Eur Biophys J (2014) 43:229–240 DOI 10.1007/s00249-014-0957-x

Original Paper

Dynamics of tRNA translocation, mRNA translocation and tRNA dissociation during ribosome translation through mRNA secondary structures Ping Xie 

Received: 26 November 2013 / Revised: 28 March 2014 / Accepted: 31 March 2014 / Published online: 22 April 2014 © European Biophysical Societies’ Association 2014

Abstract  The ribosome can translate through the duplex region or secondary structure of mRNA. Recent singlemolecule experimental data showed that downstream mRNA secondary structures have more sensitive effects on deacylated tRNA dissociation from the E site than on tRNA translocation in the 50S subunit. However, it is unclear how the downstream mRNA secondary structure can affect the tRNA dissociation from the E site, which is distant from the secondary structure. Here, based on our proposed ribosomal translocation model, we theoretically study the dynamics of tRNA translocation in the 50S subunit, mRNA translocation and tRNA dissociation, giving quantitative explanations of the single-molecule experimental data. It is shown that the effect of the downstream mRNA secondary structure on tRNA dissociation is via the effect on mRNA translocation, while the mRNA secondary structure has no effect on the rate of deacylated tRNA dissociation from the posttranslocation state. The slow mRNA translocation, which results in slow tRNA dissociation, derives from the occurrence of the futile transition, which is induced by the energy barrier from base pair unwinding to resist the forward translocation. The reduced translation rate through the mRNA secondary structure is induced by the slow mRNA translocation rather than the slow tRNA dissociation. Keywords  tRNA dissociation · tRNA translocation · mRNA translocation · Translation · Hybrid state

P. Xie (*)  Key Laboratory of Soft Matter Physics and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China e-mail: [email protected]

Introduction It has been well characterized that the ribosome, besides being able to translate through the single-stranded mRNA (Green and Noller 1997; Wintermeyer et al. 2004; Frank et al. 2007; Shoji et al. 2009; Petrov et al. 2011; Valle 2011), can also translate through the mRNA duplex or secondary structure via unwinding the duplex or structure (Takyar et al. 2005; Wen et al. 2008). Using an opticaltrapping technique, Qu et al. (2011) showed that the rate of ribosome translation through the mRNA duplex is reduced, which was explained to be induced by slow mRNA translocation as the ribosome unwinds the downstream mRNA duplex (Qu et al. 2011). Recently, by using single-molecule fluorescence energy transfer (smFRET), Chen et al. (2013a) elaborately studied the effect of downstream mRNA structures on tRNA translocation in the 50S subunit and deacylated tRNA dissociation from the E site. Unexpectedly, they found that some forms of the downstream mRNA secondary structures decrease the rates of both the tRNA translocation in the 50S subunit and deacylated tRNA dissociation from the E site, whereas others also decrease the rate of tRNA dissociation, but they have little effect on the tRNA translocation rate. Thus, it was proposed that the unwinding of the downstream mRNA secondary structures is not rigidly coupled to the tRNA translocation but is rigidly coupled to the deacylated tRNA dissociation from the E site, and the slow tRNA dissociation rather than the ribosomal translocation could be responsible for the reduced translation rate (Chen et al. 2013a). Although some suggestions for the origins of this long-range allosteric interaction between the mRNA-entry channel and the E site in the ribosome have been made (Chen et al. 2013a), the mechanism by which the downstream mRNA secondary structure affects the

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tRNA dissociation from the E site is still undetermined. It is unclear why some forms of the mRNA secondary structures have significant effects on the tRNA dissociation but have little effect on the tRNA translocation in the 50S subunit, while others have evident effects on both the tRNA translocation in the 50S subunit and deacylated tRNA dissociation from the E site. Moreover, an important, but unclear and still debated issue is, during the elongation cycle, which step determines the slower rate of translation through the duplex region than through the single-stranded region of mRNA. The purpose of this work is to address these unclear issues, providing quantitative explanations of the experimental data of Chen et al. (2013a). To this end, we modify the model of ribosome translation through the mRNA duplex proposed in the previous work (Xie 2013a), where we showed that the experimental data on the rate of ribosome translation through the mRNA duplex versus the external force to unzip the duplex can be quantitatively explained. The reduced translation rate through the mRNA duplex resulted from the occurrence of the futile transition during the forward mRNA translocation induced by the reversal ribosomal intersubunit rotation, i.e., no occurrence of mRNA translocation with the hydrolysis of a GTP by EF-G bound to the ribosome. Here, with the modified model the slow dissociation of tRNA from the E site can also be explained well, also resulting from the occurrence of the futile transition, with the downstream mRNA secondary structure having no effect on the rate of deacylated tRNA dissociation after the ribosomal complex transition to the posttranslocation state. Moreover, the puzzling experimental data, which show that some forms of the mRNA secondary structures have significant effects on the tRNA dissociation but have little effect on the tRNA translocation in the 50S subunit whereas others decrease the rates of both the tRNA translocation in the 50S subunit and deacylated tRNA dissociation from the E site, can also be explained well. Thus, our model can provide a consistent, quantitative explanation of both the experimental data of Qu et al. (2011) and those of Chen et al. (2013a). Our results indicate that it is the slow mRNA translocation rather than the slow tRNA dissociation that induces the slower rate of translation through the duplex region than through the single-stranded region of mRNA.

Methods Model of ribosomal translocation through the single‑stranded mRNA The smFRET data of Chen et al. (2011a) showed that addition EF-G to pretranslocation ribosomal complexes

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containing Cy3-labeled ribosomal protein L11 and Cy5labeled peptidyl-tRNA (L11-tRNA FRET pair) results in a rapid halt in fluctuations of FRET fluorescence, indicating that EF-G binding locks L11 and the peptidyl-tRNA in a fixed conformation, corresponding to either the classical or hybrid state, and the fixed conformation is kept for a period of time prior to translocation. This implies that after EF-G binding to the hybrid state, it takes a period of time for the hybrid state to transit to the posttranslocation state. However, the available biochemical data showed that after the pretranslocation complex bound with EF-G transits to the hybrid state, the translocation occurs very quickly, with a rate of about 35 s−1 (Savelsbergh et al. 2003). Thus, it is argued here that in the experiments of Chen et al. (2011a), there are at least two hybrid states bound with EF-G, called state H and state H1 here (see Fig. 1), and after EF-G binds, state H transits to state H1, from which the ribosomal unlocking and then the mRNA translocation occur very quickly. This argument is consistent with the cryo-EM data of Fischer et al. (2010) showing that three hybrid states are observed, called the pre3, pre4 and pre5 states; in the pre5 state, the peptidyl-tRNA moiety is located in nearly the same position of the 50S subunit as in the posttranslocation state, while in the pre3 and pre4 states, the peptidyl-tRNA moiety is located in a different position of the 50S subunit from that in the pre5 state. Thus, state H1 here could correspond to the pre5 state while state H to the pre3 or pre4 state. Consequently, in the smFRET experiments with ribosomal complexes containing Cy3-labeled ribosomal protein L11 and Cy5-labeled peptidyl-tRNA (Chen et al. 2011a, 2013a), the FRET efficiency in state H1 should be nearly equal to that in the posttranslocation state, while the FRET efficiency in state H should correspond to that in the hybrid state without EF-G. Based on the above discussion, the model for ribosomal translocation through the single-stranded mRNA under the experimental conditions of Chen et al. (2011a, 2013a) is shown in Fig. 1. Before EF-G binding, the pretranslocation ribosomal complex fluctuates spontaneously back and forth between the classical non-rotated (state C0) and hybrid (state H0) states, with the two states being in dynamic equilibrium with each other and the majority being in the hybrid state (Blanchard et al. 2004; Cornish et al. 2008; Fei et al. 2008; Moazed and Noller 1989; Valle et al. 2003; Zavialov and Ehrenberg 2003). EF-G can bind to both pretranslocation states (Chen et al. 2011a, 2013b). After EF-G binds to state H0, transition from state H to state H1 occurs. After EF-G binds to state C0, transition from state C to state H occurs first. Then, the transition from state H to state H1 occurs. In state H1, the rapid forward rotation of the 30S head relative to the 30S body could induce the ribosomal unlocking, detaching the mRNA-tRNA complex from the decoding center

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Fig. 1  Schematic representation of the model of tRNA–mRNA translocation through the single-stranded mRNA (see text for detailed description). Inside box corresponds to the transitions at saturating concentrations of EF-G.GTP. In the model, state C0, state H0, state C and state H have high L11-tRNA FRET efficiencies (0.4–0.8),

while state H1, state H2 and state POST have a low L11-tRNA FRET efficiency (0.2) in the smFRET experiments of Chen et al. (2011a, 2013a); state C0 and state C have a low L1-tRNA FRET efficiency, while other states have a high L1-tRNA FRET efficiency in the smFRET experiments of Chen et al. (2013a) and Fei et al. (2008)

(Schuwirth et al. 2005) (state H2). The subsequent rapid reverse ribosomal rotation induces mRNA translocation, with state H2 transiting to state POST. Note that this translocation model of Fig. 1 is similar to that proposed before (Xie 2014), where transitions from state H to state H1 to state H2 to state POST were simply treated as one transition from state H to state POST. It is noted that in the model in Fig. 1, state C0, state H0, state C and state H have high efficiencies (0.4–0.8) of FRET between the peptidyl-tRNA and large ribosomal protein L11, while other states have a low efficiency (0.2) of FRET between the peptidyl-tRNA and L11 in the smFRET experiments of Chen et al. (2011a, 2013a). In addition, state C0 and state C have a low efficiency of FRET between deacylated tRNA and L1, while other states have a high efficiency of FRET between deacylated tRNA and L1 in the smFRET experiments of Chen et al. (2013a) and Fei et al. (2008).

Model of ribosomal translocation through the duplex region of mRNA during the translation elongation cycle Based on the translocation model through the single-stranded mRNA shown in Fig. 1, the model for translocation through the duplex region of mRNA during the elongation cycle is shown in Fig. 2. We begin our description with the moment just after the peptidyl transfer, i.e., with the peptidyl-tRNA in the A/A state and deacylated tRNA in the P/P state (state C0, Fig. 2a). After the rapid binding of EF-G.GTP at saturating concentration, which is followed by rapid GTP hydrolysis [with a rate of about 250 s−1 (Savelsbergh et al. 2003; Wintermeyer et al. 2004)], the forward ribosomal intersubunit rotation occurs (state H, Fig. 2b). Then state H transits to state H1. In state H1, the ribosomal unlocking occurs (state H2), which is followed by the rapid reverse ribosomal rotation. As discussed in the previous work (Xie 2013a), due to the com(50S) petition between the effect of the finite affinity (EPE ) of the

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Fig. 2  Model of tRNA–mRNA translocation through the duplex region of mRNA (see text for detailed description). In the model, state C0, state H and state F1 (inside boxes) have high L11-tRNA FRET efficiencies (0.4–0.8), while other states have a low L11tRNA FRET efficiency (0.2) in the smFRET experiments of Chen et al. (2013a); state C0 and state F1 have a low L1-tRNA FRET efficiency, while other states have a high L1-tRNA FRET efficiency in

the smFRET experiments of Chen et al. (2013a). As the rate constants k3, k4 and k7 as saturating EF-G.GTP are much larger than other rate constants, the transitions with these rate constants are nearly unresolved in the smFRET experiments. Note that in the model proposed before (Chen et al. 2013a), only the effective transition (from state C0 to state H to state POST to state I) is considered

50S E site for deacylated tRNA and the 50S P site for the peptidyl-tRNA (Lill et al. 1989; Feinberg and Joseph 2001), which stabilizes the two tRNAs in the 50S E and P sites, and the effect of the free-energy change (Ebp) resulting from the unwinding of the downstream three mRNA base pairs as well as the annealing of the upstream several base pairs, which prevents the 30S subunit from moving downstream along the mRNA, the reverse ribosomal rotation would induce either an effective transition or a futile transition. The effective transition refers to the movement of the two tRNAs, which are coupled with the mRNA, from the 30S A and P sites to the 30S P and E sites while keeping the two tRNAs fixed to the 50S P and E sites (state POST, Fig. 2e); the futile transition refers to the movement of the two tRNAs from the 50S P and E sites to the 50S A and P sites, while keeping the two tRNAs fixed (50S) to the 30S A and P sites (state F1, Fig. 2f). Note that if EPE is larger (smaller) than Ebp, the occurrence probability of the

effective transition is larger (smaller) than that of the futile transition. After the ribosomal unlocking, Pi is also released rapidly and independently of the reverse ribosomal rotation (Savelsbergh et al. 2003). After transition to the non-rotated conformation (either state POST, Fig. 2e, or state F1, Fig. 2f), the mRNA channel in the 30S subunit becomes tight again, as proposed by Frank and Agrawal (2000). Note that both state POST and state F1 are bound with EF-G.GDP. In state POST (Fig. 2e), the ribosome becomes relocked, and EF-G.GDP is then released. After the binding of the aminoacyltRNA.EF-Tu.GTP ternary complex (state I, Fig. 2i) and then the peptidyl transfer, the next elongation cycle would proceed. State F1 (Fig. 2f) is in the classical non-rotated pretranslocation conformation. As EF-G.GDP facilitates the transition to and stabilizes the hybrid state of the pretranslocation ribosomal complex (Zavialov et al. 2005;

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Spiegel et al. 2007)1, state F1 would transit to state F2 (Fig.  2g), from which EF-G.GDP is released (state F3, Fig. 2h). Then, after the rapid binding of EF-G.GTP at saturating concentration, which is followed by rapid GTP hydrolysis, state F3 becomes state H1 again. It is noted here that since state POST is the posttranslocation state, while state F2 is the pretranslocation state, the rate of EF-G.GDP release from state POST should be different from that from state F2. It is considered here that only after transition to the posttranslocation state can the deacylated tRNA be dissociated from the ribosome, as indicated in Fig. 2 (green arrow and words). Moreover, in the model of Fig. 2, state C0, state H and state F1 (inside boxes) have high L11-tRNA FRET efficiencies (0.4–0.8), while other states have a low L11-tRNA FRET efficiency (0.2) in the smFRET experiments of Chen et al. (2011a, 2013a). In addition, state C0 and state F1 have a low L1-tRNA FRET efficiency, while other states have a high L1-tRNA FRET efficiency in the smFRET experiments of Chen et al. (2013a) and Fei et al. (2008). Equations for mean times of translocation and deacylated tRNA dissociation Biochemical data showed that the rate of the ribosomal unlocking is k3  = 35 s−1 (Savelsbergh et al. 2003).  Previous calculations showed that the time, T4 = 1 k4, of the reverse ribosomal rotation after the ribosomal unlocking is very short, on the time scale  of 10 ms (Xie −1 2013b). With k3  = 35 s and T4 = 1 k4  ≈ 10 ms, we obtain that the time of transitions from state H1 to state H2 to state F1 (Fig. 2) is τ ≈ 40 ms, which is nearly unresolved in the smFRET experiments (Chen et al. 2011a, 2013a; Fei et al. 2008). Thus, if the futile transition (from state H2 to state F1) occurs, the experimentally observed 1   Zavialov et al. (2005) found that EF-G.GDP catalyzes movement of tRNAs into the hybrid state. Experimental data of Spiegel et al. (2007) also showed that binding of EF-G.GDP in the presence of fusidic acid stabilizes the hybrid state. It is noted however that in the experiments of Spiegel et al. (2007), no stable EF-G.GDP binding to the pretranslocation state was detected in the absence of fusidic acid. This can be understood by considering that EF-G.GDP alone has a lower affinity than EF-G.GDPNP to the pretranslocation state, as demonstrated by the recent single-molecule data of Chen et al. (2013b), showing that EF-G.GDP binds infrequently, whereas EFG.GDPNP binds frequently to the pretranslocation state. However, Chen et al. (2013b) showed that EF-G.GDP has a much higher affinity to the pretranslocation state than to the posttranslocation state. Note also that studies by Wilden et al. (2006) have concluded that EF-G.GDP translocates a pretranslocation complex at rates similar to those of EF-G.GDPNP. As explained by Zavialov et al. (2005), this is due to the contamination of GDP with trace amounts of GTP. Indeed, Spiegel et al. (2007) found that commercially available GDP preparations contain significant levels of GTP, and they further showed that when using GDP that was purified by ion-exchange chromatography, no mRNA translocation was detected.

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decrease of L11-tRNA FRET efficiency during the elongation cycle would occur at the moment when state F2 appears. If the effective transition (from state H2 to state POST) occurs, the experimentally observed decrease in L11-tRNA FRET efficiency would occur at the moment when state POST appears. In other words, if the futile transition occurs, the experimentally observed lifetime of high L11-tRNA FRET is approximately  efficiency  (F11) calculated by TH→L = 1 k1 + 1 k2 + 1 k5, and if the effective transition occurs, the lifetime of high L11tRNA FRET is approximately calculated by  efficiency  (E11) TH→L = 1 k1 + 1 k2. Experimental data of Chen et al. (2011a) showed that if EF-G binds to the classical nonrotated pretranslocation state, after a period of time in the order of seconds, the translocation occurs rapidly via formation of the transient state H with a very short lifetime of only 86 ± 4 ms, having an L11-tRNA FRET efficiency in between the classical and posttranslocation states. Thus,  it is implied that 1 k2 is very small during the elongation cycle shown in Fig. 2. Consequently, we have the following approximation. If the futile transition occurs, the lifetime of high L11-tRNA FRET efficiency is approximately   (F11) calculated by TH→L = 1 k1 + 1 k5, and if the effective transition occurs, the lifetime of high L11-tRNA FRET  (E11) efficiency is approximately calculated by TH→L = 1 k1  . Similarly, since 1 k2 is very small during the elongation cycle in Fig. 2, we have the following results for the lifetime of low L1-tRNA FRET efficiency. If the futile transition occurs, the lifetime of low L1-tRNA FRET  efficiency  (F1) is approximately calculated by TL→H = 1 k1 + 1 k5 , and if the effective transition occurs, the lifetime of low (E1) L1-tRNA FRET efficiency is calculated by TL→H = 1 k1. Thus, both the lifetime of high L11-tRNA FRET efficiency and that of low L1-tRNA FRET efficiency can be approximately calculated by the same equation. Denoting by PE the probability of the occurrence of effective transition induced by one reverse ribosomal rotation, the mean lifetime of high L11-tRNA FRET efficiency (or low L1-tRNA FRET efficiency) during one elongationcycle  shown in Fig. 2  can be calculated by n TH→L = 1 k1 + ∞ k5, i.e., − P (1 ) E n=1

TH→L =

1 1 − PE 1 + . k1 k5 PE

(1)

Similarly, from Fig. 2 we obtain that the mean time for mRNA translocation in Fig. 2 is approximately calculated by   1 1 1 − PE 1 + + . TT = (2) k1 k5 k6 PE The mean time for deacylated tRNA dissociation from the ribosome is derived as follows. If the futile transition occurs, since at saturating EF-G.GTP k7 is much larger than k6, the time for deacylated tRNA dissociation after

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L1-tRNA FRET efficiency increases to the high value is approximately calculated by (F)

Td

=

1 1 + , k6 kd

(3)

where kd is the rate constant of deacylated tRNA dissociation from the E site of the ribosome in posttranslocation state (state POST). If the effective transition occurs, the time for deacylated tRNA dissociation after L1-tRNA FRET efficiency increases to the high value is approximately calculated by (E)

Td

=

1 . kd

(4)

Therefore, during one elongation cycle the mean time for deacylated tRNA dissociation after L1-tRNA FRET efficiency increases to the high value has the form ∞ ∞ (E)  (F)  PE (1 − PE )n + Td Td = T d (1 − PE )n, i.e., n=0

1 + Td = kd



n=1

1 1 + k6 kd



1 − PE . PE

(5)

Equations for time distributions of translocation and deacylated tRNA dissociation To study the lifetime distribution of high L11-tRNA FRET efficiency (or low L1-tRNA FRET efficiency) and the time distribution of deacylated tRNA dissociation, we study the temporal evolutions of state probability in Fig. 2. We denote by P1, P2, P3, P4, P5, P6 and P7 the probability of state C0, state H, state H1, state H2, state POST, state F1 and state F2, respectively. Considering that transition rates k2, k3 and k4 are much larger than k1 and k5 (see above), the lifetime distribution of high L11-tRNA FRET efficiency (or low L1-tRNA FRET efficiency) can be calculated by using the following equations:

dP1 (t) = −k1 P1 (t), dt

(6)

dP5 (t) = PE k1 P1 (t), dt

(7)

fH→L (t) = PE k1 P1 (t) + k5 P6 (t).

(10)

The time distribution for deacylated tRNA dissociation after L1-tRNA FRET efficiency increases to the high value can be approximately calculated as follows. Denoting by Pd the probability of the state with deacylated tRNA dissociation, the temporal evolution of Pd after L1-tRNA FRET efficiency increases to the high value can be approximately calculated by the following equations:

dP5 (t) = PE k6 P7 (t) − kd P5 (t), dt

(11)

dP7 (t) = (1 − PE )k6 P7 (t) − k6 P7 (t), dt

(12)

dPd (t) = kd P5 (t), dt

(13)

where kd is the rate constant of deacylated tRNA dissociation from the E site of the ribosome in the posttranslocation state. The initial conditions at t  = 0 are as follows: P5(0) = PE, P7(0) = 1 − PE, Pd(0) = 0. From Eqs. (11)– (13), the time distribution for deacylated tRNA dissociation after L1-tRNA FRET efficiency increases to the high value is calculated by fd (t) = dPd (t) dt, i.e.,

fd (t) = kd P5 (t).

(14)

From Eqs. (1) and (6)–(10), it is seen that the experimentally observed mean lifetime and lifetime distribution of high L11-tRNA FRET efficiency (or low L1-tRNA FRET efficiency) are only dependent on PE, k1 and k5. From Eqs. (5) and (11)–(14), it is seen that the experimentally observed mean time and time distribution for deacylated tRNA dissociation after L1-tRNA FRET efficiency increases to the high value are only dependent on PE, k6 and kd. From Eq. (2), it is seen that the experimentally observed mean time for mRNA translocation is dependent on PE, k1, k5 and k6. Results

dP6 (t) = (1 − PE )k1 P1 (t) − k5 P6 (t), dt

(8)

dP7 (t) = k5 P6 (t), dt

(9)

with the initial conditions at t  = 0 being as follows: P1(0)  = 1, P5(0)  =  P6(0)  =  P7(0)  = 0. From Eqs. (6)– (9), the lifetime distribution of high L11-tRNA FRET

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efficiency (or low L1-tRNA FRET efficiency) is calculated  by fH→L (t) = dP5 (t) dt + dP7 (t) dt, which is rewritten as

Choice of parameter values From the equations given in the above section, we see that there are five parameters that need to be determined: PE, k1, k5, k6 and kd. Before we present our calculation results, we first discuss the choice of these parameter values. The available biochemical data (Wintermeyer et al. 2004)

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Fig. 3  mRNA secondary structures used in the smFRET experiments of Chen et al. (2013a), with the 5′ base of the arginine codon defined as +1. Y, E, R, F and V indicate the tyrosine, glutamate, arginine, phenylalanine and valine codons, respectively. Base pairs are indicated in black, unpaired bases in red. Boxes and arrows indicate the mutations. During the first elongation cycle, tRNAGlu and tRNAArg

are bound to the 30S P and A sites, respectively, in the pretranslocation state (PREER), while during the second elongation cycle, tRNAArg and tRNAPhe are bound to the 30S P and A sites, respectively, in the pretranslocation state (PRERF). As the experimental data indicated (Chen et al. 2013a), the base of the stem loop is 13 mRNA nt downstream from the 5′ end of the P site just before unwinding

showed that after transition to the posttranslocation state, EF-G.GDP dissociates from the E-site with a rate of about 20 s−1. As discussed above, the rate of EF-G.GDP release from the posttranslocation state (state POST) should be different from that from the pretranslocation state (state F2). Previous analyses showed that the rate of EF-G.GDP release from state F2 is smaller than that from state POST (Xie 2013a). Moreover, recent single-molecule data showed that the lifetime of EF-G.GDP bound to the pretranslocation hybrid state is more than tenfold that bound to the posttranslocation state (Chen et al. 2013b). Thus, we take the rate of EF-G.GDP release from state F2 to be 1/ tenfold of that from state POST, i.e., we take k6  = 2 s−1 throughout our calculations in this work. In addition, we take kd  = 0.63 s−1 throughout our calculation, which is consistent with the experimental data obtained by Chen et al. (2011b). We take the effective transition probability PE to be a variable parameter. However, it is noted that PE = 1 when ribosome translates through the single-stranded mRNA because no resistance to forward translocation is present, and PE decreases as the stability of the mRNA secondary structure is enhanced. As available smFRET data showed (Cornish et al. 2008), different tRNAs bound to the A and P sites have very different effects on the rates of ribosomal intersubunit rotation. Thus, values of k1 and k5 could be different for different tRNAs. Similarly, it is expected that EF-G.GTP should have different effects from EF-G.GDP on the rate of ribosomal intersubunit rotation. Thus, the value of k1 could be different from that of k5. We will adjust

values of k1 and k5 to make the calculated results consistent with the available experimental data. Since the purpose of the present work is mainly to explain the smFRET experimental data of Chen et al. (2013a), where mRNA secondary structures mPL, mSL13, mSL-14, mSL-15, mPK, mPK-G3C G4C, mPK-U7C, mPK-SL, etc., were used, for convenience of reading, we redraw these secondary structures in Fig. 3, and some experimental data are shown again in Table 1. As the experimental data indicated (Chen et al. 2013a), the base of the stem loop is 13 mRNA nt downstream from the 5′ end of the P site just before unwinding; we will discuss our results with this. Mean times of translocation and deacylated tRNA dissociation As shown above, our model gives the results that for both the effective and futile transitions, the lifetimes of high L11-tRNA FRET efficiency and low L1-tRNA FRET efficiency during one elongation cycle can be approximately calculated by   the same equations, i.e., (F11) (F1) TH→L ≈ TL→H ≈ 1 k1 + 1 k5 for the futile transition  (E11) (E1) and TH→L ≈ TL→H ≈ 1 k1 for the effective transition. This is consistent with the smFRET data showing that the lifetimes of the 2 → 3 conversions from both L11-tRNA and L1-tRNA FRET signals have nearly the same values (Chen et al. 2013a) (see Table 1). Here, the lifetime of the 2  → 3 conversion in the smFRET experiments of Chen et al. (2013a) is defined as the lifetime of high L11-tRNA

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Table 1  Some smFRET experimental data reproduced from Supplementary Table 2 of Chen et al. (2013a) FRET pairs

2 → 3

2 → 3

3 → 4

5 → 6

6 → 7

(kH→L)

(kH→L)

(k(obs) ) d

(kH→L)

(kd

L11-tRNA

L1-tRNA

L1-tRNA

L11-tRNA

L1-tRNA

mPL (s−1) mPL mSL-13 mSL-14 mSL-15 mPK mPK-G3C G4C mPK-U7C

0.31 ± 0.02 1.00 ± 0.06 1.10 ± 0.04 1.07 ± 0.07 0.94 ± 0.08 1.00 ± 0.12 1.10 ± 0.11 0.97 ± 0.18

0.43 ± 0.02 1.00 ± 0.04 1.10 ± 0.05 1.21 ± 0.13 0.96 ± 0.04 1.21 ± 0.13 1.15 ± 0.23 0.92 ± 0.18

0.63 ± 0.04 1.00 ± 0.06 0.53 ± 0.05 0.42 ± 0.04 0.34 ± 0.03 0.94 ± 0.06 1.07 ± 0.07 0.94 ± 0.11

0.48 ± 0.02 1.00 ± 0.05 0.47 ± 0.06 0.55 ± 0.04 0.50 ± 0.04 0.91 ± 0.04 0.88 ± 0.11 0.84 ± 0.06

0.38 ± 0.01 1.00 ± 0.04 0.46 ± 0.06 0.48 ± 0.08 0.49 ± 0.02 0.39 ± 0.03 1.08 ± 0.14 0.55 ± 0.11

mPK-SL

0.97 ± 0.03

1.00 ± 0.09

0.89 ± 0.10

0.88 ± 0.07

0.46 ± 0.07

(obs)

)

  (obs) Note that the rates of 2 → 3 conversion and 3 → 4 conversion correspond to kH→L = 1 TH→L and kd = 1 Td, respectively, defined in this work during the first elongation cycle (see the  definition in the legend of Fig. 3), and the rates of 5 → 6 conversion and 6 → 7 conversion cor (obs) respond to kH→L = 1 TH→L and kd = 1 Td, respectively, defined in this work during the second elongation cycle (see the definition in the legend of Fig. 3)

FRET efficiency or of low L1-tRNA FRET efficiency during the first elongation cycle (see the definition in the legend of Fig. 3). Moreover, as the smFRET data of Fei et al. (2008) showed, both the pretranslocation hybrid state (whether bound with or not bound with EF-G) and the posttranslocation state have the same value (0.84) of high L1-tRNA FRET efficiency. It is thus deduced that in the smFRET experiments of Chen et al. (2013a), the 2 → 3 conversion should characterize the transition to either one pretranslocation hybrid state (i.e., the complete movement of two tRNAs to the 50S P and E sites) or the posttranslocation state, but should not be rigidly coupled to the transition to the posttranslocation state (i.e., the mRNA translocation). The deduction is also consistent with our model (see Fig. 2). In the following, we will present more detailed explanations of the smFRET data of Chen et al. (2013a). Using Eqs. (1), (2) and (5), we calculate the mean lifetime of high L11-tRNA FRET efficiency (or low L1-tRNA FRET efficiency), TH→L, the mean time for mRNA translocation, TT, and the mean time for deacylated tRNA dissociation after L1-tRNA FRET efficiency increases to the high value, Td, respectively. In Fig. 4, we show the   calculated results of rates , k = 1 T k = 1 TT and H→L H→L T  (obs) kd = 1 Td versus the effective transition probability PE with two different sets of values for k1 and k5, where k1 = 0.43 s−1 and k5 = 10 s−1 in Fig. 4a and k1 = 0.65 s−1 and k5 = 0.92 s−1 in Fig. 4b. Figure 4a corresponds to the results during the first elongation cycle in the experiments of Chen et al. (2013a), while Fig. 4b to the results during the second elongation cycle (see the definition in the legend of Fig. 3).

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Fig. 4  Calculated results of the mean transition rate (kH→L) from the high to low L11-tRNA FRET efficiency (or from the low to high L1-tRNA FRET efficiency), mean mRNA translocation rate (kT) and (obs) mean rate (kd ) of deacylated tRNA dissociation after L1-tRNA FRET efficiency increased to the high value during one elongation cycle as functions of the effective transition probability (PE) induced by reverse ribosomal rotation. a With k1 = 0.43 s−1 and k5 = 10 s−1, corresponding to the first elongation cycle containing 2 → 3 and 3 → 4 conversions. b With k1 = 0.65 s−1 and k5 = 0.92 s−1, corresponding to the second elongation cycle containing 5 → 6 and 6 → 7 conversions

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Table 2  Summary of the theoretical data with mPL, mSL-13, mSL-14 and mSL-15 secondary structures

First elongation cycle

Second elongation cycle

PE

kH→L

(obs) kd

mPL(s−1) mPL mSL-13 mSL-14

1 1 0.73 0.49

0.43 1 0.98 0.96

0.63 1 0.52 0.42

0.43 1 0.91 0.79

0.67 0.67 0.35 0.35

0.48 1 0.56 0.56

0.38 1 0.47 0.47

0.43 1 0.51 0.51

mSL-15

0.40

0.93

0.34

0.72

0.35

0.56

0.47

0.51

From Fig. 4a (corresponding to the first elongation (obs) cycle), we see that at PE  = 1, we have kd   = 0.63 s−1 −1 and kH→L  = 0.43 s (see Table 2), which are consistent with the smFRET data for rate constants of 3 → 4 and 2 → 3 conversions with mPL secondary structure, respectively (see Table 1). Here, the lifetime of the 3 → 4 conversion in the smFRET experiments is defined as that of high L1-tRNA FRET efficiency during the first elongation cycle. It is noted that with mPL (see Fig. 3), during the first elongation cycle, which contains 3 → 4 and 2 → 3 conversions, only a weak A:U base pair (+14) has to unwind as the ribosome translocates downstream by one codon; moreover, because there is no base pair immediately upstream of the A:U base pair, no annealing effect is present to resist the forward translocation. Thus, only a very small free energy has to be overcome as the ribosome moves downstream by one codon. This is nearly equivalent to the ribosome translation through the single-stranded mRNA. Consequently, the effective transition probability PE is expected to be nearly equal to 1. In addition, from (obs) Fig. 4a we see that at PE = 0.73, values of kd and kH→L become 0.33 s−1 and 0.42 s−1, which are about 0.52-fold and 0.98-fold those at PE  = 1, respectively (see Table 2). (obs) These values of kd and kH→L at PE = 0.73 are consistent with the smFRET data for rate constants of 3 → 4 and 2 → 3 conversions with mSL-13 secondary structure, (obs) respectively (see Table 1). At PE  = 0.49, values of kd and kH→L become 0.26 s−1 and 0.41 s−1, which are about 0.42-fold and 0.96-fold those at PE  = 1, respectively (see (obs) Table  2). These values of kd and kH→L at PE  = 0.49 are consistent with the smFRET data for rate constants of 3  → 4 and 2 → 3 conversions with mSL-14 secondary structure, respectively (see Table 1). At PE = 0.40, values (obs) of kd and kH→L become 0.21 s−1 and 0.40 s−1, which are about 0.34-fold and 0.93-fold those at PE = 1, respectively (obs) (see Table 2). These values of kd and kH→L at PE = 0.40 are consistent with the smFRET data for rate constants of 3  → 4 and 2 → 3 conversions with mSL-15 secondary structure, respectively (see Table 1). The above results indicate that during the first elongation cycle, mSL-13, mSL-14 and mSL-15 secondary structures give effective transition probability PE  = 0.73, 0.49 and 0.40, respectively (see Table 2). It is noted that during

kT

PE

kH→L

kd

kT

(obs)

the first elongation cycle, for mSL-13, mSL-14 and mSL15 secondary structures (see Fig. 3), three G:C base pairs (+12, +13, +14) need to unwind as the ribosome translocates downstream by one codon. Thus, a large free energy has to be overcome as the ribosome moves downstream by one codon, causing the effective transition probability PE to be evidently smaller than 1. Moreover, since during the first elongation cycle there are zero, one and two G:C pairs, respectively, which are immediately upstream of the three G:C base pairs (+12, +13, +14) that need to unwind, the annealing effect to resist the forward translocation for mSL-15 is larger than that for mSL-14, which is larger than that for mSL-13. These are consistent with our results showing that the effective transition probability PE for mSL-15 is smaller than that for mSL-14, which is smaller than that for mSL-13. In addition, from Fig. 4a we see that at PE  = 1 (corresponding to mPL), kT  = 0.43 s−1, while at PE  = 0.73 (corresponding to mSL-13), 0.49 (corresponding to mSL14) and 0.40 (corresponding to mSL-15), kT  = 0.39 s−1, 0.34 s−1 and 0.31 s−1, which are 0.91-, 0.79- and 0.72-fold that at PE  = 1 (corresponding to mPL), respectively (see Table 2). These results indicate that although the rate kH→L is kept nearly unchanged with the change of the mRNA secondary structure, the rate kT is changed evidently, but with the magnitude of the change in kT being smaller than (obs) that of the change in kd (see Table 2). These predicted results can be easily tested by future experiments. From Fig. 4b (corresponding to the second elongation (obs) cycle), we see that at PE = 0.67, we have kd  = 0.38 s−1 −1 and kH→L  = 0.48 s (see Table 2), which are consistent with the smFRET data for rate constants of 6 → 7 and 5 → 6 conversions with mPL secondary structure, respectively (see Table 1). Here, the lifetime of the 5 → 6 conversion in the smFRET experiments is defined as that of high L11-tRNA FRET efficiency, and the lifetime of the 6 → 7 conversion is defined as that of high L1-tRNA FRET efficiency during the second elongation cycle. Moreover, from (obs) Fig. 4b we see that at PE = 0.35, values of kd and kH→L −1 −1 become 0.18 s and 0.27 s , which are about 0.47-fold and 0.56-fold those at PE = 0.67, respectively (see Table 2). (obs) These values of kd and kH→L at PE = 0.35 are consistent with the smFRET data for rate constants of 6 → 7 and

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5  → 6 conversions, respectively, with mSL-13, mSL-14 and mSL-15 secondary structures (see Table 1). It is noted that in the experiments of Chen et al. (2013a), in the second elongation cycle the 5 → 6 conversion with mPL secondary structure corresponds to the mRNA translocation by unwinding of three A:U base pairs (+15, +16, +17) (see Fig. 3). Moreover, there is the annealing effect resulting from one A:U base pair that is immediately upstream of the three A:U base pairs (+15, +16, +17) to be unwound. Both effects would resist the forward translocation, inducing the effective transition probability to be smaller than 1, which is consistent with our theoretical value of PE  = 0.67 (see Table 2). The 5 → 6 conversion with mSL-13, mSL-14 and mSL-15 secondary structures requires unwinding of three G:C base pairs (+15, +16, +17) (see Fig. 3); moreover, there are several G:C pairs that are immediately upstream of the three G:C base pairs (+15, +16, +17) to be unwound, giving PE  = 0.35 evidently smaller than PE  = 0.67 with mPL secondary structure (see Table 2). By comparison, in the first elongation cycle the 2 → 3 conversion with mSL-15 secondary structures requires unwinding of three G:C base pairs (+12, +13, +14), and two G:C base pairs are immediately upstream of the three G:C base pairs (+12, +13, +14) that need to unwind, which is similar to the case of the 5 → 6 conversion with mSL-13, mSL-14 and mSL-15 secondary structures. Thus, the 5 → 6 conversion should give a probability of effective transition that is close to the 2 → 3 conversion, which is consistent with our results showing that the 5 → 6 conversion gives PE  = 0.35, while the 2 → 3 conversion gives PE = 0.40 (see Table 2). In addition, from Fig. 4b we see that at PE = 0.67 (corresponding to mPL), kT = 0.43 s−1, whereas at PE = 0.35 (corresponding to mSL-13, mSL-14 and mSL-15), kT = 0.22 s−1, which is about 0.51-fold that at PE = 0.67 (corresponding to mPL) (see Table 2). These predicted results indicate that the magnitude of the change in kT is nearly the same as that of the change in kH→L, which can be easily tested by future experiments. In the above, we give quantitative explanations of the experimental data with mPL and mSLs. Now, we give qualitative explanations of the experimental data with mPK and its variants. First, we consider the first elongation cycle containing 3 → 4 and 2 → 3 conversions. The smFRET data showed that the mPK structures have small effects (obs) on kd and kH→L relative to mPL during this elongation cycle (see Table 1). In our model, this corresponds to the case that the structures have small effects on the effective transition probability PE during the first elongation cycle. For mPK-G3C G4C (see Fig. 3), since no base pair is present in the immediately upstream position of the two G:C base pairs (+12, +13) and in the immediately downstream position of the two G:C base pairs (+12, +13) there are

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two unpaired bases, the isolated two G:C base pairs (+12, +13) can be easily unwound by the thermal noise, implying that there is a very small resistance to the forward translocation. This has nearly no effect on the effective (obs) transition probability PE; thus, both kd and kH→L are kept unchanged. For other mPK structures such as mPK, mPK-U7C and mPK-SL (see Fig. 3), although the translocation requires unwinding of three mRNA base pairs (+12, +13, +14), there is no annealing effect, which could give a small resistance to the forward translocation, inducing the effective transition probability PE to be slightly smaller than that for mPL. Thus, as seen from Fig. 4a, kH→L is kept (obs) unchanged, while kd is decreased slightly, consistent with the smFRET data (see Table 1). Then, we consider the second elongation cycle containing 6 → 7 and 5 → 6 conversions. For mPK-G3C G4C (see Fig. 3), the translocation requires unwinding of only two weak A:U base pairs (+16, +17), implying that there is very small resistance to the forward translocation. This has a very small effect on the effective tran(obs) and kH→L are kept sition probability PE. Thus, both kd nearly unchanged, consistent with the smFRET data (see Table  1). For mPK, mPK-U7C and mPK-SL (see Fig. 3), the translocation requires unwinding of three mRNA base pairs (+15, +16, +17); moreover, the annealing effect is present, which results from several G:C pairs immediately upstream of the three base pairs (+15, +16, +17) that need to unwind. This would make the effective transition probability PE evidently smaller than that with mPL. For exam(obs) ple, we see from Fig. 4b that at PE = 0.40, values of kd and kH→L are 0.21 s−1 and 0.32 s−1, respectively, which are about 0.55-fold and 0.67-fold those (0.38 s−1 and 0.48 s−1) at PE = 0.67 for mPL, which are close to the smFRET data (see Table 1). Time distributions of translocation and deacylated tRNA dissociation To study the time distributions fH→L (t) and fd(t), we first solve Eqs. (6)–(9) and (11)–(13) numerically by using the Runge–Kutta method to obtain the temporal evolutions of the state probabilities and then use Eqs. (10) and (14) to calculate fH→L (t) and fd(t). In Fig. 5 we show two distributions fH→L (t) with different values of PE, k1 and k5, where in Fig. 5a we take PE = 0.73, k1 = 0.43 s−1 and k5 = 10 s−1, corresponding to 2 → 3 conversion with mSL-13 secondary structure in the experiments of Chen et al. (2013a) (see above for Fig. 4a), and in Fig. 5b, we take PE = 0.67, k1 = 0.65 s−1 and k5 = 0.92 s−1, corresponding to 5 → 6 conversion with mPL secondary structure in the experiments (see above for Fig. 4b). It is seen that the distributions fH→L (t) can be approximately fitted by using the single-exponential

Eur Biophys J (2014) 43:229–240

Fig. 5  Calculated results (black dots) for lifetime distribution of the high L11-tRNA FRET efficiency (or the low L1-tRNA FRET efficiency), fH→L (t). Lines (red) are fits to the single-exponential function, fH→L (t) = C exp (−kH→L t), where C is a constant. a With PE = 0.73, k1 = 0.43 s−1 and k5 = 10 s−1, corresponding to 2 → 3 conversion with mSL-13 secondary structure in experiments of Chen et al. (2013a). b With PE  = 0.67, k1  = 0.65 s−1 and k5  = 0.92 s−1, corresponding to 5 → 6 conversion with mPL secondary structure in experiments of Chen et al. (2013a)

function, fH→L (t) = C exp (−kH→L t), where C is a constant. This is consistent with the smFRET data of Chen et al. (2013a). The time distributions for deacylated tRNA dissociation, fd(t), which correspond to cases in Fig. 5, are shown in Fig. 6, where Fig. 6a corresponds to 3 → 4 conversion with the mSL-13 secondary structure in the experiments of Chen et al. (2013a), and Fig. 6b corresponds to 6 → 7 conversion with mPL secondary structure in the experiments. From Fig.  6, it is seen that the distributions fd(t) also approximately have the single-exponential forms, also consistent with the smFRET data of Chen et al. (2013a).

Discussion As mentioned in the Introduction, the recent smFRET data showed that some forms of the downstream mRNA secondary structures decrease the rates of both the tRNA translocation in the 50S subunit and deacylated tRNA dissociation from the E site, whereas other forms also decrease

239

Fig. 6  Calculated results (black dots) for the time distribution of deacylated tRNA dissociation from the E site, fd(t), with k6  = 2 s−1 and kd = 0.63 s−1. Lines (red) are fits to the single-exponential function. a PE = 0.73, which corresponds to 3 → 4 conversion with the mSL-13 secondary structure in experiments of Chen et al. (2013a). b PE = 0.67, which corresponds to 6 → 7 conversion with the mPL secondary structure in experiments of Chen et al. (2013a)

the rate of tRNA dissociation, but they have little effect on the tRNA translocation rate (Chen et al. 2013a). Based on these experimental data, Chen et al. (2013a) argued that the unwinding of mRNA secondary structures is more closely coupled to the E-site tRNA dissociation than to the tRNA translocation; thus, the slow tRNA dissociation could be responsible for the reduced translation rate (Chen et al. 2013a; Qu et al. 2011). However, how the downstream mRNA secondary structure can influence tRNA dissociation from the E site that is distant from the secondary structure is unclear. By contrast, to explain the experimental data on the average rate of ribosome translation through the mRNA duplex versus the external force to unzip the duplex, Qu et al. (2011) proposed that the unwinding of the downstream mRNA secondary structure reduces the mRNA translocation, which is responsible for the reduced translation rate. In our previous work (Xie 2013a), based on the model similar to that in Fig. 2, we provided a quantitative explanation of the experimental data of Qu et al. (2011). In the model, the ribosome uses only one active mechanism (mechanical unwinding) rather than two active mechanisms (open-state stabilization and mechanical unwinding), as proposed by Qu et al. (2011), to unwind the mRNA duplex. The reduced translation rate through the duplex

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region is induced by the occurrence of the futile transition. Moreover, the ribosomal pausing associated with the –1 frameshifting at the slippery sequence of mRNA has also been explained well by considering the occurrence of the futile transitions (Xie 2013c). Here, in this work, based on a similar model (Fig. 2), we show that the experimental data of Chen et al. (2013a) on the effect of the downstream mRNA secondary structure on the tRNA translocation in the 50S subunit and tRNA dissociation can also be explained well. The unwinding of the downstream mRNA secondary structure is rigidly coupled to the mRNA translocation, but is not rigidly coupled to the tRNA dissociation. The slow tRNA dissociation is also induced by the occurrence of the futile transition. In summary, we show that the reduced rate of ribosome translation through the duplex region of mRNA derives from the reduced mRNA translocation rate, which in turn results from the occurrence of the futile transition rather than the reduced rate of deacylated tRNA dissociation from the E site. The secondary structure of the downstream mRNA has no effect on the deacylated tRNA dissociation from the E site after the ribosomal complex transition to the posttranslocation state. The slow dissociation of tRNA from the E site also derives from the occurrence of the futile transition; thus, the dissociation has no effect on the translation rate. Acknowledgments  This work was supported by the National Natural Science Foundation of China (grant No. 11374352).

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