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Biomolecular dynamics: order–disorder transitions and energy landscapes

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2012 Rep. Prog. Phys. 75 076601

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Rep. Prog. Phys. 75 (2012) 076601 (29pp) doi:10.1088/0034-4885/75/7/076601

Biomolecular dynamics: order–disordertransitions and energy landscapesPaul C Whitford1, Karissa Y Sanbonmatsu2 and Jose N Onuchic1

1 Center for Theoretical Biological Physics, Department of Physics, Rice University, 6100 Main, Houston,TX 77005-1827, USA2 Theoretical Biology and Biophysics, Theoretical Division, Los Alamos National Laboratory, MS K710,Los Alamos, NM 87545, USA

Received 21 February 2012, in final form 27 April 2012Published 28 June 2012Online at stacks.iop.org/RoPP/75/076601

AbstractWhile the energy landscape theory of protein folding is now a widely accepted view forunderstanding how relatively weak molecular interactions lead to rapid and cooperative proteinfolding, such a framework must be extended to describe the large-scale functional motionsobserved in molecular machines. In this review, we discuss (1) the development of the energylandscape theory of biomolecular folding, (2) recent advances toward establishing a consistentunderstanding of folding and function and (3) emerging themes in the functional motions ofenzymes, biomolecular motors and other biomolecular machines. Recent theoretical,computational and experimental lines of investigation have provided a very dynamic picture ofbiomolecular motion. In contrast to earlier ideas, where molecular machines were thought tofunction similarly to macroscopic machines, with rigid components that move along a fewdegrees of freedom in a deterministic fashion, biomolecular complexes are only marginallystable. Since the stabilizing contribution of each atomic interaction is on the order of the thermalfluctuations in solution, the rigid body description of molecular function must be revisited. Anemerging theme is that functional motions encompass order–disorder transitions and structuralflexibility provides significant contributions to the free energy. In this review, we describe thebiological importance of order–disorder transitions and discuss the statistical–mechanicalfoundation of theoretical approaches that can characterize such transitions.

(Some figures may appear in colour only in the online journal)

This article was invited by R H Austin.

Contents

1. Introduction: the biological context for physicalquestions 2

2. Development of energy landscape theory for proteinfolding 22.1. Spin glasses 42.2. Parallels between glasses and protein folding 52.3. Energy landscape theory and the principle of

minimal frustration 53. Modeling approaches for protein folding 6

3.1. Lattice models for protein folding 63.2. Off-lattice models for protein folding 8

4. Theoretical approaches to investigateconformational rearrangements 114.1. Normal mode analysis 124.2. Multi-basin structure-based models 13

4.3. Relationship between free-energy barriers andkinetics 17

4.4. Computational considerations 175. Roles of order–disorder transitions in biomolecular

functional dynamics 185.1. Conformational rearrangements: cracking 185.2. Biomolecular association: fly-casting 195.3. Protein recognition: global unfolding and

refolding 205.4. Riboswitch signaling: substrate-assisted

ordering 205.5. Molecular machines: entropically-guided

rearrangements 216. Closing remarks 22Acknowledgments 23References 23

0034-4885/12/076601+29$88.00 1 © 2012 IOP Publishing Ltd Printed in the UK & the USA

Rep. Prog. Phys. 75 (2012) 076601 P C Whitford et al

1. Introduction: the biological context for physicalquestions

The biomolecular origins of cellular dynamics occur over awide range of length and time scales. Each biomoleculehas specific functional properties that are governed by itsenergetics. Two well-studied classes of biomolecules (atthe functional level) are proteins and nucleic acids. Bothare linear polymer chains composed of specific sequencesof monomers called ‘residues.’ Proteins are composed ofamino-acid residues (figure 1(a)), of which there are twentynaturally occurring types. Ribonucleic acid (RNA, figure 1(b))chains and deoxyribonucleic acid chains (DNA) are composedof sugar-phosphate backbones and four types of nucleic acidside chains (A,U,C,G in RNA and A,T,C,G in DNA). Whilethe primary role of DNA is to carry genetic information,RNA molecules can adopt well-ordered three-dimensionalconfigurations, perform chemistry [1, 2] and function asmolecular machines [3–5]. In fact, in recent years it has beenshown that the majority of the human genome is transcribedinto RNA but not translated in protein, suggesting that non-coding RNAs may regulate many cellular processes. Due to thefunctional importance of protein and RNA structural dynamics,this review will focus on their dynamics, though the conceptspresented may be applicable to other biomolecular systems.

Gene expression in the cell includes two fundamentalsteps: transcription and translation. Transcription is theprocess by which DNA is read and a complementary messengerRNA (mRNA) molecule is generated (figure 2). In prokaryotes(single-celled organisms), mRNA is directly translated byribosomes to produce proteins. In eukaryotes (multicellularorganisms), precursor-mRNA (pre-mRNA) is produced duringtranscription. In eukaryotes, pre-mRNA is further modified bythe cell (e.g. splicing [6]), which yields mature mRNA that istranslated by ribosomes into protein sequences. In addition tothis central dogma for the flow of genetic information, manyfeedback channels exist in the cell. For example, proteinsand RNAs often regulate transcription and translation [7–13],as well as aid in heritable changes in gene expression bycoordinating the structural reorganization of biomolecularassemblies [14, 15].

During transcription and translation the resulting linearpolymers initially adopt unstructured (or partially structured)ensembles of configurations. This is followed by aconformational search for the lowest-energy configuration.Since these polymers can have hundreds (even thousands) ofresidues (where each residue has several degrees of freedom)the accessible phase space of these molecules is large. Thevastness of the available phase space led to the proposalof the so-called ‘Levinthal’s Paradox.’ [16] That is, if apolymer chain has N residues and each residue has M

available conformations, the number of possible configurationsof the polymer chain will be MN . If each residue hasonly two possible conformations, then a 100 residue chainwould have 2100 ≈ 1030 possible configurations. However,functional proteins and RNA molecules rapidly find a specific‘native’ three-dimensional structure (often on sub-secondtimescales). If the search process were completely random,

and a configuration could be sampled every picosecond, thenthe average folding time would be 1

2 1030 × 10−12 s ≈1018 s≈1010 years. While considering the excluded volume ofa biomolecule, or by assuming that proteins are initiallycollapsed [17], may reduce this counting problem, thesearguments alone cannot account for the apparent eighteenorders of magnitude discrepancy in timescales. Accordingly,the search process must not be random. Identifying howproteins have evolved to fold to specific structures, withrapid rates, is often referred to as the ‘protein foldingproblem.’ While the theoretical biological physics communityhas focused largely on protein folding, similar considerationsapply for the folding of RNA [18].

To rationalize the kinetic disparities presented byLevinthal, the ‘funneled’ energy landscape concept wasintroduced [19]. Combined with The Principle of MinimalFrustration [20–22], this theoretical framework for the energylandscape of biomolecular dynamics has demonstrated thatin order for proteins to be foldable, there must exist anensemble of routes by which the molecule may reachthe folded configuration. Folding then proceeds as adiffusive process over a relatively smooth energy landscape(i.e. one that lacks large traps), and the ensemble ofpossible configurations continuously narrows as the nativeconfiguration is approached.

Since folding and functional dynamics occur on the sameenergy landscape, the principles developed to understandprotein folding must extend to, and account for, biomolecularfunction. Many multi-domain proteins, such as AdenylateKinase and kinesin, undergo conformational rearrangementsduring their function [23–27]). Accordingly, the energylandscapes of these molecules facilitate both, rapid foldingand functional dynamics. Since the energy landscape (i.e.the Hamiltonian3) is determined by the chemical compositionof the molecule, polymer sequences encode the energetics ofthe native configurations and large-scale functionally relevantrearrangements, making it desirable to have a consistenttheoretical framework for understanding both.

In this review, we provide a brief historical perspective(1970–2000) on the development of the energy landscapetheory of biomolecular motion, as developed in the context ofglass-forming systems and protein folding. Next, we discussrecent work (2000–2010) to extend protein folding conceptsand methods to describe functional dynamics of biomolecules.The focus of this discussion is on the use of theoreticalmodels of energy landscapes that allow us to characterize therelationship between structure, folding and function. We willfinish by discussing the role of order–disorder transitions inbiomolecular functional dynamics.

2. Development of energy landscape theory forprotein folding

Grounded in statistical mechanics, the energy landscape theoryof biomolecular dynamics [20–22, 28] has provided a generalframework for understanding the folding and functional

3 In this review, ‘Hamiltonian’ will refer to the potential energy contributiononly.

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Figure 1. The diversity of protein and RNA residues. (top) Protein chains are composed of amino-acid residues, which contain carbon(orange spheres), nitrogen (blue), oxygen (red), sulfur (yellow) and hydrogen (white) atoms. Each amino acid has a common set of C, N, Oand H atoms (boxed). The range of chemical compositions of residues is depicted by the atomic structures of 5 (of the 20 naturallyoccurring) different amino acids. (bottom) RNA chains are composed of four type of nucleic acids. Each nucleic acid has a common set ofbackbone atoms (left). Canonical Watson–Crick base pairs are formed between A-U and C-G pairs (right). Structural figures were preparedwith VMD [277].

Figure 2. Gene expression in the cell. The two fundamental steps of gene expression are transcription and translation. (a) In prokaryotes,messenger RNA (mRNA) is transcribed and then read by ribosomes to produce proteins. (b) In eukaryotes, precursor-mRNA is producedduring transcription. pre-mRNA is modified and transported in the cell and then mature mRNA is read by ribosomes. There are many typesof post-transcriptional modifications that may be performed on pre-mRNA, though only splicing is depicted. During splicing, specificsequence of mRNA (introns) are removed from the pre-mRNA. (c) Once a protein is synthesized by the ribosome, it folds to thelowest-energy ‘native’ ensemble.

properties of biomolecules. For Levinthal’s paradox tomanifest, the energy landscape of a biomolecule must berelatively random. That is, if the landscape is ‘rough’4

(figure 3), the Levinthal search problem could arise. When thestabilizing energy gap δE between the folded and unfoldedensembles is large, relative to the energetic roughness �E,the landscape can be described as an energetic ‘funnel.’ [19]In a funneled energy landscape, configurations are locally

4 i.e. the energetic barriers separating locally connected configurations arelarge, relative to the overall energy gap between the folded and unfoldedensembles.

connected (i.e. are kinetically accessible through simplemoves) to states that are slightly more native and to those thatare slightly less native. The degree of native content is alsocorrelated with the stabilizing energy, resulting in a stabilizingenergy gap that is much larger than the energetic barriersseparating locally connected states. In such a landscape,the diffusive walk is guided by the stabilizing energy andconnectivity of states, and Levinthal’s paradox is averted.

In many proteins, folding may be described as a pseudo-first order phase transition [22, 29] where the system goesfrom a disordered ensemble to an ordered ensemble by

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Figure 3. Energy landscapes of biomolecular folding. (a) When theenergy landscape of a system is rough (red), the system must searchrandomly for the lowest-energy configuration and glass-likedynamics may arise. Biomolecules have evolved to have a largeenergy gap between the unfolded and folded ensembles δE, relativeto the energetic roughness �E. ‘Funneled’ energy landscapes (blue)enable rapid folding of biomolecules in vivo. (b) An alternaterepresentation of a funneled landscape that is extended to accountfor functional rearrangements. Functional biomolecules oftenpossess multiple basins of attraction and there are manymechanisms by which functional transitions may occur. Cracking,hinge-motion and large-scale unfolding (orange) encompassdifferent degrees of molecular disorder and they result in differentialforms of connectivity between the functional and native ensembles.

transitioning through partially folded configurations that arenot deep minima on the free-energy surface. The ensembleof configurations that separate the endpoints is known as thetransition state ensemble (TSE). Phase transitions manifestthroughout nature, suggesting that analogies between proteinsand other physical system (such as spin glasses) may beidentified. In spin systems, when the energetic roughnessis small, the systems go through phase transitions to orderedstates. Similar to glass-forming systems, when the energeticroughness in a protein is large (relative to the energy gap), thesystem is described as ‘frustrated’ [22, 30–34]. When there isa large degree of frustration, the fluctuations in energy thatarise from non-zero temperatures are insufficient to escapefrom local minima and the timescales of the system’s dynamicsdiverge.

2.1. Spin glasses

To introduce the relationship between order–disordertransitions, glassy dynamics and energetic frustration, whichare fundamental aspects of the energy landscape frameworkfor biomolecular dynamics, we will discuss these properties inthe context of spin glasses [35–37]. This is not intended to bean exhaustive exploration of spin glasses and their dynamics.Rather, we will focus on features of order–disorder systemsthat will assist in our understanding of the energy landscapesof biomolecules.

The textbook example for discussing the statisticalmechanics of phase transitions is the Ising model [38]. TheIsing model represents a ferromagnetic material as beingcomposed of a lattice of discrete sites (in any number ofdimensions), where each site adopts a spin value of +1 or−1 (figure 4). In its original formulation [39], each spin iscoupled to the adjacent spins and the energy for an arbitrary-dimensional system is given by

H = −∑{ij}

Jij sisj , (1)

where∑

{ij} denotes the sum over all adjacent spins (si , sj )and Jij is the magnetic interaction energy of spins si and sj .When Jij > 0 (ferromagnetic interaction) for every pair, thelowest-energy configuration of the system corresponds to allspins being aligned. If Jij > 0, the system is described as‘unfrustrated’ since the interactions are consistent. That is,every microscopic interaction may be satisfied in the lowest-energy configuration of the system and, at low temperatures,the magnitude of the average magnetization 〈m〉 approaches 1.

Ising systems of two or more dimensions (and Jij > 0)undergo second-order phase transitions between disordered(〈m〉 = 0) and ordered states (|〈m〉| ≈ 1) [38] at the so-calledcritical temperature TC. For T < TC, the thermal fluctuationsare sufficiently small that the stabilizing interactions arecollectively satisfied and the majority of the spins align ina single direction. For TC > T > 0 the system adoptsa ferromagnetic state, though transient anti-ferromagneticconfigurations are sampled. If Jij is not constant for allpairs, then the probability of anti-ferromagnetic orientationsare heterogeneously distributed across the system. Due to theBoltzmann weighting of configurations, this feature is obvious,though it is noted since biologically relevant parallels will bedrawn later.

In 1975, Sherrington and Kirkpatrick (SK) introducedan extended and exactly solvable Ising model that exhibitsa glass phase [35]. Rather than have Jij �= 0 for only theadjacent pairs, the Jij coupling constants are defined for allspin pairs (i.e. infinite-range interactions), and the probabilitydistribution of the constants is given by

p(Jij ) = [(2π)1/2J ]−1 exp[−(Jij − J0)2/2J 2]. (2)

In the limit J → 0, the SK spin system reduces to theunfrustrated Ising model, and Jij = J0 for all pairs. AsJ → ∞ the interactions are completely random and the systemdoes not exhibit long-range order (〈m〉 → 0). For large J

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Figure 4. Spin systems and glass transitions. (a) A two-dimensional Ising spin model. (b) Schematic of the phase diagram for the SKspin-glass model. In the SK spin-glass, three states can exist: paramagnetic, ferromagnetic and glass. The balance of energetic roughness(J ), ferromagnetic coupling (J0) and temperature T determines which state the system adopts.

(relative to J0), the system is energetically frustrated, and notall interactions can be satisfied simultaneously.

There are several general features of the SK Ising modelthat are most relevant when describing the energy landscapesof biomolecules. Figure 4 shows the phase diagram forthe SK model, in terms of the normalized quantities J0 =NJ0 and J = N1/2J , where N is the number of spins inthe system. The axes correspond to ratios of competingenergetic parameters: kBT/J and J0/J . kBT/J is theratio between thermally accessible energetic fluctuations andthe heterogeneity of the interaction weights. For large T ,individual spins have sufficient thermal energy to take anyvalue and the system enters a paramagnetic state (〈m〉 → 0).At low T , the system can enter a ferromagnetic state, or aspin-glass state. In the ferromagnetic state the individualspins are aligned (|〈m〉| → 1). In the spin-glass state,individual spins may align locally, but there is no long-range ordering. For low values of kBT/J , J0/J determineswhether the ferromagnetic or glass phase is adopted, whereJ0 is the average coupling and J is the dispersion of thecoupling constants. When J0/J > 1, the system mayadopt a ferromagnetic or paramagnetic state. If J0 is largerelative to J , then the majority of the system’s interactionswill be ferromagnetic, and every microscopic ferromagneticinteraction may be satisfied in a global ferromagnetic state.If the dispersion is large, relative to the average coupling(i.e. J0/J < 1), the system is composed of a mixture offerromagnetic and anti-ferromagnetic interactions, and it isdescribed as ‘energetically frustrated.’ In frustrated systems,not all interactions can be satisfied simultaneously (i.e. whereeach microscopic interaction is a minimum). Accordingly, ifJ0/J and kBT/J are small, the system will enter a glassy state.Hallmark features of glassy systems are a lack of long-rangeordering (〈m〉 → 0) and diverging relaxation times. Divergingrelaxation times are due to interactions being satisfied locally,though the low-energy configurations are highly degenerate,which leads to competition of local structure formation. For amore detailed discussion on the thermodynamic properties ofthe SK model, we suggest the original manuscript [35].

Glass transitions are not isolated to Ising models. They areubiquitous in nature and have been investigated computation-ally [40–42], theoretically [43–46] and experimentally [47–49]for a wide variety of systems. Here, we have limited the dis-cussion to the general relationships between energetics, order–disorder transitions and glass dynamics, in order to guide ourdescription of biomolecular dynamics.

2.2. Parallels between glasses and protein folding

While the energetics of biomolecules are not simply composedof spin–spin coupling interactions, analogous behaviorsbetween biomolecular folding and glass transitions may beidentified [49]. Most proteins spontaneously fold underphysiological conditions, and they do so in relatively shorttimescales (milliseconds). Similarly, in the Ising model, if thedispersion in the interactions is small, then a paramagnetic-to-ferromagnetic transition occurs without the formation oflong-lived kinetic traps. In the SK spin model, when1 < J0/J < 1.25, there are three accessible states.Consistent with this, in proteins, the adopted state isdetermined by the temperature-to-dispersion ratio kBT/J . Theparamagnetic : ferromagnetic : glass states thus correspond tounfolded:folded:misfolded ensembles. In polymers, glassydynamics may emerge at low temperatures [50], where theglassy ‘state’ is composed of many competing configurationsand interconversion times between them is slow. Finally, for agiven value of J0/J (between 1.0 and 1.25), these systemshave well-defined order–disorder transition temperatures(paramagnetic–ferromagnetic transition, or folding–unfoldingtransition) and a glass transition temperatures (ferromagnetic-glass or folded to glass/misfolded).

2.3. Energy landscape theory and the principle of minimalfrustration

The similarities between glass-forming systems and proteinfolding led to the proposal of the principle of minimalfrustration [19, 51, 52]. That is, naturally occurring proteinsequences are able to fold on short timescales, which indicates

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these sequences encode energetics for which the foldingtemperature Tf (i.e. the temperature at which the unfolded andfolded ensembles are equally probable) is larger than the glasstransition temperature Tg. Further, the principle states thatnature has selected for protein sequences that maximize theratio between Tf and Tg, which allows proteins to fold rapidlyin vivo (i.e. Tg < Tcell < Tf ). Tf/Tg in proteins parallels J0/J

in the SK spin-glass model. J0 is the average coupling strengthof the spins, which is analogous to the average stabilizingenergetic contribution of each native interaction in a protein. Inthe words of Go [53], J0 is similar to the degree of ‘consistency’between the native interactions in a protein. These consistentinteractions lead to an energy gap δE between the disorderedand ordered states (figure 3). J is the dispersion in the strengthof the spin interactions and it is correlated with the degree offrustration in the system. In proteins, frustration manifestsas stabilizing non-native interactions [34]. According tothese corresponding features of proteins and glasses, theprinciple of minimal frustration states that by maximizingthe ratio between the stabilizing energy gap δE and theenergetic roughness �E (which determines the ratio Tf/Tg),nature has selected for protein sequences that are able to foldsufficiently fast for biological function.

3. Modeling approaches for protein folding

Over the last 20 years, many theoretical and computationalstudies have investigated the protein folding problem. Ofparticular relevance to this review are studies that haveextended the energy landscape framework for biomolecularfolding and function. We will focus on advances that haveemployed structure-based models (SBMs) [54–56], since thesemodels are founded on the principle of minimal frustration.A common theme throughout these investigations is that byaccounting for simple geometrical considerations, such asexcluded volume and chain connectivity, dynamic features ofprotein folding and function may be accurately described. Thatis, once a protein has evolved to have a dominant energeticbasin (or several basins), the geometry of the protein is theprimary determinant of the rate/mechanism.

Energy landscape theory provides a general physicalframework for understanding the energetics of protein folding,which allows for many classes of theoretical models to beemployed. Early work used lattice models of proteins,where each residue is described as a single bead that movesbetween lattice vertices. With such models, the interplaybetween global collapse, connectivity of states and energeticswas elucidated. Subsequent work employed more easilytransferable coarse-grained off-lattice models. Recently, wehave developed models that combine all-atom resolution withthe energy landscape framework. To clarify the utility of eachapproach, we will provide an overview of how each class ofmodels has advanced our understanding of protein folding. Byconsidering the advantages and limitations of each approach,future investigations may strategically utilize the models thatare best suited for each question.

Figure 5. Lattice model for protein dynamics. In lattice models ofproteins, the residues (spheres) move between adjacent latticepositions. Similar to proteins in solution, backbone connectivity(grey) leads to sequential residues being adjacent in space. In thismodel, the degree of hydrophobicity is indicated by the spherecolors. The sequence shown is taken from Leopold et al and itrepresents a maximally-compact configuration [19].

3.1. Lattice models for protein folding

Inspired by the glass-theoretical roots of the principle ofminimal frustration, early simulation models for proteinfolding also used lattice representations of polymer chains toexplore the relationship between energetic frustration, kineticsand structural mechanisms [19, 57, 58]. In these models, thepolymer is treated as a connected chain, where each amino-acid residue is a link that transits across a discrete set of latticepositions (figure 5). The functional form of the Hamiltonianused in many studies is similar to an Ising spin model:

E = −∑{ij}

ε(si, sj )�(i, j) +∑

k

δ(|�rk − �rk+1|). (3)

The chemical identity of the primary residue sequence S

is defined as s1, . . . , sN , ε(si, sj ) is the coupling constantbetween residue types, �(i, j) = 1 if the residues are onadjacent sites and �(i, j) = 0 otherwise. The summation over{ij} is the sum over all residue pairs (not a sum over all latticesites). The second term sums over all residues that are adjacentin sequence, ensuring the connectivity of the backbone (greylines in figure 5) is maintained. δ = 0 if the residues are onadjacent sites and δ = ∞ if they are not. Changing notationreveals the similarities between this model and the Ising model:

E = −∑{ij}

J (R(i), R(j))OiOj +N−1∑k=1

δ(|�rk − �rk+1|), (4)

where {ij} is now defined for all adjacent lattice sites. Oi

is the occupancy of each lattice site (1 if occupied and 0 ifunoccupied), J (R(i), R(j)) defines the coupling between thetypes of residues in the sites, and R(i) is the type of residue atsite i. R(i) carries the same type of information as si . However,R(i) is the type of residue at the lattice site i and si is the residuetype in sequence position i. Accordingly, R(i) is not defined if

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site i is unoccupied, in which case Oi is also equal to zero. Inthis specific formulation, there are no long-range interactions,though alternate formulations have included a combination oflong-range and short-range interactions [59, 60].

The only differences between the Ising spin Hamiltonianand protein lattice Hamiltonian are the connectivity restriction(δ function in equations (3) and (4)) and the intrinsic timedependence of the site–site coupling constants J (R(i), R(j))5.While these complications have impeded efforts to find ananalytic solution for the model, the relatively small numberof discrete configurations on the lattice allows all compactconfigurations to be explicitly enumerated for small systems(i.e. a 27-mer) [29, 30, 61–63].

3.1.1. Utility of lattice models. While lattice modelsemploy a simplified representation of the atomic structureand energetics, they possess several features that allow for ageneral understanding of polymer folding to emerge. First,there is specificity in the sequence of the monomers. Notonly do individual beads (residues) have unique energeticcontent, but adjacent residues (in sequence) are constrained tobe on adjacent vertices (i.e. covalent geometry is maintained).Another physically valid constraint in the model is thatmonomers cannot pass through one another. In an Isingsystem, it is possible for adjacent lattice sites to simultaneouslyexchange values (e.g. position i goes from spin −1 to +1 whenposition i+1 goes from +1 to −1). In lattice models of proteins(and in real proteins) the repulsive nature of atomic nuclei leadsto an excluded volume that prevents this from occurring.

The low computational cost of Ising-like lattice modelsfor protein folding has allowed for extensive exploration of thebalance between energetic stability, entropic contributions andenergetic frustration in protein folding [19, 64–68], and haselucidated new strategies for protein design [69]. One of themost significant results stemming from lattice model studieswas the folding funnel paradigm for biomolecular folding.This concept emerged from investigation into the energetic,thermodynamic and kinetic differences between ‘foldable’and ‘unfoldable’ polypeptide sequences. To rationalize theapparent paradox implied by the arguments of Levinthal,Leopold et al [19] used a lattice model to simulate foldingfor two protein sequences. While the beads were notassigned energetic values based on a particular polypeptidesequence, the energetic parameters were designed, suchthat one sequence was foldable. That is, the sequencewas designed to meet the following criteria: (1) 20 uniquebead ‘flavors’ were used, which reflects the 20 naturallyoccurring residue types in proteins, (2) the overall strengthof each bead–bead interaction was on the scale of a fewkBT and (3) the potential energy had a dominant energeticbasin. This small set of parametric requirements describe an‘ideal’ protein that has a unique native configuration. Thenon-foldable sequence was randomly assigned bead flavors,representing a random heteropolymer that does not have aunique native configuration. While both sequences do haveglobal energetic minima, it was shown that the folding time of

5 Since residues move across the lattice, R(i) changes with time.

the unfoldable sequence was far greater than in real proteins.This is consistent with the arguments of Levinthal, wherethe search for the native configuration on a rough energysurface is also random, and the likelihood of finding the nativeconfiguration becomes vanishingly small. In contrast, for thefoldable sequence, folding could be described as a diffusiveprocess over an initially broad ensemble of configurations (i.e.the unfolded ensemble), and the kinetically connected sub-ensembles decrease in size as the native basin is approached.In particular, the energy landscape resembled a ‘funnel,’ wherethe native ensemble resides at the bottom (figure 3).

3.1.2. How to describe folding: reaction coordinates. Priorto the advent of lattice models for protein folding, kineticmodeling of protein folding involved a traditional biochemicaldescription, where the unfolded and folded basins are separatedby a single transition state. Since the free-energy hadnot been evaluated or measured directly at that time, thelandscape was considered in terms of a generalized, idealreaction coordinate, where approximations are necessary toconnect the folding rates and free-energy barriers. Sincelattice models provide pseudo-atomic resolution of the foldingdynamics, they necessitated rigorous methods for characteringthe collective motion of biomolecular processes. That is,the energy landscape of a system is defined in terms of 3N

degrees of freedom, where N is the number of atoms. In theselandscapes, the free-energy barrier of folding is described bythe TSE that separates the unfolded and folded ensembles.One of the objectives of statistical mechanics is to describe thecollective dynamics of complex systems along a low number ofreaction coordinates, or order parameters. In the case of proteinfolding, the first questions to address were: are there simplestructural reaction coordinates that can capture the essentialproperties of a folding process? And, what are the appropriatecoordinates?

There are several requirements for an appropriate reactioncoordinate. First, for a given system configuration, thecoordinate must be uniquely defined. Second, each basinand TSE must have distinct values of the coordinate. Third,local moves in Cartesian space must map continuouslyonto the reaction coordinate. If this requirement is notsatisfied, then this coordinate cannot describe diffusive motion,since discontinuities will result in the projection of ballistic-like tunneling events in that coordinate space. For agiven coordinate, the potential of mean force and diffusioncoefficient must also be consistent with the kinetics ofinterconversion between the local basins (see relationshipbetween free-energy barriers and kinetics below). Finally,by definition, a configuration in the TSE must be equallylikely to proceed to either of the adjacent free-energy minima,and the coordinate must be able to identify a TSE with thisproperty. Since lattice models provided the first opportunity touse atomistic-based reaction coordinates, two intuitive reactioncoordinates were introduced: the radius of gyration Rg and thefraction of native contacts Q. Following the initial studiesthat employed these coordinates, many subsequent studieshave investigated the relationships between diffusion, barrierheights and kinetics along them, in both lattice and off-lattice

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Figure 6. Multiple representations of proteins. The protein Chymotrypsin Inhibitor 2 (CI2) (pdb entry: 2ci2) shown in (a) stickrepresentation; (b) all-atom representation (excluding hydrogens) that depicts the excluded volume of the atoms and (c) Cα representationthat is often used in molecule simulations. Each residue type is give a unique color, highlighting the heterogeneity of protein sequences.

models [70–78]. In the context of off-lattice models, ithas been shown that Q can satisfy the above requirementswhen describing folding of two-state proteins (i.e. two basinsseparated by a single barrier) [79]. In contrast, Rg is able todistinguish between the folded and unfolded configurations,but it is less able to distinguish between native and near-nativeconfigurations. However, by considering the folding as afunction of both coordinates, folding may be well described ascollapse (decrease in Rg) followed by reorganization (increasein Q) [19, 29, 57, 64, 65, 80–83].

3.2. Off-lattice models for protein folding

3.2.1. Cα models. While lattice models enabled a generalcharacterization of heteropolymer collapse, real proteins donot reside on lattices. Rather, the coordinates of proteinstake continuous values in three-dimensional space. Toprovide a continuous description of the phase space, simplifiedmodels that employ an off-lattice representation have beenexplored [31, 54, 84–92]. Early off-lattice models employeda coarse-grained representation of the polymer, where eachresidue was represented as a single sphere (figure 6(c)). Withan off-lattice representation, it is possible to study specificproteins and not only model systems. That is, while it ispossible to design lattice configurations that resemble thestructures of specific proteins, the off-lattice description istransferable to any protein, which has enabled the explorationof the relationship between chain connectivity and excludedvolume on the folding properties of many proteins.

The principle of minimal frustration states that theenergy gap between the unfolded ensemble and foldedensemble must be large, relative to the energetic roughness(i.e. frustration). Inspired by this, many off-latticemodels employ forcefields that are completely specific to thenative, folded configuration. These unfrustrated forcefieldsrepresent the limiting case of the principle of minimalfrustration. Similar to an ideal gas theory, these unfrustratedmodels represent a low-order approximation to a molecule’senergetics, and thus provides a baseline framework uponwhich the role of energetic perturbations and roughnessmay be better understood [54, 81, 93–96]. For example,this approach has been used to illuminate the effects of

cellular crowding [97] and desolvation [81] on protein folding,folding under non-equilibrium conditions [87, 98–101], co-translational folding [102] and aggregation processes [103].

Clementi et al introduced one of the most popular off-lattice models for protein folding, which is referred to as aSBM, or a Go-like model [54]. In its initial formulation,the folded configuration is defined as the lowest-energyconfiguration. Each residue–residue interaction formed inthe native configuration is assigned an attractive Lennard-Jones-like interaction, and the minimum energy correspondsto having the residue–residue distances that are found inthe native configuration. Adjacent residues are restrainedby harmonic interactions, which account for the covalentbackbone geometry of the system. The torsion angles formedby residues i, i + 1, i + 2, i + 3 are given cosine functions, eachwith a minimum corresponding to the native configuration.The functional form of this potential is

V =bonds∑

i

εir (r

i − rio)

2 +angles∑

i

εiθ (θ

i − θ io)

2

+dihedrals∑

i

εiφ

{[1 − cos(φi − φi

o)]

+1

2[1 − cos(3(φi − φi

o))]

}

+contacts∑

ij

εij

C

[5

(σ ij

r

)12

− 6

(σ ij

r

)10]

+non-contacts∑

ij

εij

NN

(σ

ij

NN

r

)12

, (5)

where εir = 100, εi

θ = 20, εiφ = 1, ε

ij

C = 1, εij

NN =1 and σ

ij

NN = 4.0 Å. rio, θ i

o, φio and σ ij are given the

values found in the native structure. The bond and angleterms maintain the chain-like connectivity of the polymer.The 10–12 potential is used to model each native contact.While in this implementation the interactions are completelyadditive, other off-lattice models include non-additive (or, 3-body) interactions [104, 105], which is sometimes necessaryto recover the proper degree of folding cooperatively.

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In this model, since the solvent is not explicitlyrepresented, the stabilizing 10–12 interactions are solvent-averaged effective interactions. That is, the nativeconfiguration has evolved to be the lowest free-energyconfiguration. Therefore, the native interactions are effectivelystabilized, even though direct residue–residue interactions maybe unfavorable. For example, it is possible for a nativeconfiguration to have two positively charged residues neareach other, even though the direct interaction between likecharges is repulsive. However, since these residues are inclose proximity in the lowest free-energy configuration, thepotential of mean force between them has a minimum at thenative distance. According to this logic, the native interactionsin these forcefields are ‘effective’ interactions that are averagedover the contributions of water and non-native interactions.

SBMs have forcefully demonstrated that chain connectiv-ity and excluded volume significantly restrict the possible fold-ing routes of biopolymers. This restriction of phase space canlead to a feature known as ‘topological frustration’ [31, 106–111]. Topological frustration may be characterized as the re-quirement of particular subunits to fold first, regardless of therelative stabilities. That is, it is possible that formation of par-ticular native interactions will impose strong structural con-straints on the system, leaving insufficient space for the chainto reorganize and form additional native interactions. Topo-logical frustration is not energetic in the classical sense, whereattractive interactions lead to large escape barriers from non-native configurations. Rather, similar to how excluded volumemay lead to caged dynamics, where large barriers are asso-ciated with reorganization in collapsed polymers [112], in atopologically frustrated fold the early formation of specific in-teractions can introduce energetic barriers. If the ‘incorrect’contacts form prematurely, the excluded volume of each atomwill lead to high-energy barriers that impede folding. There-fore, it would be more likely that the protein will return tothe unfolded ensemble than continue folding. Topologicallyfrustrated proteins must search for the appropriate initial in-teractions that are able to facilitate additional native-structureformation.

To illustrate ways in which topological frustration canmanifest, we compare the folding mechanisms of two smallproteins: Protein A and SH3 (figure 7). Protein A is a three-helix bundle protein and SH3 is composed of several betasheets. In the folding of protein A, the TSE has ratherhomogeneous structural content [113]. This homogeneity ofthe transition state ensemble suggests that the order of structureformation is not crucial. Protein A is composed of α-helicesthat are formed by many interactions between residues that areclose in sequence. Each turn of the helix results in interactionsbetween residue i and i + 4. In addition to helical interactions,the interface between the helices introduces non-localinteractions. By inspecting the native structure, one can easilyenvision how individual turns of the helix may be relativelydisordered while some interface interactions are maintained.In contrast, SH3 has a very polarized TSE, where some regionsare highly ordered and others are largely disordered.

Two classes of proteins that are considered highlytopologically frustrated are knotted proteins [114, 115] andthe cytokine Interleukin family of proteins [108, 110, 116, 117]

Figure 7. Protein structures vary in complexity. (a) Protein A (pdb:1BDD) is composed of three α-helices and is considered a simplefold. (b) The SH3 domain of Src (pdb: 1FMK) is composed of twoadjacent β-hairpins, which leads to more complex folding dynamics.Protein L (pdb: 2ptl) (panel c) and and G (panel d) (pdb: 3gp1) sharea common overall fold, but they exhibit different folding dynamicsin solution. Inspection of the packing of residues in the core of theprotein (e), (f ) shows that the fine details of these interactions aredifferent. Side chains are shown explicitly for residues that are incontact between the helix and the sheet. These packing effects leadto unique folding dynamics of the proteins [122].

(figure 8). A knotted structure may be defined by the followingcriteria: if the terminal ends of the chain are pulled in oppositedirections, it will not reach a fully extended structure. Rather,pulling results in a compact knot. Knots have been identified ina number of proteins and recent work has employed simplifiedmodels to characterize how complex native structures can bereached during folding [118, 119]. Entropic considerationssuggest that the most favorable way to fold would be to firstform a loose loop and then thread the terminal end of the proteinthrough the loop. Subsequent local rearrangements then leadto a completely folded configuration. If the loop were to close(i.e. fully form) early in the folding process, then there wouldbe insufficient space for the threading process to occur, whichhighlights the topological difficulties associated with foldingthese proteins.

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Figure 8. Topologically frustrated structures. Two classes ofproteins that are considered to be of ‘complex’ folds are knottedproteins (top, stereo view. PDB: 2EFV) and the Interleukin familyof proteins (bottom, stereo view. PDB: 6I1B). In the knotted protein,folding involves the formation of a loop and then threading of theterminal end through the loop. If the loop closes prematurely, thenthe threading process will be inhibited by excluded volume effects.In the Interleukin family, there is structural competition betweenregions during folding. If the C-terminal trefoil folds early, theprotein ‘backtracks’ and partially unfolds before full folding occurs.

The excluded volume of individual atoms can also lead tobacktracking in proteins [91, 107, 120, 121], including knottedones [118, 119]. Backtracking is defined as a phenomenonwhere native structure forms, breaks and reforms duringforward progress of the global folding process. In the case ofIL-1β, when residues in the C-terminal trefoil region possessnative structural content early, as folding continues to proceed,this region temporarily unfolds prior to it reaching a fullyfolded configuration. As discussed above, one hallmark oftopological frustration is that the protein returns to the unfoldedensemble if the ‘wrong’ interactions form prematurely. In thecase of IL-1β and backtracking, the unfolding is localized,where specific structural units reversibly unfold and refold asthe forward global folding process takes place.

3.2.2. All-atom models. Following the progression fromlattice to off-lattice Cα models, the next logical step was toextend biomolecular models to all-atom (AA) representations6.In all-atom SBMs [56, 122–128], every heavy atom (and, insome cases, hydrogen) is explicitly represented. Similar to itsCα counterpart, every interaction in the native configurationis stabilizing, which ensures the native structure is the lowest-energy configuration. Adopting this representation has twoadvantages that are immediately evident. First, by includingall atoms, the excluded volume of the atomic structure isexplicitly accounted for, which allows one to characterize the

6 Cβ models have also been studied, but there are more significant structuraldifferences when moving to an all-atom representation, and we will focusattention there.

relationship between the details of atomic packing and thefunctional/folding properties. Through comparison with a Cα

model, it was shown that side-chain packing can significantlyinfluence the folding of some proteins. Specifically, proteinsthat have spatially symmetric backbone configurations canhave two or more apparently equivalent folding pathways,when described by a Cα model. A second motivation forusing an AA model is that non-specific atomic interactions (i.e.interactions that are not formed in the native configuration)may be explicitly included, where the native-centric SBMwould serve as a baseline model. That is, one may partitionthe effects of individual energetic contributions on the foldingprocess (or other molecular processes) by comparing thedynamics in an unfrustrated model and a model where non-specific attractive interactions are included.

While the cooperative transition associated with proteinfolding encompasses a large change in configurational entropyof the backbone, the contribution of side-chain entropy isless clear. We recently extended the Cα SBM to an all-atom representation and detected a partial separation betweenbackbone ordering and side-chain ordering [56]. Thesemodels yield a peak in the specific heat curve, indicativeof a phase transition between the disordered (unfolded) andordered (folded) ensembles, which is coincident with backboneordering. However, this transition is accompanied by onlypartial ordering of the side chains. Figure 9 shows a typicaltrajectory for a single protein, described by the radius ofgyration Rg (which measures chain collapse), the fraction ofall-atom native contacts formed QAA (which characterizesside-chain packing) and the fraction of Cα − Cα contactsformed QCα

(which measures tertiary structure formation).Both 〈QAA〉 and 〈QCα

〉 undergo sharp transitions at Tf

(figure 9(b)), but 〈QCα〉 is larger than 〈QAA〉. This separation

of backbone and side-chain ordering has also been noted inrecent explicit-solvent simulations [129], suggesting it is arobust feature of protein folding. Figure 9(d) shows whichresidues exhibit ordered backbones with disordered side chainsin the folded ensemble (highlighted in yellow). It is strikingthat disorder in the side chains is not homogeneous betweenall residues. Even though the residue–residue contactsindicate that the tertiary structure of the molecule is formed,specific regions have more highly disordered side chains,which is similar to anisotropies in ferromagnetic-phase spin-glass systems. The disorder accessible in a folded proteinraises the question: since folding does not necessitate theordering of all side chains, does side-chain mobility fulfilla functional purpose? Loop residues are often involvedin protein recognition processes [130–132]. The predictedflexibility of side-chain conformations in folded proteins isconsistent with the notion that loop sequences may evolve inresponse to functional pressure that is independent of pressureto folding efficiently. Loop flexibility may also allow forconformational adaptability, which may enable proteins toparticipate in parallel signaling pathways. An alternativerole of side-chain flexibility is that balancing order–disordertransitions in the side chains may be a way in which proteinsmodulate the free-energy barriers of interconversion betweenfunctional conformations [133–135].

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Figure 9. Side-chain ordering and backbone collapse during protein folding. (a) A simulated folding trajectory of CI2, shown as a functionof the radius of gyration Rg (blue), fraction of native all-atom contacts formed Qall-atom (black) and fraction of native Cα contacts formedQCα . Rg measures backbone collapse, which decreases when Qall-atom and QCα increase. QCα reaches larger values than Qall-atom indicatingthat the backbone is ordered, while the side chains are not in their native configurations. (b) 〈Qall-atom〉 and 〈QCα 〉 as function of T indicatethat both coordinates capture the global folding transition. (c) As temperature is reduced below Tf , there is continuous movement of thenative basin to higher values of Qall-atom. (d) Residues in SH3 that have a larger fraction of Cα contacts formed than all-atom contacts aredepicted by yellow spheres.

All-atom models have highlighted modes by which side-chain dynamics may guide the folding process. For example,the experimentally characterized disparate folding propertiesof protein L and protein G (two proteins that share thesame global fold, figure 7) could be accounted for by subtledifferences in packing interactions of the side chains [122].Since these β-sheet structures are symmetric at a coarse (i.e.residue) level, the details of the atomic packing (betweenthe α helix and β-sheet) significantly influence the predictedfolding rates and mechanisms. Recently, it was also shownhow orientational information contained in side-chain packingreduces the prevalence of malformed knots in proteins [119].Specifically, at the residue level, the energetic penalty of‘crossing’ mistakes is small, which can lead to the nativeconfiguration becoming kinetically inaccessible.

Similar to studies with Cα models, when using all-atommodels, one may ask: what dynamic properties are robust, andwhich may be more easily manipulated in order to gain precisefunctional control? Using an all-atom SBM, we investigatedthe effects of energetic perturbations on the folding propertiesof several well-studied proteins: CI2, SH3 and Protein A [56].Unsurprisingly, the thermodynamic quantities were sensitiveto the energetic details. Consistent with the theoretical workof Portman and Wolynes [136], these models demonstratedhow the balance between chain persistence length and non-local stabilizing interactions impacts the free-energy barriersof folding. That is, as the persistence length is increased,the free-energy barriers also increase. Consistent with the

findings from Cα models [137, 138], the folding mechanismswere largely robust, though there was a small degree ofvariation. Protein A, which is considered a simple fold, wasmost sensitive to the distribution of energetics, while SH3was most robust. This is consistent with the SH3 fold beingtopologically frustrated [106]. In topologically frustratedfolds, the phase space that the protein may sample en-routeto the native configuration is reduced, relative to a simple fold.The fact that the SH3 folding mechanism was only marginallyperturbed by changes in energetics is consistent with this highlyrestricted phase space.

4. Theoretical approaches to investigateconformational rearrangements

Recent work has expanded the energy landscape approachbeyond folding, in order to describe large-scale conformationaltransitions associated with function. Often, biomoleculesdo not possess a single well-defined native configuration.Rather, there are competing basins of attraction and thebalance between the conformations governs the biologicalfunctions of the molecule [139–141]. Some enzymes undergoconformational rearrangements upon substrate binding [142,143], taking the molecule from an enzymatically inertstate to a conformation that is competent for chemistry.These rearrangements can even be rate limiting in thecatalytic cycles of enzymes [144]. In molecular machines,processive functions require conformational cycling through

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functional configurations, such as the ‘walking’ of kinesinon microtubules [145] and the elongation cycle of theribosome [146–148]. Molecular machines that are notprocessive may also utilize large-scale rearrangements duringfunction (e.g. the spliceosome [6]).

At first glance, it may be tempting to characterizefunctional rearrangements as rigid-body movements aboutwell-defined ‘hinge’ regions, since macroscopic machinesfunction in this manner. While this approximation may beaccurate in some cases [149], applying this framework toall biological systems can be misleading. In macroscopicmachines, the rigid components are ‘infinitely’ rigid (ordersof magnitude stronger than the man-made hinges). Theinteractions that maintain macroscopic rigid bodies areextremely strong, relative to thermal energy, which allowsmacroscopic machines to maintain their structural integrityover a wide range of temperatures and conditions. Therefore,rigid macroscopic components undergo negligible changes inentropy during function. In contrast, biomolecules are onlymarginally stable, where a 10–20% increase in temperatureoften results in high-entropy unfolding. Further, in molecularmachines, the weak interactions (i.e. on the scale of thermalfluctuations) that stabilize the folded configurations alsogovern the dynamics of functional motions. Since foldingand function take place on a single energy landscape, we mustexpand our energy landscape approaches for folding to accountfor functional dynamics.

In this section, we will discuss current theoreticaland computational approaches that are used to explorebiomolecular functional energy landscapes. A challenge inthese investigations is that in order for these molecules tointerconvert between stable configurations, the native basinmust not be uniquely defined. Since the folding and functionallandscapes of a biomolecule are one and the same, the funneledenergy landscape for folding [19] has to be extended to includecompeting basins. Similar to studies on protein folding, simplemodels allow for a low-order approximation to the energetics,which enables one to identify which features of the functionaldynamics are most robust and which are most responsive tochanges in energetics and cellular/experimental conditions. Bypartitioning the robust features, one may design new strategiesto precisely target and regulate biomolecular dynamics.

4.1. Normal mode analysis

In normal mode analysis (NMA), the potential energy isexpanded about a particular minimum to second order, and thelocal motions are described as a superposition of orthogonaloscillatory motions [150]. In solution, biomolecules exhibitoverdamped dynamics and their trajectories do not followharmonic modes with precise frequencies. However, theenergy local to the minimum may be expanded in terms ofthe modes:

E =∑

n

1

2ω2

nq2n, (6)

where the normal coordinate of mode n, qn, has a characteristicfrequency ωn. Each eigenvector of the Hessian matrix (i.e.the normal mode) corresponds to a single degree of freedom,

and the equipartition theorem dictates that each mode has anaverage energy of 1/2kBT . It then follows that 〈q2

n〉 ∝ ω−1n ,

such that large-scale motions occur along the lowest-frequencymodes. Interestingly, normal mode analysis of proteins hasrepeatedly demonstrated that low-frequency modes correspondto global deformations [151–153] that are in functionallyrelevant directions [154].

A commonly used potential energy function used forNMA of proteins is the Tirion potential [155]. Similar toSBMs, the Tirion potential defines the native configurationsas the lowest-energy configuration and atoms that are neareach other are assigned harmonic potentials, where the minimaare assigned based on the initial configuration. Consistentwith SBMs for folding, since the initial structure is typicallya stable configuration, the Tirion potential provides a low-order approximation to the landscape about that minimum.Additionally, this simple approximation has been shown toaccurately reproduce the local fluctuations about stable proteinconfigurations [133, 143, 156–158].

Low-frequency motions are largely determined by theglobal structure, and not specific energetic details [154].While individual interactions are important for the stabilityof the molecule, the connectivity of bonded and non-bonded interactions restricts the possible fluctuations, suchthat many Hamiltonians yield similar global motions abouta chosen basin [159]. These low-frequency motions inproteins and molecular complexes are also often correlatedwith functionally relevant motions [160]. In other words,the global structure determines the accessible local motionsand these motions correlate with the large-scale changesrequired for function. Therefore, this provides a directconnection between molecular structure and function. Theribosome (figure 10) is an excellent example that illustratesthis relationship. Analysis of local-basin fluctuations havesuggested that the movement of ‘stalk’ regions is lowerin energy than most other motions. While initial studiesemployed NMA with harmonic potentials [153, 158, 161],subsequent studies found similar results using all-atomstructure-based simulations [163] and all-atom explicit-solventsimulations [164]. These independent classes of theoreticalmodels employ different assumptions and parameterizations.The fact that the fluctuations are so similar confirms that thedominant contributor to the local fluctuations is the overallstructure of the molecular complex.

Since the local fluctuations described by NMA often cor-relate with functional motions, ongoing efforts are extendingthese methods to uncover the pathways between specified end-points [133, 156, 165, 166]. Miyashita et al developed an it-erative NMA approach to obtain a sequence of structures thatrepresent a low-energy pathway between two functional con-figurations of the protein adenylate kinase (AdK) (figure 11).To do so, they first defined a Tirion potential for the initialconfiguration, calculated the normal modes and then displacedthe structure along several low-frequency modes (where move-ment along each mode was proportional to the overlap with thedisplacements between endpoint configurations). After mak-ing a small displacement (such that the harmonic approxima-tion was valid), the Tirion potential was re-defined for the new

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Figure 10. The Ribosome. (Center) Atomic structure of an in-tact 70S ribosome (∼150 000 non-hydrogen atoms). Transfer RNA (tRNA,P-site tRNA shown in red) molecules read messenger RNA (mRNA) through interactions on the ‘small’ (30S) subunit and add the incomingamino acid to the growing protein chain located in the ‘large’ (50S) subunit. An additional tRNA (i.e. the A-site tRNA) is buried deep insideof the ribosome. During protein synthesis, the 50S ‘stalks’ assist movement of tRNA molecules. The L11 stalk (right) aids the entry ofincoming tRNA molecules and the L1 stalk (left) facilitates the disassociation of tRNA molecules. NMA, simulations with SBMs andsimulations with explicit solvent suggest that the stalks are very flexible (i.e. have low-energy displacements), relative to other componentsof the ribosome.

Figure 11. Functional configurations of adenylate kinase. Adk iscomposed of three domains: LID (red), NMP (blue) and core (grey).(a) When a ligand is not present, the dominant configuration has thedomains arranged in a more extended fashion. (b) Upon ligandbinding, the molecule adopts a compact conformation, where theligand is isolated and chemistry is permitted. AP5A is a bi-substrateanalogue that mimics the natural substrates of Adk: ATP and AMP.

configuration, the modes were re-calculated and the systemwas displaced again. This process was repeated to generatea sequence of configurations along the modes that have thehighest overlap with the rearrangement. In this approach, onlya few low-frequency modes were necessary to describe themajority of the global displacements. Following these initialstudies, adenylate kinase has become the paradigm protein sys-tem for testing new concepts and methods in protein conforma-tional transitions [133, 156, 165, 167–172]. A common findingin these investigations is that the global motions are describedwell by low-frequency motions, though high-frequency mo-tions are necessary to precisely organize catalytic sites.

4.2. Multi-basin structure-based models

A physicist’s approach to a new scientific question is to makethe lowest-order approximation to the system’s energetics,

take that expansion to its limits of applicability and thenexpand to higher order. In this respect, the lowest-order approximation to a systems’ dynamics is a staticstructure, which represents the average coordinates insideof a basin 〈�ri〉. The first-order correction would be theisotropic dispersion in the coordinates 〈��r2

i 〉, which canbe estimated from experimentally accessible Debye–Wallerfactors [173]. The dispersion may also be described in termsof anisotropic motions, where 〈��x2

i 〉, 〈��y2i 〉 and 〈��z2

i 〉 areindependent. The anisotropic dispersion characterizes theshape of the landscape about the average structure [173]and it can be obtained from normal mode analysis or fromx-ray crystallographic refinement methods [174, 175]. Whilethe correlation between anisotropic motions and functionalrearrangements has allowed NMA to identify low-energypaths between basins of attraction, when atomic displacementsinvolve the breaking and formation of interactions, theharmonic approximation becomes increasingly inaccurate. Toexpand the description of the landscape to include large-scaleconformational rearrangements, recently developed modelingapproaches represent the potential energy landscape as havinga defined number of experimentally identified basins ofattraction [176–178]. While there is a range of molecularmodeling approaches available to study conformationalrearrangements, we will focus our attention on investigationsthat use multiple structure-based potentials to define asingle multi-basin potential. For those interested inalternative approaches, such as the use of semi-empiricalforcefields [179, 180], there are many reviews available thatmay be of interest [181–184].

Multi-basin SBMs allow one to explore the robustnessof functional dynamics and develop energetic descriptions ofthe process that are quantitatively consistent with availableexperimental knowledge. In contrast to the application ofSBMs to protein folding, where there is a single dominantbasin (i.e. the native ensemble), these approaches extendthe description of the landscape to include multiple basinsof attraction. When there are multiple well-characterized

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Figure 12. Macroscopic and microscopic mixing. When combing two potential energy functions, one may combine them term-by-term, orin a global fashion. (a) Two hypothetical potential energy surfaces, where each curve depicts the total energy in each potential, given interms of an arbitrary coordinate. (b) Two 10–12 interactions that are defined for the same atom–atom pair. This depiction represents a pairthat is 6 Å in one configuration and 10 Å in the second. (c) A macroscopically mixed (Boltzmann) Hamiltonian, using the two potentials in(a) as input. Different values of the mixing temperature Tmixing and weighting factor Ci are shown. (d) A microscopically mixed atom–atompotential energy function that includes minima at both 6 and 10 Å. This representation uses a Gaussian function for the minimum at 10 Å.

conformations associated with a particular biomolecule, themodel must possess a minimum for each conformation(figure 12). The general strategy when building these models isto construct a structure-based forcefield for each configurationand then systematically merge the energetic terms into a singlepotential energy function.

Multi-basin approaches may be partitioned into twobroad categories, which we will refer to as ‘microscopic’and ‘macroscopic’ mixing. To illustrate the differences,we will first consider the functional form of a single-basinstructure-based forcefield (equation (5)). Each experimentalconformation can be used to uniquely define a structure-based energy function VX. A multi-basin Hamiltonian isthen obtained by combining the information stored in theseforcefields. Since the bonded functions (ro, θo, χo) are basedon the covalent geometry of the molecules, and the chemicalcomposition does not change during the processes describedhere, we will assume that the bonded functions are identicalfor all conformations. For ease of discussion, we will alsoneglect the changes in dihedral angles (φo), which reducesthe relevant contribution to the non-local contact energies ofpotential X, V contacts

X .In a microscopic-mixing model, the individual energetic

terms are combined term-by-term, which allows thecombined potential energy function to be decomposed in thefollowing way:

V contactsmixed =

∑i

fi(ri), (7)

where fi(ri) is a function that depends on a single degreeof freedom (such as the distance between a specific atom–atom pair ri). We consider an ideal protein that has twoconformations and the only difference in these conformationsis that atom pairs 1 : 20 and 30 : 60 (the numbers represent thesequence of the interacting atoms) are separated by differentdistances. In this example, the two degrees of freedom thatdetermine the energetics of the system are the atom–atomdistances r1,20 and r30,60. In a microscopic-mixing model, thesum of the terms could be rewritten as

V contactsmixed = f1,20(r1,20) + f30,60(r30,60). (8)

In a macroscopically mixed model, none of the energetic termsmay be expressed as a function of a single degree of freedom.For our hypothetical system, the energy could then only beexpressed as

V contactsmixed = f (r1,20, r30,60). (9)

In the macroscopic-mixing case, the energetic contribution ofeach degree of freedom is explicitly coupled to the energeticcontribution of every other degree of freedom.

The microscopic and macroscopic mixing modelsrepresent the limiting ways by which to combine energeticsfrom two forcefields. In principle, there is a continuum ofpartially microscopically mixed models, though they are not ascommonly used as other methods of mixing and they will not bediscussed further. For a given functional transition, one mustconsider the differences between the endpoint structures when

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determining whether a microscopic or macroscopic mixingmodel is most appropriate. While there are no establishedrules that dictate the use of each approach, we will outlinesome general considerations and implications for each.

4.2.1. Microscopic-mixing models. In microscopic-mixingmodels, the non-bonded energetics of the system may beexpressed as a sum of functions that each depend on onlyone atom–atom distance (equation (7)). While microscopicmixing of multiple SBMs has been implemented in a variety ofways [138, 177, 185], we will describe two classes of modelsthat will illustrate the strengths and limitations of each. Forthis discussion, we will focus on Adk, a three-domain proteinthat is known to populate at least two different configurationsduring function: the apo and ligand-bound forms (figure 11).

The first-microscopic mixing protocol uses ‘mutuallyexclusive’ contact interactions [168, 177]. In a mutuallyexclusive potential, the native atomic interactions for eachconformation are combined, such that the total interactionbetween atoms i and j is identical to the potential in one ofthe single-basin potentials. The apo (ligand-free) and holo(ligand-associated) forms of Adk have 531 and 565 nativecontacts [177]. For contacts that are common to both structures(i.e. same residue pairs at roughly the same distances), thereis no need for mixing. For the unique contacts, one variantof a mutually exclusively mixed forcefield would contain theunion of the apo contacts Qapo and the contacts unique to

the holo form Quniqueholo . One interpretation of this potential

energy surface is that there is an energetic basin correspondingto the apo form, and ‘functional frustration’ (Qunique

holo ) allowsthe holo form to be significantly populated. Since theprotein has evolved to populate both of these conformations(i.e. these conformations are basins on the landscape), thisdescribes the system as having competing internal interactionsthat favor either the apo, or ligand-bound, conformation.Another interpretation of this forcefield is that Adk has evolvedthe apo configuration to be the global basin and ligandbinding introduces additional minima between holo atom pairs.From this perspective, there is energetic competition betweenligand binding and the evolutionarily selected energeticminimum of the isolated protein. Computational work thathas employed non-structure-based forcefields [171, 186, 187],single-molecule experimental investigations [188] and x-raycrystallographic studies [189] have shown that in the absenceof a ligand, Adk exhibits large-scale fluctuations that span allthe way to closed configurations. These findings, in additionto the framework provided by the mixed-basin model, suggestthat the apo basin is dominant and that residual stabilizinginteractions of the holo atom pairs lead to partial sampling ofthe closed form.

In terms of the energy landscape of Adk, the aboveexample suggests that minimally frustrated proteins maycontain residual frustration that contributes specifically to thefunctional dynamics. With advances in computing capabilitiesand algorithms [190–192], a question that should now betractable is: does the energetic roughness required for functioninfluence folding mechanisms and rates, or are the foldingproperties robust? In other words, are function and folding

governed by disparate features on the energy landscape? Whilewe are unaware of a theoretical investigation that has explicitlyprobed this aspect of folding and function in Adk, ongoingwork has aimed to identify regions of residual frustration,which may provide insights into the overlap of functional andfolding energetics [34].

A second type of microscopic mixing has utility whencontacts are formed in both conformations of the system,but the atom–atom distances of these pairs are significantlydifferent (i.e. the differences cannot be rationalized as local-basin fluctuations). For example, a contact between residue10 and 35 may be formed in both conformations, but the atompairs may be separated by 6 and 10 Å in the two conformations(figure 12(b)). In these cases, it may be necessary to includea single potential energy function between the two atoms thathas minima at both 6 and 10 Å (figure 12(d)). This scenario ismore frequently encountered in coarse-grained models (suchas Cα models for proteins), where side-chain reorganizationcan lead to changes in the distance between Cα atoms ofresidues that are in contact. For example, it is possible for twoTryptophan residues (figure 1) to be stacked, or be interactingin a extended orientation. At a coarse-grained level, theresidues are considered ‘in-contact’ in both conformations,though the distances between the Cα atoms may change by∼10 Å. If both configurations are stable, the potential of meanforce between the residues must have two minima and thistype of multi-basin atom–atom interaction protocol would beappropriate.

When microscopically mixing N Hamiltonians (each withMi minima), the total number of minima in the mixed potentialis not necessarily equal to

∑Ni Mi . Since the functional form

of a microscopically mixed potential energy function can berewritten as a sum of functions, each of which only dependson a single degree of freedom ri , minima on the energy surfacemust only satisfy:

∂

∂xk

Emixed = ∂

∂xk

∑ij

Econtactsij (rij )

= ∂

∂xk

∑i

Econtactsik (rik) = 0 (10)

for all xk , where Econtactsij (rij ) is the function that describes

the total interaction potential between atom-pair ij . While,in principle, this condition may only be satisfied in theendpoint configurations, that is not necessarily the case. ForAdk, microscopic-mixing models have suggested at leasttwo additional minima [177] that are not captured by amacroscopic mixing scheme [169]. That is, two domainsof Adk (LID and NMP) are known to interconvert betweenopen and closed configurations [193, 194]. Microscopic-mixing models suggest that two possible intermediate basinsmay also exist, where one domain is closed in eachbasin. Similar configurations have been identified throughx-ray crystallographic analysis [195] and theoretical analysisindicates these partially closed configurations may increase thefidelity [156] and rate of catalysis [171].

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4.2.2. Macroscopic mixing models. In macroscopic mixingmodels the minima are unambiguously defined a priori andall energetic contributions of each potential are coupled to oneanother. The simplest macroscopic mixing model would bedefined by a simple ‘if’ condition:

H macromixed (�x) =

{H1(�x) if H1(�x) < H2(�x),

H2(�x) if H1(�x) > H2(�x),(11)

where H1(�x) and H2(�x) are the potential energy functionsbeing mixed and the coordinates of the system are given by�x. By definition, the combined potential energy functiontakes on the value of the lowest-energy surface at eachpoint in configuration space. This approach may begeneralized to N potential energy functions as H(�x) =min(H1(�x), . . . , HN(�x)). Since the molecules are on a singlepotential energy surface HX at each point, the set of minimaon this combined landscape is simply the union of the minimain the individual Hi

7.There are several considerations when using a

macroscopically mixed Hamiltonian. First, when integratingthe equations of motion, the first derivative must be defined forall possible configurations, which necessitates methods thatcontinuously transition between the potentials. Second, if oneseeks to identify the robust features of functional dynamics, theversatility of macroscopic mixing models can be of utility. Forexample, by using macroscopically mixed multi-basin modelsthat are capable of explicit modulation of the barrier heightsand the depths of each basin, one may determine the extentto which the observed structural mechanisms are determinedby the precise distribution of stabilizing energetic interactions.To illustrate the utility of these approaches, we will describetwo commonly implemented macroscopic-mixing algorithms:the ‘Boltzmann weighting’ and ‘superposition’ approaches.

In the Boltzmann weighting approach [176], the mixedHamiltonian Hmixed (constructed from Hamiltonians Hi) isdefined according to the following relationship:

exp(−Hmixedβmixing) =∑

exp(−(Hi − Ci)βmixing). (12)

βmixing = 1/(kBTmixing) and Ci is an offset thatincreases/decreases the relative weight of Hi in the mixed-basin forcefield. Here, the barrier height and depth of eachbasin are governed by the parameters Tmixing and Ci . Fora particular configuration, if (H1 − C1) − (H2 − C2) =10kBTmixing, then H2 would be weighted by a factor of e10

relative to H1. For example, in Adk, a particular experimentalmeasurement may suggest that the open conformation is morestable by several kBT , a condition that may be imposed byadjusting the values of Ci [169]. Tmixing is the ‘mixingtemperature,’ which is unrelated to the simulated temperature.Tmixing modulates the height of the barrier separating H1 andH2 by defining how sharp the transition is between potentials.For large values of Tmixing, the relative weight of each potentialchanges less with the difference between the potentials, andlower barriers are to be expected. Figure 12 shows an

7 This assumes that for each set of coordinates that defines a minimum�xmini , the following condition is satisfied: min(H1(�xmin

i ), . . . , HN(�xmini )) =

Hi(�xmini ). In practice, this is normally imposed by shifting the energy of the

input potentials appropriately [176, 196].

example of how two functions may be combined with theBoltzmann weighting approach. For extensive discussion onspecific applications of Boltzmann-weighted mixed potentials,we suggest additional reading [169, 176, 197].

In the Superposition approach [178, 196], a smoothpotential, H S

mixed, is defined as the eigenvalue of thecharacteristic equation:(

H1 �

� H2 + C2

) (c1

c2

)= HS

mixed

(c1

c2

), (13)

where � is the coupling constant between H1 and H2, and(c1, c2) is the eigenvector. � governs the barrier height(similar to Tmixing in Boltzmann mixing), and C2 controls therelative weights of the input potentials. Solving for the secularequation yields the lowest-energy solution:

H Smixed = H1 + H2 + C2

2−

√(H1 − H2 − C2

2

)2

+ �2.

(14)

Similar to the Boltzmann weighting approach, the superpo-sition approach ensures a smooth transition (i.e. defined firstderivative) between potential energy surfaces and it allows therelative depths of each basin to be directly modulated.

4.2.3. Micro- versus Macroscopic mixing. Considerationmust be given when deciding which mixing method bestdescribes a particular transition. In the macroscopicmixing models, explicit declarations allow for straightforwardcalibration of some thermodynamic properties. Although,in these models, the ‘all-or-nothing’ contribution of eachpotential is a restrictive constraint on the dynamics andmay appear non-physical. In contrast, the flexibility ofmicroscopic-mixing approaches may necessitate a variety ofqualitative decisions to be made, as well.

Each approach provides unique insights into the physicsof the system. When we used a microscopic-mixing modelfor AdK [177], we first asked whether or not a mixed set ofcontacts would provide sufficient information to encode thedesired endpoints as energetic minima, while also exhibitingappropriate structural fluctuations about the endpoints. Sincewe were able to satisfy these conditions, the microscopicmodel provided an energetic framework to interpret theorigins of Adk’s functional transitions. However, withthis approach, the free-energy barrier (potential of meanforce as a function of inter-domain distance) was on theorder of 1–2kBT , which may be a low estimate (describedbelow). In contrast, in a macroscopically mixed potentialfor Adk, it is possible to artificially increase the barrierheight between the conformations. While the macroscopicallymixed potential has not yielded essential intermediate basinsin earlier implementations (likely due to limitations onthe available phase space in macroscopic mixing models),these approaches may be extended beyond the two-basinrepresentation. Combing these approaches, one may usemicroscopic models to identify stable configurations that resultfrom mixing and then use macroscopic models (with more thantwo minima) to tune the barrier heights further. Together, this

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layered approach may establish an overall picture of the energylandscape that is consistent with experimental knowledge andalso provides an energetic/entropic characterization of thedynamics.

4.3. Relationship between free-energy barriers and kinetics

It is worth taking a moment to discuss the relationship betweenfree-energy barriers and rates in biological molecules, sincethis is often where theoretical predictions and experimentalkinetic measurements intersect. The barrier height of aconformational change may be related to the mean first passagetime 〈τ 〉 via the relationship [20, 198]:

〈τ 〉 =∫ ρfinal

ρinitial

dρ

∫ ρ

0dρ ′ exp[(G(ρ) − G(ρ ′))/kBT ]

D(ρ),

(15)

where G(ρ) is the free energy as a function of the reactioncoordinate ρ, D(ρ) is the diffusion in ρ-space, and ρfinal

and ρinitial are the values of the coordinate for the endpointconfigurations. When the diffusion coefficient in ρ-space isnot available, and the free-energy profile is dominated by asingle barrier of height �G, then it is useful to approximatethis relationship as [199]

〈τ 〉 = 1

Cexp(�G/kBT ). (16)

C may be interpreted as the ‘attempt frequency’ of barriercrossing. While we are not aware of an estimate ofC for conformational transitions in Adk, estimates areavailable for protein folding [200, 201], RNA folding [18] andrearrangements in the ribosome [202]. These independenttheoretical, experimental and computational studies suggestC is ∼1–10 µs−1 for many folding and functional transitions.Since conformational transitions in Adk occur on a smallerlength-scale than tRNA rearrangements inside of the ribosome,it would be reasonable to expect C to be larger for Adk’stransitions. The corresponding 〈τ 〉 values would then representan upper estimate (for a given �G). For example, free-energy barriers of 1–2kBT [177] and C = 1 µs−1 correspondto 〈τ 〉 ∼ 10 µs. Kinetic measurements indicate that theseconformational transitions are rate limiting and that they occuron timescales of hundreds of microseconds [144], which isslightly slower than the rates predicted by the microscopic-mixing model. However, an order of magnitude in timescaleonly corresponds to a 2.8 kBT difference in barrier height andthe observed deviation may be considered minor.

4.4. Computational considerations

Simulations allow one to enumerate the configuration spaceof a molecule, from which thermodynamic and kineticproperties may be calculated for a given model. Withinfinite computing time, one could exploit the ergodicity ofthese systems and simply wait for the full phase space tobe sampled with a Boltzmann distribution. However, theenergy landscapes of biomolecules are complex, and oftenpossess many competing energetic basins. Therefore, when

using conventional molecular dynamics (or Monte Carlo)simulations, the majority of the calculation will be spentsampling the low-energy basins, and barrier-crossing eventswill be rare. Since the transition states between stable basinslimit the overall functional dynamics, significant efforts havebeen dedicated to overcoming the sampling issues associatedwith high barriers. Some general strategies for characterizingtransition states are to enhance the overall sampling of thesimulation [203–206], increase the sampling along particularcoordinates [207], or to specifically search for the transitionstates [208–213].

In enhanced sampling approaches, system perturbationsare introduced to increase the sampling in particular regionsof phase space and then statistical-mechanical relationshipsare used to recover the physical quantities of the unperturbedsystem. For example, when using replica exchange methods,multiple copies of the system are simultaneously simulated,and each image is simulated with a unique HamiltonianHi , or at a unique temperature Ti . During the simulationsreplicas are probabilistically interconverted (i.e. the copy atTi goes to temperature Ti+1 and the replica at Ti+1 goes toTi). The system at low temperature may then ‘jump’ tothe higher-energy configurations that are sampled at highertemperatures. Since these jumps are non-physical, the kineticsof the system are not directly observable. However, ifthe jumps between temperatures satisfy detailed balance, acanonical ensemble will be sampled. While this methodhas been very successful for biomolecular folding [214–220],where the relevant transition is temperature induced, thereare limitations to its applicability. For example, whenstudying the interconversion between competing energeticbasins, performing simulations at high temperature can lead toglobal unfolding of the system. More generally, temperaturechanges can be associated with phase transitions that are notrelevant to the process of interest (e.g. global folding, orevaporation of explicit water). When applied to appropriaterearrangements, replica exchange methods can significantlyincrease the statistics of a given process.

An alternative to calculating the full range of possibleconfigurations is to use path sampling approaches [208–213].Rather than allow the system to freely search all phase space,these algorithms search for the lowest-energy pathway thatconnects the predefined endpoints. To search for these low-energy paths, the system is described by a set of images, whereadjacent images are coupled though energetic restraints, orconstraints. As the calculation proceeds, the set of imagesevolves with time, such that the set of images convergeonto the lowest energy (or lowest free-energy) pathway.These methods are highly effective for describing transitionsthat possess a dominant low-energy pathway. When thereare large changes in entropy as a function of the progresscoordinate, then sampling the high-entropy regions has thepotential to be computationally challenging. In nudged elasticband methods [210–213, 221], the replicas are coupled toone another, which restricts the relative mobility of adjacentimages. If the given rearrangement has a high-entropysubstate, then the harmonic coupling between images can limitthe characterization of that space. To overcome this issue,

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these methods often use harmonic approximations to the localdensity of states, in order to estimate the local contribution tothe entropy. For the reader interested in the detailed use of theseapproaches, we suggest additional reading [222]. If there aremultiple competing paths, then these estimations of entropywould not be accurate, since the harmonic approximation tothe density of states would no longer be valid. In stringmethods [209–212, 223], rather than couple each image to theothers, the images are only required to be equispaced alonga predefined set of reaction coordinates. As the calculationproceeds, the images are evolved until the lowest-energy pathis found (in the defined coordinate space). Since each image isallowed to fluctuate in directions perpendicular to the reactioncoordinates, the local phase space may be fully sampled. Thus,string methods can provide a more complete characterization ofthe minimum free-energy pathways than elastic band methods.

5. Roles of order–disorder transitions inbiomolecular functional dynamics

Largely inspired by the seminal works of Haldane andPauling [224, 225], a conventional view of biomolecularfunctional dynamics is that there are rigid regions of residuesand rearrangements are facilitated by highly flexible ‘hinge’regions [27, 149, 165, 226, 227]. This description is oftenenticing due to its similarities with macroscopic machines.However, in contrast to macroscopic machines, where therigidity of components spans orders of magnitude (i.e. hingesare often lubricated, leading to ‘flat’ potential energy functions,and moving components are composed of solid-phase metals),the weak interactions that maintain biomolecular stabilityare the same interactions that facilitate functional dynamics.Since the non-bonded energetic interactions that maintainbiomolecular structure are normally on the scale of a few kBT ,there is no reason to assume that a hinge/rod description willaccount for the full range of biomolecular functional dynamics.

Here, we discuss a variety of recent findings that expandour picture of biomolecular functional dynamics beyond thesimple hinge/rod picture. Of particular interest is the biologicalrelevance of order–disorder transitions. While energeticinteractions are essential for dynamics, biomolecules mayalso exploit large changes in entropy through local, or global,folding/unfolding events. In doing so, biomolecules may tunetheir functional dynamics by evolving differential stabilitiesof particular subunits. To illustrate the biological role ofheterogeneous biomolecular stability and flexibility, we willdiscuss several classes of order–disorder transitions that havebeen implicated in the functional dynamics of polypeptide andnucleic acid chains.

5.1. Conformational rearrangements: cracking

Early studies that employed iterative normal mode analysisto characterize the conformational transitions of AdK [133,156, 165, 228] suggested that biomolecules may partiallyunfold and refold during conformational transitions. Thatis, it was predicted that internal ‘strain’ energy accumulatesduring conformational rearrangements, and this strain is

Figure 13. Cracking during functional rearrangements. Iterativenormal mode analysis suggests large levels of strain energyaccumulate (black) during conformational rearrangements. Torelieve strain that is localized to specific residues, biomolecules maycrack, or partially unfold and refold. By unfolding, a large numberof unfolded configurations are accessible (red), which increases theentropy of the transition state ensemble and reduces the free-energybarrier associated with the transition (blue). X-ray crystallographicdata and NMR measurements [27] probe the fluctuations local to aparticular energetic basin (yellow lines), and the local excitationscan have dynamics that are correlated with large-scalerearrangements [134].

heterogeneously distributed across the protein. Not onlywas strain isolated to particular residues, but the level ofstrain energy would have been sufficient to denature theprotein (>20 kcal mol−1). This high strain would also resultin large free-energy barriers that would lead to biologicallyirrelevant timescales (i.e. transitions would occur on hours,not milliseconds). To circumvent these high-strain states,it was proposed that these residues may ‘crack,’ or locallyunfold and refold during conformational transitions [133].By cracking, residues would release stabilizing interactions,enter higher-entropy unfolded states and then refold to thealternate configuration. The increase in entropy would reducethe free-energy barriers of rearrangement, and would increasethe kinetic rates (figure 13) [134]. At first glance, thecracking mechanism may appear too complex to increasethe rates of interconversion. Although, as discussed above,the kinetic prefactor for protein folding is ∼1 µs−1. Sincecracking is a smaller-scale process than folding of an entireprotein, one should expect the timescale associated withthe cracking transition to be less than 1 µs (assuming thatinterconversion between order–disorder does not introducean additional significant free-energy barrier). The abilityto transition between order and disorder on such timescalesmakes it clear that cracking should be a kinetically accessiblemechanism by which conformational rearrangements mayoccur.

When the cracking mechanism is employed, the balancebetween order and disorder for individual residues willdetermine the functional dynamics. Therefore, proteinfunctional dynamics may be tuned by selecting for amino-acidsequences that have appropriate local stabilities. Analogousto the SK Ising model, each residue is uniquely coupledto the rest of the protein, which allows residues to have

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differential propensities to spontaneously unfold, even undernative conditions [229, 230].

Following the original prediction of cracking, manystudies have aimed to use more complex modeling approachesto characterize the location and scale of cracking in Adk.With a microscopically mixed multi-basin Hamiltonian, wefound that the degree of cracking in Adk is largest inthe highest-strain regions that were predicted by Miyashitaet al [177]. In contrast, simulations of Adk that employedmacroscopic mixing models have suggested a lower-degreeof cracking [169]. As discussed above, this attenuated levelof cracking may be the result of the stringent restrictionsimposed by macroscopic mixing models. These studies haveemployed coarse-grained models that represent each residueas a single bead, located at the position of the Cα atom.Therefore, in these studies, cracking was measured in termsof large-scale changes in backbone dynamics. Models thatexplicitly include all-atoms have demonstrated that proteinfolding transitions may be described as a superposition ofpartially separable transitions: backbone collapse and side-chain rearrangements [56, 129, 231]. This separation ofbackbone and side-chain ordering suggests that the detailsof cracking may be more subtle than originally envisioned.Specifically, cracking may also involve the local disorderingof side chains, with a more minor contribution from backbonecracking [186]. Together, these studies suggest that multi-basin models that employ atomic resolution will reveal a broadrange of cracking events that span from individual residue sidechains, to regions of consecutive residues.

Cracking has also been identified in the motor proteinkinesin. Kinesin molecules are able to ‘walk’ on microtubules,which enables transport within the cell [26]. During each‘step,’ kinesin binds ATP, undergoes relative displacement of≈80 Å of the domains [145], hydrolyzes ATP and releasesADP. Simulation studies that employed time-dependentHamiltonians showed that the power stroke in the kinesin headdomain may be facilitated by cracking [232, 233]. Hyeon et alshowed that a simple model for substrate directed walking isthat ATP binding shifts the effective potential energy surfaceof the leading domain, such that it prefers to place the trailingdomain ‘in front.’ This leads to an internal conflict of domainenergetics, which leads to partial disordering of the linkerregion that connects them. If flexibility of the linker were toohigh, then strain could not be transferred between the domains.Conversely, if the linker were rigid, it would not be capableof undergoing the conformational rearrangements associatedwith walking.

The potential importance of cracking has motivated theinvestigation of its role in other proteins’ conformationaldynamics, as well. Itoh and Sasai developed a dual-basinstatistical-mechanical model and found evidence of crackingin the C-terminal tail of calmodulin [135]. Similarly, Tripathiand Portman developed a variational model to describeconformational rearrangements in the N-terminal domain ofcalmodulin and found that specific regions undergo significantchanges in flexibility during rearrangements [234], whichis consistent with the cracking paradigm. Using a coarse-grained model, Hyeon et al also provided evidence of cracking

during functional rearrangements in protein kinase A [235].Experimentally, biophysical measurements have implicatedcracking during the functional dynamics of AdK [229, 236].Peng et al used instantaneous normal mode (INM) calculationsto guide conformational rearrangements in a simulation ofAdk [237] and found that immediately preceding domainrearrangements, the INMs were dissimilar to those foundin either of the end-state ensembles. This suggests thatsimple low-energy, rigid-body rearrangements are insufficientto account for Adk’s functional dynamics and that crackingmay occur. Finally, Okazaki et al surveyed multiple proteinsand showed that cracking appears to be a common feature inmultiple protein rearrangements [196].

5.2. Biomolecular association: fly-casting

The balance of many, often competing, kinetic processes isrequired for proper functional dynamics of the cell [238,239]. The highly non-equilibrium nature of cellular dynamicsrequires that signaling interactions be kinetically accessibleand regulated. If bimolecular association events are simplythree-dimensional random walks, the only way to modulate thekinetics of association would be to evolve proteins of differentshapes and sizes, thereby changing their diffusion coefficients.However, evolutionary pressure has led to nearly every aspectof the cell being tunable, including association kinetics.

To address the apparently limited tunability of bimolecularassociation, Shoemaker et al proposed the ‘fly-casting’mechanism [240], where proteins partially unfold whenisolated and the full structure forms upon substrate binding.Using a free-energy functional approach, they demonstratedthat if part of the protein can unfold, and the unfolded formof the protein has a marginal affinity for the substrate, theexponential increase in phase space available for substratecapture can lead to increased rates of association. Itwas later demonstrated through molecular simulations [241,242] that these changes in flexibility can impact the ratesof bimolecular association events. Bioinformatic analysisof protein–RNA interactions have also suggested that theelectrostatic composition of biomolecules have evolved toguide disordered regions during fly-casting events [243].

In addition to using fly-casting dynamics for bimolecularassociation, recent work has elaborated on the role ofintrinsically disordered regions during DNA ‘searching.’ Thatis, the molecular flexibility that enables fly-casting may alsofacilitate ‘sliding’ motions [95], where the protein ‘scans’ theDNA when looking for the lowest-energy binding site [244].During these searches, proteins may utilize a combination ofsliding and ‘hopping’ between sites. During hopping, theprotein dissociates from the DNA and reattaches itself viathe fly-casting mechanism. The balance between flexibility,disorder and atomic energetics determines the extent to whicheach mechanism is utilized, which ultimately determines if thegenetic messages may be expressed at the appropriate times.

Similar to cracking, the fly-casting mechanism providesa means by which a balance of local stabilities dictates thefunctional dynamics of a protein. However, in contrast tocracking, where the endpoints are stable configurations, in fly

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Figure 14. Riboswitch folding and ligand recognition. (a) Secondary structure of the SAM-1 riboswitch. The P1 helix is formed by theterminal residues, making it a non-local helix [125]. (b) Average fraction of native P1 contacts formed (〈QP1〉) as a function of the fractionof all native contacts being formed (Qtotal) is shown in blue. Potential of mean force shown in green. The largest barrier in the pmfcorresponds to folding of the P1 helix. (c) Same as (b), but in the presence of the SAM ligand. The average fraction of ligand interactions〈Qligand〉 is shown in magenta. Ligand binding induces earlier folding of P1 and it reduces the associated barrier. (d) Structuralrepresentation of the final step of folding, when the SAM ligand is present. All secondary structure, along with some tertiary structure isformed, and the ligand binds prior to P1 structure formation.

casting the folded configuration only needs to be stable uponsubstrate binding.

5.3. Protein recognition: global unfolding and refolding

While fly-casting and cracking involve partial unfolding ofsubstructures (figure 3(b)), in some cases, global order–disorder transitions may be linked to functional dynamics. Therepressor of primer (Rop) dimer8 is formed by association oftwo alpha-helical protein chains that bind to ‘kissing’ RNAhairpins, in order to stabilize their interactions. Based onexperimental kinetic data [245], it had been suggested thatRop and its mutants can adopt two symmetrically-relatedcompeting conformations during function [246]. To studythe dynamics of this process, a multi-basin potential energyfunction was employed to simulate the transitions betweenthese competing configurations [247]. Since a microscopicallymixed Hamiltonian was employed, it would have been possiblefor the protein to undergo this rearrangement through asequence of local displacements, while the monomers remainin association with each other. Despite the energetic flexibilityafforded by the microscopic model, these simulations indicatethat the Rop dimer globally unfolds and then refoldsduring its conformational rearrangements. These predictionsfrom simulations were later verified in single-moleculeexperiments [248]. Consistent with cracking and fly-castingdynamics, this global denaturation is afforded by the marginalstability of proteins under physiological conditions. Global,or local increases in stability could make unfolding kineticallyunfavorable, and functional regulation in Rop would then notbe possible.

5.4. Riboswitch signaling: substrate-assisted ordering

Riboswitch aptamer domains are untranslated regions ofmRNA that, in response to ligand binding, adopt well-defined three-dimensional configurations, whereas alternateensembles of configurations are accessed in the absence ofligand. These conformational changes can even include large-scale unfolding and refolding of secondary structure, similar

8 Rop is also sometimes called Rom (RNA 1 modulator).

to the operation of the Rop dimer. The balance between theseligand-bound and ligand-free ensembles directly regulatestranscription and translation [249, 250] (figure 14(a)). Sincethe identification of riboswitches in bacteria [251], therehas been vigorous investigation into the relationship betweenligand binding, structure formation and gene expression. Afrequently adopted framework for interpreting experimentaldata is that all secondary structure (i.e. helices), and sometertiary structure, is formed prior to substrate binding. Forsystems where this assumption holds, the role of ligandbinding is relegated to facilitating minor rearrangements.That form of operation is reminiscent of allosteric proteinfunction, where tertiary structure is usually fully formed andconformational transitions associated with ligand binding arelimited to the repositioning of domains. However, bacterialriboswitches fold with the ligand in solution, suggestingthat ligand binding may alter the dynamics of the globalreorganization/folding processes, in addition to coordinatingfinal atomic repositioning.

Simulations with unfrustrated models have shown thatentropic bottlenecks during folding can be directly perturbedby ligand association. In the case of the SAM-1 riboswitch,entropic contributions disfavor the formation of the P1 helix(figure 14(b)), which is the step associated with the largest free-energy barrier during folding [125]. This large difference inconfigurational entropy can be attributed to the connectivityof the helices. The strands composing the P2, P3 and P4helices are each connected by a few-residue ‘turn,’ and theP1 helix is formed by the terminal residues in the domain(figure 14(a)). While the P2/3/4 turns are composed of 4residues, the P1 strands are connected by a linker of ∼80residues, hence the designation of ‘non-local helix.’ Thus,the configurational entropy of the unfolded P1 helix is muchlarger than the other helices. By simulating the process in thepresence of the ligand, it was shown that this rate-limiting stepis accelerated by ligand binding, suggesting that ligand bindingcan have a thermodynamic and kinetic influence on aptamerformation. Consistent with this prediction, recent chemicalprobing data has shown that in the absence of ligand, the P1helix exhibits high mobility, and this mobility is attenuated

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by substrate binding [252]9. Single-molecule experimentsindicate that ligand binding most strongly affects the dynamicsof the P1 helix [253], which highlights the potential of SAMto direct P1 folding.

Coarse-grained simulations of a purine riboswitch alsofound that the non-local helix is least stable and itsformation is rate limiting [254]. Similarly, simulations ofthe preQ1 riboswitch showed that the folding mechanismscan be redirected by substrate binding [255]. Single-molecule experiments have also revealed that ligand bindinghas a focused effect on the non-local helix in a purineriboswitch [256]. Each of these studies demonstrate the rolethat ligand binding has on folding, suggesting that substraterecognition and folding are concomitant.

Similar to protein recognition, substrate-assisted foldingin RNA may encompass fly-casting-like dynamics. Asdiscussed above, the principal effect of the fly-castingmechanism is to accelerate substrate binding rates byincreasing the capture radius for the substrate. Riboswitchstudies suggest that there may be a symbiotic relationshipbetween folding and ligand association: fly-casting increasesassociation rates and ligand association increases thebiomolecular folding rates. Similar to fly-casting proteins,if aptamer domain formation is too fast, or stable, thenthere would be a separation of timescales between ligandbinding and riboswitch folding. In that scenario, ligandbinding may increase the folding rate slightly, but thepresence/absence of ligand would only represent a minorperturbation. Together, these studies corroborate a generaltheme in molecular biophysics: biomolecular complexes haveevolved to be marginally stable, allowing them to accesshigher-entropy states and increase the available phase spacefor regulation.

5.5. Molecular machines: entropically-guidedrearrangements

The final example of the role of entropy in biomolecularfunctional dynamics that we will discuss is found in theelongation cycle of the ribosome. During translation, mRNA isread by the ribosome through the use of transfer RNA (tRNA)molecules, which produces proteins with specific amino-acid sequences. The decoding process involves individualaminoacyl-tRNA (aa-tRNA) molecules that form 3-base-pair interactions (i.e. codon–anticodon interactions) with themRNA, where each base pair is composed of two (adenine-uracil) or three (guanine-cytosine) hydrogen bonds. Whenaccounting for competition with water, the hydrogen bondsof base pairs provide ∼1–2kBT of stabilizing energy, each.Since four types of nucleic acids compose these base-pairinteractions, there are 43 = 64 unique messages that may beencoded by a single mRNA codon. With only 64 possible waysto encode 20 different naturally occurring amino acids, whereall possible base-pair interactions are on comparable energeticscales, thermodynamic differences between cognate (correct

9 It should be noted that this study was for an isolated aptamer, whichlacked the expression platform. However, the dynamic effect of the ligandis anticipated to be similar in the context of the full riboswitch.

tRNA-mRNA pairs) and near-cognate (1 base-pair mismatch,such as a U on the tRNA and a G on the mRNA) can onlyaccount for an error rate of about 1 in 100. Since the errorrate of the ribosome is approximately 1 in 103–104, additionalproofreading mechanisms must be employed [257].

To explain the fidelity of ribosome dynamics, Hopfieldproposed the ‘kinetic proofreading’ mechanism [258]. Inthis mechanism, by loading a marginally-stable intermediatevia a driven process (such as GTP hydrolysis), thedifferences in kinetic accessibility of the initial and finalconfigurations may be greater than the correspondingdifferences in thermodynamic stability. Kinetic proofreadingwas proposed as a general strategy by which molecularprocesses may increase the fidelity of a particular substrateselection step, though here we will only discuss it incontext of ribosome fidelity [259]. In the ribosome,elongation factor-Tu (EF-Tu) uses the energy stored in GTPto load the ribosome with aa-tRNA molecules [260–262]in an energetically stained A/T configuration [163, 263].Upon EF-Tu release, cognate (i.e. correct) aa-tRNAs enterthe ribosome (≈90 Å displacement) and adopt the ‘A/A’configuration, while non-cognate aa-tRNAs are rejected. Inthe kinetic proofreading framework, the A/T configurationis loaded by EF-Tu, but both the A/A and dissociatedconfigurations are more stable. The A/T-to-A/A transition(called ‘accommodation’) can then serve as the proofreadingstep in translation. Consistent with the kinetic proofreadingmechanism, theoretical considerations have demonstrated thatkinetics and fidelity impose competing limitations on ribosomedynamics [264]. If the accommodation step were too fast(averaged over all tRNA species), large energetic factors wouldbe necessary to remove the incorrect tRNAs from the ribosome.However, no such factors are required for rejection. In contrast,if accommodation were too slow, then the dissociation of tRNAfrom the ribosome would be overly likely, even for cognatetRNAs.

Investigations into the balance of kinetics, energetics andfidelity of accommodation [265] have revealed atomic detailsof the endpoints [146, 261, 266], the TSEs [260, 267, 268]and rates [265] of accommodation. These studies haveprovided many insights into the details of accommodation,including the role of disorder in the incoming aa-tRNAmolecule. Using an SBM, we showed that the flexibility ofthe single-stranded 3′-CCA end (figure 15) may lead to largechanges in the configurational entropy during accommodation(figure 15), which also facilitates a multi-step accommodationprocess [268]. Upon dissociation of EF-Tu, the 3′-CCAend can adopt a disperse set of configurations within theA/T ensemble, which leads to an increase in configurationalentropy. As the aa-tRNA enters the ribosome, the initialaccommodation sub-step is aa-tRNA elbow movement towardan A/A-like configuration. This is followed by aa-tRNAarm movement and finally 3′-CCA end movement into thecatalytic center of the ribosome (the peptidyltransferase center,PTC). During the elbow-association step, the accessibleconfigurations of the 3′-CCA end appear to be constant,which indicates a negligible change in configurational entropyof the 3′-CCA end during this transition. As the tRNA

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Figure 15. Entropy changes in the 3′-CCA end of aa-tRNA during accommodation. Atomic models of tRNA (A-site tRNA in yellow, P-sitetRNA in red) and mRNA (green) at different points during accommodation. Amino acids attached to the ends of tRNA are shown in blue.(a) Atomic model of the A/A configuration of tRNA, based on cryo-EM densities. Three reaction coordinates for describingaccommodation (RElbow, RArm, R3′ ) are indicated. (b/c) Intermediate populations predicted from simulations [268] suggest that transientinteractions are formed with the A loop (pink) and H89 (gray) of the large subunit. (d) Endpoint of accommodation, as determined fromx-ray crystallography. (e) Pseudo-dihedral angle �2 describes the configuration of the 3′-CCA end as it enters the center of the ribosome.(f ) During accommodation, the range of accessible configurations of the 3′-CCA end decreases, indicative of an entropic penalty foraccommodation.

arm moves into the A site, movement of the 3′-CCA endis partially restricted (figure 15), suggesting that a decreasein configurational entropy resists such motions. Finally, asthe 3′-CCA end enters the PTC, it becomes highly ordered,as required for the chemical step (peptide bond formation)to occur [3]. This further decrease in configurationalentropy represents an additional entropic penalty duringaccommodation.

This order–disorder–order transition in the 3′-CCA endof the aa-tRNA can have profound biological implications.Specifically, changes in configurational entropy of theuniversally conserved (across all tRNA species) 3′-CCA endmay provide an entropic counterweight that ensures 3′-CCAentry into the PTC is the last step during accommodation.Since the incoming amino acid is attached to the 3′-CCA end,once it accommodates the amino acid it may be added to thegrowing peptide chain. By favoring the late entry of the 3′-CCA end, this entropic contribution can facilitate the use ofadditionally proofreading ‘machinery’ prior to incorporating apotential mistake.

While previous simulations showed that flexibility in the3′-CCA end is necessary for it to enter the PTC [267], this wasthe first time that configurational entropy of the 3′-CCA endwas explicitly identified as contributing to the global tRNAdynamics. There are many signatures of a dynamic 3′-CCAend in tRNA which may be found in the literature. Forexample, the 3′-CCA end of tRNA molecules are often notresolved in crystallographic structures [269], or the dispersion

in their coordinates is large (as measured by the Debye–Wallerfactors) [270]. In cryogenic electron microscopy studies thereis often a low density obtained for the 3′-CCA end [271],indicating that the 3′-CCA end structure is composed ofa diverse set of configurations, relative to other molecularcomponents. The 3′-CCA end also interacts with manydistinct binding partners [272], including synthetases [273],EF-Tu [274] and the ribosome [275], which necessitates thatits conformation be adaptable.

These studies highlight the balance between stabilizingenergetic interactions (that favor accommodation) anddestabilizing entropic contributions (that resist the process).If flexibility of the 3′-CCA end were increased, it would bepossible for accommodation to become too slow, and correcttRNA molecules would be rejected. Although, if 3′-CCAflexibility were decreased, it may not be possible for thetRNA molecules to navigate through the narrow corridor ofthe ribosome [267, 276].

6. Closing remarks

Grounded in statistical mechanics, the energy landscapeframework has enabled the quantitative characterization ofbiomolecular dynamics at the atomic level. Here, we havedescribed the historical context surrounding the developmentof this framework, which originated in protein folding. Sincethen, it has been repeatedly extended to address biomolecularfunctional processes of increasing complexity. An exciting

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consequence of the energy landscape description is that thereis an interplay between order–disorder transitions and thefunctional dynamics of biomolecules. From a wide rangeof studies, there are many examples demonstrating how themarginal stability of biomolecules allows them to adjust theirunfolding properties as a mode for tuning the functionaldynamics. Accordingly, these deceptively complex molecularassemblies possess an extremely large range of phase spaceavailable for function. In the future, we anticipate that thistheme will be echoed in assembly/disassembly processes,transient formation of excited states and large-scale collectivemotions in the cell. Eventually, the molecular description willreach scales that overlap with systems-level biology, whichrepresents one of the next major challenges for the biologicalphysics community.

Acknowledgments

PCW would like to thank Professor Alexander Schug(Karlsruher Institut fur Technologie) and Dr Jeffrey Noel(UCSD) for critical feedback during manuscript preparation.This work was supported by the Center for TheoreticalBiological Physics sponsored by the NSF (Grant PHY-0822283) and by NSF- MCB-1214457, in addition to supportfrom NIH Grant R01-GM072686. JNO is a CPRIT Scholarin Cancer Research sponsored by the Cancer Prevention andResearch Institute of Texas.

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