In Silico Modelling and Analysis of Ribosome Kinetics and aa-trna Competition

Transcription

1 In Silico Modelling and Analysis o Ribosome Kinetics and aa-trna Competition D. Bošnački 1 T.E. Pronk 2 E.P. de Vink 3 Dept. o Biomedical Engineering, Eindhoven University o Technology Swammerdam Institute or Lie Sciences, University o Amsterdam Dept. o Mathematics and Computer Science, Eindhoven University o Technology Abstract We present a ormal analysis o ribosome kinetics using probabilistic model checking and the tool Prism. We compute dierent parameters o the model, like probabilities o translation errors and average insertion times per codon. The model predicts strong correlation to the quotient o the concentrations o the socalled cognate and near-cognate trnas, in accord with experimental indings and other studies. Using piecewise analysis o the model, we are able to give an analytical explanation o this observation. 1 Introduction The translation mechanism that synthesizes proteins based on mrna sequences is a undamental process o the living cell. Conceptually, an mrna can be seen as a string o codons, each coding or a speciic amino acid. The codons o an mrna are sequentially read by a ribosome, where each codon is translated using an amino acid speciic transer-rna (aa-trna), building one-by-one a chain o amino acids, i.e. a protein. In this setting, aa-trna can be interpreted as molecules containing a so-called anticodon, and carrying a particular amino acid. Dependent on the pairing o the codon under translation with the anticodon o the aa-trna, plus the stochastic inluences such as the changes in the conormation o the ribosome, an aa-trna, arriving by Brownian motion, docks into the ribosome and may succeed in adding its amino acid to the chain under construction. Alternatively, the aa-trna dissociates in an early or later stage o the translation. Since the seventies a vast amount o research has been devoted, unraveling the mrna translation mechanism and related issues. By now, the overall process o translation is reasonably well understood rom a qualitative perspective. The translation 1 Supported by FP6 LTR ESIGNET. 2 Funded by the BSIK project Virtual Laboratory or e-science VL-e. 3 Corresponding author, evink@win.tue.nl. 23

2 process consists o around twenty small steps, a number o them being reversible. For the model organism Escherichia coli, the average requencies o aa-trnas per cell have been collected, but regarding kinetics relatively little is known exactly. Over the past ew years, Rodnina and collaborators have made good process in capturing the time rates or various steps in the translation process or a small number o speciic codons and anticodons [14, 17, 18, 9]. Using various advanced techniques, they were able to show that the binding o codon and anticodon is crucial at a number o places or the time and probability or success o elongation. Based on these results, Viljoen and co-workers started rom the assumption that the rates ound by Rodnina et al. can be used in general, or all codon-anticodon pairs as estimates or the reaction dynamics. In [7], a complete detailed model is presented or all 64 codons and all 48 aa-trna classes or E. coli, on which extensive Monte Carlo experiments are conducted. In particular, using the model, codon insertion times and requencies o erroneous elongations are established. Given the apparently strong correlation o the ratio o so-called near-cognates vs. cognate and pseudo-cognates, and near-cognates vs. cognates, respectively, it is argued that competition o aa-trnas, rather than their availability decides both speed and idelity o codon translation. In the present paper, we propose to exploit abstraction and model checking o continuous-time Markov chains (CTMCs) with Prism [13, 10]. The abstraction conveniently reduces the number o states and classes o aa-trna to consider. The tool provides built-in perormance analysis algorithms and path chasing machinery, relieving its user rom mathematical calculations. More importantly, rom a methodological point o view, the incorporated CSL-logic [2] allows to establish quantitative results or parts o the system, e.g. or irst-passage time or a speciic state. Such piecewise analysis proves useul when explaining the relationships suggested by the data collected rom the model. Additionally, in our case, the Prism tool enjoys rather avourably response times compared to simulation. Related work The present investigation started rom the Monte-Carlo experiments o mrna translation reported in [7]. A similar stochastic model, but based on ordinary dierential equations, was developed in [11]. It treats insertion times, but no translation errors. The model o mrna translation in [8] assumes insertion rates that are directly proportional to the mrna concentrations, but assigns the same probability o translation error to all codons. Currently, there exist various applications o ormal methods to biological systems. A selection o recent papers rom model checking and process algebra includes [16, 4, 5]. More speciically pertaining to the current paper, [3] applies the Prism modelchecker to analyze stochastic models o signaling pathways. Their methodology is presented as a more eicient alternative to ordinary dierential equations models, including properties that are not o probabilistic nature. Also [10] employs Prism on various types o biological pathways, showing how the advanced eatures o the tool can be exploited to tackle large models. Organization o the paper Section 2 provides the biological background, discussing the mrna translation mechanism. Its Prism model is introduced in Section 3. In Section 4, it is explained how error probabilities are obtained rom the model and why they correlate with the near-cognate/cognate raction. This involves adequate estimates o speciic stochastic subbehaviour. Insertion times are the subject o Section 5. There too, it is illustrated how the quantitative inormation o parts o the systems is instrumental in deriving the relationship with the ratio o pseudo-cognate and near-cognates vs. cognates. 4 Acknowledgments We are grateul to Timo Breit, Christiaan Henkel, Erik Luit, 4 An appendix presents supplementary igures and data. 24

3 Jasen Markovski, and Hendrik Viljoen or ruitul discussions and constructive eedback. 2 A kinetic model o mrna translation In nature, there is a ixed correspondence o a codon and an amino acid. This is the well-known genetic code. Thus, an mrna codes or a unique protein. However, the match o a codon and the anticodon o a trna is dierent rom pair to pair. The binding inluences the speed o the actual translation. 5 Here, we give a brie overview o the translation mechanism. Our explanation is based on [17, 12]. Two main phases can be distinguished: peptidyl transer and translocation. The peptidyl transer phase runs through the ollowing steps. aa-trna arrives at the A-site o the ribosome-mrna complex by diusion. The initial binding is relatively weak. Codon recognition comprises (i) establishing contact between the anticodon o the aa-trna and the current codon in the ribosome-mrna complex, and (ii) subsequent conormational changes o the ribosome. GTPase-activation o the elongation actor EF-Tu is largely avoured in case o a strong complementary matching o the codon and anticodon. Ater GTP-hydrolysis, producing inorganic phosphate P i and GDP, the ainity o the ribosome or the aa-trna reduces. The subsequent accommodation step also depends on the it o the aa-trna. Next, the translocation phase ollows. Another GTP-hydrolysis involving elongation actor EF-G, produces GDP and P i and results in unlocking and movement o the aa-trna to the P-site o the ribosome. The latter step is preceded or ollowed by P i - release. Reconormation o the ribosome and release o EF-G moves the trna, that has transerred its amino acid to the polypeptide chain, into the E-site o the ribosome. Further rotation eventually leads to dissociation o the used trna. At present, there is little quantitative inormation regarding the translation mechanism. For E. coli, a number o speciic rates have been collected [17, 9], whereas some steps are known to be relatively rapid. The undamental assumption o [7], that we also adopt here, is that experimental data ound by Rodnina et al. or the UUU and CUC codons, extrapolate to other codons as well. However, urther assumptions are necessary to ill the overall picture. In particular, Viljoen proposes to estimate the delay due to so-called non-cognate aa-trna, that are blocking the ribosomal A-site, as 0.5ms. Also, accurate rates or the translocation phase are largely missing. Again ollowing [7], we have chosen to assign, i necessary, high rates to steps or which data is lacking. This way these steps will not be rate limiting. 3 The Prism model The abstraction o the biological model as sketched in the previous section is twoold: (i) Instead o dealing with 48 classes o aa-trna, that are identiied by the their anticodons, we use our types o aa-trna distinguished by their matching with the codon under translation. (ii) We combine various detailed steps into one transition. The irst reduction greatly simpliies the model, more clearly eliciting the essentials o the underlying process. The second abstraction is more a matter o convenience, though it helps in compactly presenting the model. For a speciic codon, we distinguish our types o aa-trna: cognate, pseudocognate, near-cognate, non-cognate. Cognate aa-trnas have an anticodon that strongly couples with the codon. The amino acid carried by the aa-trna is always the right 5 See Figure 2 and Figure 3 in the appendix. 25

4 one, according to the genetic code. The binding o the anticodon o a pseudo-cognate aa-trna or a near-cognate aa-trna is weaker, but suiciently strong to occasionally result in the addition o the amino acid to the nascent protein. In case the amino acid o the aa-trna is, accidentally, the right one or the codon, we call the aa-trna o the pseudo-cognate type. I the amino acid does not coincide with the amino acid the codon codes or, we speak in such a case o a near-cognate aa-trna. 6 The match o the codon and the anticodon can be very poor too. We reer to such aa-trna as being non-cognate or the codon. This type o aa-trna does not initiate a translation step at the ribosome. The Prism model can be interpreted as the superposition o our stochastic automata, each encoding the interaction o one o the types o aa-trna. The automata or the cognates, pseudo-cognates and near-cognates are very similar; the cognate type automaton only diers in its value o the rates rom those or pseudo-cognates and nearcognates, while the automata or pseudo-cognates and or near-cognates only dier in their arrival process. The automaton or non-cognates is rather simple. Below, we are considering average transition times and probabilities or reachability based on exponential distributions. Thereore, ollowing common practice in perormance analysis, there is no obstacle to merge two subsequent sequential transitions with ratesλandµ, say, into a combined transition o rateλµ/(λ+µ). This way, an equivalent but smaller model can be obtained. However, it is noted, that in general, such a simpliication is not compositional and should be taken with care. For the modeling o continuous-time Markov chains, Prism commands have the orm [label]guard rate : update ;. In short, rom the commands whose guards are ulilled in the current state, one command is selected proportional to its relative rate. Subsequently, the update is perormed on the state variables. So, a probabilistic choice is made among commands. Executing the selected command results in a progress o time according to the exponential distribution or the particular rate. We reer to [13, 10] or a proper introduction to the Prism modelchecker. Initially, control resides in the common start state s=1 o the Prism model with our boolean variablescogn,pseu,near andnonc set to alse. Next, an arrival process selects one o the booleans that is to be set to true. This is the initial binding o the aa-trna. The continuation depends on the type o aa-trna: cognate, pseudocognate, near-cognate or non-cognate. In act, a race is run that depends on the concentrationsccogn,cpseu,cnear andcnonc o the our types o aa-trna and a kinetic constant k1. Following Markovian semantics, the probability in the race or cogn to be set to true (the others remaining alse) is the relative concentration ccogn/(ccogn+cpseu+cnear+cnonc). // initial binding [ ] (s=1) -> k1 * c_cogn : (s =2) & (cogn =true) ; [ ] (s=1) -> k1 * c_pseu : (s =2) & (pseu =true) ; [ ] (s=1) -> k1 * c_near : (s =2) & (near =true) ; [ ] (s=1) -> k1 * c_nonc : (s =2) & (nonc =true) ; As the aa-trna, that is just arrived, may dissociate too, the reversed reaction is in the model as well. However, control does not return to the initial state directly, but, or modelchecking purposes, irst to the state s=0 representing dissociation. At the same time, the boolean that was true is reset. Here, cognates, pseudo-cognates and near-cognates are handled with the same rate k2b. Non-cognates always dissociate as captured by the separate ratek2bx. // dissociation 6 The notion o a pseudo-cognate comes natural in our modeling. However, the distinction between a pseudo-cognate and a near-cognate is non-standard. Usually, a near-cognate reers to both type o trna. 26

5 [ ] (s=2) & ( cogn pseu near ) -> k2b : (s =0) & (cogn =alse) & (pseu =alse) & (near =alse) ; [ ] (s=2) & nonc -> k2bx : (s =0) & (nonc =alse) ; An aa-trna that is not a non-cognate can continue rom states=2 in the codon recognition phase, leading to state s=3. This is a reversible step in the translation mechanism, so there are transitions rom state s=3 back to state s=2. However, the rates or cognates vs. pseudo- and near-cognates, viz. k3bc, k3bp and k3bn, dier signiicantly (see Table 1). Note that the values o the booleans do not change. // codon recognition [ ] (s=2) & ( cogn pseu near ) -> k2 : (s =3) ; [ ] (s=3) & cogn -> k3bc : (s =2) ; [ ] (s=3) & pseu -> k3bp : (s =2) ; [ ] (s=3) & near -> k3bn : (s =2) ; The next orward transition, rom state s=3 to state s=4, is a combination o detailed steps involving the processing o GTP. The transition is one-directional, again with a signiicant dierence in the ratek3c or a cognate aa-trna and the ratesk3p and k3n or pseudo-cognate and near-cognate aa-trna, that are equal. // GTPase activation, GTP hydrolysis, EF-Tu conormation change [ ] (s=3) & cogn -> k3c : (s =4) ; [ ] (s=3) & pseu -> k3p : (s =4) ; [ ] (s=3) & near -> k3n : (s =4) ; In states=4, the aa-trna can either be rejected, ater which control moves to the state s=5, or accommodates, i.e. the ribosome reconorms such that the aa-trna can hand over the amino acid it carries, so-called peptidyl transer. In the latter case, control moves to state s=6. As beore, rates or cognates and those or pseudo-cognates and near-cognates are o dierent magnitudes. // rejection [ ] (s=4) & cogn -> k4rc : (s =5) & (cogn =alse) ; [ ] (s=4) & pseu -> k4rp : (s =5) & (pseu =alse) ; [ ] (s=4) & near -> k4rn : (s =5) & (near =alse) ; // accommodation, peptidyl transer [ ] (s=4) & cogn -> k4c : (s =6) ; [ ] (s=4) & pseu -> k4p : (s =6) ; [ ] (s=4) & near -> k4n : (s =6) ; Ater a number o movements back-and-orth between state s=6 and state s=7, the binding o the EF-G complex becomes permanent. In the detailed translation mechanism a number o (mainly sequential) steps ollows, that are summarized in the Prism model by a single transition to a inal state s=8, that represents elongation o the protein in nascent with the amino acid carried by the aa-trna. The synthesis is successul i the aa-trna was either a cognate or pseudo-cognate or the codon under translation, relected by eithercogn orpseu being true. In case the aa-trna was a near-cognate (non-cognates never pass beyond state s=2), an amino acid that does not correspond to the codon in the genetic code has been inserted. In the later case, an insertion error has occurred. // EF-G binding [ ] (s=6) -> k6 : (s =7) ; [ ] (s=7) -> k7b : (s =6) ; // GTP hydrolysis, unlocking, trna movement and Pi release, // rearrangements o ribosome and EF-G, dissociation o GDP [ ] (s=7) -> k7 : (s =8) ; 27

6 A number o transitions, linking the dissociation state s=0 and the rejection state s=5 back to the start states=1, where a race o aa-trnas o the our types commences a new, and looping at the inal states=8, complete the Prism model. // no entrance, re-entrance at state 1 [ ] (s=0) -> FAST : (s =1) ; // rejection, re-entrance at state 1 [ ] (s=5) -> FAST : (s =1) ; // elongation [ ] (s=8) -> FAST : (s =8) ; Table 1 collects the rates as gathered rom the biological literature [17, 7] and used in the Prism model above. k1 140 k3c 260 k4rc 60 k6 150 k2 190 k3p, k3n 0.40 k4rp, k4rn FAST k k2b 85 k3bc 0.23 k4c k7b 140 k2bx 2000 k3bp, k3bn 80 k4p, k4n 46.1 Table 1: Rates o the Prism model. In the next two sections, we will study the Prism model described above or the analysis o the probability or insertion errors, i.e. extension o the peptidyl chain with a dierent amino acid than the codon codes or, and o the average insertion times, i.e. the average time it takes to process a codon up to elongation. 4 Insertion errors In this section we show how the model checking eatures o Prism can be used to predict the misreading requencies or individual codons. The translation o mrna into a polypeptide chain is perormed by the ribosome machinery with high precision. Experimental measurements show that on average, only one in 10,000 amino acids is added wrongly. 7 For a codon under translation, a pseudo-cognate anticodon carries precisely the amino acid that the codon codes or. Thereore, successul matching o a pseudocognate does not lead to an insertion error. In our model, the main dierence o cognates vs. pseudo-cognates and near-cognates is in the kinetics. At various stages o the peptidyl transer the rates or true cognates dier rom the others up to three orders o magnitude. Figure 1 depicts the relevant abstract automaton, derived rom the Prism model discussed above. In case a transition is labeled with two rates, the letmost number concerns the processing o a cognate aa-trna, the rightmost number that o a pseudocognate or near-cognate. In three states a probabilistic choice has to be made. The probabilistic choice in state 2 is the same or cognates, pseudo-cognates and near-cognates alike, the ones in state 3 and in state 4 diers or cognates and pseudo-cognates or near-cognates. For example, ater recognition in state 3, a cognate aa-trna will go through the hydrolysis phase leading to state 4 or a raction o the cases (computed as 260/( )), a raction being close to 1. In contrast, or a pseudo-cognate or near-cognate aa-trna this is only. Cognates will accommodate and continue to state 6 with probability 0.736, while pseudo-cognates and near-cognates will do so 7 Our indings, see Table 4, based on the kinetic rates available are slightly higher. 28

7 Figure 1: Abstract automaton or error insertion with the small probability 0.044, the constant FAST being set to 1000 in our experiments. As the transition rom state 4 to state 6 is irreversible, the rates o the remaining transitions are not o importance here. The probability or reaching state 8 in one attempt can be easily computed by Prism via the CSL-ormula P=? [ (s!=0 & s!=5) U (s=8) {(s=2) & cogn} ]. The ormula asks to establish the probability or all paths wheresis not set to0nor5, untilshave been set to8, starting rom the (unique) state satisyings=2 & cogn. We obtain ps c= 0.508, p s p = and ps n= , with ps c the probability or a cognate to end up in state 8 and elongate the peptidyl chain without going through state 0 nor state 5; ps p and ps n the analogues or pseudo- and near-cognates, respectively. Note that these values are the same or every codon. Dierent among codons are the concentrations o cognates, pseudo-cognates and near-cognates. 8 Ultimately, the requencies c, p and n o the types o aa-trna in the cell, i.e. the actual number o molecules o the kind, determine the rates or an arrival As reported in [7], the probability or an erroneous insertion, is strongly correlated with the quotient o the number o near-cognate anticodons and the number o cognate anticodons. 9 In the present setting, this correlation can be ormally derived. We have that an insertion error occurs i a near-cognate succeeds to attach its amino acid. Thereore, P(error) = P(near & elongation elongation) p = s n ( n/tot) ps c ( c/tot)+ps p ( p /tot)+ps n ( n/tot) with tot= c + p + n, and where we have used that pn s n p c s c P(elongation)=( c /tot) ps+ c ( p /tot) ps+ p ( n /tot) ps n and that ps, p ps n pc s. Note, the ability to calculate the latter probabilities, illustrating that the approach o piecewise analysis, is instrumental in obtaining the above result. n c 5 Competition and insertion times We continue the analysis o the Prism model or translation and discuss the correlation o the average insertion time or the amino acid speciied by a codon, on the one hand, and the relative abundance o pseudo-cognate and near-cognate aa-trnas, on the other hand. The insertion time o a codon is the average time it takes to elongate the protein in nascent with an amino acid. The average insertion time can be computed in Prism using the concept o rewards (also known as costs in Markov theory). Each state is assigned a value as its reward. 8 See Table 3 in the appendix. 9 See Figure 4 in the appendix. 29

8 Further, the reward o each state is weighted per unit o time. Hence, it is computed by multiplication with the average time spent in the state. The cumulative reward o a path in the chain is deined as a sum over all states in the path o such weighted rewards per state. Thus, by assigning to each state the value 1 as reward, we obtain the total average time or a given path. For example, in Prism the CSL ormular=? [ F (s=8) ] which asks to compute the expected time to reach states=8. Recall, in states=8 the amino acid is added to the polypeptide chain. So, a script modelchecking the above ormula then yields the expected insertion time per codon. 10 A little bit more ingenuity is needed to establish average exit times, or example or a cognate to pass rom states=2 to state s=8. The point is that conditional probabilities are involved. However, since dealing exponential distributions, elimination o transition in avour o adding their rates to that o the remaining ones, does the trick. Various results, some o them used below, are collected in Table 2. (The probabilities o ailure and success or the noncognates are trivial, p x = 1 and ps x seconds.) ps c p c Ts c T c = 0, with a time per ailed attempt T x = p p s p p T p s T p p n s p n T n s T n Table 2: Exit probabilities and times (in seconds) or three types o aa-trna. Failure or exit to statess=0 ors=5; success or exit to states=8. There is a visible correlation between the quotient o the number o near-cognate aa-trna and the number o cognate aa-trna. 11 In act, the average insertion time or a codon is approximately proportional to the near-cognate/cognate ratio. This can be seen as ollows. The insertion o the amino acid is completed i states=8 is reached, either or a cognate, pseudo-cognate or near-cognate. As we have seen, the probability or the latter two is negligible. Thereore, the number o cognate arrivals is decisive. With p c and pc s being the probability or a cognate to ail, i.e. exit at states=0 ors=5, or to succeed, i.e. reach o states=8, the insertion time T ins can be regarded as a geometric series. (Note the exponent i below.) Important are the numbers o arrivals o the other aa-trna types per single cognate arrival, expressed in terms o requencies. We have T ins = i=0 (p c)i ps c ((average delay or i+1 cognate arrivals)+t s c ) = i=0 (p c)i p c s (i (T c + p ) p+ n c p c s T n T p c i=0 i (p c )i p+ n c. + n T n + x T x)+t s c c c We have used that T c and Ts c p are negligible, T equals T n, and x T x is relatively small. c Note that the estimate is not accurate or small values o p + n. Nevertheless, closer inspection show that or these values the approximation remains order-preserving. Again, the results obtained or parts o the systems are pivotal in the derivation. 6 Concluding remarks In this paper, we presented a stochastic model o the translation process based presently available data o ribosome kinetics. We used the CTMC acilities o the Prism tool. Compared to simulation, our approach is computationally more reliable (independent 10 See Table 5 in the appendix. 11 See Figure 5 in the appendix. 30

9 on the number o simulations) and has aster response times (taking seconds rather then minutes or hours). More importantly, modelchecking allowed us to perorm piecewise analysis o the system, yielding better insight in the model compared to just observing the end-to-end results with a monolithic model. Based on this, we improved on earlier observations, regarding error probabilities and insertion times, by actually deriving the correlation suggested by the data. In conclusion, we have experienced aa-trna competition as a very interesting biological case study o intrinsic stochastic nature, alling in the category o the well known lambda-phage example [1]. Our model opens a new avenue or uture work on biological systems that possess intrinsically probabilistic properties. It would be interesting to apply our method to processes which, similarly to translation, require high precision, like DNA repair, charging o the trnas with amino acids, etc. Also, using our model one could check i amino acids with similar biochemical properties substitute erroneously or one another with greater probabilities than dissimilar ones. Reerences [1] A. Arkin et al. Stochastic kinetic analysis o developmental pathway biurcation in phage lambda-inected Escherichia coli cells. Genetics, 149: , [2] C. Baier et al. Approximate symbolic model checking o continuous-time Markov chains. In Proc. CONCUR 99, pages LNCS 1664, [3] M. Calder et al. Analysis o signalling pathways using continuous time Markov chains. In Transactions on Computational Systems Biology VI, pages LNBI 4220, [4] N. Chabrier and F. Fages. Symbolic model checking o biochemical networks. In Proc. CMSB 2003, pages LNCS 2602, [5] V. Danos et al. Rule-based modelling o cellular signalling. In Proc. CONCUR, pages LNCS 4703, [6] H. Dong et al. Co-variation o trna abundance and codon usage in Escherichia coli at dierent growth rates. Journal o Molecular Biology, 260: , [7] A. Fluitt et al. Ribosome kinetics and aa-trna competition determine rate and idelity o peptide synthesis. Computational Biology and Chemistry, 31: , [8] M.A. Gilchrist and A. Wagner. A model o protein translation including codon bias, nonsense errors, and ribosome recycling. Journal o Theoretical Biology, 239: , [9] K.B. Gromadski and M.V. Rodnina. Kinetic determinants o high-idelity trna discrimination on the ribosome. Molecular Cell, 13(2): , [10] J. Heath et al. Probabilistic model checking o complex biological pathways. In Proc. CMSB 2006, pages LNBI 4210, [11] A.W. Heyd and D.A. Drew. A mathematical model or elongation o a peptide chain. Bulletin o Mathematical Biology, 65: , [12] G. Karp. Cell and Molecular Biology, 5th ed. Wiley,

10 [13] M. Kwiatkowska et al. Probabilistic symbolic model cheking with Prism: a hybrid approach. Journal on Sotware Tools or Technology Transer, 6: , See alsohttp:// [14] T. Pape et al. Complete kinetic mechanism o elongation actor Tu-dependent binding o aa-trna to the A-site o E. coli. EMBO Journal, 17: , [15] D. Parker. Implementation o Symbolic Model Checking or Probabilistic Systems. PhD thesis, University o Birmingham, [16] C. Priami et al. Application o a stochastic name-passing calculus to represent ation and simulation o molecular processes. Inormation Processing Letters, 80:25 31, [17] M.V. Rodnina and W. Wintermeyer. Ribosome idelity: trna discrimination, prooreading and induced it. TRENDS in Biochemical Sciences, 26(2): , [18] A. Savelsbergh et al. An elongation actor G-induced ribosome rearrangement precedes trna mrna translocation. Molecular Cell, 11: ,

11 Appendix: suplementary igures and data Figure 2: Kinetic scheme o peptidyl transer taken rom [7]. Figure 3: Kinetic scheme o translocation taken rom [7]. 33

12 34 // translation model stochastic // constants const double ONE=1; const double FAST=1000; // trna rates const double c_cogn; const double c_pseu; const double c_near; const double c_nonc; constdoublek1 = 140; const double k2b = 85; const double k2bx=2000; constdoublek2 = 190; const double k3bc= 0.23; const double k3bp= 80; const double k3bn= 80; const double k3c= 260; const double k3p= 0.40; const double k3n= 0.40; const double k4rc= 60; const double k4rp=fast; const double k4rn=fast; const double k4c= 166.7; const double k4p= 46.1; const double k4n= 46.1; constdoublek6 = 150; constdoublek7b = 140; const double k7 = 145.8; module ribosome s : [0..8]init1; cogn : boolinit alse; pseu : boolinit alse; near : boolinit alse; nonc : boolinit alse; // initial binding [ ] (s=1)-> k1 * c_cogn: (s =2)&(cogn =true); [ ] (s=1)-> k1 * c_pseu: (s =2)&(pseu =true); [ ] (s=1)-> k1 * c_near: (s =2)&(near =true); [ ] (s=1)-> k1 * c_nonc: (s =2)&(nonc =true); [ ] (s=2)& ( cogn pseu near)-> k2b : (s =0)& (cogn =alse)& (pseu =alse)& (near =alse); [ ] (s=2)& nonc-> k2bx:(s =0)&(nonc =alse); // codon recognition [ ] (s=2)&(cogn pseu near ) -> k2 : (s =3); [ ] (s=3)&cogn -> k3bc : (s =2); [ ] (s=3)&pseu -> k3bp : (s =2); [ ] (s=3)&near -> k3bn : (s =2); // GTPase activation, GTP hydrolysis, reconormation [ ] (s=3)&cogn -> k3c : (s =4); [ ] (s=3)&pseu -> k3p : (s =4); [ ] (s=3)&near -> k3n : (s =4); // rejection [ ] (s=4)&cogn -> k4rc : (s =5)& (cogn =alse); [ ] (s=4)&pseu -> k4rp : (s =5)& (pseu =alse); [ ] (s=4)&near -> k4rn : (s =5)& (near =alse); // accommodation, peptidyl transer [ ] (s=4)&cogn -> k4c : (s =6); [ ] (s=4)&pseu -> k4p : (s =6); [ ] (s=4)&near -> k4n : (s =6); // EF-G binding [ ] (s=6)-> k6 : (s =7); [ ] (s=7)-> k7b : (s =6); // GTP hydrolysis, unlocking, // trna movement and Pi release, // rearrangements o ribosome and EF-G, // dissociation o GDP [ ] (s=7)-> k7 : (s =8); // no entrance, re-entrance at state 1 [ ] (s=0)-> FAST*FAST: (s =1); // rejection, re-entrance at state 1 [ ] (s=5)-> FAST*FAST: (s =1); // elongation [ ] (s=8)-> FAST*FAST: (s =8); endmodule rewards true : 1; endrewards

13 codon cognate pseudo- near- non- codon cognate pseudo- near- noncognate cognate cognate cognate cognate cognate UUU GUU UUC GUC UUG GUG UUA GUA UCU GCU UCC GCC UCG GCG UCA GCA UGU GGU UGC GGC UGG GGG UGA GGA UAU GAU UAC GAC UAG GAG UAA GAA CUU AUU CUC AUC CUG AUG CUA AUA CCU ACU CCC ACC CCG ACG CCA ACA CGU AGU CGC AGC CGG AGG CGA AGA CAU AAU CAC AAC CAG AAG CAA AAA Table 3: Frequencies o cognate, pseudo-cognate, near-cognate and non-cognates or E. coli as molecules per cell [6].

14 UUU CUU UUC CUC UUG e-4 CUG e-4 UUA CUA UCU e-10 CCU UCC CCC UCG CCG UCA e-4 CCA UGU e-4 CGU e-10 UGC CGC e-4 UGG CGG UGA e-4 CGA e-4 UAU e-10 CAU UAC CAC UAG CAG UAA e-4 CAA GUU e-10 AUU GUC AUC GUG e-4 AUG GUA AUA GCU e-10 ACU e-10 GCC ACC GCG e-4 ACG GCA ACA GGU e-10 AGU e-4 GGC e-4 AGC GGG e-10 AGG GGA AGA GAU AAU GAC AAC GAG e-4 AAG GAA AAA Table 4: Probabilities per codon or erroneous elongation 36

15 UUU CUU GUU AUU UUC CUC GUC AUC UUG CUG GUG AUG UUA CUA GUA AUA UCU CCU GCU ACU UCC CCC GCC ACC UCG CCG GCG ACG UCA CCA GCA ACA UGU CGU GGU AGU UGC CGC GGC AGC UGG CGG GGG AGG UGA CGA GGA AGA UAU CAU GAU AAU UAC CAC GAC AAC UAG CAG GAG AAG UAA CAA GAA AAA Table 5: Estimated average insertion time per codon in seconds 37

16 Figure 4: Correlation o n c ratio and error probabilities Figure 5: Correlation o p+ n c ratio and average insertion times 38