quantitative connection between ensemble thermodynamics and … · 2015-12-29 · quantitative...

Quantitative Connection between Ensemble Thermodynamics andSingle-Molecule Kinetics: A Case Study Using Cryogenic ElectronMicroscopy and Single-Molecule Fluorescence Resonance EnergyTransfer Investigations of the RibosomeColin D. Kinz-Thompson,†,∥ Ajeet K. Sharma,‡,∥,¶ Joachim Frank,⊥,§ Ruben L. Gonzalez, Jr.,†

and Debashish Chowdhury*,‡

†Department of Chemistry, Columbia University, New York, New York 10032, United States‡Department of Physics, Indian Institute of Technology, Kanpur 208016, India⊥Howard Hughes Medical Institute, Department of Biochemistry and Molecular Biophysics, Columbia University, New York, NewYork 10027, United States§Department of Biology, Columbia University, New York, New York 10027, United States

ABSTRACT: At equilibrium, thermodynamic and kinetic information can be extractedfrom biomolecular energy landscapes by many techniques. However, while static,ensemble techniques yield thermodynamic data, often only dynamic, single-moleculetechniques can yield the kinetic data that describe transition-state energy barriers. Herewe present a generalized framework based upon dwell-time distributions that can be usedto connect such static, ensemble techniques with dynamic, single-molecule techniques,and thus characterize energy landscapes to greater resolutions. We demonstrate the utilityof this framework by applying it to cryogenic electron microscopy (cryo-EM) and single-molecule fluorescence resonance energy transfer (smFRET) studies of the bacterialribosomal pre-translocation complex. Among other benefits, application of thisframework to these data explains why two transient, intermediate conformations of thepre-translocation complex, which are observed in a cryo-EM study, may not be observedin several smFRET studies.

1. INTRODUCTION

Biomolecular machines operate on energy landscapes withtransition-state energy barriers which range from ∼kBT to theenergy of covalent bonds.1−3 Characterizing the wells andbarriers which comprise these energy landscapes is importantfor understanding the thermodynamics and kinetics ofbiomolecular machines, and how these thermodynamics andkinetics can be modulated in order to regulate the activities ofthese machines.4−9 However, due to the stochasticity inherentto these processes, as well as the transient and/or rare nature ofstates separated by low transition-state energy barriers,extremely sensitive techniques are often required to obtainthe level of detail necessary to adequately describe suchsystems.10,11 Sufficiently sensitive ensemble techniques, such ascryogenic electron microscopy (cryo-EM), can measure static,equilibrium-state populations, and this provides information onthe relative energy differences between distinct states. However,because of the vanishingly small probability of observing atransition state, techniques such as cryo-EM are not able tocharacterize the transition-state energy barriers responsible formuch of the regulation of biomolecular processes.12 For-tunately, dynamic, time-dependent, single-molecule techniques,such as single-molecule fluorescence resonance energy transfer(smFRET), can directly monitor the kinetics of these processes,

and allow the characterization of the transition-state energybarriers with theories such as transition-state theory orKramers’s theory.13 smFRET is a particularly powerfultechnique for connecting single-molecule kinetics to ensemblethermodynamics obtained from cryo-EM in that the FRETefficiency (EFRET) obtained from the smFRET experiments canbe correlated to structures obtained from the cryo-EMexperiments. Despite this significant advantage over manyother single-molecule techniques, like all techniques, smFRETapproaches often suffer from limitations to spatial and temporalresolution, and also often require structural information todevelop biologically informative signals.14 Therefore, for anyparticular system, static, equilibrium-state, ensemble techniquesand dynamic, time-dependent, single-molecule techniquesprovide complementary approaches for studying the underlyingbiological processes. Nonetheless, given the current limitationsin their application, the pictures they provide may not always becongruous.

Special Issue: Biman Bagchi Festschrift

Received: December 25, 2014Revised: March 15, 2015Published: March 18, 2015

Article

pubs.acs.org/JPCB

© 2015 American Chemical Society 10888 DOI: 10.1021/jp5128805J. Phys. Chem. B 2015, 119, 10888−10901

pubs.acs.org/JPCB

http://dx.doi.org/10.1021/jp5128805

The bacterial ribosome is one example of a biological systemthat has been well studied by both ensemble and single-molecule techniques, although the associated energy landscaperemains only coarsely defined.15,16 Responsible for translatingmessenger RNAs (mRNAs) into their encoded proteins, theribosome is composed of a large and a small subunit (50S and30S in bacteria, respectively). During the elongation stage oftranslation, the ribosome undergoes consecutive rounds of anelongation cycle, in which it successively adds amino acids tothe nascent polypeptide chain in the order dictated by thesequence of the mRNA. In the first step of the elongation cycle,the mRNA-encoded aminoacyl-transfer RNA (aa-tRNA) isdelivered to the aa-tRNA binding (A) site of the ribosome inthe form of a ternary complex (TC) that is composed of theribosomal guanosine triphosphatase (GTPase) elongationfactor (EF) Tu, guanosine triphosphate (GTP), and aa-tRNA.17−20 Upon delivery of the mRNA-encoded aa-tRNAinto the A site, peptide bond formation results in transfer of thenascent polypeptide chain from the peptidyl-tRNA bound atthe ribosomal peptidyl-tRNA binding (P) site to the aa-tRNAat the A site, generating a ribosomal pre-translocation (PRE)complex carrying a newly deacylated tRNA at the P site and anewly formed peptidyl-tRNA, extended by one amino acid, atthe A site.21−23 Subsequently, the ribosome must translocatealong the mRNA, moving the newly deacylated tRNA from theP site to the ribosomal tRNA exit (E) site and the newlyformed peptidyl-tRNA from the A site to the P site.15,18,24−30

While translocation can occur spontaneously, albeit slowly, invitro,31 it is accelerated by orders of magnitude in vivo throughthe action of EF-G, another ribosomal GTPase.18,30

Prior to translocation and in the absence of EF-G, at leastthree individual structural elements of the PRE complexundergo thermally driven conformational fluctuations: (i) theP- and A-site tRNAs fluctuate between their classical P/P and

A/A configurations and their hybrid P/E and A/P config-urations (where, relative to the classical P/P and A/Aconfigurations, the hybrid P/E and A/P configurations arecharacterized by the movement of the acyl acceptor ends of theP- and A-site tRNAs from the P and A sites of the 50S subunitinto the E and P sites of the 50S subunit, respectively); (ii) theribosome fluctuates between its non-rotated and rotatedsubunit orientations (where, relative to the non-rotated subunitorientation, the rotated subunit orientation is characterized by acounterclockwise rotation of the 30S subunit relative to the 50Ssubunit when viewed from the solvent-accessible side of the30S subunit);32 and (iii) the L1 stalk of the 50S subunitfluctuates between its open and closed conformations (where,relative to the open L1 stalk conformation, the closed L1 stalkconformation is characterized by movement of the L1 stalk intothe inter-subunit space such that it can make a direct contactwith the hybrid P/E-configured tRNA) (Figure 1).33 Because ofthe stochastic nature of thermally driven processes, the tRNAs,ribosomal subunits, and L1 stalk within an ensemble of PREcomplexes will asynchronously fluctuate between thesetransiently populated states in the absence of EF-G. Whilethis structural heterogeneity impedes ensemble studies of thesedynamics, they have been successfully characterized by single-molecule methods.29,34−44

Remarkably, smFRET studies performed by Fei and co-workers have observed PRE complexes fluctuating between twodiscrete states: (i) global state 1 (GS1), characterized byclassically configured tRNAs, non-rotated subunits, and anopen L1 stalk, and (ii) global state 2 (GS2), characterized byhybrid-configured tRNAs, rotated subunits, and a closed L1stalk.34,35 The observation by Fei and co-workers that the PREcomplex fluctuates between just two states in the smFRETstudies is consistent with numerous subsequent smFRETstudies from several other groups in which the tRNAs,

Figure 1. Cartoon schematic mechanism of PRE complex fluctuations. After peptide-bond formation, the PRE complex fluctuates between MS I/GS1 and MS II/GS2, passing through IS1 and IS2, until EF-G-catalyzed translocation occurs.

The Journal of Physical Chemistry B Article

DOI: 10.1021/jp5128805J. Phys. Chem. B 2015, 119, 10888−10901

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ribosomal subunits, or L1 stalk elements of PRE complexes arealso observed to fluctuate between just two states, correspond-ing to the classical and hybrid tRNA configurations, the non-rotated and rotated subunit orientations, or the open andclosed L1 stalk conformations, respectively.34,35,37,38,40,44,45

Furthermore, the initial observation by Fei and co-workersthat the PRE complex fluctuates between GS1 and GS2 isconsistent with the more recent observation that fluctuations ofthe tRNAs between their classical and hybrid configurations,the ribosomal subunits between their non-rotated and rotatedorientations, and the L1 stalk between its open and closedconformations are physically coupled, and coordinated by theribosome in order to maximize and regulate the efficiency oftranslocation.44 smFRET studies reveal that the thermody-namics and kinetics of the equilibrium between GS1 and GS2are sensitive to (i) the presence, identity, and acylation status ofthe P-site tRNA;29,34−42 (ii) the presence and acylation statusof the A-site tRNA;29,34−42 (iii) the binding of EF-G;34,35,37−40

(iv) Mg2+ concentration;41 (v) temperature;43 (vi) the bindingof ribosome-targeting antibiotic inhibitors of transloca-tion;41,44,46 and (vii) the perturbation of inter-subunit rotationvia disruption of specific ribosomal inter-subunit interac-tions.42,44 Collectively, these studies have provided deepinsights into the roles that the P- and A-site tRNAs, EF-G,antibiotics, cooperative conformational changes, and allosteryplay in regulating translocation.Cryo-EM studies of PRE complexes performed by

Agirrezabala and co-workers have directly observed two statesthat are presumably the structural equivalents of GS1 and GS2,termed macrostate 1 (MS I) and macrostate 2 (MS II),respectively.32,47−49 More recent molecular dynamics simu-lations of cryo-EM-derived structural models of MS I and MS IIsupport this presumption, and found that, in the studies by Feiand co-workers, the EFRET values observed in GS1 and GS2 areconsistent with the simulations of MS I and MS II,respectively.50 However, in addition to MS I and MS II, amore recent analysis of the cryo-EM data set reported byAgirrezabala et al. has revealed the presence of two additionalstates that are presumably intermediate between MS I/GS1 andMS II/GS2 along the reaction coordinate.51 Neither of thesetwo intermediate states, referred to here as intermediate state 1(IS1) and intermediate state 2 (IS2), nor any others, weredetected in the smFRET studies by Fei and co-workers34,35,44,45

or several other groups.37,38,40 In contrast, we note thatsmFRET studies of PRE complexes by Munro and co-workersidentified and characterized two additional states thatpresumably lie along the reaction coordinate between MS I/GS1 and MS II/GS2.29,42 However, these studies relied heavilyon the use of smFRET data collected using ribosomes in whicha substitution mutation disrupts a critical ribosome−tRNAinteraction, and consequently causes the P-site tRNA in theresulting intermediate states to adopt conformations that arevery different from those observed in either IS1 or IS2 in PREcomplexes formed using wild-type ribosomes.51

Since the smFRET experiments by Fei and co-workers andthe cryo-EM experiments by Agirrezabala and co-workersinterrogate PRE complexes composed of wild-type ribosomes(i.e., without mutations that disrupt ribosome−tRNA inter-actions), it is highly likely that IS1 and IS2 are also present inthe smFRET data, but that, given the spatial resolution, timeresolution, and/or signal-to-background ratio (SBR) of thesmFRET experiments, IS1 and IS2 do not produce largeenough changes in EFRET to be distinguished from GS1 or GS2.

By connecting the results from static, equilibrium-state,ensemble experiments, such as cryo-EM, with the resultsfrom dynamic, time-dependent, single-molecule experiments,such as smFRET, through a theoretical framework, thesehypotheses can be tested, and the energy landscape where thePRE complex exists can be characterized more precisely. Here,we present such a framework based upon equilibrium-stateprobabilities and dwell-time distributions (see refs 52 and 53,and references therein). This framework is general and can beapplied to various other ensemble and single-moleculetechniques; we use a linear kinetic model but emphasize thatthe equations can also be derived for other models. As anillustrative case study, we apply this generalized framework toanalyze the data obtained from the cryo-EM and smFRETstudies by Agirrezabala et al. and Fei et al., respectively. Indoing so, we connect the distribution of the MS I, MS II, IS1,and IS2 states of the PRE complex observed by cryo-EM to thetransition rates between the GS1 and GS2 states observed bysmFRET.

2. METHODS2.1. Dwell-Time Distribution Framework for N-State

Markov Chain. Consider two distinct chemical states, 1 andN, of the system connected linearly by a number of on-pathwayintermediate states, 2 through N − 1, with transitions fromstate i to state j occurring at rate αij,

··· −α

α

α

α

α

α

α

α

− −

− −

−

−H Ioo H Ioo H Ioooooooo H IoooooN N1 2 1

N N

N N

N N

N N

21

12

32

23

1, 2

2, 1

, 1

1,

(1)

If Pμ(t) is the probability of finding the system in chemicalstate μ at time t, then the time evolution of these probabilities isgoverned by a set of coupled master equations. The steady stateof this set of equations, corresponding to the constraints ∂Pμ/∂t= 0, yields the equilibrium-state occupation probabilities Pμ

eq ofpopulating each state. The distribution of times taken by thesystem to reach one terminus from the other can be calculatedby modifying the original kinetic scheme, imposing anabsorbing boundary at the destination state. If the destinationstate is N, then the corresponding modified kinetic schemewould be

··· − ⎯ →⎯⎯⎯⎯α

α

α

α

α

α α

− −

− − −H Ioo H Ioo H Ioooooooo N N1 2 1

N N

N N N N

21

12

32

23

1, 2

2, 1 1,

(2)

By writing master equations for this new scheme, andadopting the method of Laplace transform, we analyticallycalculated the probability, f p(t) dt, that the system, initially atstate 1, reaches state N in the time interval between t and t + dt.By evaluating the first moment of this distribution, the meantime for this transition, ⟨tp⟩, can be calculated. An analogousprocess can be performed to calculate the probability, f r(t) dt,that the system, initially in state N, reaches state 1 in the timeinterval between t and t + dt, and hence obtain the mean timefor this transition, ⟨tr⟩.The two expressions for the mean transition times between

the termini, ⟨tp⟩ and ⟨tr⟩, form a system of equations with all2N − 2 αij as variables. Ratios of the Pμ

eq define relationshipsbetween the rate constants αij; so, if the equilibrium-stateprobabilities are known, substitution of these ratios into theexpressions for ⟨tp⟩ and ⟨tr⟩ reduces the number of degrees offreedom in the system of equations. With an experimentalmeasure of the mean transition time between the terminalstates, the system of equations can be solved for the αij rate



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constants. The four-state model (two intermediate states) issolved in Appendix A, as is the derivation of the expression forthe variances of tp and tr. Equivalent expressions for the three-state model (one intermediate state) are given in Appendix B.2.2. smFRET Simulations. We simulated 100 EFRET versus

time trajectories with a linear, three-state kinetic scheme.Dwell-times prior to transitions to other states wereexponentially distributed according to the appropriate rateconstants. In each EFRET versus time trajectory, the value ofEFRET corresponding to each state was randomized by choosingr from a normal distribution, and calculating EFRET = (1 + (r/R0)

6)−1, where R0 is the Forster radius. For each EFRET versustime trajectory, R0 was also randomly chosen from a normaldistribution. Noise reflecting a reasonable SBR for the total-internal reflection fluorescence (TIRF) microscope used in thesmFRET experiments (i.e., σ = 0.05) was also added to eachEFRET versus time trajectory. More details can be found inAppendix C.

3. RESULTS AND DISCUSSION3.1. General Framework. Because of their complexity,

biomolecular systems are often investigated with multipletechniqueseach with its individual strengths and weaknesses.However, different techniques occasionally yield disparatemechanistic pictures that must ultimately be resolved. Onesituation in which this problem manifests itself is when a static,equilibrium-state technique such as cryo-EM detects on-pathway intermediates, but a dynamic, time-dependenttechnique such as smFRET does not. This situation couldarise if the dynamic technique is not sensitive enough todistinguish the intermediate state from other states of thebiomolecular system. In order to reconcile such contrastingmeasurements, we need to estimate the lifetime of the transientintermediates if these, indeed, exist. In an effort to get theseestimates we consider a linear kinetic pathway with on-pathwayintermediates, such as in eq 1, though the framework presentedhere can easily be extended to include off-pathwayintermediates. Note that for an N-state linear kinetic schemethere are 2N − 2 rate constants αij. Therefore, in principle, thenumerical values of all the individual αij could be obtained if2N − 2 independent algebraic equations satisfied by these rateconstants were available. As we argue now, except for somesmall values of N, the rate constants αij are usuallyunderdetermined by the available experimental information.Time-dependent smFRET experiments are typically analyzed

with a hidden Markov model (HMM).14,54−57 Among otherthings, such an analysis yields HMM-idealized state versus timetrajectories from which a distribution of lifetimes in a particularstate can be calculated. If sufficiently transient, on-pathwayintermediates between the initial and final state exist, thedistribution of idealized lifetimes will not appear significantlydifferent from what it would be in the absence of theintermediate states (e.g., an exponential distribution for arandom transition with a time-independent probability ofoccurrence). In such a case, the simplest model that thesmFRET data supports is that of a transition with nointermediate states; so, assuming Markovian transitions, themean lifetimes obtained from the HMM-idealized trajectorieswould be taken to be the inverses of the effective rate constantsfor the transitions between the initial and final states. Incontrast, the analytical expressions for the mean lifetimes spenttraveling between the terminal states, via intermediate states, ofan N-state kinetic scheme contain the rate constants αij that

describe the direct transitions between the intermediate states(see Appendices A and B). Therefore, equating the meanlifetimes for the forward and reverse transitions between theterminal states that were inferred from dynamic, single-molecule experiments with the corresponding theoreticallycalculated mean lifetimes yields two algebraic equations thatinvolve 2N − 2 rate constants αij.Thus, in practice, except for the trivial case of N = 2, the

information available in the form of the effective rates offorward and reverse transitions between the two terminal stateswould be inadequate to determine all the 2N − 2 rate constantsαij that describe the full kinetic mechanism. Obviously, forlarger values of N, the number of degrees of freedom must bereduced further by acquiring additional experimental informa-tion. This extra information comes from the equilibrium-stateexperiments. Including information about the equilibrium-stateprobabilities for the N states provides N − 1 additionalindependent equations (the constraint of normalization of theprobabilities, i.e., their sum must be equal to unity, reduces thenumber from N to N − 1). So, for N = 3, one would have justenough information to write down four independent equationssatisfied by the four rate constants in the three-state model.However, for N = 4, we have fewer equations than the numberof unkowns and, therefore, in the absence of any otherinformation, one of the rate constants would remain a freeparameter. Any one of the six rate constants can be selected asthe free parameter. Then, as we will show later in this section,varying the selected free parameter allows one to enumerate allthe solutions which are consistent with the data, and therebyimpose lower and/or upper bounds on the magnitudes of therates. In case of higher values of N, the analytical expressionsfor the variance of tp and tr, reported in the Appendices, can beutilized for further reduction of the number of degrees offreedom if the corresponding experimental data becomesavailable in the future.

3.2. Model System: The Bacterial Pre-translocationalComplex. Agirrezabala and co-workers collected cryo-EM dataon PRE complexes containing tRNAfMet in the P site, and fMet-Trp-tRNATrp in the A site.49 Using ML3D, a maximum-likelihood-based classification method, particles from this dataset were more recently grouped into six classes.51 Theconclusions of this study strongly suggest that three of theclasses represent MS I and MS II−MS II, being comprised oftwo structurally similar classes. Additionally, the authorspropose that two of the other classes represent on-pathwayintermediate states (IS1 and IS2, respectively) between MS Iand MS II. As the remaining class represents PRE complexesthat are missing a tRNA in the A site, it is therefore ignored.Thus, the model of PRE dynamics proposed by this study is

α

α

α

α

α

αH Ioo H Ioo H Ioo1 2 3 4

21

12

32

23

43

34

(3)

where states 1, 2, 3, and 4 represent MS I, IS1, IS2, and MS II,respectively, and are distributed as shown in Table 1.Similarly, smFRET experiments performed by Fei and co-

workers monitored PRE complexes as they transitioned, drivenby thermal energy, between two global conformational states,GS1 and GS2, which correspond structurally to MS I andMS II, respectively.34,35 By monitoring the relative change inthe distance between the P-site tRNA and the ribosomalprotein L1 within the L1 stalk of the 50S subunit, the smFRETsignal developed by Fei and co-workers probably reports uponthe tRNA motions along the pathway proposed by Agirrezabala



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and co-workers. However, these smFRET measurements wereperformed for several PRE complexes of variable composition.Of these complexes, perhaps the one most relevant to the workperformed by Agirrezabala and co-workers is the PREfM/Fcomplex carrying a tRNAfMet in the P site and a fMet-Phe-tRNAPhe, rather than a fMet-Trp-tRNATrp, in the A site. Sincethe identities of the A-site dipeptidyl-tRNA in the twoexperiments differ, this could potentially lead to tRNA-dependent differences in the populations and lifetimes of thevarious states and, consequently, the rates of transitionsbetween these states. Indeed, smFRET studies have shownthat the lifetimes of GS1 and GS2 do depend on thepresence34,35 and acylation status (i.e., deacylated tRNAPhe

versus Phe-tRNAPhe versus fMet-Phe-tRNAPhe) of the A-sitedipeptdyl-tRNA.36,41 It should be noted, however, that theeffect of the identity of the A-site dipeptidyl-tRNA itself (i.e.,tRNAs other than tRNAPhe) has not yet been tested bysmFRET. In addition to the difference in the identities of the A-site dipeptidyl-tRNA in the cryo-EM and smFRET studies, theMg2+ concentrations employed in the two studies differ. Thecryo-EM studies were performed at [Mg2+] = 3.5 mM, and thesmFRET studies were performed at [Mg2+] = 15 mM.Previously, smFRET studies have demonstrated that changesto the Mg2+ concentration over this range affect the populationsand lifetimes of the GS1 and GS2 states.41 With this in mind, itis likely that the equilibrium-state populations observed in thecryo-EM experiments and the corresponding state occupanciesin the smFRET experiments are disparate. Nonetheless, despitetheir experimental differences, these cryo-EM and smFRETstudies are the most experimentally similar cryo-EM andsmFRET studies of wild-type bacterial PRE complexes thathave been reported in the literature. Therefore, as a case study,we have chosen to quantitatively compare these two particularstudies in order to demonstrate the application of the generalframework developed in section 3.1.The transition rates between GS1 and GS2 reported using

the PREfM/F complex for the L1-tRNA donor−acceptor labelingscheme were kGS1→GS2 = 2.8 ± 0.2 s−1 and kGS2→GS1 = 3.0 ± 0.4s−1.35 Given that no evidence of intermediate states wasobserved, this suggests that any intermediates states, if theyexist, might be very transient relative to the time resolutionwith which the smFRET data were acquired. Indeed, there is alimitation to the time resolution with which smFRET data canbe acquired with the electron-multiplying charge-coupleddevice (EMCCD) cameras that are typically used as detectorsin TIRF microscopy-based smFRET experiments. Transitionsthat are faster than the EMCCD camera’s acquisition rate (20s−1 in the work of Fei et al.)35 result in time averaging of theEFRET and the recording of a single, artifactual data point thatappears at the time-averaged value of the EFRET between thestates involved in the rapid fluctuations. This is a well-documented feature of smFRET data analysis, which we term“blurring”.54 This effect is further compounded by the fact thatcurrent state-of-the-art computational methods used to analyze

the smFRET data cannot distinguish between artificial, short-lived (i.e., one data point) “states” resulting from blurring andactual, short-lived (i.e., one data point) states resulting from thesampling of true intermediate states.54 With such an analysis,the true molecular states become hidden among the “blurred”states.Reanalysis of the original PREfM/F data using the software

package ebFRETa state-of-the-art, HMM-based analysismethod for smFRET data56,57yields a better estimate ofthe transition rates. This is because ebFRET uniquely enablesanalysis of the entire ensemble of individual EFRET versus timetrajectories, instead of analyzing them in the traditional,isolated, one-by-one manner. The two-state rates inferred bymeans of ebFRET are similar to, though perhaps more accuratethan, those reported originally by Fei and co-workers: kGS1→GS2= 2.0 ± 0.2 s−1 and kGS2→GS1 = 2.8 ± 0.1 s−1 (see Table 2).

Interestingly, application of ebFRET reveals that the smFRETPREfM/F data are best described by a five-state model. However,further analysis indicates that the three additional states areprobably artifacts of blurring, because they are negligiblypopulated, have extremely transient lifetimes, and occur at anEFRET that is in between the EFRET values of the two well-defined states.

3.3. Four-State Model of PRE Complex Dynamics. Thedynamics of PRE complexes were analyzed using the generalframework presented in section 3.1, where the experimentaldata summarized in Tables 1 and 2 were used as constraints forthe linear, four-state kinetic scheme shown in eq 3. For thiskinetic scheme, the mean time needed for the forwardtransition from the terminal state 1 to the terminal state 4,⟨tp⟩, is given by

ααα

α αα α α

αα α

⟨ ⟩ = + + + + +⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥t

11

11

1p

12

21

23

21 32

23 34 23

32

34 34

(4)

(see Appendix A for the full derivation). The correspondingmean time for the reverse transition from state 4 to state 1, ⟨tr⟩,is given by

ααα

α αα α α

αα α

⟨ ⟩ = + + + + +⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥t

11

11

1r

43

34

32

34 23

32 21 32

23

21 21

(5)

(see Appendix A for the full derivation). The probabilities forthe occupation of the four states at equilibrium are given by

α α αα α α α α α α α α α α α

=+ + +

P1eq 43 32 21

43 32 21 12 43 32 12 23 43 12 23 34(6)


=+ + +

P2eq 12 43 32

43 32 21 12 43 32 12 23 43 12 23 34(7)

Table 1. Summary of the Ribosomal PRE ComplexesObserved by Agirrezabala and Co-workers51

state index μ state class Pμeq

1 MS I 2 0.2312 IS1 4A 0.1313 IS2 4B 0.1404 MS II 5/6 0.498

Table 2. Summary of the Rates of Transition between GS1and GS2 Observed by Fei and Co-workers35 and in ThisWork

kGS1→GS2 (s−1) kGS2→GS1 (s

−1) ref

2.8 ± 0.2 3.0 ± 0.4 352.0 ± 0.2 2.8 ± 0.1 this work



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=+ + +

P3eq 12 23 43

43 32 21 12 43 32 12 23 43 12 23 34(8)


=+ + +

P4eq 12 23 34

43 32 21 12 43 32 12 23 43 12 23 34(9)

with the normalization condition ∑μ=14 Pμ

eq = 1 (see Appendix Afor the full derivation). Using the equations for ⟨tp⟩ and ⟨tr⟩(eqs 4 and 5, respectively), and the equations for Pμ

eq (eqs 6−9),a plot was generated of all the rate constants αij as functions ofan independent α43 (Figure 2).

Notably, for some values of the independent rate constant,solutions for the dependent rate constants are negative. Whilethis is a consistent solution of the model, only the values whereall rate constants are positive are physically relevant solutions.The boundaries to this region where all rate constants arepositive therefore represent the upper and lower bounds on therate constants for the four-state model that are consistent withboth the cryo-EM and the smFRET studies.Interestingly, Figure 2 depicts the upper and lower bounds

for α34 and α43, but only lower-bound cutoffs for the other α’s.These other rate constants all asymptotically converge topositive infinity; α12 and α21 increase with increasing α43, whileα23 and α32 compensate by decreasing to their lower bound.Plotting these results as a function of an independent α12 or α23yields the same boundsthere is only a narrow window whereall α’s are consistent with the cryo-EM and smFRET data. Withsuch boundaries on the individual rate constants, one canestimate the EMCCD camera acquisition rate that would beneeded to distinctly observe the transient states of interest.3.4. Three-State Pre-translocation Model. Since the

number of equations available in our four-state model is five,whereas the number of unknown rate constants is six, we couldonly express five rate constants in terms of the sixth one. Incontrast, because we can reduce the number of states from fourto three (vide inf ra), in this subsection we use the fourcorresponding independent equations to extract the absolute

values of the four rate constants associated with the three-statemodel,

α

α

α

αH Ioo H Ioo1 2 3

21

12

32

23

(10)

As explained in Appendix C, structural analysis stronglysuggests that the L1−tRNA distance in IS1 is insufficientlydifferent from that of MS I so as to result in an EFRET that issignificantly different than that of MS I. Thus, MS I and IS1 canbe combined into a single state, state 1, thereby reducing thefour-state model into a three-state kinetic scheme as shown ineq 10, where states 2 and 3 correspond to IS2 and MS II,respectively.The expressions for ⟨tp⟩, ⟨tr⟩, and Pμ (μ = 1, 2, 3) are given

by

ααα α

⟨ ⟩ = + +⎡⎣⎢

⎤⎦⎥t

11

1p

12

21

23 23 (11)

ααα α

⟨ ⟩ = + +⎡⎣⎢

⎤⎦⎥t

11

1r

32

23

21 21 (12)

α αα α α α α α

=+ +

P1eq 21 32

21 32 12 32 23 12 (13)


=+ +

P2eq 12 32

21 32 12 32 23 12 (14)


=+ +

P3eq 23 12

21 32 12 32 23 12 (15)

where the normalization condition is∑μ=1eq Pμ = 1 (see Appendix

B for detailed derivations). These expressions, together with thecorresponding experimental data, are utilized to write downfour independent equations. The rate constants computed bysolving those four equations are shown in Table 3.

Interestingly, this calculation suggests that the rate-limitingsteps for the foward and reverse reactions are IS2→MS II andMS II→IS2, respectively, while interconversion between MS I/IS1 and IS2 occurs relatively rapidly. These rates for MS I/IS1to IS2 interconversion are approximately the same as or fasterthan the 20 s−1 EMMCD camera acquisition rate of the originalsmFRET data from Fei et al.; additionally, the change in theL1−tRNA distance between the MS I/IS1 and IS2 states isrelatively small (∼80 Å to 64 Å), resulting in a correspondinglysmall difference in EFRET (∼0.15 to 0.40), so any separation ofMS I/IS1 and IS2 that might have been observed in the

Figure 2. Rate constants for the four-state model as a function of α43.The gray region contains the solutions where all rate constants arepositive. Both axes are log-scaled, so any incompatible rates (negativevalued) are not shown. The points where rate constants switch frombeing negative to positive valued are denoted with a black vertical line.Horizontal dashed lines denote upper and lower bounds for particularrate constants.

Table 3. Rate Constants for PREfM/F Using a Linear, Three-State Kinetic Scheme Where MS I and IS1 Have BeenCombined into the First Statea

α max kL1−tRNA (s−1) mean kL1−tRNA ± 1σ (s−1)

α12 18.1 23.3 ± 22.7α21 46.8 52.5 ± 33.6α23 5.90 5.89 ± 0.42α32 1.66 1.66 ± 0.12

aError from the ebFRET-estimated and smFRET-determined rateconstants, and the counting error from the cryo-EM study, werepropagated into distributions of the α. These distributions are strictlynot normal distributions, although α23 and α32 are approximatelynormal.



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smFRET data would likely have been obscured in the HMManalysis process by camera blurring arising from interconver-sion rates that are similar to the acquisition rate. The fastinterconversion and small expected changes in EFRET suggestthat MS I, IS1, and IS2 might have originally been interpretedas a “single”, averaged state in the analysis of the smFRET data.3.5. Synthetic smFRET Time Series. To investigate how

ebFRET would treat this “single”, averaged smFRET state,synthetic time series simulating a three-state PREfM/F complexwere constructed guided by the analysis above. Estimates forEFRET were based upon the cryo-EM structures reported byAgirrezabala and co-workers (see Appendix C), and the kineticscheme and associated rate constants employed are those insection 3.4. Since the rate constants for the transition betweenstates MS I/IS1 and IS2 are of the same order of magnitude asthe frame rate of this simulation, traditional smFRET dataanalysis of these synthetic data provides insight into whetherblurring could have obscured any transient, intermediate statesin the data reported by Fei and co-workers. Typically, suchobfuscation begins to manifest when dwell-times in a state ofinterest approach the same order of magnitude as the EMCCDcamera acquisition time, because of errors in estimating thelengths of the dwell-times.54

The synthetic smFRET data set was constructed by carryingout simulations of EFRET versus time trajectories where each“single ribosome” had randomized simulation parameters asdescribed in Appendix C. This probabilistic approach accountsfor experimental variation (e.g., uneven illumination in thefield-of-view), as well as ensemble variations (e.g., staticdisorder from a small sub-population of ribosomes lacking an

A-site dipeptidyl-tRNA). An example of a synthetic EFRET vstime trajectory is shown in Figure 3, where the ensemble meanvalues of EFRET were ∼0.16, 0.40, and 0.74 for states MS I/IS1,IS2, and MS II, respectively.With regard to the distribution of EFRET values observed from

any ensemble of EFRET versus time trajectories, blurring wouldresult in a shift of some of the density of the equilibrium-stateoccupancy probability distribution to an intermediate, averagedvalue between the blurred states. Deviation of a histogram ofthe observed, simulated EFRET in the synthetic data set from thedistribution predicted by the equilibrium-state occupancies ofthe linear, three-state model therefore can be used tocharacterize the amount of blurring present in the syntheticdata. We modeled the normalized histogram of the syntheticensemble with normal distributions weighted by theirrespective equilibrium-state probability, Pμ

eq (Figure 4A). Themean of each state was distributed according to the distributionof static EFRET for that state in each of the synthetic time series(Figure 4C). This approach accurately reflects a non-blurredhistogram of EFRET (Figure 5). Deviations that occur aretherefore due to blurring or, if they had been simulated in thissynthetic data set, could have been due to the presence ofunaccounted-for states. Notably, a large portion of the MS I/IS1 density in Figure 4A is relocated into the region betweenMS I/IS1 and IS2. This is a direct manifestation of blurring. Bycollapsing MS I/IS1 and IS2 into one averaged state (Figure4B,C), we find that the data are much better described by onlytwo states (Figure 4D). In this case, the artifactual, blurred,averaged state overwhelms any distinction between the MS I/IS1 and IS2 states, whereas when the simulation is performed

Figure 3. Example synthetic EFRET versus time trajectory. The gray, yellow, and blue horizontal lines denote the EFRET means used to simulate MS I/IS1, IS2, and MS II for this time series, respectively. The red horizontal line is the equilibrium-state-weighted average of the means of the MS I/IS1and IS2 states.

Figure 4. (A) Histogram of synthetic PRE complex data (gray) modeled with three non-blurred states. (B) Histogram of synthetic PRE complexdata (gray) modeled with two non-blurred states. The “Avg. State” is a weighted combination of MS I/IS1 and IS2. (C) Histograms and probabilitydistributions of the static EFRET value means generated for the synthetic PRE complex data set. (D) Plots of the model probability densities minus thenormalized histograms from panels A and B. The sum-of-squares values of these residuals are 10.03 and 4.02 for the three-state and two-statemodels, respectively, suggesting that the blurred, synthetic, PRE complex data are better represented by a two-state model.



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with an acquisition rate that is significantly faster than thetransitions of interest (2000 s−1), these states are well-resolved(Figure 5).This blurred, synthetic, three-state ensemble of EFRET versus

time trajectories was then analyzed with ebFRET. Interestingly,ebFRET overestimated the true number of kinetic states. Mostlikely this is due to the fact that the dwell-times in states MS I/IS1 and IS2 are too transient relative to the simulation’s“acquisition rate”, because ebFRET is able to accurately inferthe model parameters from the same data with a moreappropriate acquisition rate of 2000 s−1 (Figure 5). As a result,the blurred, synthetic data points are modeled by ebFRET asdistinct “states”even though these states do not actually existon the “energy landscape” of the simulated PREfM/F complex.

4. CONCLUSIONWe have established a general, theoretical framework thatintegrates equilibrium-state cryo-EM observations with dynam-ic, time-dependent smFRET observations. This framework isnot limited exclusively to cryo-EM and smFRET; any techniquethat provides static, equilibrium-state populations can beintegrated with any other technique that provides dynamic,time-dependent transition rates. The analysis reported heresuggests that states IS1 and IS2, previously identified by cryo-EM, are not detected by smFRET because their short lifetimesresult in transitions that are too fast to be characterized, giventhe acquisition rate of the EMCCD camera and the limitationsof the manner in which hidden Markov models areimplemented. Moreover, structural analyses indicate that theL1−tRNA smFRET signal is unable to yield distinguishablesignals for MS I and IS1, given the typical SBR of TIRFmicroscope-based smFRET measurements. Quantitative anal-ysis of the experimental data, based on the analytical theorypresented here, provides a possible explanation for why the

PRE complex intermediates observed by cryo-EM (i.e., IS1 andIS2) escaped detection in the smFRET studies by Fei et al.34,35

and, possibly, other groups.37,38,40 Conversely, application ofthis framework to smFRET data in which intermediate stateshave been detected, but have been difficult to assign to specificPRE complex structures,29,42 should allow researchers todetermine whether and how such intermediates correspondto IS1 and/or IS2. Perhaps more importantly, we can nowpredict the lifetimes and corresponding rates of transitions intoand out of IS1 and IS2, which are crucial for designing futurekinetic experiments. This work therefore resolves a discrepancyin the field and opens a path for performing and analyzingfuture experiments. Furthermore, we hope to use thistheoretical framework to make predictions regarding futureexperiments in which the cryo-EM and smFRET data would becollected under more comparable conditions. This theoreticalframework will also be useful in guiding future experimentalexplorations which include, for example, stabilization of IS1 orIS2 (or any other intermediates that may be ultimatelyidentified) using different tRNAs or mutant ribosomes.

■ APPENDICES

A. Four-State ModelThe theoretical results reported here are based on approachessimilar to those followed in refs 52 and 53 for analyticalcalculations of the distribution of the dwell-times of a ribosome.We use the kinetic scheme

α

α

α

α

α

αH Ioo H Ioo H Ioo1 2 3 4

21

12

32

23

43

34

(16)

where integer indices 1, 2, 3, and 4 represent a discretechemical state and αij denotes the transition probability per unittime (i.e., rate constant) for the i→j transition. If Pμ(t) is theprobability of finding the system in chemical state μ at time t,then the time evolution of these probabilities is governed by thefollowing master equations:

α α= − +P t

tP t P t

d ( )d

( ) ( )112 1 21 2 (17)

α α α α= − + +P t

tP t P t P t

d ( )d

( ) ( ) ( ) ( )212 1 23 21 2 32 3 (18)

α α α α= − + +P t

tP t P t P t

d ( )d

( ) ( ) ( ) ( )323 2 32 34 3 43 4 (19)

α α= −P t

tP t P t

d ( )d

( ) ( )434 3 43 4 (20)

Now we calculate the time-independent occupation probability,Pμeq, of each of these states by finding the equilibrium-state

solutions of eqs 17−20,α α α

α α α α α α α α α α α α=

+ + +P1

eq 43 32 21

43 32 21 12 43 32 12 23 43 12 23 34(21)


=+ + +

P2eq 12 43 32

43 32 21 12 43 32 12 23 43 12 23 34(22)


=+ + +

P3eq 12 23 43

43 32 21 12 43 32 12 23 43 12 23 34(23)

Figure 5. Histograms of the same ensemble of synthetic EFRET vs timetrajectories rendered with different time resolutions. The left-mostpeak in the gray histogram, which was used in Figure 4, is well-resolvedinto two separate peaks when the synthetic data are rendered with a100× faster acquisition rate (green histogram). The three-state modeldistribution (black curve) is that from Figure 4, which was modeledusing only the initial simulation parameters. Analysis of the faster timeresolution time series (green histogram) using ebFRET accuratelyestimated all four transition rates (k12 = 17.8 ± 0.6 s−1, k21 = 47.2 ± 1.6s−1, k23 = 6.4 ± 0.6 s−1, k32 = 1.82 ± 0.17 s−1), as well as thedistribution of EFRET means (μ1 = 0.16, σ1 = 0.035, μ2 = 0.41, σ2 =0.067, μ3 = 0.74, σ3 = 0.058) and the noise parameter (σnoise = 0.050).



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=+ + +

P4eq 12 23 34

43 32 21 12 43 32 12 23 43 12 23 34(24)

We also calculate the distribution of the time spenttransitioning from chemical states 1−4 for the first time bymodifying the original kinetic scheme into

⎯→⎯α

α

α

α αH Ioo H Ioo1 2 3 4

21

12

32

23 34

and writing the master equations according to this new scheme:

α α= − +P t

tP t P t

d ( )d

( ) ( )112 1 21 2 (25)

α α α α= − + +P t

tP t P t P t

d ( )d

( ) ( ) ( ) ( )212 1 21 23 2 32 3 (26)

α α α= − +P t

tP t P t

d ( )d

( ) ( ) ( )323 2 32 34 3 (27)

α=P t

tP t

d ( )d

( )434 3 (28)

These equations can be re-written in terms of the followingmatrix notations:

=tt

tP

MPd ( )

d( )

(29)

where P(t) is a column matrix whose elements are P1(t), P2(t),and P3(t), and

α α

α α α α

α α α

=

−

− +

− +

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥M

0

( )

0 ( )

12 21

12 21 23 32

23 34 32 (30)

Now, by introducing the Laplace transform of the probability ofkinetic states,

∫ =μ μ

∞−P s P t t( ) ( ) e dst

0 (31)

the solution of eq 29 in Laplace space is

= − −s sP I M P( ) ( ) (0)1(32)

The determinant of the matrix sI−M is a third-orderpolynomial,

− = + + +−s a s a s a s aI M( ) 13

32

21 0 (33)

where

=a 13 (34)

α α α α α= + + + +a2 12 21 23 32 34 (35)

α α α α α α α α α α

α α

= + + + +

+

a1 21 34 21 32 23 34 12 23 12 34

12 32 (36)

α α α=a0 12 23 34 (37)

We can solve eq 32 by using the initial conditions

= = = =P P P P(0) 1 and (0) (0) (0) 01 2 3 4 (38)

Suppose that the probability of transitioning from chemicalstate 1 to state 4 in the time interval between t and t + Δt isf p(t)Δt. Then,

Δ = Δf t t P t( ) ( )p4 (39)

Therefore, we find

ω= =f tP t

tP t( )

d ( )d

( )p 434 3 (40)

Taking the Laplace transform of eq 40 gives

ω = f s P s( ) ( )p34 3 (41)

Now we can calculate an analytical expression for P3(s) bysolving eq 32. Inserting this expression into eq 41 yields

α α αω ω ω

=+ + +

f ss s s

( )( )( )( )

p 12 23 34

1 2 3 (42)

where ω1, ω2, and ω3 are the solutions of the equation

ω ω ω− + − =a a a 032

21 0 (43)

Taking the inverse Laplace transform of eq 42 gives

α α αω ω ω ωα α α

ω ω ω ωα α α

ω ω ω ω

=− −

+− −

+− −

ω

ω

ω

−

−

−

f t( )( )( )

e

( )( )e

( )( )e

t

t

t

p 12 23 34

1 2 1 3

12 23 34

2 1 2 3

12 23 34

3 1 3 2

1

2

3

(44)

Now, solving for the first moment of this distribution,

∫⟨ ⟩ =∞

t tf t t( ) dp0

p(45)

gives

ααα

α αα α α

αα α

⟨ ⟩ = + + + + +⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥t

11

11

1p

12

21

23

21 32

23 34 23

32

34 34

(46)

Similarly, the second moment is

∫⟨ ⟩ = =−∞

t t f t ta a a

a( ) d

2( )p

2

0

2 p 12

0 2

02

(47)

Analogously, one can also obtain the exact formula for thedistribution of the time spent transitioning from chemical state4 to 1:

α α α

α α α

α α α

=Ω − Ω Ω − Ω

+Ω − Ω Ω − Ω

+Ω − Ω Ω − Ω

−Ω

−Ω

−Ω

f t( )( )( )

e

( )( )e

( )( )e

t

t

t

r 43 32 21

1 2 1 3

43 32 21

2 1 2 3

43 32 21

3 1 3 2

1

2

3

(48)

Here, Ω1, Ω2, and Ω3 are the solutions of the equation

Ω − Ω + Ω − =b b b 03 22 1 0 (49)

where

α α α=b0 43 32 21 (50)



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α α α α α α α α α α

α α

= + + + +

+

b1 34 21 34 23 32 21 43 32 43 21

43 23 (51)

α α α α α= + + + +b2 43 34 32 21 23 (52)

Now, solving for the first moment of this distribution gives

∫ ααα

α αα α

ααα α

⟨ ⟩ = = = + +

+ + +

∞ ⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

t tf t tbb

( ) d1

1

11

1

r0

r 1

0 43

34

32

34 23

32 21

32

23

21 21 (53)

and solving for the second moment gives

∫⟨ ⟩ = =−∞

t t f t tb b b

b( ) d

2( )r

2

0

2 r 12

0 2

02

(54)

Assuming that the experimentally observed, fractional pop-ulation of chemical state μ, χμ, represents the equilibrium-statesolutions, Pμ

eq, ratios of χ can be used to write relationshipsbetween several α’s:

αχχ

α αχχ

α αχχ

α= = =211

212 32

2

323 34

4

343

(55)

The expectation values for the time spent transitioning fromchemical states 1 to 4 (⟨tp⟩), and transitioning from chemicalstates 4 to 1 (⟨tr⟩), are assumed to be equivalent to the inversesof the experimentally observed transition rates between the twostates of a two-state model (k12 and k21). Using theseexpressions and the experimentally observed two-state rates,making substitutions with eq 55, and rearranging yields thefollowing system of equations:

α α α

χχ

χ χ χχ

= · + · + ·

= + =+ +⎛

⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

kC C

C C

11

1 1 1;

1 ,

12 121

232

43

11

22

1 2 3

4 (56)

α α α

χ χ χχ

χ χχ

= · + · + ·

=+ +

=+⎛

⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

kC C

C C

1 1 11

1;

,

213

124

23 43

34 3 2

14

4 3

2 (57)

which has fewer constraints than degrees of freedom. Toproceed, we solve the system of equations keeping one degreeof freedom independent, moving that term to the left-hand sideof the equation, and treating it as part of the constraints.Independent α12. We solve the system of equations for an

independent α12 by matrix inversion:

α

α

α

α

= →

− ·

− ·

= ⇒

= −

⎡

⎣

⎢⎢⎢⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤

⎦

⎥⎥⎥⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥k

kC

C C

CB AX X

A B

11

1

1 1 1

1

112 12

213

12

1 2

4

23

43

1 (58)

where

=· −

·−

−−

⎡⎣⎢⎢

⎤⎦⎥⎥C C C

C

C CA

1( 1 )

11

1 2 4

2

4 1

which yields the following rate constants:

α = independent12

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟21

1

212

αα

α α=

−· − − · −

C C C k kk k k C k C k k

( )( )[1 ( ) ( )]23

1 2 4 12 21 12

21 12 12 21 2 12 12 3 12 21

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟32

2

323

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟34

4

343

αα

α α=

−− · − + · −

C C C k kC k k k C k C k k

( )( )[ ( ) ( )]43

1 2 4 12 21 12

4 21 12 12 21 1 12 12 3 12 21

Independent α23. Similarly, for an independent α23,

αα

α α=

−· − − · −

C C k kk C k k C k C k k

(1 )( )[1 ( ) ( )]12

2 3 12 21 23

21 23 1 12 21 2 12 23 4 12 21

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟21

1

212

α = independent23

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟32

2

323

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟34

4

343

αα

α α=

−− · − + · −

C C k kC k C k k k C k k

(1 )( )[ ( ) 1 ( )]43

2 3 12 21 23

3 21 23 1 12 21 12 23 4 12 21

Independent α43. Finally, for an independent α43,

αα

α α=

−· − − · −

C C C k kC k C k k C k k k

( )( )[ ( ) ( )]12

4 1 3 12 21 43

4 21 43 2 12 21 1 12 43 12 21

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟21

1

212

αα

α α=

−− · − + · −

C C C k kC k C k k k k k

( )( )[ ( ) 1 ( )]23

4 1 3 12 21 43

3 21 43 2 12 21 12 43 12 21

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟32

2

323

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟34

4

343



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α = independent43

B. Three-State ModelGiven the following linear, three-state model,

α

α

α

αH Ioo H Ioo1 2 3

21

12

32

23

the time evolution of the probability, Pμ(t), will be governed by

α α= − +P t

tP t P t

d ( )d

( ) ( )112 1 21 2 (59)

α α α α= − + +P t

tP t P t P t

d ( )d

( ) ( ) ( ) ( )212 1 21 23 2 32 3 (60)

α α= −P t

tP t P t

d ( )d

( ) ( )323 2 32 3 (61)

We calculate the equilibrium-state probabilities,α α

α α α α α α=

+ +P1

eq 21 32

21 32 12 32 23 12 (62)


=+ +

P2eq 12 32

21 32 12 32 23 12 (63)


=+ +

P3eq 23 12

21 32 12 32 23 12 (64)

Then, we also calculate the distribution of the time spent by amolecule transitioning from chemical states 1 to 3. For thispurpose, we modify the scheme into

⎯→⎯α

α αH Ioo1 2 3

21

12 23

and write the master equations according to this new scheme:

α α= − +P t

tP t P t

d ( )d

( ) ( )112 1 21 2 (65)

α α α= − +P t

tP t P t

d ( )d

( ) ( ) ( )212 1 21 23 2 (66)

α=P t

tP t

d ( )d

( )323 2 (67)

As in the four-state result, we can solve these equations with theLaplace transform method to yield

α αω ω

α αω ω

=−

+−

ω ω− −f t( ) e et tp 12 23

2 1

12 23

1 2

1 2

(68)

where ω1 and ω2 are the solutions of the equation

ω ω α α α α α− + + + =( ) 0212 21 23 12 23 (69)

Now, solving for the first moment,

ααα α

⟨ ⟩ = + +⎡⎣⎢

⎤⎦⎥t

11

1p

12

21

23 23 (70)

and the second moment,

⟨ ⟩ =−

tc c

c2( )

p2 1

20

02

(71)

whereα α=c0 12 23 (72)

α α α= + +c1 12 21 23 (73)

Similarly, we can also calculate the distribution of the timespent transitioning from chemical states 3 to 1,

α α α α=

Ω − Ω+

Ω − Ω−Ω −Ωf t( ) e et tr 21 32

2 1

21 32

1 2

1 2

(74)

where Ω1 and Ω2 are the solutions of the equation

α α α α αΩ − Ω + + + =( ) 0232 23 21 32 21 (75)

In this case, we calculate the first moment,

ααα α

⟨ ⟩ = + +⎡⎣⎢

⎤⎦⎥t

11

1r

32

23

21 21 (76)

and the second moment,

⟨ ⟩ =−

td d

d2( )

r2 1

20

02

(77)

where

α α=d0 32 21 (78)

α α α= + +d1 32 23 21 (79)

Assuming that the experimentally observed, fractional pop-ulation of state μ, χμ, represents the equilibrium-state solutions,Pμeq, ratios of χ can be used to write relationships between

several α:

αχχ

α αχχ

α= =and211

212 23

3

232

(80)

The expectation values for the time spent transitioning fromchemical states 1 to 3 (⟨tp⟩), and from chemical states 3 to 1(⟨tr⟩), are assumed to be equivalent to the inverses of theexperimentally observed transition rates between the two finalstates of a two-state model, k12 and k21. Using these expressionsand experimentally observed rates, substituting eq 80, and thenrearranging yields the following system of equations:

α αχχ

⟨ ⟩ = = · + · ≡ +⎛⎝⎜⎜

⎞⎠⎟⎟t

kC C

11

1 1; 1p

12 121

321

1

3 (81)

α αχ χ

χ⟨ ⟩ = = · + · ≡

+⎛⎝⎜⎜

⎞⎠⎟⎟t

kC C

1 11

1;r

212

12 322

3 2

1 (82)

which can be solved as for the four-state model,

α

α

α

α

= → = ⇒

= =−

·

· − ·

− · + ·

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡⎣⎢

⎤⎦⎥⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

k

k

C

C

C C

kC

k

Ck k

B AX X

1

1

1

1

1

1

1

11

(1 )

11 1

11

1

12

21

1

2

12

32

12

32

1 2

122

21

212 21 (83)

to yield the three-state model rate constants,

α =−· − ·

C C k kk C k

(1 )( )(1 )12

1 2 12 21

21 1 12



10898


αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟21

1

212

αχχ

α=⎛⎝⎜⎜

⎞⎠⎟⎟23

3

232

α =−

− · − ·C C k k

C k k(1 )( )( 1 )32

1 2 12 21

2 21 12

C. smFRET SimulationsIn order to simulate PRE complex dynamics with transientintermediates, we estimated the values of EFRET for each PREcomplex conformational state. We estimated EFRET for the L1−tRNA labeling scheme by measuring the distances between theβ-carbon of the threonine at position 202 of the L1 protein ofthe 50S ribosomal subunit and the sulfur of the thiouridine atposition 8 of tRNAfMet from the atomic-resolution, moleculardynamics flexible fitting of the classes of PRE complexesdeposited in the Protein Data Bank by Agirrezabala and co-workers (Table 4).51 While the absolute accuracy of these

estimates is likely imprecise, it is reasonable to interpret therelative distances as an informative measure of the relativeEFRET values. As the distances measured ignore anyforeshortening due to the space occupied by the fluorophoreand its hydrocarbon linker, subsequent analysis compensatedby overestimating R0 = 60 Å. Notably, all the classes measuredyielded distinct values of EFRET except for classes 2 and 4A(MS I and IS1, respectively). Since the distances between thelabeling sites on the L1 protein and the P-site tRNA are 78 Å(EFRET ≈ 0.17) and 81 Å (EFRET ≈ 0.14) for classes 2 and 4A,respectively, MS I and IS1 are most likely indistinguishable,given the SBR of the TIRF-based smFRET measurements usedby Fei and co-workers.34,35 As such, we chose to group MS Iand IS1 together into state 1, while IS2 corresponded to state 2,and MS II corresponded to state 3.From these EFRET estimates, Markovian transitions along a

linear, three-state kinetic scheme were then simulated for 100state versus time trajectories. Each state versus time trajectorywas 50 s in length, and they were eventually transformed intodiscrete EFRET versus time trajectories, where each data point isthe mean EFRET value during a 50 ms time period. For eachstate versus time trajectory, the distances between the donorand acceptor fluorophores in the ith state, ri, were randomizedfor each time series with a normal distribution, (μ = ri, σ = 2Å). The EFRET of each state was then calculated as EFRET = (1 +(r/R0)

6)−1, where R0 is the Forster radius (a parameterdependent upon the identity of donor and acceptorfluorophores, as well their local environments). R0 wasrandomized for each time series within a reasonable range forthe Cy3-Cy5 FRET donor−acceptor pair with a normaldistribution, (μ = 60 Å, σ = 2 Å). The EFRET versus time

trajectory was then discretized by calculating the average valueof EFRET during sequential 50 ms long time periods. Noise wasadded to the EFRET versus time trajectories that was normallydistributed at each data point with a standard deviation of0.05a reasonable SBR for data collected on the TIRFmicroscope used in the smFRET experiments.In order to model the histogram of the simulated EFRET

versus time trajectories, we used a Gaussian mixture modelwhere each state is modeled to contribute as a normaldistribution centered at the respective mean EFRET value for thatstate, and is weighted by the equilibrium-state probability forthat state (see Appendix B). To account for the simulatedheterogeneity in the ensemble of synthetic EFRET versus timetrajectories, the mean EFRET value of each state wasmarginalized out by integrating over the joint-probabilitydistribution of the normal distribution of EFRET and a betadistribution of the mean EFRET observed in that state (Figure4C) with parameters determined by a maximum likelihoodestimate from the exact simulated EFRET means. For the “Avg.State” distribution (cf. Figure 4B), the distribution of EFRETmeans that was employed was beta distributed with a linearcombination of the parameters of the mean EFRET distributionsfrom states 1 and 2 (Figure 4C). The standard deviation usedfor the normal distribution of EFRET values for each state wastaken exactly as the standard deviation used to add noise to thesynthetic EFRET versus time trajectories.As described in section 3.5, this model of the observed EFRET

value histograms is for a temporally-resolved histogram withoutany blurring present. Performing the same simulation describedabove, but with a 0.5 ms acquisition time period yields anaccurately modeled set of EFRET versus time trajectories (Figure5). Furthermore, analyzing this data with ebFRET yields anaccurately estimated number of states, rate constants,distribution of EFRET means, and noise parameter.Analysis of the 50 ms time resolution data with ebFRET

found the most evidence for a five-state kinetic modelall ofwhich were significantly populated. Since the synthetic data wassimulated with a three-state kinetic model, the ebFRET analysisis not consistent with the original simulation. Additionally, therate constants inferred by ebFRET for a three-state model fromthe simulated data do not match the original simulationparameters. The rate constants inferred by ebFRET for thetwo-state model were kGS1→GS2 = 1.41 ± 0.05 s−1 and kGS2→GS1= 1.41 ± 0.05 s−1, but these differ from those learned from thedata of Fei and co-workers (cf. section 3.2). This suggests thatthe original simulation parameters are not consistent with theexperimental data from the smFRET study of Fei and co-workers.34,35 Most likely, the discrepancy is due to experimentaldifferences between the smFRET and cryo-EM studies thatwere compared using the general framework presented here.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: [email protected].

Present Address¶A.K.S.: Chemistry Department, University Park, PA 16802,United States.

Author Contributions∥C.D.K.-T. and A.K.S. contributed equally.

NotesThe authors declare no competing financial interest.

Table 4. Distances and Approximate FRET Efficiencies ofPREfM/F Ribosomes from Cryo-EM Structures

class r(L1−tRNA) (Å) EFRET, R0 = 60 Å

2 78 0.174A 81 0.144B 64 0.415 43 0.896 55 0.62



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mailto:[email protected]


■ ACKNOWLEDGMENTSThe authors thank Dr. Jingyi Fei for comments on themanuscript, and Ms. Melissa Thomas and Dr. Wen Li forassistance in the preparation of the figures. This work issupported in part by the U.S. Department of Energy Office ofScience Graduate Fellowship Program (DE-AC05-06OR23100) (C.D.K.-T.); the Council of Scientific andIndustrial Research (A.K.S.); the Howard Hughes MedicalInstitute and NIH-NIGMS grants (R01 GM29169 and R01GM55440) (J.F.); a Burroughs Wellcome Fund CABS Award(CABS 1004856), an NSF CAREER Award (MCB 0644262),an NIH-NIGMS grant (R01 GM084288), an American CancerSociety Research Scholar Grant (RSG GMC-117152), and aCamille Dreyfus Teacher-Scholar Award (R.L.G.); and the Prof.S. Sampath Chair Professorship and a J.C. Bose NationalFellowship (D.C).

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