Extended HKH Model - CiteSeerX
Accuracy of tRNA Charging and Codon:Anticodon Recognition: Relative Importance for Cellular Stability A. Kowald and T.B.L. Kirkwood Department of Mathematical Biology National Institute for Medical Research The Ridgeway, Mill Hill, London NW7 1AA England
Running title: Stability of Cellular Translation May, 1992
Abstract Cellular homeostasis and the mechanisms which control homeostasis are important for understanding such fundamental processes as ageing and the origin of life. Several models have studied the importance of accurate protein synthesis for cellular stability, but these models have not considered the complexities of the translation process in any detail. Here we develop a new model which describes the interplay between aatRNA synthetases, the cellular pool of charged tRNA's and the process of codon:anticodon recognition. We also take the processive character of the ribosomes into account. In common with previous work, our model predicts that the cellular translation apparatus can either be stable or deteriorate progressively with time. However, because our model explicitly describes di erent subreactions of the overall translation process, we are also able to assess the relative importance of accurate tRNA charging and codon:anticodon recognition for cellular stability. It appears that the tRNA charging by the aatRNA synthetases plays the key role in controlling the long term stability of the cell. Ribosomal errors are less important because error-prone ribosomes, being processive, produce mainly inactive proteins which do not contribute to error propagation within the translation machinery
1 Introduction Translation of genetic information from DNA into proteins is accomplished by a molecular machinery which is composed largely of its own products, namely proteins. Because of this, Orgel (1963) pointed out that there exists the possibility of a cyclic propagation and ampli cation of errors which eventually could render a cell inviable. Errors in protein synthesis occur at low but detectable rates (Kirkwood et al., 1984). An error in synthesising a protein which is itself involved in protein synthesis may produce a molecule which is more error-prone than the correct, error-free form. Such a molecule may make yet more errors in the proteins it helps to synthesize. In this way, it is conceivable that errors can trigger an autocatalytic cycle in which more and more errors are produced in successive rounds of protein synthesis, until eventually the system will break down. It is clear that such an error catastrophe cannot be the unavoidable fate of cells, or lifeforms based on genetic translation would be hopelessly unstable. However, it has been suggested that error feedback, perhaps in a generalised process also involving transcription errors and DNA replication errors, might underly the cellular breakdown which is thought by many to cause the ageing of higher animals (Orgel, 1973; Kirkwood et al., 1984; Holliday, 1986). It is also an interesting question how early life forms crossed the hurdle of potential instability in the translation process. The role of protein synthesis errors in ageing remains unclear. Alterations in protein synthesis and protein turnover rates with ageing are consistent with an increase in the production of abnormal molecules, yet whether these abnormal molecules are the result of error propagation remains controversial (Holliday, 1986; Kurland, 1986; Dice & Go , 1987; Rosenberger et al., 1991). One source of confusion stems from the experimental diculties of measuring protein errors and protein feedback (Kirkwood & Holliday, 1986). Another stems from our incomplete understanding of the theoretical dynamics of error propagation. Whether or not error propagation turns out to be of practical
importance in present day biological systems, it is valuable for our understanding of molecular processes to know how errors can be ampli ed or damped in a self-renewing translation apparatus. Several theoretical models have attempted to desribe the process of error propagation in protein synthesis and to examine its potential relevance to ageing and to the origins of life. Orgel (1963) originally suggested that error ampli cation would automatically result from the feedback of mistakes, but in a correction of this view (Orgel, 1970), he presented the rst formal model. In his model he simpli ed the translation apparatus to a single equation which represented the average accuracy of one generation of molecules as a linear function of the accuracy of the preceding generation of molecules. Depending on the magnitude of the linear coecient, which describes the feedback of mistakes from one generation to the next, Orgel's model generated the important conclusion that the cell could either deteriorate to an error catastrophe or maintain a stable existence. Since 1970, various more detailed models of error propagation have been described. Ho man (1974) introduced the idea of a polypeptide 'adaptor' which combines the role of the aminoacyl-tRNA (aatRNA) synthetases and tRNA in a single conceptual entity. It was assumed that within the adaptor some residues were required for activity and some for speci city. With these assumptions Ho man calculated an equation for the average accuracy qt of the generation t adaptors in terms of the accuracy of the generation t-1 adaptors, i.e. qt = f (qt 1 ). This model showed that the stability of the translation system is not static, as in Orgel's (1970) model, but dynamic. A sudden shift of the accuracy below a certain threshold, perhaps occurring by chance, can push a previously stable system onto an unstable pathway. In formulating his model, however, Ho man made one crucial assumption which had the e ect that erroneous adaptors necessarily lose most of their activity. This does not seem biologically justi ed, and a modi ed model by Kirkwood & Holliday (1975) surmounted this limitation. Goel
& Ycas (1975) described a further model of this general type (see also Goel & Islam, 1977), but their model contained some restrictive assumptions which severely limit its realism as a general model for feedback of errors in protein synthesis (see Kirkwood, 1980). Blomberg et al. (1985) proposed a model which incorporates an error damping term that is not explicitly de ned in terms of the properties of normal and erroneous molecules, and for this reason the model cannot be used satisfactorily to analyse the stability properties of the translation system without further elaboration. For the purposes of examining the scope for error propagation within the cellular translation system and for identifying those parameters which most strongly in uence the stability of protein synthesis accuracy, the combined Ho man-Kirkwood-Holliday (HKH) model is presently the most useful. However, all the models to date, including the HKH model, fail to consider the multi-step nature of the protein synthetic pathways. Several distinct operations involving accurate molecular recognition are required for the incorporation of the right amino acid into the growing polypeptide chain, and the simple models do not allow for these. The models also do not allow for the fact that some parts of the translation machinery, in particular the ribosomes, are processive { that is, a single ribosome handles the synthesis of a complete protein {, whereas other components such as the aatRNA synthetases a ect accuracy on an amino acid by amino acid basis. In this paper we present a model that allows for the separate contributions of errors in tRNA charging and in codon:anticodon matching to the overall feedback of errors in protein synthesis. We also allow for the processive action of ribosomes. This permits us to identify more speci cally the parameters of the translation process which are the most critical for preventing error feedback. Since the new model is an extension of the HKH model we begin by summarizing the main features of the HKH model.
2 The Ho man-Kirkwood-Holliday Model In the HKH model each generation of the translation apparatus is de ned to consist of a set of polypeptide adaptors, which combine the functions of all components of the protein synthesizing machinery. There is a subset of m amino acid residues, whose correct insertion is vital for activity and a further non-overlapping subset of n residues which is important for speci city. Incorrect insertion of one or more of the m residues results in total loss of activity, while incorrect insertion of one or more of the n residues results in a total loss of speci city. Errors in any remaining residues are assumed to be of no importance. The attention of the HKH model is restricted to only those adaptors which contain no errors in the m sites, as inactive adaptors do not contribute to translation. The adaptors in generation t consist of two types, a fraction qtn 1 which contain no errors in the n sites, and a fraction 1 qtn 1 which have lost their speci city. qt 1 is here the average accuracy of the generation t 1 adaptors. Furthermore it is assumed that a normal adaptor makes correct insertions S times faster than each of the 1 possible incorrect insertions, being the number of di erent amino acids and S being the speci city of a normal adaptor. The residual activity of an erroneous adaptor, i.e. the activity given as a fraction of the activity of a normal adaptor, was termed R. With these assumptions the average accuracy qt determined as the rate of making correct insertions divided by the total rate of making insertions, both correct and incorrect, can shown to be n (S + 1)R] + (S + 1)R qt = qqnt 1[(S t 1 S + 1)(1 R) + (S + 1)R
(1)
In a plot of qt against qt 1 this equation generates a sigmoid shape (Figure 1). If the curve crosses qt = qt 1 , as shown for curves (b) and (c) in Figure 1, a region of stability exists, and the accuracy will normally converge to qstable .
However, if the curve does not cross the line qt = qt 1 , or if some intrinsic or extrinsic factor reduces the accuracy below qthreshold, then the accuracy decreases steadily from generation to generation and an error catastrophe is unavoidable. For given values of and n, the values of S and R determine stability. Increasing S and/or reducing R tend to generate a more stable system. This is understandable since increasing the speci city of normal adaptors reduces their intrinsic error rate, while reducing the residual activity of erroneous adaptors e ectively damps the feedback of errors. Figure 2 illustrates, for = 20 and n = 1, 2, 5, 10 the dependence of stability or instability on the values of R and S. The curves represent boundaries between stable and unstable regions of the R-S parameter space. The curves rise with increasing S while the value of S is low, but approach a plateau for values of S above 103 104 . This is also true for n = 1 but in this case R is about 10 when the curve reaches the plateau. These values of S correspond to error rates for normal adaptors (( 1)=(S + 1)) in the range of about 0.002 - 0.02. Since the error rates for protein synthesis, as measured experimentally (Kirkwood et al., 1984), are lower than this, a realistic value for S will put the system in the right-hand region of Figure 2 where the dominant parameter which determines stability is R. Thus the HKH model can yield useful insights into the stability of translation and it has several advantages in comparison to other models, yet it has still the problem of treating the translation process as a single reaction. Furthermore, like the other models it does not take account of the fact that the ribosome is processive.
3 The Extended-Ho man-Kirkwood-Holliday (EHKH) Model The new model considers error propagation in a system which consists of the synthetases which charge tRNA's with amino acids, the pools of charged tRNA's (aatRNA's), and
the codon:anticodon selection at the ribosomes (Figure 3). As in the HKH model it is assumed that the translation machinery consists of discrete and distinguishable generations.
3.1 aatRNA Synthetases The synthetases are a close analogue of the adaptors in the HKH model. The aatRNA synthetases select amino acids from the available pool, combine them with tRNA's and are then released for further charging reactions. It is assumed that there is a subset mS of amino acid residues in each synthetase whose correct insertion is vital for activity and a further non-overlapping subset nS of residues vital for normal speci city. An error in one or more of the nS sites destroys speci city of the synthetase. Furthermore it is supposed that a normal synthetase makes correct reactions SPS times faster than each of the 1 possible incorrect reactions, being the number of different amino acids and SPS being the speci city of a normal synthetase. The residual activity of an erroneous synthetase i.e. the activity given as a fraction of the activity of a normal synthetase, is termed RAS . With these assumptions the rates for normal and erroneous synthetases of correctly and erroneously charging tRNA's are as given in Table 1 and may be explained as follows. kS denotes the rate at which a correct synthetase correctly charges tRNA's. SPS is de ned so that the average rate at which a correct synthetase selects one of the possible wrong amino acids is kS =SPS . If the number of di erent amino acids in the system is there are ( 1) wrong amino acids, so the total rate of producing erroneously charged tRNA's by correct synthetases is ( 1 )kS =SPS . For erroneous synthetases it is assumed that charging with the correct amino acid occurs at the same rate as charging with each of the incorrect amino acids and that the total rate of charging by erroneous synthetases is RAS times the total rate of charging by correct synthetases. The expressions given in Table 1 for the charging rates by erroneous
synthetases were calculated from these requirements.
3.2 Ribosomes Ribosomes facilitate correct codon:anticodon matching in protein synthesis and help to maintain the correct reading frame by translocation of the peptidyl-tRNA complex in relation to the mRNA at each elongation step. In this model frameshift errors are not considered and the attention is focused on the role of the ribosome in promoting correct codon:anticodon matching. As for the synthetases it is assumed that a subset nR of amino acid residues in the ribosomal proteins is required to be correctly inserted for normal speci city, and that a subset mR of residues is required to be correctly inserted for normal activity. An erroneous ribosome is one which has the subset mR of residues correctly inserted (hence it is not inactive), but which contains an error in one or more of the nR subset of residues. Since the ribosomes do not themselves select aatRNA's, but instead facilitate the exploitation of the intrinsic speci city of codon:anticodon matching provided by WatsonCrick base pairing, it is supposed that the speci city of an erroneous ribosome is not abolished, but has some value SP'R which is di erent from the speci city SPR of a normal ribosome. It will be assumed that in general SPR0 < SPR , although it is conceivable that some errors in ribosomal proteins may lead to increased speci city, and the model allows both possibilities to be considered. Speci city of a ribosome is de ned in the same way as speci city of a synthetase, in that if kR is the rate of correct codon:anticodon matching on a normal ribosome then each of the possible incorrect matchings occurs at a rate kR =SPR . The number of possible codon:anticodon matchings needs to be considered quite carefully since wobble in the third codon position allows tRNA's with di erent anticodons to match a single codon. This number could be set to be the number of di erent tRNA's which is intermediate between the number of amino acids and the number of
sense codons. However, for simplicity this number is taken to be , the number of di erent amino acids, since the behaviour of the model is not strongly dependent on this assumption. In e ect, it is assumed that there is a single species of tRNA for each amino acid and that codons coding for the same amino acid are equivalent. It is also supposed that the total activity of an erroneous ribosome may di er from that of a correct ribosome by the factor RAR which may be less than or greater than one, and which will be termed the residual activity of an erroneous ribosome. From these assumptions Table 2 can be derived which gives the rates of correct and incorrect anticodon selection for normal and erroneous ribosomes. Finally, when applying these rates, the processiveness of ribosomes has to be taken into account. This means that, unlike the synthetases, when polymerizing a protein a given ribosome performs a multi-step reaction.
3.3 The Combined Model For a combined model the rates given in Table 1 have to be used to determine the fractions of correctly and incorrectly charged tRNA's, and then the rates given in Table 2 have to be used to determine the fractions of correctly and incorrectly inserted amino acids into proteins, and in particular into the synthetases and ribosomes of the currently produced generation of the translation apparatus. To do this the fractions of correct and erroneous synthetases and ribosomes in the preceding generation, which is the generation currently carrying out the translation process, have to be taken into account. De ne Sc;t, Se;t , Si;t to be the fractions of correct, erroneous and inactive synthetases, respectively, in generation t, and Rc;t, Re;t , Ri;t to be similar fractions for the ribosomes (Sc;t + Se;t + Si;t = 1, Rc;t + Re;t + Ri;t = 1). De ne Tt to be the fraction of correctly charged tRNA's in generation t. Then since the fraction of correctly charged tRNA is the amount of correctly charged
tRNA divided by the total amount of charged tRNA, both correct and incorrect, the rates in Table 1 can be used to obtain:
Tt =
SPS + 1) Sc;t 1 kS + Se;t 1 kS RAS (SP S 1 ) + Se;t 1 kS RAS (1 + 1 ) Sc;t 1 kS (1 + SP SPS S
from which follows:
Tt =
Sc;t 1 SPSSP+S 1 + Se;t 1 RA S Sc;t 1 + Se;t 1 RAS
(2)
Note that the rate term kS cancels out in this expression, since the fraction of correctly charged tRNA depends only on the relative values of the rates of charging in Table 1. The contribution to accuracy resulting from the operation of the ribosomes is now considered. Because the accuracy of codon:anticodon matching may be de ned as the rate of correct matching divided by the total rate of matching, both correct and incorrect, accuracies qc and qe exist for correct and erroneous ribosomes respectively. These are given by: qc =
SPR SPR + 1
(3)
qe =
SPR0 SPR0 + 1
(4)
Note that the accuracies qc, qe are for single steps of protein synthesis and recall that the ribosomes are processive. For a protein containing m residues whose correct insertion is required for activity and n residues whose correct insertion is required for speci city, the fractions Pc, Pe, Pi of correct, erroneous and inactive molecules, respectively, made by a processive ribosome with a single-step accuracy of q, and assuming for the time being that all tRNA's are correctly charged, will be:
Pc = qn+m
(5)
Pe = qm (1 qn)
(6)
Pi = 1 qm
(7)
If a mixture of correct and erroneous ribosomes exists, working at di erent rates, as given in Table 2, these equations must be modi ed. In this case Pc , Pe and Pi also depend on the amount of correct and erroneous ribosomes and their activities. n+m + Re;t 1 qn+m RAR e Pc = Rc;t 1 Rqc + R RA c;t 1 e;t 1 R m n m n R q (1 q ) + R e (1 qe ) RAR Pe = c;t 1 c R c + Re;t 1 qRA c;t 1 e;t 1 R m + Re;t 1 (1 qem ) RAR Pi = Rc;t 1 (1 R qc ) + R RA c;t
1
e;t
1
R
(8) (9) (10)
Finally the fact that not all tRNA's are correctly charged has to be taken into account to allow for errors to arise in the nal protein either by mischarging of tRNA or by incorrect codon:anticodon matching. (There is also a small possibility that errors occurring at both these steps may cancel out.) To take account of both kinds of errors together the rates of correct anticodon selection by normal and erroneous ribosomes (see Table 2) are denoted A1 and A2 , respectively, and the corresponding rates of incorrect anticodon selection are designated A3 and A4 . Table 3 gives the rates of correct and incorrect amino acid incorporation allowing for the fraction Tt of correctly charged tRNA. Now equations (3) and (4) can be modi ed using Table 3 to give expressions for the single step accuracy of amino acid incorporation of correct and erroneous ribosomes allowing for the fraction of correctly charged tRNA's. These are: R 1) Tt + 1 qc = (SPSP R+ 1
(11)
0 R 1) Tt + 1 qe = (SPSP 0 R+ 1
(12)
Substitution of these expressions in equations (8), (9) and (10) gives the fraction of correct, erroneous and inactive protein produced by the translation apparatus. By substituting n = nS , m = mS and Pc = Sc;t , Pe = Se;t , Pi = Si;t recurrence relations are obtained from which the fractions of correct, erroneous and inactive synthetases can be calculated as a function of time. Similarly, by substituting n = nR ,
m = mR and Pc = Rc;t , Pe = Re;t , Pi = Ri;t the changes in the fractions of the di erent types of ribosomes can be calculated . This provides the required model.
4 Simulation Methods and Results The EHKH model is more complex than the HKH model (the parameters are listed in the appendix) and its behaviour cannot be summarized in a simple plot of the accuracy of the generation t translation apparatus against the generation t 1 (as in Figure 1). One way to examine the behaviour of the model is to plot the temporal development of the system, as is done in Figure 4, which gives two examples of simulations in one of which the translation process is stable (a) and in the other unstable (b). Of greater interest, however, is to study which combination of parameters give rise to stable and unstable behaviours. This has been done by using planes in parameter space and by plotting the borderline between stability and instability, as is done in Figures 5 to 8. Points in the parameter space to the left of the curves represent an unstable system, whereas points in the parameter space to the right of the curve represent a stable system. The criterion used to de ne stability was that the level of correctly charged tRNA must not fell below 1:1=. According to equation (2) this level (Tt ) can never fall below 1=, so a combination of parameters is regarded as stable when the resulting level of correctly charged tRNA remains 10% or more above the absolute minimum. The absolute minimum itself (1=) cannot be used as a criterion
for stability, because it is approached asymptotically and this would lead to the wrong conclusion that the whole parameter space is stable. Figure 2 showed a plane of parameter space for the HKH model, and illustrated the stability of the HKH model depending on the residual activity of erroneous adaptors (R) and the speci city (S ) of correct adaptors for a varying number of speci city sites (n). If the number of di erent amino acids, , is regarded as a xed parameter, then stability in the HKH model is described completely by this diagram. Figure 5 represents the equivalent parameter space for the synthetase component of the new model. The residual activity of the synthetases (RAS ) has replaced R and the speci city of the synthetases (SPS ) has replaced S . By setting the residual speci city of the erroneous ribosome SP'R equal to SPR , error propagation in the ribosomal part of the model is inhibited and only tRNA charging can contribute to an error accumulation. There is a close similarity between Figures 2 and 5 indicating that the adaptors of the HKH model are similar in behaviour to the synthetases in the EHKH model, as might be expected. Figure 6 examines the in uence of the residual activity of erroneous ribosomes, RAR , on the stability of the ribosomal part of the model. Because a completely inhibited error propagation in the part of the model which deals with the charging of tRNA would prevent instability entirely, a value of 0:05 for RAS was used in this simulation. Figure 7 shows the e ect of varying n in the SP'R , SPR parameter plane. With increasing n the border of stability moves towards higher values of SPR . This is especially pronounced for values of SP'R between 102 {104 . For n = 15 this e ect can no longer be compensated for by an increase of SPR . Figure 8, displaying the SPS , SPR plane, examines the part of the model which describes the charging of tRNA as well as the ribosomal part. Increasing the residual activity of erroneous synthetases reduces the area of stability in the upper right corner.
5 Discussion This paper describes a new model of error propagation in the translation machinery of a cell. In comparison to previous models it shows several advantages. It is the rst model which splits the translation process into several connected reactions, namely the charging of tRNA and the actual protein polymerization step at the ribosomes, which itself depends on the level of correctly charged tRNA. This connection enables the model to assess the importance of these subprocesses for the overall error propagation. The fact that the model deals explicitly with the processiveness of the ribosome is an important point, because the processiveness of the ribosome dampens the ribosomal contribution to the error feed back to a large degree. An erroneous ribosome polymerizes all amino acids of a new protein (ribosome or synthetase) but if it has a decreased speci city it will normally introduce so many errors in this particular protein, that the vast amount of protein synthesized will be inactive. This prevents an ampli cation of the error level. The parameter plane shown in Figure 5 is especially interesting for two reasons. First the close similarity to Figure 2 shows that the HKH model corresponds most closely to the tRNA charging part of the new model. Second, Figure 5 clearly shows the in uence of the residual activity of the erroneous aatRNA synthetases (RAS ). This is the only parameter of the EHKH model which can force the system into stability or instability independently of the other parameters. Processiveness of the ribosomes explains the shape of the border line in Figure 7 when varying the residual speci city of erroneous ribosomes SP'R . As can be seen from equations (5 - 7) for very low values of SP'R the main products of erroneous ribosomes are inactive proteins which do not contribute to the error propagation. If SP'R is enhanced to a similar level as the speci city of the correct ribosome, SPR , error propagation in the ribosomal part of the translation process is limited because the erroneous
ribosomes mainly produce correct proteins. But as can be derived from equation 6 the production of erroneous ribosomes reaches a maximum when the accuracy is equal to
pn m=(n + m). If SP'
R
is chosen so that qe is equal to this critical accuracy a high
speci city SPR is needed to stabilize the system. This explains the sensitivity of the model for values of SP'R between 102 {104 . The results of Figure 6 are a direct consequence of this phenomenon. For extremely large or small values of SP'R stability is almost independent of the residual activity of erroneous ribosomes, whereas for some medium range values of SP'R the system depends strongly on the activity of erroneous ribosomes. Notice that, although in Figure 6 and 7 parameters exclusively of the ribosomal part are investigated, error propagation in the tRNA charging process is not totally inhibited (RAS > 0 ) because this would automatically force the system to be stable. Figure 8 shows that in a system with error propagation in both reaction pathways, tRNA charging and codon:anticodon recognition, the speci cities of ribosomes and aatRNA synthetases are of equal importance. A region of stability can be expanded either by decreasing SPS or by decreasing SPR . For a given set of parameters there are minimum requirements for SPS and SPR which cannot be compensated for by increasing the second parameter. The EHKH model can also be used to consider the relevance of \error coupling" to translational stability. Error coupling is an expression which describes the idea that the occurrence of a missense error can raise the probability of a subsequent reading frame error (Kurland, 1986). If translation continues it will very soon encounter a nonsense codon which terminates the translation process. If this really happens in vivo and the ribosome is especially designed to couple missense and frame shift errors, then this raises the question about the function of this process. There are two feasible explanations. Firstly, it can be a mechanism to prevent error propagation by coupling a missense error to the termination of translation. Or
it can be an energy saving strategy, because it prevents the cell for making erroneous and inactive proteins whose synthesis is costly and whose subsequent degradation also requires energy. The results of the EHKH model suggest that there is no need for a further error damping mechanism in the processive ribosomal part of the translation reaction, because the most error prone part is the charging of tRNA. If error coupling occurs, the EHKH model suggests that this is mainly for energy saving reasons. The new model is derived from the HKH model and a comparison with its predecessor shows that it veri es the main results of the HKH model. The system can be in a stable or unstable state depending on the chosen parameters, but this is a dynamical behaviour in that a stable cell can be pushed onto the unstable path by a sudden change of the fractions of the constituents of the translation machinery (aatRNAc ; Rc ; Re ; Sc ; Se ). Furthermore the speci city and the residual activity of the erroneous aatRNA synthetases are the important factors which determine the fate of a cell. The most important single factor is the residual activity of the aatRNA synthetases which has to be below a certain threshold for the cell to be stable. This threshold is modulated by the speci cities SPR and SPS . Planes of parameter space are a convenient way to examine several parameters in one diagram. Whereas the HKH model is described by just one of those planes, several are needed to account for the increased complexity of the new model. This increased complexity is mainly due to the increased number of parameters. However, because each of these parameters has a biological meaning this is advantageous as it allows a more precise identi cation of the parts of the information transfer machinery which are important for error propagation. The model represents an important improvement but leaves room for future work. The model calculates the level of correct, erroneous and inactive enzymes in terms of fractions of the overall protein content but does not consider absolute enzyme concen-
trations. If an error catastrophe occurs within a cell, not only the fractions of correctly synthesized proteins decreases but also the absolute concentrations of all proteins are likely to decline, which eventually will endanger the viability of the cell. Furthermore it is presumed that all constituents of the generation t 1 (enzymes and aatRNA's) are replaced by the molecules of the next generation. This simpli cation leads to problems if the total synthetic capacity of the cell declines. Under those conditions the complete replacement of the members of one generation by the members of the following generation becomes less realistic. Consequently the development of a continuous timescale model which will use absolute concentrations is currently in progress. We thank Dr R.F. Rosenberger for helpful discussions.
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Appendix Parameters and Variables of the EHKH Model The EHKH model contains several variables and parameters which are summarized in the following table. For parameters, the default value which is used for simulations is given. For variables the possible range is given.
nS
=
10
Number of speci city sites in the aatRNA synthetases.
mS
=
10
Number of activity sites in the aatRNA synthetases.
kS
:
SPS
= 10000 Speci city of correct aatRNA synthetases.
Activity of correct aatRNA synthetases.
RAS =
0.05
Residual activity of erroneous aatRNA synthetases.
Sc;t
=
0-1
Fraction of correct aatRNA synthetases in generation t.
Se;t
=
0-1
Fraction of erroneous aatRNA synthetases in generation t.
Si;t
=
0-1
Fraction of inactive aatRNA synthetases in generation t.
nR
=
10
Number of speci city sites in the ribosome.
mR
=
10
Number of activity sites in the ribosome.
kR
:
Activity of correct ribosomes.
SPR = 10000 Speci city of the correct ribosome. SP0R =
1
Residual speci city of the erroneous ribosome.
RAR =
1
Residual activity of erroneous ribosome.
Rc;t
=
0-1
Fraction of correct ribosomes in generation t.
Re;t
=
0-1
Fraction of erroneous ribosomes in generation t.
Ri;t
=
0-1
Fraction of inactive ribosomes in generation t.
Tt
=
0-1
Fraction of correctly charged tRNA's.
=
20
Number of di erent amino acids.
qc
=
0-1
Accuracy of correct ribosomes.
qe
=
0-1
Accuracy of erroneous ribosomes.
Figure 1
1.0 0.9
Decreasing R
0.8 0.7
0.5 0.4
q(threshold)
0.3 0.2
(c)
(b)
(a)
q(stable)
q(t)
0.6
0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 q(t-1)
Figure 2
1.O
R
n=1 O.8
n=2 O.6
O.4 n=5 O.2 n=10 O.O 1O O
1O 1
1O 2
1O 3
1O 4
1O 5
1O 6
1O 7
S
Figure 3
mRNA
Ribosomes
Amino acids + tRNA’s
Synthetases
aatRNA’s
a)
1
0
Fraction
Figure 4
0 Time in Generations
100
b) 1 0
Fraction 0 Time in Generations
100
Figure 5
1.O
RAs
n=1 O.8
n=2 O.6
O.4
n=5
O.2
n=10 O.O 1O O
1O 1
1O 2
1O 3
1O 4
1O 5
1O 6
1O 7
SPs
Figure 6
1.0 (a)
(b)
(c)
RAR
0.8 0.6 (d) 0.4 0.2 0.0 10 3
10 4
10 5 SPR
10 6
10 7
Figure 7
10 7 10 6
(a)
(b)
(c)
SP’R
10 5 10 4 10 3 10 2 10 1 10 0 10 0
10 1
10 2
10 3
10 4 SPR
10 5
10 6
10 7
Figure 8
10 7 10 6
(a)
(b)
10 2
10 3
(c)
SPS
10 5 10 4 10 3 10 2 10 1 10 0 10 0
10 1
10 4 SPR
10 5
10 6
10 7
Table 1: Rates of correct and incorrect charging of tRNA's by normal and erroneous synthetases (see text for de nition of parameters).
Normal synthetases Erroneous synthetases Rate of correct tRNA charging Rate of incorrect tRNA charging
kS
S + RAS kS SPSP S
kS ( 1) SP S
S + ( 1)RAS kS SPSP S
1
1
Table 2: Rates of correct and incorrect anticodon selection on normal and erroneous ribosomes (see text for de nitions of parameters).
Normal ribosome Rate of correct anticodon selection Rate of incorrect anticodon selection
Erroneous ribosome
kR
R (SPR + RAR kR SP SPR (SPR +
( 1) SPkRR
R + 1 ( 1) RAR kR SPRSP(SP R +
0
0
1) 1)
0
1)
amino acid insertion
Rate of incorrect
amino acid insertion
Rate of correct
A3 Tt + A1 (1 Tt ) + A3 (1 Tt )
A1 Tt + A3 1 T1t
Normal ribosomes
2 1
A4 Tt + A2 (1 Tt ) + A4 (1 Tt )
A2 Tt + A4 1 T1t
Erroneous ribosomes
2 1
Rates of correct and incorrect amino acid incorporation on normal and erroneous ribosomes allowing for the fraction of correctly charged tRNA.
Table 3:
Figure Legends Figure 1: q-diagram representation of the Ho man-Kirkwood-Holliday model. Curves (a), (b) and (c) have been obtained by calculating Equation 1 for R=0.5, R=0.1 and R=0.01 respectively and show the stabilizing e ect of decreasing R (residual activity of erroneous adaptors). For curve (c) the values of qstable and qthreshold have been indicated. For further details see text. For all three curves S =1000, =20 and n=5 have been used. Figure 2: The diagram shows the parameter space of the HKH model. The region left to a speci c curve represents an area in parameter space which results in an unstable system while parameter values to the right result in a stable system. = 20. Figure 3: Reactions described by the EHKH model. Dashed lines represent the action of enzymes and solid lines show the conversion of one molecular species in another.
Figure 4: The time course of Tt (||), Rc = Sc (- - - -), Re = Se ( ) and Ri = Si( ) is shown for a stable (a) and an unstable (b) simulation of the EHKH model. Figure 5: The RAS {SPS plane in the parameter space of the EHKH model. The region to the left of a speci c curve represents instability, the region to the right of a curve stability.
=20, m=10, RAR =1, SPR=10000, SP'R =10000. Figure 6: The RAR {SPR plane in the parameter space of the EHKH model. The region to the left of a speci c curve represents an area of instability, the region to the right stability. Parameter settings are: =20, m=10, n=15, RAS = 0:05, SPS = 10000. Di erent curves are generated by varying SPR0 as follows: (a) SPR0 = 10000, (b) SPR0 = 50, (c)
SPR0 = 3000 and (d) SPR0 = 1000. Figure 7: The SPR0 {SPR plane in the parameter space of the EHKH model. The area to the left of the curves represents instability, the region to the right stability. Parameters used are: =20, m=10, RAS = 0:05, RAR = 1, SPS = 10000. Di erent curves are generated by varying n as follows: (a) n=3, (b) n=10 and (c) n=15. Figure 8: The SPS {SPR plane in the parameter space of the EHKH model. An almost perfect symmetry is generated in this diagram which shows stability depending on the speci cities of the aatRNA-synthetases (SPS ) and the ribosomes(SPR ). Parameters used are:
=20, m=10, n=10, RAR = 1, SPR0 = 1. Di erent curves are generated by varying RAS as follows: (a) RAS = 0:001 (b) RAS = 0:01 and (c) RAS = 0:1.
Running title: Stability of Cellular Translation May, 1992
Abstract Cellular homeostasis and the mechanisms which control homeostasis are important for understanding such fundamental processes as ageing and the origin of life. Several models have studied the importance of accurate protein synthesis for cellular stability, but these models have not considered the complexities of the translation process in any detail. Here we develop a new model which describes the interplay between aatRNA synthetases, the cellular pool of charged tRNA's and the process of codon:anticodon recognition. We also take the processive character of the ribosomes into account. In common with previous work, our model predicts that the cellular translation apparatus can either be stable or deteriorate progressively with time. However, because our model explicitly describes di erent subreactions of the overall translation process, we are also able to assess the relative importance of accurate tRNA charging and codon:anticodon recognition for cellular stability. It appears that the tRNA charging by the aatRNA synthetases plays the key role in controlling the long term stability of the cell. Ribosomal errors are less important because error-prone ribosomes, being processive, produce mainly inactive proteins which do not contribute to error propagation within the translation machinery
1 Introduction Translation of genetic information from DNA into proteins is accomplished by a molecular machinery which is composed largely of its own products, namely proteins. Because of this, Orgel (1963) pointed out that there exists the possibility of a cyclic propagation and ampli cation of errors which eventually could render a cell inviable. Errors in protein synthesis occur at low but detectable rates (Kirkwood et al., 1984). An error in synthesising a protein which is itself involved in protein synthesis may produce a molecule which is more error-prone than the correct, error-free form. Such a molecule may make yet more errors in the proteins it helps to synthesize. In this way, it is conceivable that errors can trigger an autocatalytic cycle in which more and more errors are produced in successive rounds of protein synthesis, until eventually the system will break down. It is clear that such an error catastrophe cannot be the unavoidable fate of cells, or lifeforms based on genetic translation would be hopelessly unstable. However, it has been suggested that error feedback, perhaps in a generalised process also involving transcription errors and DNA replication errors, might underly the cellular breakdown which is thought by many to cause the ageing of higher animals (Orgel, 1973; Kirkwood et al., 1984; Holliday, 1986). It is also an interesting question how early life forms crossed the hurdle of potential instability in the translation process. The role of protein synthesis errors in ageing remains unclear. Alterations in protein synthesis and protein turnover rates with ageing are consistent with an increase in the production of abnormal molecules, yet whether these abnormal molecules are the result of error propagation remains controversial (Holliday, 1986; Kurland, 1986; Dice & Go , 1987; Rosenberger et al., 1991). One source of confusion stems from the experimental diculties of measuring protein errors and protein feedback (Kirkwood & Holliday, 1986). Another stems from our incomplete understanding of the theoretical dynamics of error propagation. Whether or not error propagation turns out to be of practical
importance in present day biological systems, it is valuable for our understanding of molecular processes to know how errors can be ampli ed or damped in a self-renewing translation apparatus. Several theoretical models have attempted to desribe the process of error propagation in protein synthesis and to examine its potential relevance to ageing and to the origins of life. Orgel (1963) originally suggested that error ampli cation would automatically result from the feedback of mistakes, but in a correction of this view (Orgel, 1970), he presented the rst formal model. In his model he simpli ed the translation apparatus to a single equation which represented the average accuracy of one generation of molecules as a linear function of the accuracy of the preceding generation of molecules. Depending on the magnitude of the linear coecient, which describes the feedback of mistakes from one generation to the next, Orgel's model generated the important conclusion that the cell could either deteriorate to an error catastrophe or maintain a stable existence. Since 1970, various more detailed models of error propagation have been described. Ho man (1974) introduced the idea of a polypeptide 'adaptor' which combines the role of the aminoacyl-tRNA (aatRNA) synthetases and tRNA in a single conceptual entity. It was assumed that within the adaptor some residues were required for activity and some for speci city. With these assumptions Ho man calculated an equation for the average accuracy qt of the generation t adaptors in terms of the accuracy of the generation t-1 adaptors, i.e. qt = f (qt 1 ). This model showed that the stability of the translation system is not static, as in Orgel's (1970) model, but dynamic. A sudden shift of the accuracy below a certain threshold, perhaps occurring by chance, can push a previously stable system onto an unstable pathway. In formulating his model, however, Ho man made one crucial assumption which had the e ect that erroneous adaptors necessarily lose most of their activity. This does not seem biologically justi ed, and a modi ed model by Kirkwood & Holliday (1975) surmounted this limitation. Goel
& Ycas (1975) described a further model of this general type (see also Goel & Islam, 1977), but their model contained some restrictive assumptions which severely limit its realism as a general model for feedback of errors in protein synthesis (see Kirkwood, 1980). Blomberg et al. (1985) proposed a model which incorporates an error damping term that is not explicitly de ned in terms of the properties of normal and erroneous molecules, and for this reason the model cannot be used satisfactorily to analyse the stability properties of the translation system without further elaboration. For the purposes of examining the scope for error propagation within the cellular translation system and for identifying those parameters which most strongly in uence the stability of protein synthesis accuracy, the combined Ho man-Kirkwood-Holliday (HKH) model is presently the most useful. However, all the models to date, including the HKH model, fail to consider the multi-step nature of the protein synthetic pathways. Several distinct operations involving accurate molecular recognition are required for the incorporation of the right amino acid into the growing polypeptide chain, and the simple models do not allow for these. The models also do not allow for the fact that some parts of the translation machinery, in particular the ribosomes, are processive { that is, a single ribosome handles the synthesis of a complete protein {, whereas other components such as the aatRNA synthetases a ect accuracy on an amino acid by amino acid basis. In this paper we present a model that allows for the separate contributions of errors in tRNA charging and in codon:anticodon matching to the overall feedback of errors in protein synthesis. We also allow for the processive action of ribosomes. This permits us to identify more speci cally the parameters of the translation process which are the most critical for preventing error feedback. Since the new model is an extension of the HKH model we begin by summarizing the main features of the HKH model.
2 The Ho man-Kirkwood-Holliday Model In the HKH model each generation of the translation apparatus is de ned to consist of a set of polypeptide adaptors, which combine the functions of all components of the protein synthesizing machinery. There is a subset of m amino acid residues, whose correct insertion is vital for activity and a further non-overlapping subset of n residues which is important for speci city. Incorrect insertion of one or more of the m residues results in total loss of activity, while incorrect insertion of one or more of the n residues results in a total loss of speci city. Errors in any remaining residues are assumed to be of no importance. The attention of the HKH model is restricted to only those adaptors which contain no errors in the m sites, as inactive adaptors do not contribute to translation. The adaptors in generation t consist of two types, a fraction qtn 1 which contain no errors in the n sites, and a fraction 1 qtn 1 which have lost their speci city. qt 1 is here the average accuracy of the generation t 1 adaptors. Furthermore it is assumed that a normal adaptor makes correct insertions S times faster than each of the 1 possible incorrect insertions, being the number of di erent amino acids and S being the speci city of a normal adaptor. The residual activity of an erroneous adaptor, i.e. the activity given as a fraction of the activity of a normal adaptor, was termed R. With these assumptions the average accuracy qt determined as the rate of making correct insertions divided by the total rate of making insertions, both correct and incorrect, can shown to be n (S + 1)R] + (S + 1)R qt = qqnt 1[(S t 1 S + 1)(1 R) + (S + 1)R
(1)
In a plot of qt against qt 1 this equation generates a sigmoid shape (Figure 1). If the curve crosses qt = qt 1 , as shown for curves (b) and (c) in Figure 1, a region of stability exists, and the accuracy will normally converge to qstable .
However, if the curve does not cross the line qt = qt 1 , or if some intrinsic or extrinsic factor reduces the accuracy below qthreshold, then the accuracy decreases steadily from generation to generation and an error catastrophe is unavoidable. For given values of and n, the values of S and R determine stability. Increasing S and/or reducing R tend to generate a more stable system. This is understandable since increasing the speci city of normal adaptors reduces their intrinsic error rate, while reducing the residual activity of erroneous adaptors e ectively damps the feedback of errors. Figure 2 illustrates, for = 20 and n = 1, 2, 5, 10 the dependence of stability or instability on the values of R and S. The curves represent boundaries between stable and unstable regions of the R-S parameter space. The curves rise with increasing S while the value of S is low, but approach a plateau for values of S above 103 104 . This is also true for n = 1 but in this case R is about 10 when the curve reaches the plateau. These values of S correspond to error rates for normal adaptors (( 1)=(S + 1)) in the range of about 0.002 - 0.02. Since the error rates for protein synthesis, as measured experimentally (Kirkwood et al., 1984), are lower than this, a realistic value for S will put the system in the right-hand region of Figure 2 where the dominant parameter which determines stability is R. Thus the HKH model can yield useful insights into the stability of translation and it has several advantages in comparison to other models, yet it has still the problem of treating the translation process as a single reaction. Furthermore, like the other models it does not take account of the fact that the ribosome is processive.
3 The Extended-Ho man-Kirkwood-Holliday (EHKH) Model The new model considers error propagation in a system which consists of the synthetases which charge tRNA's with amino acids, the pools of charged tRNA's (aatRNA's), and
the codon:anticodon selection at the ribosomes (Figure 3). As in the HKH model it is assumed that the translation machinery consists of discrete and distinguishable generations.
3.1 aatRNA Synthetases The synthetases are a close analogue of the adaptors in the HKH model. The aatRNA synthetases select amino acids from the available pool, combine them with tRNA's and are then released for further charging reactions. It is assumed that there is a subset mS of amino acid residues in each synthetase whose correct insertion is vital for activity and a further non-overlapping subset nS of residues vital for normal speci city. An error in one or more of the nS sites destroys speci city of the synthetase. Furthermore it is supposed that a normal synthetase makes correct reactions SPS times faster than each of the 1 possible incorrect reactions, being the number of different amino acids and SPS being the speci city of a normal synthetase. The residual activity of an erroneous synthetase i.e. the activity given as a fraction of the activity of a normal synthetase, is termed RAS . With these assumptions the rates for normal and erroneous synthetases of correctly and erroneously charging tRNA's are as given in Table 1 and may be explained as follows. kS denotes the rate at which a correct synthetase correctly charges tRNA's. SPS is de ned so that the average rate at which a correct synthetase selects one of the possible wrong amino acids is kS =SPS . If the number of di erent amino acids in the system is there are ( 1) wrong amino acids, so the total rate of producing erroneously charged tRNA's by correct synthetases is ( 1 )kS =SPS . For erroneous synthetases it is assumed that charging with the correct amino acid occurs at the same rate as charging with each of the incorrect amino acids and that the total rate of charging by erroneous synthetases is RAS times the total rate of charging by correct synthetases. The expressions given in Table 1 for the charging rates by erroneous
synthetases were calculated from these requirements.
3.2 Ribosomes Ribosomes facilitate correct codon:anticodon matching in protein synthesis and help to maintain the correct reading frame by translocation of the peptidyl-tRNA complex in relation to the mRNA at each elongation step. In this model frameshift errors are not considered and the attention is focused on the role of the ribosome in promoting correct codon:anticodon matching. As for the synthetases it is assumed that a subset nR of amino acid residues in the ribosomal proteins is required to be correctly inserted for normal speci city, and that a subset mR of residues is required to be correctly inserted for normal activity. An erroneous ribosome is one which has the subset mR of residues correctly inserted (hence it is not inactive), but which contains an error in one or more of the nR subset of residues. Since the ribosomes do not themselves select aatRNA's, but instead facilitate the exploitation of the intrinsic speci city of codon:anticodon matching provided by WatsonCrick base pairing, it is supposed that the speci city of an erroneous ribosome is not abolished, but has some value SP'R which is di erent from the speci city SPR of a normal ribosome. It will be assumed that in general SPR0 < SPR , although it is conceivable that some errors in ribosomal proteins may lead to increased speci city, and the model allows both possibilities to be considered. Speci city of a ribosome is de ned in the same way as speci city of a synthetase, in that if kR is the rate of correct codon:anticodon matching on a normal ribosome then each of the possible incorrect matchings occurs at a rate kR =SPR . The number of possible codon:anticodon matchings needs to be considered quite carefully since wobble in the third codon position allows tRNA's with di erent anticodons to match a single codon. This number could be set to be the number of di erent tRNA's which is intermediate between the number of amino acids and the number of
sense codons. However, for simplicity this number is taken to be , the number of di erent amino acids, since the behaviour of the model is not strongly dependent on this assumption. In e ect, it is assumed that there is a single species of tRNA for each amino acid and that codons coding for the same amino acid are equivalent. It is also supposed that the total activity of an erroneous ribosome may di er from that of a correct ribosome by the factor RAR which may be less than or greater than one, and which will be termed the residual activity of an erroneous ribosome. From these assumptions Table 2 can be derived which gives the rates of correct and incorrect anticodon selection for normal and erroneous ribosomes. Finally, when applying these rates, the processiveness of ribosomes has to be taken into account. This means that, unlike the synthetases, when polymerizing a protein a given ribosome performs a multi-step reaction.
3.3 The Combined Model For a combined model the rates given in Table 1 have to be used to determine the fractions of correctly and incorrectly charged tRNA's, and then the rates given in Table 2 have to be used to determine the fractions of correctly and incorrectly inserted amino acids into proteins, and in particular into the synthetases and ribosomes of the currently produced generation of the translation apparatus. To do this the fractions of correct and erroneous synthetases and ribosomes in the preceding generation, which is the generation currently carrying out the translation process, have to be taken into account. De ne Sc;t, Se;t , Si;t to be the fractions of correct, erroneous and inactive synthetases, respectively, in generation t, and Rc;t, Re;t , Ri;t to be similar fractions for the ribosomes (Sc;t + Se;t + Si;t = 1, Rc;t + Re;t + Ri;t = 1). De ne Tt to be the fraction of correctly charged tRNA's in generation t. Then since the fraction of correctly charged tRNA is the amount of correctly charged
tRNA divided by the total amount of charged tRNA, both correct and incorrect, the rates in Table 1 can be used to obtain:
Tt =
SPS + 1) Sc;t 1 kS + Se;t 1 kS RAS (SP S 1 ) + Se;t 1 kS RAS (1 + 1 ) Sc;t 1 kS (1 + SP SPS S
from which follows:
Tt =
Sc;t 1 SPSSP+S 1 + Se;t 1 RA S Sc;t 1 + Se;t 1 RAS
(2)
Note that the rate term kS cancels out in this expression, since the fraction of correctly charged tRNA depends only on the relative values of the rates of charging in Table 1. The contribution to accuracy resulting from the operation of the ribosomes is now considered. Because the accuracy of codon:anticodon matching may be de ned as the rate of correct matching divided by the total rate of matching, both correct and incorrect, accuracies qc and qe exist for correct and erroneous ribosomes respectively. These are given by: qc =
SPR SPR + 1
(3)
qe =
SPR0 SPR0 + 1
(4)
Note that the accuracies qc, qe are for single steps of protein synthesis and recall that the ribosomes are processive. For a protein containing m residues whose correct insertion is required for activity and n residues whose correct insertion is required for speci city, the fractions Pc, Pe, Pi of correct, erroneous and inactive molecules, respectively, made by a processive ribosome with a single-step accuracy of q, and assuming for the time being that all tRNA's are correctly charged, will be:
Pc = qn+m
(5)
Pe = qm (1 qn)
(6)
Pi = 1 qm
(7)
If a mixture of correct and erroneous ribosomes exists, working at di erent rates, as given in Table 2, these equations must be modi ed. In this case Pc , Pe and Pi also depend on the amount of correct and erroneous ribosomes and their activities. n+m + Re;t 1 qn+m RAR e Pc = Rc;t 1 Rqc + R RA c;t 1 e;t 1 R m n m n R q (1 q ) + R e (1 qe ) RAR Pe = c;t 1 c R c + Re;t 1 qRA c;t 1 e;t 1 R m + Re;t 1 (1 qem ) RAR Pi = Rc;t 1 (1 R qc ) + R RA c;t
1
e;t
1
R
(8) (9) (10)
Finally the fact that not all tRNA's are correctly charged has to be taken into account to allow for errors to arise in the nal protein either by mischarging of tRNA or by incorrect codon:anticodon matching. (There is also a small possibility that errors occurring at both these steps may cancel out.) To take account of both kinds of errors together the rates of correct anticodon selection by normal and erroneous ribosomes (see Table 2) are denoted A1 and A2 , respectively, and the corresponding rates of incorrect anticodon selection are designated A3 and A4 . Table 3 gives the rates of correct and incorrect amino acid incorporation allowing for the fraction Tt of correctly charged tRNA. Now equations (3) and (4) can be modi ed using Table 3 to give expressions for the single step accuracy of amino acid incorporation of correct and erroneous ribosomes allowing for the fraction of correctly charged tRNA's. These are: R 1) Tt + 1 qc = (SPSP R+ 1
(11)
0 R 1) Tt + 1 qe = (SPSP 0 R+ 1
(12)
Substitution of these expressions in equations (8), (9) and (10) gives the fraction of correct, erroneous and inactive protein produced by the translation apparatus. By substituting n = nS , m = mS and Pc = Sc;t , Pe = Se;t , Pi = Si;t recurrence relations are obtained from which the fractions of correct, erroneous and inactive synthetases can be calculated as a function of time. Similarly, by substituting n = nR ,
m = mR and Pc = Rc;t , Pe = Re;t , Pi = Ri;t the changes in the fractions of the di erent types of ribosomes can be calculated . This provides the required model.
4 Simulation Methods and Results The EHKH model is more complex than the HKH model (the parameters are listed in the appendix) and its behaviour cannot be summarized in a simple plot of the accuracy of the generation t translation apparatus against the generation t 1 (as in Figure 1). One way to examine the behaviour of the model is to plot the temporal development of the system, as is done in Figure 4, which gives two examples of simulations in one of which the translation process is stable (a) and in the other unstable (b). Of greater interest, however, is to study which combination of parameters give rise to stable and unstable behaviours. This has been done by using planes in parameter space and by plotting the borderline between stability and instability, as is done in Figures 5 to 8. Points in the parameter space to the left of the curves represent an unstable system, whereas points in the parameter space to the right of the curve represent a stable system. The criterion used to de ne stability was that the level of correctly charged tRNA must not fell below 1:1=. According to equation (2) this level (Tt ) can never fall below 1=, so a combination of parameters is regarded as stable when the resulting level of correctly charged tRNA remains 10% or more above the absolute minimum. The absolute minimum itself (1=) cannot be used as a criterion
for stability, because it is approached asymptotically and this would lead to the wrong conclusion that the whole parameter space is stable. Figure 2 showed a plane of parameter space for the HKH model, and illustrated the stability of the HKH model depending on the residual activity of erroneous adaptors (R) and the speci city (S ) of correct adaptors for a varying number of speci city sites (n). If the number of di erent amino acids, , is regarded as a xed parameter, then stability in the HKH model is described completely by this diagram. Figure 5 represents the equivalent parameter space for the synthetase component of the new model. The residual activity of the synthetases (RAS ) has replaced R and the speci city of the synthetases (SPS ) has replaced S . By setting the residual speci city of the erroneous ribosome SP'R equal to SPR , error propagation in the ribosomal part of the model is inhibited and only tRNA charging can contribute to an error accumulation. There is a close similarity between Figures 2 and 5 indicating that the adaptors of the HKH model are similar in behaviour to the synthetases in the EHKH model, as might be expected. Figure 6 examines the in uence of the residual activity of erroneous ribosomes, RAR , on the stability of the ribosomal part of the model. Because a completely inhibited error propagation in the part of the model which deals with the charging of tRNA would prevent instability entirely, a value of 0:05 for RAS was used in this simulation. Figure 7 shows the e ect of varying n in the SP'R , SPR parameter plane. With increasing n the border of stability moves towards higher values of SPR . This is especially pronounced for values of SP'R between 102 {104 . For n = 15 this e ect can no longer be compensated for by an increase of SPR . Figure 8, displaying the SPS , SPR plane, examines the part of the model which describes the charging of tRNA as well as the ribosomal part. Increasing the residual activity of erroneous synthetases reduces the area of stability in the upper right corner.
5 Discussion This paper describes a new model of error propagation in the translation machinery of a cell. In comparison to previous models it shows several advantages. It is the rst model which splits the translation process into several connected reactions, namely the charging of tRNA and the actual protein polymerization step at the ribosomes, which itself depends on the level of correctly charged tRNA. This connection enables the model to assess the importance of these subprocesses for the overall error propagation. The fact that the model deals explicitly with the processiveness of the ribosome is an important point, because the processiveness of the ribosome dampens the ribosomal contribution to the error feed back to a large degree. An erroneous ribosome polymerizes all amino acids of a new protein (ribosome or synthetase) but if it has a decreased speci city it will normally introduce so many errors in this particular protein, that the vast amount of protein synthesized will be inactive. This prevents an ampli cation of the error level. The parameter plane shown in Figure 5 is especially interesting for two reasons. First the close similarity to Figure 2 shows that the HKH model corresponds most closely to the tRNA charging part of the new model. Second, Figure 5 clearly shows the in uence of the residual activity of the erroneous aatRNA synthetases (RAS ). This is the only parameter of the EHKH model which can force the system into stability or instability independently of the other parameters. Processiveness of the ribosomes explains the shape of the border line in Figure 7 when varying the residual speci city of erroneous ribosomes SP'R . As can be seen from equations (5 - 7) for very low values of SP'R the main products of erroneous ribosomes are inactive proteins which do not contribute to the error propagation. If SP'R is enhanced to a similar level as the speci city of the correct ribosome, SPR , error propagation in the ribosomal part of the translation process is limited because the erroneous
ribosomes mainly produce correct proteins. But as can be derived from equation 6 the production of erroneous ribosomes reaches a maximum when the accuracy is equal to
pn m=(n + m). If SP'
R
is chosen so that qe is equal to this critical accuracy a high
speci city SPR is needed to stabilize the system. This explains the sensitivity of the model for values of SP'R between 102 {104 . The results of Figure 6 are a direct consequence of this phenomenon. For extremely large or small values of SP'R stability is almost independent of the residual activity of erroneous ribosomes, whereas for some medium range values of SP'R the system depends strongly on the activity of erroneous ribosomes. Notice that, although in Figure 6 and 7 parameters exclusively of the ribosomal part are investigated, error propagation in the tRNA charging process is not totally inhibited (RAS > 0 ) because this would automatically force the system to be stable. Figure 8 shows that in a system with error propagation in both reaction pathways, tRNA charging and codon:anticodon recognition, the speci cities of ribosomes and aatRNA synthetases are of equal importance. A region of stability can be expanded either by decreasing SPS or by decreasing SPR . For a given set of parameters there are minimum requirements for SPS and SPR which cannot be compensated for by increasing the second parameter. The EHKH model can also be used to consider the relevance of \error coupling" to translational stability. Error coupling is an expression which describes the idea that the occurrence of a missense error can raise the probability of a subsequent reading frame error (Kurland, 1986). If translation continues it will very soon encounter a nonsense codon which terminates the translation process. If this really happens in vivo and the ribosome is especially designed to couple missense and frame shift errors, then this raises the question about the function of this process. There are two feasible explanations. Firstly, it can be a mechanism to prevent error propagation by coupling a missense error to the termination of translation. Or
it can be an energy saving strategy, because it prevents the cell for making erroneous and inactive proteins whose synthesis is costly and whose subsequent degradation also requires energy. The results of the EHKH model suggest that there is no need for a further error damping mechanism in the processive ribosomal part of the translation reaction, because the most error prone part is the charging of tRNA. If error coupling occurs, the EHKH model suggests that this is mainly for energy saving reasons. The new model is derived from the HKH model and a comparison with its predecessor shows that it veri es the main results of the HKH model. The system can be in a stable or unstable state depending on the chosen parameters, but this is a dynamical behaviour in that a stable cell can be pushed onto the unstable path by a sudden change of the fractions of the constituents of the translation machinery (aatRNAc ; Rc ; Re ; Sc ; Se ). Furthermore the speci city and the residual activity of the erroneous aatRNA synthetases are the important factors which determine the fate of a cell. The most important single factor is the residual activity of the aatRNA synthetases which has to be below a certain threshold for the cell to be stable. This threshold is modulated by the speci cities SPR and SPS . Planes of parameter space are a convenient way to examine several parameters in one diagram. Whereas the HKH model is described by just one of those planes, several are needed to account for the increased complexity of the new model. This increased complexity is mainly due to the increased number of parameters. However, because each of these parameters has a biological meaning this is advantageous as it allows a more precise identi cation of the parts of the information transfer machinery which are important for error propagation. The model represents an important improvement but leaves room for future work. The model calculates the level of correct, erroneous and inactive enzymes in terms of fractions of the overall protein content but does not consider absolute enzyme concen-
trations. If an error catastrophe occurs within a cell, not only the fractions of correctly synthesized proteins decreases but also the absolute concentrations of all proteins are likely to decline, which eventually will endanger the viability of the cell. Furthermore it is presumed that all constituents of the generation t 1 (enzymes and aatRNA's) are replaced by the molecules of the next generation. This simpli cation leads to problems if the total synthetic capacity of the cell declines. Under those conditions the complete replacement of the members of one generation by the members of the following generation becomes less realistic. Consequently the development of a continuous timescale model which will use absolute concentrations is currently in progress. We thank Dr R.F. Rosenberger for helpful discussions.
References Blomberg, C., Johansson, J. & Liljenstro m, H. (1985). Error propagation in E.
coli protein synthesis. J. theor. Biol. 113, 407{423. Dice, J. F. & Goff, S. A. (1987). Error catastrophe and aging: Future directions of
research. In: Modern Biological Theories of Aging (Warner, H., Butler, R., Sprott, R. & Schneider, E. L., eds). Raven Press. New York. Goel, N. S. & Islam, S. (1977). Error catastrophe in and the evolution of the protein
synthesizing machinery. J. theor. Biol. 68, 167{182. Goel, N. S. & Ycas, M. (1975). The error catastrophe hypothesis with reference to
aging and the evolution of the protein synthesizing machinery. J. theor. Biol. 54, 245{282. Hoffman, G. W. (1974). On the origin of the genetic code and the stability of the
translation apparatus. J. molec. Biol. 86, 349. Holliday, R. (1986). Genes, proteins and cellular ageing. Van Nostrand Reinhold.
New York. Kirkwood, T. B. L. & Holliday, R. (1975). The stability of the translation appa-
ratus. J. molec. Biol. 97, 257{265. Kirkwood, T. B. L. & Holliday, R. (1986). Ageing as a consequence of natural
selection. In: The biology of human ageing (Bittles, A. & Collins, K., eds), pp. 1{15. Cambridge University Press. Kirkwood, T. B. L., Holliday, R. & Rosenberger, R. F. (1984). Stability of
the cellular translation process. Int. Rev. Cyt. 92, 93{132. Kirkwood, T. B. L. (1980). Error propagation in intracellular information transfer.
J. theor. Biol. 82, 363{382.
Kurland, C. G. (1986). The error catastrophe: a molecular fata morgana. BioEssays
6, 33{35. Orgel, L. E. (1963). The maintenance of the accuracy of protein synthesis and its
relevance to ageing. Proc. natn. Acad. Sci. U.S.A. 49, 517{521. Orgel, L. E. (1970). The maintenance of the accuracy of protein synthesis and its
relevance to ageing: A correction. Proc. natn. Acad. Sci. U.S.A. 67, 1476. Orgel, L. E. (1973). Ageing of clones of mammalian cells. Nature, Lond. 243, 441. Rosenberger, R. F., Gounaris, E. & Kolettas, E. (1991). Mechanisms responsi-
ble for the limited lifespan and immortal phenotypes in cultured mammalian cells. J. theor. Biol. 148, 383{392.
Appendix Parameters and Variables of the EHKH Model The EHKH model contains several variables and parameters which are summarized in the following table. For parameters, the default value which is used for simulations is given. For variables the possible range is given.
nS
=
10
Number of speci city sites in the aatRNA synthetases.
mS
=
10
Number of activity sites in the aatRNA synthetases.
kS
:
SPS
= 10000 Speci city of correct aatRNA synthetases.
Activity of correct aatRNA synthetases.
RAS =
0.05
Residual activity of erroneous aatRNA synthetases.
Sc;t
=
0-1
Fraction of correct aatRNA synthetases in generation t.
Se;t
=
0-1
Fraction of erroneous aatRNA synthetases in generation t.
Si;t
=
0-1
Fraction of inactive aatRNA synthetases in generation t.
nR
=
10
Number of speci city sites in the ribosome.
mR
=
10
Number of activity sites in the ribosome.
kR
:
Activity of correct ribosomes.
SPR = 10000 Speci city of the correct ribosome. SP0R =
1
Residual speci city of the erroneous ribosome.
RAR =
1
Residual activity of erroneous ribosome.
Rc;t
=
0-1
Fraction of correct ribosomes in generation t.
Re;t
=
0-1
Fraction of erroneous ribosomes in generation t.
Ri;t
=
0-1
Fraction of inactive ribosomes in generation t.
Tt
=
0-1
Fraction of correctly charged tRNA's.
=
20
Number of di erent amino acids.
qc
=
0-1
Accuracy of correct ribosomes.
qe
=
0-1
Accuracy of erroneous ribosomes.
Figure 1
1.0 0.9
Decreasing R
0.8 0.7
0.5 0.4
q(threshold)
0.3 0.2
(c)
(b)
(a)
q(stable)
q(t)
0.6
0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 q(t-1)
Figure 2
1.O
R
n=1 O.8
n=2 O.6
O.4 n=5 O.2 n=10 O.O 1O O
1O 1
1O 2
1O 3
1O 4
1O 5
1O 6
1O 7
S
Figure 3
mRNA
Ribosomes
Amino acids + tRNA’s
Synthetases
aatRNA’s
a)
1
0
Fraction
Figure 4
0 Time in Generations
100
b) 1 0
Fraction 0 Time in Generations
100
Figure 5
1.O
RAs
n=1 O.8
n=2 O.6
O.4
n=5
O.2
n=10 O.O 1O O
1O 1
1O 2
1O 3
1O 4
1O 5
1O 6
1O 7
SPs
Figure 6
1.0 (a)
(b)
(c)
RAR
0.8 0.6 (d) 0.4 0.2 0.0 10 3
10 4
10 5 SPR
10 6
10 7
Figure 7
10 7 10 6
(a)
(b)
(c)
SP’R
10 5 10 4 10 3 10 2 10 1 10 0 10 0
10 1
10 2
10 3
10 4 SPR
10 5
10 6
10 7
Figure 8
10 7 10 6
(a)
(b)
10 2
10 3
(c)
SPS
10 5 10 4 10 3 10 2 10 1 10 0 10 0
10 1
10 4 SPR
10 5
10 6
10 7
Table 1: Rates of correct and incorrect charging of tRNA's by normal and erroneous synthetases (see text for de nition of parameters).
Normal synthetases Erroneous synthetases Rate of correct tRNA charging Rate of incorrect tRNA charging
kS
S + RAS kS SPSP S
kS ( 1) SP S
S + ( 1)RAS kS SPSP S
1
1
Table 2: Rates of correct and incorrect anticodon selection on normal and erroneous ribosomes (see text for de nitions of parameters).
Normal ribosome Rate of correct anticodon selection Rate of incorrect anticodon selection
Erroneous ribosome
kR
R (SPR + RAR kR SP SPR (SPR +
( 1) SPkRR
R + 1 ( 1) RAR kR SPRSP(SP R +
0
0
1) 1)
0
1)
amino acid insertion
Rate of incorrect
amino acid insertion
Rate of correct
A3 Tt + A1 (1 Tt ) + A3 (1 Tt )
A1 Tt + A3 1 T1t
Normal ribosomes
2 1
A4 Tt + A2 (1 Tt ) + A4 (1 Tt )
A2 Tt + A4 1 T1t
Erroneous ribosomes
2 1
Rates of correct and incorrect amino acid incorporation on normal and erroneous ribosomes allowing for the fraction of correctly charged tRNA.
Table 3:
Figure Legends Figure 1: q-diagram representation of the Ho man-Kirkwood-Holliday model. Curves (a), (b) and (c) have been obtained by calculating Equation 1 for R=0.5, R=0.1 and R=0.01 respectively and show the stabilizing e ect of decreasing R (residual activity of erroneous adaptors). For curve (c) the values of qstable and qthreshold have been indicated. For further details see text. For all three curves S =1000, =20 and n=5 have been used. Figure 2: The diagram shows the parameter space of the HKH model. The region left to a speci c curve represents an area in parameter space which results in an unstable system while parameter values to the right result in a stable system. = 20. Figure 3: Reactions described by the EHKH model. Dashed lines represent the action of enzymes and solid lines show the conversion of one molecular species in another.
Figure 4: The time course of Tt (||), Rc = Sc (- - - -), Re = Se ( ) and Ri = Si( ) is shown for a stable (a) and an unstable (b) simulation of the EHKH model. Figure 5: The RAS {SPS plane in the parameter space of the EHKH model. The region to the left of a speci c curve represents instability, the region to the right of a curve stability.
=20, m=10, RAR =1, SPR=10000, SP'R =10000. Figure 6: The RAR {SPR plane in the parameter space of the EHKH model. The region to the left of a speci c curve represents an area of instability, the region to the right stability. Parameter settings are: =20, m=10, n=15, RAS = 0:05, SPS = 10000. Di erent curves are generated by varying SPR0 as follows: (a) SPR0 = 10000, (b) SPR0 = 50, (c)
SPR0 = 3000 and (d) SPR0 = 1000. Figure 7: The SPR0 {SPR plane in the parameter space of the EHKH model. The area to the left of the curves represents instability, the region to the right stability. Parameters used are: =20, m=10, RAS = 0:05, RAR = 1, SPS = 10000. Di erent curves are generated by varying n as follows: (a) n=3, (b) n=10 and (c) n=15. Figure 8: The SPS {SPR plane in the parameter space of the EHKH model. An almost perfect symmetry is generated in this diagram which shows stability depending on the speci cities of the aatRNA-synthetases (SPS ) and the ribosomes(SPR ). Parameters used are:
=20, m=10, n=10, RAR = 1, SPR0 = 1. Di erent curves are generated by varying RAS as follows: (a) RAS = 0:001 (b) RAS = 0:01 and (c) RAS = 0:1.
