absorbed bythe donor will lead to energy transfer to the acceptor, called the FRET efficiency, E, is given bykF/(kF + kD), which (with Equation 1) leads to the relation (Figure 1)

E(r) = R60

R60 + r6

. 2.

The Forster radius is thus the characteristic distance that results in a transfer efficiency of 1/2. E isdetermined experimentally either by counting the number of emitted donor and acceptor photons[E = nA/(nA + nD), including the necessary corrections (34)] or by measuring fluorescencelifetimes, for example, of the donor in the presence (τDA) and absence (τD) of the acceptor (E =1 − τDA/τD for a single fixed distance).

Because the molecules are observed individually (either from bursts of photons originatingfrom single molecules diffusing through the confocal observation volume, or in fluorescence timetraces of immobilized molecules), structural heterogeneity can be resolved if the underlying con-formations differ in transfer efficiency and if their interconversion dynamics are sufficiently slowcompared to the time resolution of the measurement. A simple example is the equilibrium betweenfolded and unfolded molecules freely diffusing in solution, resulting in separate subpopulations inthe transfer efficiency histogram (Figures 1c and 3a) if they interconvert on a timescale slowerthan approximately a millisecond, the typical diffusion time through the confocal volume. In thiscase, the events corresponding to unfolded molecules can be singled out, and the properties ofthe unfolded state can be investigated without overlap from the signal of folded molecules (and ofmolecules lacking an active acceptor dye). The same idea can be extended to systems with morethan two subpopulations (8, 121). For a more detailed introduction to the technical aspects andthe versatility of multiparameter detection and analysis, we refer the reader to recent reviews (136and 146).

Quantifying Distance Distributions

An important aspect that needs to be taken into account for unfolded proteins and IDPs is thatwe 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 unfoldedmolecules, we still have to factor in the conformational heterogeneity within the unfolded-state

www.annualreviews.org • Polymer Physics of Unfolded Proteins and IDPs 209

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05

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Shot noise: thevariation in count ratesabout fixed means dueto the discrete natureof the signal. Insingle-moleculeForster resonanceenergy transfer(FRET), itscontribution is oftenimportant because ofthe relatively smallnumber of photonsdetected per unit time

Excluded volume:describes the net two-body interactionbetween monomers ina chain, includingsteric repulsion andcharge- orsolvent-mediatedinteractions

Persistence length:a measure of polymerstiffness, defined as thecharacteristic lengthalong the chain overwhich orientationalcorrelations betweensegments decay

ensemble. Dynamics in proteins above the microsecond timescale usually involve persistent tertiarystructure or cooperative interactions (41). In the absence of such folded or misfolded structures, thedynamics within unfolded or intrinsically disordered states are thus expected to be fast comparedto the typical photon detection rates of ∼103–105 s−1 (137); this case results in a narrow, shotnoise–dominated peak in the transfer efficiency histogram (56, 57). If the fluorophores reorientrapidly compared to the donor fluorescence lifetime, which is often the case for unfolded proteinsand IDPs, as indicated by fluorescence anisotropy measurements (83, 113), κ2 can be assumed toaverage to 2/3, and the mean transfer efficiency of the unfolded state, 〈E〉, results as

〈E〉 =∫ lc

cE(r)P (r) dr, 3.

where c is the distance of closest approach of the dyes, lc is the contour length of the polypeptidesegment probed, and P(r) is a suitably normalized probability density function (Figure 1). [If theserequirements 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 unfoldedstate in a transfer efficiency histogram can be analyzed only in terms of a single parameter, its meanFRET efficiency, 〈E〉, and Equation 3 is thus underdetermined unless additional information aboutthe functional form of P(r) is available. Different approaches have been chosen to obtain suitabledistance distributions. The simplest approximation is the use of polymer-physical models to reflectthe underlying chain statistics—for example, the Gaussian chain (67, 109, 138, 142), worm-likechain (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 parameterin 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 ofthe chain, and that the mean of the distribution is near the Forster radius, these distributions usuallyprovide very similar results in terms of average chain dimensions (109). Alternatively, molecularsimulations (at various levels of coarse graining) have been employed to obtain information aboutthe shape of P(r) (11, 81, 100, 110, 116, 117, 139) and to assess deviations from simple polymermodels (116, 139) (Figure 2).

Additional experimental information about P(r) is available from the analysis of fluorescencelifetimes (67, 90, 147) (Figure 1): Because the dynamics of disordered states (for the chain lengthsaccessible in single-molecule FRET) are much slower than the fluorescence lifetimes of thedyes (∼1 to 4 ns), the distribution of distances results in a distribution of fluorescence inten-sity 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 va-lidity of specific functional forms of P(r), ideally by a global analysis of both donor and acceptor

←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−Figure 1Experimental observables and data analysis in single-molecule Forster resonance energy transfer spectroscopy of unfolded-statedimensions and dynamics. (a) The signal from individual molecules labeled with donor (D) and acceptor (A) fluorophores freelydiffusing through the confocal volume is detected and (b,f ) analyzed in terms of the distance-dependent transfer efficiency betweendonor and acceptor. (c) Transfer efficiency histograms illustrate the separation of subpopulations (here, folded state at high E, unfoldedstate at intermediate E, and donor-only population at E ≈ 0). (d ) The combination with fluorescence lifetime analysis (e) providesadditional 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 timesare 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 distanceprobability density function, P(r). K0(r) is the rate matrix that describes the distance-dependent kinetics of interconversion between theelectronic states (b), and p(r,t) is the vector of their populations.


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Gaussian chain

Worm-like chain (WLC)

Sanchez

Self-avoiding walk (SAW)

1 3 4

lc

lp

7 1 1−ϕln(1−ϕ)

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0 20 40 60 80 100 120 140

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〈r 2〉1/2

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π3/2e–αα–3/2 (1+ 3/α +15/(4α2))

4lp(1–(r/lc)2)

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P(r) =

P(r) = P(r |Rg)P(Rg,ε,RgΘ)dRg

P(Rg,ε,RgΘ) = Z –1R6g exp –

R2g

R2gΘ

P(r |Rg) = δ · Rg δ · Rg δ · Rg δ · Rg

lc /2

Rc

⌠⌡

0.020

0.015

0.010

0.005

0.000

P(r)

0.020

0.015

0.010

0.005

0.000

P(r)

r(Å)

0 20 40 60 80 100 120 140

r(Å)

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, lpthe persistence length, ε the mean interaction energy between amino acids); all other parameters are known independently. Rg� is theroot-mean-squared radius of gyration of the � state (see the section Chain Dimensions and Scaling Laws), and φ is the volume fractionof 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 thedenatured-state end-to-end distance distributions of protein L at (b) 2.4 M and (c) 6 M guanidinium chloride (GdmCl) using aCα-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 thedonor fluorescence lifetime versus transfer efficiency from count rates (Figure 1), in which theexistence of a distance distribution resulting from rapidly interconverting conformations showsup as a deviation from the linear dependence of τDA on E expected for a single fixed distance (59,78, 147).

Residual secondary structure or the formation of long-range interactions can lead to deviationsfrom the distance distributions based on simple models or simulations, even if global aspects such as

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Stokes radius: theradius of a sphere that,according to theStokes-Einsteinrelation, has the samediffusion coefficient asthe molecule ofinterest

Fluorescencecorrelationspectroscopy (FCS):a method used toquantify the dynamicsof molecular systems,such as theirtranslational diffusionor conformationalchanges, via theresulting fluctuationsin fluorescenceemission rates

Reconfigurationtime: the relaxationtime of theautocorrelationfunction of thedistance between twomonomers in a chain

length scaling are preserved (48, 116). Experimentally, these effects can be quantified by mappingdifferent segments of the protein (by varying the positions of the fluorophores) and testing theconsistency with the models employed (67, 70, 110, 116, 147), for example, in terms of the scalingof 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, andalthough hydrodynamic properties can be difficult to relate to polymer models quantitatively, theyprovide complementary information about changes in chain dimensions and length scaling (70, 71,97, 144). Finally, combining single-molecule FRET with complementary methods sensitive to thepresence of structure, such as circular dichroism spectroscopy (68) or NMR (under conditions inwhich interference with the signal from folded or structured states can be excluded), will becomeincreasingly important.

Quantifying Chain Dynamics

Single-molecule fluorescence can be used to probe conformational dynamics over a wide range oftimescales (137). A benefit of single-molecule experiments is the possibility of extracting informa-tion on dynamics, even from equilibrium measurements, without requiring a synchronization ofthe system by perturbation methods, as frequently employed in ensemble experiments. For pro-teins, 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 subpopulationsin FRET efficiency histograms (25, 58, 68, 138), and from fluorescence trajectories of immobi-lized molecules (25, 26, 87, 88, 121, 124, 125). In this way, both the equilibrium distributionsand the kinetics can often be obtained from the same measurement. Alternatively, nonequilib-rium 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 nanosecondfluorescence correlation spectroscopy (nsFCS) combined with FRET (55, 112, 113). Distancefluctuations between the donor and the acceptor fluorophores then result in fluctuations in fluo-rescence intensity on the timescale of interconversion between the different configurations of thepolypeptide chain. For relating the measured decay time of the fluorescence correlation curvesto the reconfiguration time of the chain, the relative motion of the two labeled positions can berepresented as a diffusive process in a potential of mean force that corresponds to the equilibriumdye-to-dye distance distribution of the unfolded protein (Figure 1). In this simplified descrip-tion, the dynamics of the system are determined (a) by the shape of the potential [i.e., the freeenergy as a function of the distance between the dyes, G(r), which is obtained directly from thedistance 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). Thisrelaxation time corresponds to the reconfiguration time, τ r, of the chain segment between thedyes.

In summary, the single-molecule approach described here can provide two essential quanti-ties that characterize the global properties of unfolded or intrinsically disordered proteins: in-tramolecular distance distributions (from transfer efficiency and fluorescence lifetime measure-ments) and reconfiguration dynamics (from fluorescence correlation analysis). These are keyquantities in the statistical mechanics of polymers (20, 30, 38, 50) and thus make the resultsamenable to the application (and even the testing) of concepts in polymer physics, with the goalof providing a coherent and quantitative framework for the behavior of unfolded proteins andIDPs.


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POLYMER PROPERTIES OF UNFOLDEDAND DISORDERED PROTEINS

Denaturants and the Collapse Transition

Polymers respond to changes in solvent quality by a change in chain dimensions: In poor sol-vent, where intrachain interactions dominate over chain-solvent interactions, they collapse intocompact globules; in good solvent, where chain-solvent interactions are favorable, they populaterather 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 guani-dinium chloride (GdmCl), however, proteins and peptides are highly soluble and (in the absenceof disulfides) form expanded unfolded states (80, 127). Correspondingly, the chain dimensions offolded globular proteins, quantified, e.g., in terms of the radius of gyration, Rg, exhibit a depen-dence on the number of peptide segments in the chain, N, that is in good agreement with thescaling behavior expected for compact globules, Rg ∝ N1/3 (36, 159, 168). For unfolded proteinsin high concentrations of denaturant, measurements of intrinsic viscosity, NMR, and small-anglescattering 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 structureor populated folding intermediates, denatured states are expected to exhibit a compaction akinto heteropolymer collapse when the solvent quality (here modulated by the denaturant concen-tration) is reduced (2, 20, 27, 71, 117). This behavior has been demonstrated particularly clearlyby the molecular transfer model (117) (Figure 3b), which combines simulations in the absenceof denaturants with experimentally determined transfer free energies of side chain and backbonemoieties from water to denaturant solution (153). However, detecting this behavior in ensembleexperiments had been challenging, as the corresponding signal change is hard to separate fromthe folding transition. The separation of folded and unfolded subpopulations in single-moleculeFRET experiments (33) led to the identification of this unfolded-state compaction in terms ofa continuous shift in transfer efficiency (138, 142) (Figure 3a); it has since been observed formany proteins (65, 137) and is in agreement with a range of other experiments, theory, and sim-ulations (71, 94, 107, 117, 155, 156, 158, 174, 181). However, the detectability of unfolded-statecompaction in SAXS experiments has remained controversial (175).

Particularly interesting is the analysis of denaturant-dependent unfolded-state compaction interms of the theory of coil-globule transitions (35, 61, 131), which allows chain compaction or ex-pansion 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 expressionfor the probability density function of Rg in the form of a Boltzmann-weighted Flory-Fisk distri-bution (Figure 2). Ziv & Haran (180) analyzed a large set of results in which the mean transferefficiency 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〉 andcomputing the free energy of the chain based on the Sanchez theory, they found that the dif-ference between the free energy of unfolded proteins and their most collapsed states dependsapproximately linearly on the concentration of GdmCl and that the change of this free energywith denaturant concentration is close to the equilibrium m value, the derivative of the free energydifference 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 originatesfrom the free energy change of collapse, thus providing a link between the polymeric propertiesof the unfolded state and protein folding mechanisms (180).

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0 2 4 6 8

0.5

1.0

1.5

ε (k BT

)E

GdmCl concentration (M)

GdmCl or urea (M)

CspTm

a b

c d

ε

0 4 8 120

10

20

×10–2

0

150

7

0

5

0.9

1

0.7

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0

5

00.0 0.5 1.0

20

2.0 M

3.5 M

1.5 M

0.0 MN

umbe

r of e

vent

s (×

100)

0

10 5.0 M

8.0 M

0

5 0.8 M

〈E 〉

P(r)

r (nm)

Figure 3Denaturant-dependent unfolded-state compaction. (a) Changes in unfolded-state dimensions can bedetected in single-molecule Forster resonance energy transfer (FRET) experiments owing to the separationof folded and unfolded populations, here shown for cold shock protein (Csp) as a function of guanidiniumchloride (GdmCl) concentration (67, 138). (b) Denaturant dependence of the experimental transferefficiencies 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 theoryenables the analysis of single-molecule FRET experiments and the underlying changes in chain dimensionsin terms of their mean-field interaction energy ε as a function of denaturant concentration (180) and (d ) wasused by Ziv & Haran (180) to relate ε for the unfolded states of many small proteins to their equilibriumunfolding m values. Panel b modified from Reference 117 with permission, copyright 2008, NationalAcademy 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 bythe mean interaction energy between amino acids as expressed in the Sanchez theory (Figure 2),comprises different contributions, including hydrogen bonding and hydrophobic and electrostatic


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0 10 20 30 40 50 600

10

20

30

40

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60

70EVFRCLJ

| j− i|

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Denaturant concentration (M)

R g (n

m)

R g (n

m)

〈rij〉 (

Å)

a b c

d

1

2

3

5

7

10

10 100

ν = 0.58

1,000

N

νννννννν = 0.58

2.82.44.03.6

3.2

2.84.8

4.4

4.0

3.6

3.2

2.8

0 2 4 6 8

Folded

disorderedSwollen

coilDisordered

globulef– f+

Swollencoil

Intrinsically

Hydropathy

ν = 0.34

hCyp

Figure 4Effect of sequence composition on chain dimensions. (a) Diagram of states to classify predicted conformational properties ofintrinsically disordered proteins (IDPs) as a function of the fraction of positively ( f+) and negatively ( f−) charged residues, and thehydropathy 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 (opencircles) 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 GdmClbinding and electrostatic repulsion are indicated as continuous and dashed gray lines, respectively. The colored squares indicate thechanges in Rg upon addition of 1 M KCl. (c) Averaged interresidue distances, 〈Rij〉, versus sequence separation, | j − i |, based onsimulations for 10 naturally occurring IDPs (symbols), illustrate the dependence of chain dimensions on the content of charged residuesand their distribution in the sequence (29). The solid lines are reference curves corresponding to an excluded volume chain (EV, red ), aFlory random coil (FRC, black), and a compact globule [Lennard–Jones (LJ), blue]. (d ) Rg for unfolded cyclophilin as a function of thenumber of bonds, N, at different GdmCl concentrations from 0 M (blue) to 6 M (red ) (70). Colored dashed lines are fits according toRg = Rg0 Nν . Gray circles are RG values determined for unfolded proteins by small-angle X-ray scattering (SAXS) (84). Open bluecircles are RG values of denatured proteins under native conditions determined with SAXS (163). Black solid lines are fits of the datataken from Reference 84 and of monomeric folded proteins from the Protein Data Bank with fits of the form Rg ∝ Nν ; the resultingscaling exponents, ν, are indicated. Panel a modified from Reference 164, panel b modified from Reference 109, panel c modified fromReference 29, and panel d modified from Reference 70.

interactions. For IDPs, charge interactions are expected to be particularly pronounced, as theiramino acid composition is strongly biased toward low hydrophobicity and high charge content (97,162). The conformational properties of proteins, especially their degree of expansion, as a functionof these contributions can be classified in terms of a state diagram (29, 97, 164) (Figure 4a). Oneway of treating the underlying interactions of charges within the chain quantitatively is through

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Polyelectrolyte:a macromolecule inwhich a substantialfraction of themonomers containionizable or ionicgroups

Polyampholyte:a polyelectrolyte thatcontains bothpositively andnegatively chargedgroups

Polyampholytetheory: describes theeffect of chargeinteractions on theconformations of apolymer, includingboth repulsiveinteractions betweencharges of the samesign and attractiveinteractions betweencharges of oppositesign

polyampholyte theory (37), which encompasses both repulsive interactions between charges of thesame 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 withprobability 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 distancesassumed to follow Gaussian chain statistics. The effect of the electrostatic interactions on chaindimensions can then be treated as a contribution to an effective excluded volume, v∗b3. Introducingthe expansion factor α as the ratio of an effective segment length b1 and the real segment lengthb, one can then describe the chain dimensions using the expressions

α5 − α3 = 43

(3

2π

)3/2

N 1/2ν∗, where α = b1

b, 4.

and ν∗b3 = νb3 + 4π lB( f − g)2

κ2− π l2

B( f + g)2

κ. 5.

The second term of the right-hand side in Equation 5 accounts for repulsive interactions due to thenet charge of the polypeptide, resulting in an increase of v∗b3, as in related forms of polyelectrolytetheory (62). The third term leads to a reduction of v∗b3 through attractive interactions betweencharges of opposite sign. lB is the Bjerrum length and 1/κ the Debye length.

Muller-Spath et al. (109) applied this theory to a series of unfolded and intrinsically disorderedproteins for which the importance of the net charge was qualitatively obvious from the differentresponse 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 allchains, but the screening of charged residues in the IDPs by GdmCl caused a pronounced collapseof 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 equalnumbers of acidic and basic amino acids, exhibited signs of polyampholyte behavior: The additionof salt led to an expansion rather than a collapse, indicating that attraction between charge residuesof opposite sign causes compaction. The response of the entire set of proteins was described wellby the theory of Higgs and Joanny, combined with a simple binding model (96) accounting fordenaturant-induced chain expansion (109). The result demonstrated that charge interactions candominate 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 patterningwithin the sequence (29) or the formation of persistent local or long-range structure (44, 93) canlead to deviations from this relatively simple behavior (Figure 4c). For instance, segregation ofcharges along the sequence can lead to structure formation, and simulations or more advancedtheoretical approaches are required for a quantitative treatment of these effects (29, 132).

Chain Dimensions and Scaling Laws

Denaturants and charge interactions are only two of the factors that have a pronounced influenceon the dimensions of denatured proteins and IDPs. Given the quasi-continuous spectrum of theiramino acid compositions (162) and the broad range of relevant solution conditions, a quantitativeway of classifying the degree of expansion or collapse of unfolded and intrinsically disorderedproteins is essential (Figure 4a). That some denatured proteins are neither as expanded as fullyunfolded states nor as compact as folded proteins was already realized in the 1980s. Both equi-librium and kinetic intermediates in protein folding with such “molten globule” properties wereobserved (123), and the scaling of chain dimensions with the length of the polypeptide was found


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Scaling laws: describethe scale invarianceobserved for manynatural phenomenaand are frequentlyused in polymerphysics, for example,to quantify the scalingof chain dimensionswith chain length

� conditions:conditions underwhich attractiveinteractions within thechain counterbalancethe steric repulsionbetween monomers,and the polymerfollows the statistics ofan ideal chain to goodapproximation

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 com-paction quantitatively to the effective interaction energies between the residues via the Sancheztheory of coil-globule transitions described above (131, 142, 180). Doing so requires knowledgeof the dimensions of the chain at its � point, where chain-chain and chain-solvent interactions ofa 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 classicway of determining length scaling in polymers—is difficult for proteins. Hofmann et al. (70) thusmade use of the finding that previously determined values of the prefactor, Rg0, in scaling laws ofthe type Rg = Rg0 Nν vary only from ∼0.2 nm to ∼0.3 nm, even when comparing polypeptidesin high concentrations of denaturants and folded globular proteins (48, 84, 168, 176), and canbe interpolated in this range (70). With this constraint, the resulting scaling exponents, ν, can bequantified for a broad range of proteins (Figure 4d ) and depend sensitively on their sequence com-position, net charge, and denaturant concentration. This approach also allows the � conditionsto be identified based on the chain dimensions. Interestingly, protein sequences with a moderatenet charge and average hydrophobicity were found to be close to � conditions in water (70). Ithas been suggested that folding from the � state provides an ideal evolutionary compromise byfavoring 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 single-molecule FRET experiments of an FG-repeat nucleoporin, a connection between chain collapseand the capacity to form hydrogels has been proposed (103). In summary, length scaling exponentsare thus an appealing way of quantifying the degree of expansion of unfolded proteins and IDPsbecause they are independent of the length of the individual polypeptide (160) and can be usedto reflect differences in effective solvent quality brought about both by differences in amino acidcomposition (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 some-what counterintuitive. Typical homopolymers in organic solvent, for example, expand monotoni-cally with increasing temperature (149), as expected if thermal energy increases compared with at-tractive intrachain interactions (131). For proteins, however, laser temperature-jump experimentshad 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 hadshown increasing radii of hydration as temperature was reduced (92). The separation of foldedand unfolded subpopulations in single-molecule FRET experiments enabled an investigation ofthis issue over a wider range of temperatures. For Csp, a continuous collapse of the unfolded statebetween ∼280 K and 340 K was observed (114). Similarly, denatured frataxin collapsed over abroad range of temperatures, spanning both the cold- and the heat-denatured state (4).

Although this temperature-induced compaction is suggestive of hydrophobic interactions as adriving force (92, 129), highly charged IDPs exhibit the most pronounced collapse (114, 172), in-dicating the importance of additional contributions related to changes in solvation free energy as afunction of temperature. A critical role of the solvent contribution is supported by molecular sim-ulations of unfolded proteins with different water models (12, 114). Interestingly, the compactionobserved for five different unfolded or intrinsically disordered proteins could be reproduced

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0.7

0.8

0.9

λ = 6.93.81.90.9

1

0 0.1 0.2 0.3 0.4

SPT

1 5 10 50 100 500

3.0

3.2

3.4

3.6

Degree of polymerization0 0.5 1

0

50

Transfer efficiency

0

50

Num

ber o

f bur

sts Cytosol

Nucleus

3.8

4.0

4.2

g DD

(τ)

–1 –0.5 0 0.5 1

τ (μs)

a bFH renFH

cgcSPT

SPT

gcSPT

Renormalized FHFH

R g(ϕ

)/R g

(0)

R g (n

m)

ϕ

Short chainsShort chains Long chainsLong chains

Figure 5Effects of crowding and cellular conditions on intrinsically disordered proteins. (a) Crowding effect on the conformation of aself-avoiding-walk chain as a function of crowder volume fraction, φ, and the ratio of chain dimensions and the crowder size, λ, for solidspherical crowders based on simulations (79). An illustration of the chain and the crowding particles for λ = 1.9 and φ = 0.3 is shownon top. (b, top) Graphical representation of scaled-particle theory (SPT), Gaussian cloud model (gcSPT), Flory-Huggins theory (FH) inthe short-chain regime, and renormalized Flory-Huggins theory (renFH) in the long-chain regime. (b, bottom) Radius of gyration ofProTαC 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 (renFH) lines. Solid lines indicate the regime for which the respective theories were derived;otherwise, dashed lines are used. (c) Single-molecule Forster resonance energy transfer (FRET) measurements in live cells aftermicroinjection of labeled unfolded molecules: (top) Schematic illustration, (middle) single-cell FRET efficiency histograms ofprothymosin 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.

Molecular crowding:describes the effects ofhigh concentrations ofinert cosolutes onmolecular interactionsand conformations

semiquantitatively in implicit solvent simulations by taking into account the temperature-dependent solvation free energies of the constituent amino acids, with the solvation properties ofthe 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 andthe solvation properties of the constituent amino acids, similar in spirit to the molecular transfermodel (117).

Crowding and Excluded-Volume Screening

Ultimately, we want to understand the behavior of unfolded proteins and IDPs in a cellularenvironment. An important difference compared to typical experiments in vitro is the high con-centration of cellular components, resulting in molecular crowding effects (177). In view of theirlack of persistent structure, IDPs and unfolded proteins are expected to be particularly sensitiveto the effects of crowding (Figure 5a). Indeed, experiments indicate that some unfolded proteinsor IDPs gain structure upon crowding (32), while others do not (99, 151) but may change their di-mensions (72, 77, 101, 148). The theoretical concept most widely used for rationalizing crowding


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Semidilute regime:regime in which chainsare present atsufficiently highconcentration forthem to overlap so thatthe interactionsbetween polymer coilscan dominate theirbehavior

Excluded volumescreening: thescreening of repulsiveinteractions within achain by theinterpenetration withother polymers atconcentrations in thesemidilute regime andabove

effects is scaled-particle theory, which provides an estimate of the change in free energy requiredfor creating a cavity equivalent to the size of the IDP (or the crowded molecule in general) ina 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 be-comes entropically more unfavorable with increasing crowder concentration, thus promoting theformation of more compact configurations (79, 106, 177).

However, this treatment can fail in the presence of polymeric crowders, most obviously whenthe 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 bothIDPs and crowders in the model explicitly to account both for the interpenetration between IDPsand crowders and for the mutual overlap between crowding polymers in the semidilute regimeand 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 volumescreening effects quantitatively (Figure 5b) (148). Surprisingly, both recent single-molecule FRETmeasurements of the highly charged IDP prothymosin α (83) and ensemble FRET experimentsof polyethylene glycol (54) in live eukaryotic cells have not indicated a pronounced effect ofcrowding on chain dimensions (Figure 5c). One possible reason may be that the macromolecularconcentrations in cultured eukaryotic cells are lower (119, 169) than is commonly assumed basedon estimates for microbial cells (179). Another aspect that will require future study and thatmay be difficult to treat with simple models is how crowding effects are modulated by attractiveintermolecular interactions (105).

Chain Dynamics of Unfolded and Intrinsically Disordered Proteins

Conformational dynamics in proteins can occur over an enormous span of timescales owing to thebroad range of enthalpic and entropic contributions involved. The slowest conformational changefor an individual structured protein will usually be its global folding and unfolding, which can takeplace on timescales from microseconds to hours and beyond. Within unfolded or intrinsicallydisordered states, however, cooperative interactions linked to the formation of specific structuresare 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 chemicallyunfolded 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 withmembranes (47).

In the absence of persistent intra- or intermolecular interactions, an unfolded polypeptide chainis expected to approach dynamics that are limited by monomer diffusion through the solvent anddescribed by simple polymer models, such as the Rouse or Zimm theories (38). In the Rousetheory, chain dynamics are represented by harmonic modes, arising from the motion of beadsrepresenting individual chain segments connected by ideal springs, and whose dynamics are thuscontrolled only by the neighboring beads and the viscous drag from the solvent. For severalproteins unfolded in high concentrations of denaturant, good agreement with these models hasbeen observed, corresponding to reconfiguration times, τ r, of several tens of nanoseconds for 50-to 70-residue segments (147). Note that according to Onsager’s regression hypothesis, this timealso corresponds to the collapse time of the chain (112), in agreement with laser temperature-jump experiments (129), and is thus closely linked to the pre-exponential factor in protein folding

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Internal friction: adissipative force withina medium (in contrastto hydrodynamicfriction with thesolvent), here withinan unfolded protein,that slows down itsdynamics

kinetics (40, 112, 129, 173). In terms of an effective end-to-end diffusion coefficient (150), thesedynamics 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) typ-ically increases concomitant with chain compaction (14, 112, 147), in contrast to the behaviorexpected 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 opera-tionally as a deviation from proportionality between relaxation time and solvent viscosity, internalfriction had previously been identified in folded proteins (3) and in the context of protein foldingreactions (18, 64). Its molecular origin was mainly assigned to the large degree of desolvation inthe very compact folded or transition states and the resulting decoupling from the solvent. Forunfolded proteins, however, where the expansion of the chain relative to the folded state entailspronounced solvation even in the absence of denaturant (67, 70), the origin of internal friction isless obvious.

Internal Friction

There is a long history of theoretical concepts in polymer dynamics that address the question ofinternal friction (or internal viscosity). Early ideas go back to Kuhn & Kuhn (85), Cerf (19), and deGennes (30). A particularly simple class of models that allow internal friction to be incorporatedrigorously (23) are those of Rouse (128) and Zimm (38, 178). Internal friction can be includedin 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 itincreases all relaxation times by the same internal friction time, τ i, yielding a modified spectrumof relaxation times,

τ (n) = τRouse/n2 + τi, n = 1, 2, . . . , 6.

where τRouse is the Rouse time, corresponding to the slowest relaxation mode of the Rouse chainin the absence of internal friction, and n is the mode number. A simple common assumption isthat only the dynamics corresponding to τRouse/n2 depend on solvent viscosity, η, whereas τ i doesnot (30). Because the overall solvent-dependent relaxation time, τ s, with contributions from allrelaxation modes, is the same as τRouse 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 measuredas a function of solution viscosity and extrapolated linearly to zero viscosity, the internal frictiontime would result as the intercept. The same behavior is found for the Zimm model when inter-nal friction is included analogously, only with a mode dependence of n−2/3 compared to n−2 inEquation 6 (23).

The solvent viscosity dependence predicted by Equation 7 is in good agreement with ex-perimental observations for unfolded Csp (147), with increasing values of τ i as the denaturantconcentration is decreased (Figure 6b). This result was confirmed by an orthogonal approach notrequiring changes in viscosity: Because the chain exhibits a spectrum of relaxation modes (Equa-tion 6), the relative contribution of internal friction to the reconfiguration dynamics will dependon the length of the segments probed (82, 95). By placing the FRET dyes in different positionswithin the chain, one can determine τ i independently. Very similar results are obtained (147) usingthe Zimm model with internal friction (23). For chemically unfolded spectrin domains, significantinternal friction was present even at the highest GdmCl concentrations (14), indicating a possiblerole for residual local interactions and secondary structure formation. Notably, spectrin domains


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Figure 6Internal friction in unfolded proteins and intrinsically disordered proteins (IDPs). (a) Schematic of a Rouse chain (represented as beadscoupled by harmonic springs) with internal friction (indicated as dashpots) and terminal dyes. (b) Solvent viscosity (η) dependences ofchain reconfiguration times, τ r, of terminally labeled cold shock protein at different guanidinium chloride concentrations (indicated ineach 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 differentunfolded proteins and IDPs, as indicated by their apparent persistence length, lp, shows a trend of more internal friction for morecompact 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 rescalingthe solvent mass (140), with linear (solid line) and power-law (dashed line) fits, and a fit based on a Rouse model for contact formationincluding 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 amodified from Reference 82; panel b modified from Reference 147; panel c modified from Reference 42, copyright 2014, AmericanChemical Society; panel e modified from Reference 140, copyright 2012, American Chemical Society; and panel f modified withpermission 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 simula-tions that interactions such as hydrogen bonds (140) local in sequence contribute more stronglyto 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 thanunfolded Csp or spectrin in the absence of denaturant (147). A plot of τ i versus the dimensionsof the chains 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 internalfriction. Echeverria et al. (42) simulated unfolded Csp at different GdmCl concentrations,found remarkable agreement with the experimentally determined dynamics (Figure 6c), andconcluded from their analysis that dihedral angle transitions provide the dominant mechanism ofinternal friction. It will be interesting to further investigate the possible connection to intrachain

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interactions such as hydrogen bonding, which has been suggested to cause internal frictionbased on atomistic simulations of peptides (140) (Figure 6e). Another potentially very importantcontribution is the low sensitivity of dihedral angle isomerization to solvent viscosity, as predictedtheoretically by Portman et al. (122) and recently identified to affect protein folding dynamics byde Sancho et al. (31) in simulations (Figure 6f ).

CURRENT DEVELOPMENTS AND FUTURE CHALLENGES

Polymer concepts have proven very valuable for a physical understanding and classification ofthe structural and dynamic behavior of unfolded and intrinsically disordered proteins. This isthe case especially in the context of the global properties typically probed with single-moleculeFRET. Obviously, the applicability of simple models needs to be evaluated for each case, andin the presence of persistent secondary or tertiary structure, the role of specific interactions andprotein folding may have to be included in a physical description. For many questions, a closercombination with experimental methods complementary to single-molecule spectroscopy willthus 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 populatedexclusively, these methods can provide essential additional information, such as the local structuraland dynamic properties available from NMR. This type of integrated approach will lead to a morecomprehensive view of the behavior of unfolded proteins and IDPs (161). In regard to single-molecule spectroscopy, challenges include developing a more complete use of the broad spanof timescales accessible from correlation analysis (136, 146), taking full advantage of three-colorFRET experiments (52, 102), and the extension to intracellular measurements (1, 83).

At the same time, simulations will become increasingly important for linking experimentalobservations and chain statistics to molecular detail. The traditional focus on folded proteins forthe parameterization of the energy functions for molecular dynamics simulations has typicallyresulted 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 routineavailability 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 proteinsand IDPs even though their heterogeneous conformational ensembles are underdetermined byexperimental observables alone (75, 110, 165).

This improved quantitative understanding paves the way for addressing questions of increasingcomplexity, 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 toremain partially, largely, or possibly even completely unstructured in the bound state (157) andthus retain much of their polymer dynamics. Closely connected are the processes of liquid-liquidphase separation and coacervation (16), which have recently seen a revival in the context of non-membrane bound compartments in cells (74). A key challenge will be to bridge the gap betweenthe mesoscopic picture emerging from the concepts of phase separation (74) and the underlyingmicroscopic/molecular mechanisms (91), both of them deeply rooted in polymer physics. Methodssuch as single-molecule spectroscopy and NMR are ideally suited for characterizing the resultingheterogeneous conformational ensembles and understanding the mechanisms of complex forma-tion. In summary, a large array of problems related to the conformations and dynamics of unfoldedand intrinsically disordered proteins is waiting to be addressed, and concepts from polymer andsoft 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 thatmight be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

We thank Alessandro Borgia for discussions and for providing raw data for figures, and the mem-bers of our research group and our colleagues and collaborators for discussions over the years ontopics related to this review, especially Robert Best, William Eaton, Irina Gopich, Gilad Haran,Dima Makarov, Jeetain Mittal, Rohit Pappu, Attila Szabo, and Dave Thirumalai. Our work hasbeen supported by the Swiss National Science Foundation, the European Research Council, andthe Human Frontier Science Program.

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http://www.annualreviews.org/r/nevs

BB45-FrontMatter ARI 11 June 2016 8:55

Annual Review ofBiophysics

Volume 45, 2016Contents

Imaging Specific Genomic DNA in Living CellsBaohui Chen, Juan Guan, and Bo Huang � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Transcription Dynamics in Living CellsTineke L. Lenstra, Joseph Rodriguez, Huimin Chen, and Daniel R. Larson � � � � � � � � � � � � � �25

Cell Geometry: How Cells Count and Measure SizeWallace F. Marshall � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �49

Elastic Properties of Nucleic Acids by Single-Molecule ForceSpectroscopyJoan Camunas-Soler, Marco Ribezzi-Crivellari, and Felix Ritort � � � � � � � � � � � � � � � � � � � � � � � � �65

Design Principles of Length Control of Cytoskeletal StructuresLishibanya Mohapatra, Bruce L. Goode, Predrag Jelenkovic, Rob Phillips,

and Jane Kondev � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �85

First-Passage Processes in the GenomeYaojun Zhang and Olga K. Dudko � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 117

Protein Folding—How and Why: By Hydrogen Exchange, FragmentSeparation, and Mass SpectrometryS. Walter Englander, Leland Mayne, Zhong-Yuan Kan, and Wenbing Hu � � � � � � � � � � � � 135

Mechanisms of ATP-Dependent Chromatin Remodeling MotorsCoral Y. Zhou, Stephanie L. Johnson, Nathan I. Gamarra,

and Geeta J. Narlikar � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 153

Group II Intron Self-SplicingAnna Marie Pyle � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 183

Single-Molecule FRET Spectroscopy and the Polymer Physics ofUnfolded and Intrinsically Disordered ProteinsBenjamin Schuler, Andrea Soranno, Hagen Hofmann,

and Daniel Nettels � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 207

Globular Protein Folding In Vitro and In VivoMartin Gruebele, Kapil Dave, and Shahar Sukenik � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 233

v

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BB45-FrontMatter ARI 11 June 2016 8:55

Computational Methodologies for Real-Space Structural Refinementof Large Macromolecular ComplexesBoon Chong Goh, Jodi A. Hadden, Rafael C. Bernardi, Abhishek Singharoy,

Ryan McGreevy, Till Rudack, C. Keith Cassidy, and Klaus Schulten � � � � � � � � � � � � � � � � � 253

Self-Organization and Forces in the Mitotic SpindleNenad Pavin and Iva M. Tolic � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 279

The Radical-Pair Mechanism of MagnetoreceptionP. J. Hore and Henrik Mouritsen � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 299

Insights into Cotranslational Nascent Protein Behavior fromComputer SimulationsFabio Trovato and Edward P. O’Brien � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 345

Allosterism and Structure in Thermally Activated Transient ReceptorPotential ChannelsIgnacio Diaz-Franulic, Horacio Poblete, German Mino-Galaz, Carlos Gonzalez,

and Ramon Latorre � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 371

Indexes

Cumulative Index of Contributing Authors, Volumes 41–45 � � � � � � � � � � � � � � � � � � � � � � � � � � � 399

Errata

An online log of corrections to Annual Review of Biophysics articles may be found athttp://www.annualreviews.org/errata/biophys

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single-molecule fret spectroscopy and the polymer ......forster resonance¨ energy transfer (fret),...

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