J. Mol. Biol. (1975) 97, 257-265
LETTERS TO THE EDITOR
T h e Stability o f the T r a n s l a t i o n A p p a r a t u s A mathematical model has recently been published by Hoffman (1974) in which he considers the potential stability of information transfer in a machinery for translation that perpetuates itself, and leads to the conclusions: (1) that stable translation can be achieved by even a primitive, initially inaccurate translation machinery, and (2) that it is very unlikely that ageing in present day organisms could be due to an error catastrophe following a loss of this stability. An essential assumption of this model is that a molecule which loses fidelity in its role in information transfer also loses most of its biological activity. This is, however, in disagreement with published data. By generalizing the model and thereby making it more biologically realistic, we have reached the conclusion that the protein-error theory of ageing is based on justifiable assumptions. Hoffman (1974) recently proposed a mathematical model t h a t takes into account the potential stability of information transfer in a machinery for translation t h a t perpetuates itself. From his model he concludes t h a t stable translation can be achieved b y even a primitive, initially inaccurate translation machinery and t h a t the apparent inherent stability of the system makes it very unlikely t h a t ageing in present day organisms could be due to a loss of this stability, leading to a lethal "error catastrophe" in protein synthesis (Orgel, 1963). The crucial assumption in Hoffman's model is t h a t a molecule which loses fidelity in its role in information transfer also loses most of its biological activity. The form of the model is such t h a t the greater the accuracy of the translation process, the smaller will be the activity of such a molecule in comparison to unaltered ones. We think this assumption is quite unjustified; b y relaxing it, we can generalize the model and reach a very different conclusion with regard to the build-up of errors in proteins during ageing. Restricting attention initially to the case of polypeptide adaptors, Hoffman considers successive "generations" of adaptors, each of which is responsible for the translation of the nucleic acid to give the n e x t generation. The accuracy of translation, or probability t h a t a digit (an amino acid) is correctly translated, is denoted b y ~, so t h a t the probability t h a t an error is made is 1 - - q. Denoting the generation, i, of adaptors b y use of the subscript i, he then denotes the overall rate for correctly assigning amino acids to codons by ~ t which will be a function of q~_ t, and the average rate for generation i adaptors making wrong amino acid-codon assignments b y B~, also a function of ~ _ 1. Then the average accuracy, qt, of the generation i is given b y the rate of making correct assignments, divided b y the total rate of making assignments:
~/t = iIt(qt_l) + B,(q~-l)"
(1)
I t is supposed t h a t m amino acid residues are essential for an adaptor to function at all, and t h a t n further amino acid residues are essential for the specific recognition of the correct substrates. Attention can be restricted to those adaptors which contain 257
258
T.B.L.
K I R K W O O D AND R. H O L L I D A Y
no errors in any of the m sites, as any containing errors in these sites do not contribute to the translation and therefore cannot affect its accuracy. The viable adaptors can be divided into: a fraction q~-i in which there are no errors in the n sites, and a fraction 1--q~_l in which there are errors in one or more of the n sites and t h e s e a d a p t o r s have therefore lost their specificity. The rate at which the fraction t h a t retain their specificity makes correct assignments is denoted b y a (assignments per adaptor per second), while a dimensionless specificity, S, is defined such t h a t the average rate with which this fraction makes one of the possible misassignments b y incorporating one of the wrong amino acids in a peptide chain is given b y a/S. I f the number of different amino acids involved is ~, there axe ~(~ -- 1) wrong amino acids t h a t can be incorporated, and the total average rate of making errors by adaptors of this fraction is (h -- 1)a]S. For the fraction (1 -- q~-l) that loses its specificity, the rate of making correct assignments is assumed to drop to ~[S, the same as the rate for making any particular misassignment. On the basis of these assumptions, ~ and B~ are written as:
A'~ = q ~ _ ~ + (1 -- qItl) ~/5'
(2)
B~ = ( , ~ - 1)~/,S',
(3)
whence
(~-
1)q~t_l -}'- 1
q~ = (,S' -- 1)q~_~ + ,~"
(4)
This equation gives an S-shaped dependence of q~ on q~_ 1 in the region 0 _~ ql, q~-i R*, it is evident that the higher the value of S the slower will be the build-up of an error 18
262
T. B. L. KIRKWOOD
AND
R. HOLLIDAY
0-20
0"15
0'10
0.05
n=20
n=lO0 0
I0
I0 z
I0 "~
104
I0 ~
[0 ~
I~zG. 2. Taking X = 20, t h e curves represent, for a range of values of n, the division of pairs of values of R and 8 into those giving stable translation a n d those giving unstable translation. I n general, R m u s t be small a n d ~ large for stability to be possible.
catastrophe. Figure 3 shows the number of generations of adaptors that must elapse before an initiany accurate translation, q ~ 1, degenerates to the state q ~ 0.5. Por R > ~ / n ( ~ - - 1) stability is impossible however large S may become, although increasing the value of S continues to lengthen the time required for an error catastrophe to develop. For R < ~ / n ( ~ - - 1) increasing the value of S greatly decreases the rate of build-up of errors until stability is reached. As soon as stability is achieved, there is little advantage in further increase in S. Hoffman briefly considers the possibility that chance fluctuations in q might occasionally produce a "jump" across the stable region, so that a stable translation machinery could become unstable. He concludes that on the basis of his model the chance of this event is vanishingly small. I f we examine this question in our general° ization of his model, we find, by considering the probabilities of correct and incorrect translation of the n residues, that:
S.d.q-
"\(
(1--q-)q-
)* ,
(10)
where N is the number of viable adaptors present. For R = ~ / ( S ~ ~ - - 1) this corresponds to the expression given by Hoffman, so that for R > X](S ~- ~ -- 1) the random variation in q is increased. Furthermore, for any values of S, ~, n there exists a value of R (just less than R*) such that the stable region can be made arbitrarily small, with the result that the chance of a jump across the stable region can be appreciable. The possibility of this situation occurring in an organism is supported to some extent by the argument that in an evolving system the advantage in increasing S almost disappears just as soon as stability is achieved.
LETTERS
TO THE
EDITOR
263
200
15G
g
o ¢
"6 I00 g
o
-=
(.9
50
02
,
I
I
I
I03
104
IO5
I06
I07
S
Fro. 3. T a k i n g A = 20, n = 10, t h e curves show, for a r a n g e o f R values, t h e n u m b e r o f generations o f a d a p t o r s t h a t m u s t elapse before a n initially a c c u r a t e t r a n s l a t i o n , q = 1.0, d e g e n e r a t e s to q < 0.5. F o r R > A/n(A - - 1} s t a b i l i t y is impossible, h o w e v e r large S m a y b e c o m e , a n d t h e curves c o n t i n u e t o rise steadily w i t h increasing S. F o r -~ < A/n(A - - 1) a n S v a l u e is r e a c h e d such t h a t t h e s y s t e m is stable, a n d t h e curves b e c o m e a s y m p t o t i c a l l y vertical.
In proposing this modification of Hoffman's model we should make it clear t h a t the overall model is necessarily of limited value in t h a t the assumptions on which it rests remain open to a number of further criticisms. In particular the "all or none" assumption t h a t an adaptor containing one or more errors in the crucial sites loses all specificity is likely to prove a serious oversimplification. Nevertheless, even in the absence of experimental evidence which would allow the development of a more realistic model, we can conclude t h a t 0rgel's (1963) protein error theory of ageing is based on justifiable assumptions. First, t h a t some errors in protein synthesis result in molecules which are biologically active but which introduce ambiguity into transcription and translation, and second, t h a t the probability of an error catastrophe occurring in a given period of time depends on the initial accuracy of protein synthesis. APPENDIX To find q* and R* we have to solve the simultaneous equations
Of(q, R; 8, ;~, n)
=1
(A1)
q = f ( q , R ; 8 , )~, n),
(~)
0q
where
q'~(a -- bR) +~bR f(q, R; 8, ~, n) = q"c(1 -- R) + cR ' a - - ) t S ' b
----S + ,~-- 1, c---- h ( 8 + ~ - - 1).
264
T. B. L. K I R K W O O D AND R. H O L L I D A Y
F r o m equation (A2), q(a - cq) R = 1(q" - - -_- - - ~ " (b - - cq)'
(A3)
and substituting for R in equation (A1) gives: q,+l .c(a -- b) -- q2nc2 + q(nc(a + b) - - c(a - - b)) - - nab = 0.
(A4)
W e are seeblng a solution q = q* of this equation, where 1/h < q* < 1 and, in general, because of the S-shape of the function f, q* is close to 1. W r i t i n g q = 1 - - y and expanding q2 and q~ + 1 gives a polynomial of degree n + 1 in y for which we n o w w a n t a small positive root. B y omitting terms higher t h a n y2 we obtain a quadratic in y which we m a y solve to give: 1 - - q* ~ - - 2nc(c - - a) + ~¢/[4n2c2 (e - - a) 2 + 2n2c(c - - a)(c - - b)((n + 1)(a - - b) - - 2c)] nc((n + 1)(a - - b) - - 2c) the other r o o t being negative. R e p l a c i n g a, b, c b y the expressions in S, A, n t h a t t h e y represent and simplifying the algebra, we find t h a t this root is positive, provided t h a t (S - - l)((n -- 1)(h - - 1)/2 - - I) > A2, as will be t h e case for S >> h and nh _~ 6, a n d is equal to
(h -- l)/(C~ + h), where G = (S -- I) ( (n -- I)(A -- i) -- 1 ) , so t h a t 2 \
C*+
1
q* - - C ~ + A" To obtain an a p p r o x i m a t e expression for R*, we substitute this expression for q* in equation (A3) to give: R*~__
q*" l--q*"
.
A S+A--1
.S--C ½+1 C~
Writing
I
Oin u A
q'-~ --(5' ~ + 1 q*n 1 - - q*"
1 -Jr- v, say, then
1 (1 + v) ~ -
1 1
nv + ½n(n - - 1)v 2'
since
A--I is small for S >> A. This gives A S -- C÷ n(A -- i)" ~ -[- C i f o r S >>A.
L E T T E R S TO THE E D I T O R
265
National Institute for Biological Standards and Control Hampstead, London NW3 6RB, England
T. B. L. KIREWOOD
National Institute for Medical Research Mill Hill, London, NW7 1AA, England
R. HOLI~DAY
Received 17 March 1975 REFERENCES Apirion, D. (1966). J. Mol. Biol. 16, 285-301. Betz, J. L., Brown, P. R., Smyth, M. J. & Clarke, P. H. (1974). lqatur6 (London), 247, 261-264. Brown, P. R. & Clarke, P. It. (1972). J . •en. Microbiol. 70, 287-298. Brown, J. E., Brown, P. R. & Clarke, P. H. (1969). J . Gen. Microbiol. 57, 273-285. Fincham, J. R. S. (1966). Genetio C o m p t e r a e n ~ o n , W. A. Benjamin, New York. Goldberg, A. L. & Dice, J. F. (1974). A n n u . Rev. Biochem. 43, 835-869. Hoffman, G. W. (1974). J . Mol. Biol. 86, 349-362. Langridge, J. (1969). Mol. Gen. Genet. 105, 74-83. Lewis, C. M. & Holliday, R. (1970). Nature (London), 228, 877-880. Loftfield, R. B. & Vanderjagt, D. (1972). Bioehem. J . 128, 1353-1356. Muzyczka, N., Poland, R. L. & Bessman, M. J. (1972). J . Biol. Chem. 247, 7116-7122. Orgel, L. E. (1963). Prec. Nat. Acad. Sei., U.S.A. 49, 517-521. Printz, D. B. & Gross, S. R. (1967). Genetics, 55, 451-467. Speyer, J. F. (1965). Biochem. Biophys. Res. Commun. 21, 6-8. Speyer, J. F., Karam, J. D. & Lenny, A. B. (1966). Cold Spring Harbor Syrup. Quant. Biol. 31, 693-697. Yoshida, A., Beutler, E. & l~Io~ulsky, A. G. (1971). I n Mendelian Inheritance in M a n , pp. 565a-565p, Johns Hopkins University Press, Baltimore.
LETTERS TO THE EDITOR
T h e Stability o f the T r a n s l a t i o n A p p a r a t u s A mathematical model has recently been published by Hoffman (1974) in which he considers the potential stability of information transfer in a machinery for translation that perpetuates itself, and leads to the conclusions: (1) that stable translation can be achieved by even a primitive, initially inaccurate translation machinery, and (2) that it is very unlikely that ageing in present day organisms could be due to an error catastrophe following a loss of this stability. An essential assumption of this model is that a molecule which loses fidelity in its role in information transfer also loses most of its biological activity. This is, however, in disagreement with published data. By generalizing the model and thereby making it more biologically realistic, we have reached the conclusion that the protein-error theory of ageing is based on justifiable assumptions. Hoffman (1974) recently proposed a mathematical model t h a t takes into account the potential stability of information transfer in a machinery for translation t h a t perpetuates itself. From his model he concludes t h a t stable translation can be achieved b y even a primitive, initially inaccurate translation machinery and t h a t the apparent inherent stability of the system makes it very unlikely t h a t ageing in present day organisms could be due to a loss of this stability, leading to a lethal "error catastrophe" in protein synthesis (Orgel, 1963). The crucial assumption in Hoffman's model is t h a t a molecule which loses fidelity in its role in information transfer also loses most of its biological activity. The form of the model is such t h a t the greater the accuracy of the translation process, the smaller will be the activity of such a molecule in comparison to unaltered ones. We think this assumption is quite unjustified; b y relaxing it, we can generalize the model and reach a very different conclusion with regard to the build-up of errors in proteins during ageing. Restricting attention initially to the case of polypeptide adaptors, Hoffman considers successive "generations" of adaptors, each of which is responsible for the translation of the nucleic acid to give the n e x t generation. The accuracy of translation, or probability t h a t a digit (an amino acid) is correctly translated, is denoted b y ~, so t h a t the probability t h a t an error is made is 1 - - q. Denoting the generation, i, of adaptors b y use of the subscript i, he then denotes the overall rate for correctly assigning amino acids to codons by ~ t which will be a function of q~_ t, and the average rate for generation i adaptors making wrong amino acid-codon assignments b y B~, also a function of ~ _ 1. Then the average accuracy, qt, of the generation i is given b y the rate of making correct assignments, divided b y the total rate of making assignments:
~/t = iIt(qt_l) + B,(q~-l)"
(1)
I t is supposed t h a t m amino acid residues are essential for an adaptor to function at all, and t h a t n further amino acid residues are essential for the specific recognition of the correct substrates. Attention can be restricted to those adaptors which contain 257
258
T.B.L.
K I R K W O O D AND R. H O L L I D A Y
no errors in any of the m sites, as any containing errors in these sites do not contribute to the translation and therefore cannot affect its accuracy. The viable adaptors can be divided into: a fraction q~-i in which there are no errors in the n sites, and a fraction 1--q~_l in which there are errors in one or more of the n sites and t h e s e a d a p t o r s have therefore lost their specificity. The rate at which the fraction t h a t retain their specificity makes correct assignments is denoted b y a (assignments per adaptor per second), while a dimensionless specificity, S, is defined such t h a t the average rate with which this fraction makes one of the possible misassignments b y incorporating one of the wrong amino acids in a peptide chain is given b y a/S. I f the number of different amino acids involved is ~, there axe ~(~ -- 1) wrong amino acids t h a t can be incorporated, and the total average rate of making errors by adaptors of this fraction is (h -- 1)a]S. For the fraction (1 -- q~-l) that loses its specificity, the rate of making correct assignments is assumed to drop to ~[S, the same as the rate for making any particular misassignment. On the basis of these assumptions, ~ and B~ are written as:
A'~ = q ~ _ ~ + (1 -- qItl) ~/5'
(2)
B~ = ( , ~ - 1)~/,S',
(3)
whence
(~-
1)q~t_l -}'- 1
q~ = (,S' -- 1)q~_~ + ,~"
(4)
This equation gives an S-shaped dependence of q~ on q~_ 1 in the region 0 _~ ql, q~-i R*, it is evident that the higher the value of S the slower will be the build-up of an error 18
262
T. B. L. KIRKWOOD
AND
R. HOLLIDAY
0-20
0"15
0'10
0.05
n=20
n=lO0 0
I0
I0 z
I0 "~
104
I0 ~
[0 ~
I~zG. 2. Taking X = 20, t h e curves represent, for a range of values of n, the division of pairs of values of R and 8 into those giving stable translation a n d those giving unstable translation. I n general, R m u s t be small a n d ~ large for stability to be possible.
catastrophe. Figure 3 shows the number of generations of adaptors that must elapse before an initiany accurate translation, q ~ 1, degenerates to the state q ~ 0.5. Por R > ~ / n ( ~ - - 1) stability is impossible however large S may become, although increasing the value of S continues to lengthen the time required for an error catastrophe to develop. For R < ~ / n ( ~ - - 1) increasing the value of S greatly decreases the rate of build-up of errors until stability is reached. As soon as stability is achieved, there is little advantage in further increase in S. Hoffman briefly considers the possibility that chance fluctuations in q might occasionally produce a "jump" across the stable region, so that a stable translation machinery could become unstable. He concludes that on the basis of his model the chance of this event is vanishingly small. I f we examine this question in our general° ization of his model, we find, by considering the probabilities of correct and incorrect translation of the n residues, that:
S.d.q-
"\(
(1--q-)q-
)* ,
(10)
where N is the number of viable adaptors present. For R = ~ / ( S ~ ~ - - 1) this corresponds to the expression given by Hoffman, so that for R > X](S ~- ~ -- 1) the random variation in q is increased. Furthermore, for any values of S, ~, n there exists a value of R (just less than R*) such that the stable region can be made arbitrarily small, with the result that the chance of a jump across the stable region can be appreciable. The possibility of this situation occurring in an organism is supported to some extent by the argument that in an evolving system the advantage in increasing S almost disappears just as soon as stability is achieved.
LETTERS
TO THE
EDITOR
263
200
15G
g
o ¢
"6 I00 g
o
-=
(.9
50
02
,
I
I
I
I03
104
IO5
I06
I07
S
Fro. 3. T a k i n g A = 20, n = 10, t h e curves show, for a r a n g e o f R values, t h e n u m b e r o f generations o f a d a p t o r s t h a t m u s t elapse before a n initially a c c u r a t e t r a n s l a t i o n , q = 1.0, d e g e n e r a t e s to q < 0.5. F o r R > A/n(A - - 1} s t a b i l i t y is impossible, h o w e v e r large S m a y b e c o m e , a n d t h e curves c o n t i n u e t o rise steadily w i t h increasing S. F o r -~ < A/n(A - - 1) a n S v a l u e is r e a c h e d such t h a t t h e s y s t e m is stable, a n d t h e curves b e c o m e a s y m p t o t i c a l l y vertical.
In proposing this modification of Hoffman's model we should make it clear t h a t the overall model is necessarily of limited value in t h a t the assumptions on which it rests remain open to a number of further criticisms. In particular the "all or none" assumption t h a t an adaptor containing one or more errors in the crucial sites loses all specificity is likely to prove a serious oversimplification. Nevertheless, even in the absence of experimental evidence which would allow the development of a more realistic model, we can conclude t h a t 0rgel's (1963) protein error theory of ageing is based on justifiable assumptions. First, t h a t some errors in protein synthesis result in molecules which are biologically active but which introduce ambiguity into transcription and translation, and second, t h a t the probability of an error catastrophe occurring in a given period of time depends on the initial accuracy of protein synthesis. APPENDIX To find q* and R* we have to solve the simultaneous equations
Of(q, R; 8, ;~, n)
=1
(A1)
q = f ( q , R ; 8 , )~, n),
(~)
0q
where
q'~(a -- bR) +~bR f(q, R; 8, ~, n) = q"c(1 -- R) + cR ' a - - ) t S ' b
----S + ,~-- 1, c---- h ( 8 + ~ - - 1).
264
T. B. L. K I R K W O O D AND R. H O L L I D A Y
F r o m equation (A2), q(a - cq) R = 1(q" - - -_- - - ~ " (b - - cq)'
(A3)
and substituting for R in equation (A1) gives: q,+l .c(a -- b) -- q2nc2 + q(nc(a + b) - - c(a - - b)) - - nab = 0.
(A4)
W e are seeblng a solution q = q* of this equation, where 1/h < q* < 1 and, in general, because of the S-shape of the function f, q* is close to 1. W r i t i n g q = 1 - - y and expanding q2 and q~ + 1 gives a polynomial of degree n + 1 in y for which we n o w w a n t a small positive root. B y omitting terms higher t h a n y2 we obtain a quadratic in y which we m a y solve to give: 1 - - q* ~ - - 2nc(c - - a) + ~¢/[4n2c2 (e - - a) 2 + 2n2c(c - - a)(c - - b)((n + 1)(a - - b) - - 2c)] nc((n + 1)(a - - b) - - 2c) the other r o o t being negative. R e p l a c i n g a, b, c b y the expressions in S, A, n t h a t t h e y represent and simplifying the algebra, we find t h a t this root is positive, provided t h a t (S - - l)((n -- 1)(h - - 1)/2 - - I) > A2, as will be t h e case for S >> h and nh _~ 6, a n d is equal to
(h -- l)/(C~ + h), where G = (S -- I) ( (n -- I)(A -- i) -- 1 ) , so t h a t 2 \
C*+
1
q* - - C ~ + A" To obtain an a p p r o x i m a t e expression for R*, we substitute this expression for q* in equation (A3) to give: R*~__
q*" l--q*"
.
A S+A--1
.S--C ½+1 C~
Writing
I
Oin u A
q'-~ --(5' ~ + 1 q*n 1 - - q*"
1 -Jr- v, say, then
1 (1 + v) ~ -
1 1
nv + ½n(n - - 1)v 2'
since
A--I is small for S >> A. This gives A S -- C÷ n(A -- i)" ~ -[- C i f o r S >>A.
L E T T E R S TO THE E D I T O R
265
National Institute for Biological Standards and Control Hampstead, London NW3 6RB, England
T. B. L. KIREWOOD
National Institute for Medical Research Mill Hill, London, NW7 1AA, England
R. HOLI~DAY
Received 17 March 1975 REFERENCES Apirion, D. (1966). J. Mol. Biol. 16, 285-301. Betz, J. L., Brown, P. R., Smyth, M. J. & Clarke, P. H. (1974). lqatur6 (London), 247, 261-264. Brown, P. R. & Clarke, P. It. (1972). J . •en. Microbiol. 70, 287-298. Brown, J. E., Brown, P. R. & Clarke, P. H. (1969). J . Gen. Microbiol. 57, 273-285. Fincham, J. R. S. (1966). Genetio C o m p t e r a e n ~ o n , W. A. Benjamin, New York. Goldberg, A. L. & Dice, J. F. (1974). A n n u . Rev. Biochem. 43, 835-869. Hoffman, G. W. (1974). J . Mol. Biol. 86, 349-362. Langridge, J. (1969). Mol. Gen. Genet. 105, 74-83. Lewis, C. M. & Holliday, R. (1970). Nature (London), 228, 877-880. Loftfield, R. B. & Vanderjagt, D. (1972). Bioehem. J . 128, 1353-1356. Muzyczka, N., Poland, R. L. & Bessman, M. J. (1972). J . Biol. Chem. 247, 7116-7122. Orgel, L. E. (1963). Prec. Nat. Acad. Sei., U.S.A. 49, 517-521. Printz, D. B. & Gross, S. R. (1967). Genetics, 55, 451-467. Speyer, J. F. (1965). Biochem. Biophys. Res. Commun. 21, 6-8. Speyer, J. F., Karam, J. D. & Lenny, A. B. (1966). Cold Spring Harbor Syrup. Quant. Biol. 31, 693-697. Yoshida, A., Beutler, E. & l~Io~ulsky, A. G. (1971). I n Mendelian Inheritance in M a n , pp. 565a-565p, Johns Hopkins University Press, Baltimore.
