- Email: [email protected]
Distinction between various models for messenger decay
12
A.
L.
KOCH
APPENDIX
Distinction
between Various
Models
for Messenger
Decay
ARTHUR L. KOCH After a single initiation of the transcription of a messenger molecule, a number of ribosomes move down the message, translating it, to produce polypeptide chains. Eventually the messenger becomes inactivated in the sense that there is a last ribosome to successfully translate the message. The life span of the individual message can be defined as the length of time from initiation until the last ribosome completes the synthesis of its polypeptide chain. By this definition, both the delay time until the first polypeptide chain is completed and the life span until the last is finished would be longer for distal messages in a polycistronic message regardless of the type of inactivation process. Assume that ribosomes travel down the message in an evenly spaced fashion, then each initiation results in a delay, a ramp of finished polypeptide production and then a plateau. A series of such curves can be summed in appropriate ways to predict the curve of enzyme induction under various regima of inducer addition and withdrawal, and for various assumed models for messenger inactivation. This summation or integration is most easily carried out for the case in which all messenger molecules have the same delay and life span. The results for the transcription of the tryptophan operon are consistent withafixed delay and the life span model (Morse, Mosteller & Yanofsky, 1969). We spare the reader the tedious algebra, and report that for the step-induction type experiment where the inducer is added and never removed from the culture, the resultant curve also has three phases reflecting the delay, the transient, and the steady-state production of finished polypeptide chains : Z=B
o
Z=B-;(o+@+rt In these expressions, Z is the enzyme level at any time, t; B is the basal level of the uninduced cells; 6 is the delay time as defined in the text; (J, the life span; and r, the rate of steady-state enzyme production. In the steady-state production phase, the average lag time is (r/2) (u + 6)/r = (u + 8)/Z, or the mean of the life span and delay time. Based on this relationship, average life spans of 215 to 250 seconds for batch cultures shown in Table 1 can be computed. As an example, the dashed line in Figure 5 was calculated from the life span computed as the average together with the other paramenters taken for experiment 2 from Table 1. Also shown are the actual data from this experiment. When inducer is removed or prevented from acting during the third or steady-state
13
APPENDIX
0 0
I 60
I 120 Time Lsec)
I 180
FIG. 5. Theoretical curves for step induction. The fixed life-span random inactivation model (solid line) have been fitted to the data Both curves are fitted to the same basal values and to the same production at times from 6 to 20 min shown in Table 1.
phase of induction, same mathematical
I 24(
model (dashed line) and the of experiment 2 of the text. steady-state rate of enzyme
the kinetics of de-induction or run-off can be calculated process. This too leads to three phases as follows:
2 = z,, + r (t - to)
2 = -Go +&
by the
t,
[-2toIJ -
s2 - to2 + 2 (to + u) t - P]
2, = Zto + ;(S + u) = B + rt,
t,+a
This formulation requires enzyme production to continue unaltered for a delay time, 6, after the inducer is removed but so would any model for the inactivation of messenger that assumes that the rate of translation remains constant. I know of no published data where this assumption appears to be satisfied. In the published reports from Magasanik’s laboratory (Nakada & Magasanik, 1964; Kaempfer & Magasanik, 1967a,b), whether initiation is stopped by filtration of the inducer or phage addition, the rate of enzyme production slows well before the delay time is complete. Since a delay time will be expected with any model for message inactivation, it can be concluded that a gentle enough, precise enough method to prevent only initiation is still not available. Still, the phage addition data of Kaempfer & Magasanik (1967a,b) appear to be the best available, so in Figure 6 this model is fitted by a dashed line to data reconstructed from their published data. It should be noted that Kennel1 (1970) has presented evidence that phage infection prevents ribosome association with host messenger RNA, and thus affects translation as well. Of course, the best way to estimate a and its variation would be to study the enzyme formation resulting from initiation events starting over a very brief period of time. The method of pulse induction devised by Kepes (1963), resulting in what
14
5
6
A.
L.
7
8
KOCH
9 IO Time (min)
II
12
15
Wa. 6. Theoretical ourve~ for induoer removal: “run-off kinetios”. Fixedlife-span model (dashed lines) and random inactivation model (solid line) have been fitted to the data of Kaempfer & Magasanik (1967L). In these experiments, inducer was added at time zero and a high mulfiplioity of phage added at 6 min. These curves are fitted to their control rate and we have assumed that the rate of tmnsaription and translation are alower at 30°C (at which the experiment was conducted) than at 37°C. It w&s assumed thet 6 = 120 set, D = 240 aeo, and l/k: = 60 sea.
he called the “elementary wave”, is just this. The way his experiment was carried out was to add a carefully chosen concentration of inducer to a culture, and dilute the culture 40-fold after the 20th second. The inducer level has to be high enough to start induction instantly and low enough so that the dilution instantly blocks further initiation events. If we assume that initiation events take place continuously and exclusively during the 20-second period, then the same summing approach may be used, but in this case it leads to more elaborate equations. The equations are cumbersome and will not be given, but the equations were fitted to Kepes’ published data and are shown in Figure 7 by the dashed line. Both Kepes and Magasanik treat such data by plotting the logarithm of (2, - 2) against time. Such plots frequently give quite straight lines, and the half-times extracted from such plots have been stated to be the half-life of message. The shape of such experimental plots is highly dependent on the accuracy with which 2, is known and its value is sometimes seemingly obtained by a questionable extrapolation; i.e. testing various values until one is found that makes the logarithm plot straight. For the particular case under consideration so far, the true half-time is zero because all messengers have been presumed to have the same duration of life. Thus, it takes no time at all for the number of surviving messenger molecules to drop from l/2 to l/4 the original value. About half of the apparent half-life in this case is simply the result of the enzyme production phase. The remaining portion may either be due to heterogeneity resulting from initiations having taken place after the culture was diluted, but before the internal concentration of inducer falls, or to the clear inadequacy of the fixed life-span model. The point to be stressed,however, is that much of the rise time experimentally measured does not reflect the kinetics of messenger decay but the kinetics of translation.
APPENDIX
15
0.5
c C s ci 9 x 5 :: 5 S s (h
0.4
0.3
0.2
0.1
0
120
240
360
TimeCsec)
FIU. 7. Theoretical ourves for pulse induotion. Fixed life-span model (dashed line) and random inaotivation model (solid line) have been fitted to the data of Kepes (1963,1969). To fit his data at all, a delay time of 60 sea was assumed for the solid line, o was taken to be 190 set and the value of 2, = O-63; for the dashed line l/k = 60 880 and Zm was taken to be 0.56.
It must be made clear that under any model the half-life values from semi-logsrithmic plots are not related directly to the average life or life span becauseexactly the same curves would have been obtained for the fixed life-span model, but simply displaced horizontally, had both a and 6 been increased by the ss,meamount. Such a change in a and 8 would obviously change the life span of the average messenger and would result in an increased rate of synthesis in the cell. The analogy can be made to the case of human mortality. It happens that 95-year old men die with a 50% probability per yesr. Yet is is clearly wrong to say that the half-life of man is 1 year. This is becausehuman survival does not follow random statistics and is not a first-order processstarting from birth. Still, the fit of the exponential model to the date is very good, as can be seen from the new pulse-induction data presented in Figure 4 in the text. In this experiment, there was no ambiguity about extrapolation to 2,. In this case, the inducer ws,s removed by filtration and thus combines the methods used in Kepes’ and M&gas&k’s l&borstories. So, pragmatically, we next chooseto elaborate a model where the number of polypeptide chains produced by messengermoleculesis distributed in an exponential fashion according to eekt. If we assumethat heterogeneity effects due to cell-cell variation or to variation in rates of tmnscript or to experimental artifacts are small, then a rational basis of the Kepes semi-logarithmic plot is to assumethat a messengercan be inactivated with constant probability after each successive ribosome starts to translate the message.This could be mechanistically understood on the adoption of the “killerribosome” hypothesis of David Schlessinger (1970, personal communication-see
16
A. L. KOCH
Kuwano, Schlessinger & Apirion, 1970). Its limitations and other models will be appraised below. On this basis, a messagewould function until a ribosome of a special classwas engaged which would destroy RNA as it translated it. On this basis, many messageswould be inactivated after only a few ribosomeshad translated them, but some would produce an inordinately large number of polypeptide copies. On this model, the number of chains per messagewould approximate to a simple Poisson distribution. The solution to the problems of enzyme production for the three induction regima are shown graphically in Figures 4 to 7 fitted to the experimental data as solid lines. The formulae are : for step induction, Z=B
Ost
for the run-off case the equation given immediately above applies until t = t,, + 6, then,
-’ e-k(&6 z=B+r(to - keta’ for the third case of pulse induction for a duration r, Z=B
o
2
=
B
-
~(6
+
2
=
B
+
,,.,,. _
k)
+
re-+%k
rt
+
f
e-‘t
-
6)k
a
y
As for the previous model, we omit the derivation of these equations, since they are simply repetitious integrations of the polypeptide curves from single initiation events. The derivations will be sent to those requesting them. It can be seen that in all three casesthe second model fits the data somewhat better than the first. Although in principle the elementary wave experiment is most sensitive to the life table of the messengermolecules, valid criticism of all available data for this kind of experiment can be made : one cannot be certain that induction is not abruptly terminated or the rate of protein synthesis is altered by the filtration procedure or the drug used. Similarly, for the run-off experiment, the termination of initiation cannot be consideredeffectively mild and abrupt enough in any single case. The important point is that the step-induction experiment is not subject to the limitations of how quickly or mildly the inducer can be removed and unambiguously established that a fixed life-span model is inconsistent with experiment. Therefore, although models requiring that each copy of messagebe read by a fixed number of ribosomes(time-bomb models)have not been popular for a few years, we believe that
APPENDIX
17
the present data and calculations are the first to exclude them for the case of /3-galactosidase in E. cdi. We add that this question, to our knowledge, is completely unresolved in any eucaryotic system. In fact the equation is given above for the fixed life-span model in the hope that it will be tested in such studies. There is a second circumstance which results in the identical time-course of enzyme production as does the killer-ribosome model. This is the case where an inactivation event can take place anywhere along the length of the message as well as at any time after that portion of the message has been synthesized. The recent finding that material hybridizing as lac messenger has a longer half-life than the entity coding j?-galactosidase synthesis is consistent with this model (Schwartz, Craig & Kennell, 1970). Ribosomes that have gone past the point of scission would proceed to completion, but ribosomes attached at the time of inactivation proximally to the scission point would fail to produce completed functional polypeptide chains. This model appears more plausible since it has recently been shown (Nath & Koch, 1970,1971) that there is a fraction of the cells total peptide synthesis which is made roughly in fixed proportion of the total peptide bond synthesis, that is then degraded with a half-life of less than one hour and produced in amounts that would be expected from the random scission of messenger chains producing incomplete proteins. The rate-constant for degradation of this material is the same as for peptide chains artificially terminated by puromycin. To find the life expectancy for this model, we must compute the combined probability of the message remaining intact while the &h ribosome traverses it but becoming cut before the (n + 1)th ribosome can complete its transit. This can be done easily by reference to Figure 8. In this Figure, time is represented horizontally and position along the message vertically. If chain cuts are completely random, then the chance of inactivation at any point in the field to the right of the solid line depicting messenger synthesis is as likely as at any other point in the field; this way to view the problem
Time
8. Model for random inaativation of messenger. The synthesis of the messenger is indicated by the solid line. Different translation events are designated by the dashed lines. The space-time field can be considered to be divided into little rectangles of duration T/8 and vertical distance along a message of X/s, where T is the time for total translation and X is the entire length of message. For this diagram the rate of transcription of messenger has been assumed equal to that of translation (i.e. the step-time for transcription has been set at 3 times the step-time for translation). FIG.
2
18
A. L. KOCH
was suggestedby Dr George Minty. Let us designate ~1as the probability of inactivation per unit time and per unit distance. Then let us divide the field into little rectangles of duration l/a of the transcription time, T, and of length l/s of the length of the message,X. By making s larger and larger we will be able to consider sufficiently small rectangles so that the chance of two inactivation events is negligible. The chance of an inactivation event is t.~(T/s) (X/s). The chance of surviving in any particular rectangle is (1 - p (T/s) (X/s)). The probability of the first translation event being successfulis this expression raised to a power which is the number of rectangles between the solid and the nearest parallel dashed line. Therefore, the probability of at least one polypeptide being completed is [l p(X/s) (T/s)j@ AT/T, where AT is the time between initiation of transcription and the first event initiating translation’. If the sameinterval exists between the attachment of any successivepair of ribosomes, then the probability of n polypeptide chains being produced, but not the (w + 1)th one is:
This can be recast so that the sameexpressions inside the parenthesis also appear in the exponent:
In the limit as s gets larger and larger, this expression can be simplified by the definition of e as the limit of (1 + H)l’H as H approaches zero. The final expression for P(n) becomes: P(n) = e-PnXAT[l - e-@XAT]. Since the quantity in brackets is constant and nAT is the time after initiation of transcription that the last successfulribosome starts its transit, this model predicts results indistinguishable from the previous model. Both can be written: P(n) = (1 - P)” (P) where P is the probability of any ribosome successfully transversing a previously intact message.The secondmodel has the advantage that no special classof ribosome need be postulated. If this random soissionof the messengerRNA proves to be the correct mode of messengerdecay, then it can easily be elaborated to include those cases where transcription and translation do not take place at the samerate. It is also easy to elaborate the treatment to include regulatory regions which may be transcribed but not translated. Also, the treatment can be expanded to include a separate contribution from ribosomesmoving along a poly-cistronic messageand those attaching at the beginning of a later message.
Alpers, Baker, Boezi, Bremer,
REFERENCES D. & Tompkins, G. M. (1965). Proc. Nat. Ad. Sk., Wash. 53, 797. R. F. & Yanofsky, C. (1988). Proc. Nat. Acad. Sci., Wada. 60, 313. J. I. & Cowie, D. B. (1961). Bio@ya. J. 1, 639. H. & Yuan, D. (1908). J. Mol. Biol. 38, 163.
APPENDIX Coffman, R. L., Norris, T. E. t Koch, A. L. (1969). Fed. Proc. 28, 884. Cohen, G. N. & Monod, J. (1957). &et. Rew. 21, 169. Forchhammer, J. & Lindahl, L. (1971). J. Mol. Biol. 55, 663. Canesan, A. K. & Rotman, B. (1965). J. Mol. Biol. 16, 42. Geiduschek, E. P. & Hsselkorn, R. (1969). Ann. Rev. Biochem. 38, 647. H&e, A. (1969). Ph.D. Thesis, Indiana University, Bloomington, Indiana. Kaempfer, R. 0. & Magasanik, B. (1967a). J. Mol. BioZ. 27, 453. Kaempfer, R. 0. & Magasanik, B. (19675). J. Mol. BioZ. 27, 475. Kennell, D. (1970). J. ViroZ. 6, 208. Kepes, A. (1960). Biochirn. biophys. Acta, 40, 70. Kepes, A. (1963). Biochina. biofJhy8. Acta, 76, 293. Kepes, A. (1969). In Progress &a Biophytice and MoZemZar Biology, ed. by J.A.V. Butler & D. Noble, vol. 19, part 2, p. 201. New York: Pergamon Press. Koch, A. L. (1970). J. Theoret. BioZ. 28, 203. Koch, A. L. (1971). Adv. Micro. Phqeiol. VI, 147. Koch, A. L. & Coffman, R. L. (1970). Biotech. Bioeng. 13, 661. Koch, A. L. & Deppe, C. S. (1971). J. Mol. BioZ. 55, 649. Kuwano, M., Schlessinger, D. & Apirion, D. (1970). J. Mol. BioZ. 51, 75. L&route, F. t Stent, G. S. (1968). J. Mol. BioZ. 35, 165. Leive, L. & Kollin, V. (1967). J. Mol. BioZ. 24, 247. Ma&e, 0. & Kjeldgaard, N. 0. (1966). Cont~oZ of MacronwZedar Syntheak New York: W. A. Benjamin. Manor, H., Goodman, D. & Stent, G. S. (1969). J. Mol. BioZ. 39, 1. Morse, D. E., Baker, R. F. & Yanofsky, C. (1968). Proc. Nat. Acm?. Sci., Wash. 60, 1428. Morse, D. E., Mosteller, R. D. t Yanofsky, C. (1969). Cold Spr, Ha&. Syrqx Quunt. BioZ. 84, 725. Nakada, D. & Magasanik, B. (1964). J. Mol. BioZ. 8, 106. Nath, K. & Koch, A. L. (1970). J. BioZ. Chem. 245, 2889. Nath, K. & Koch, A. L. (1971). J. BioZ. Chem., in the press. Norris, T. E. (1970). Ph.D. Thesis, Indiana University, Bloomington, Indiana. Pardee, A. B. t Prestidge, L. (1959). Biochim. biophys. Acta, 36,645. Ravel, H. R,, Luria, S. E. & Rotman, B. (1961). Proc. Nat. Ad. SC&, Wash. 47, 1956. Roberts, R. B., Abelson, P. H., Cowie, D. B., Bolton, E. T. 8z Britten, R. J. (1957). In Studies of Bioeynthe& in Escherichia coli, 2nd ed., chap. 17, p. 313. Washington, D. C. : Carnegie Inst., Wash. Rose, J. K., Mosteller, R. R. & Yanofsky, C. (1970). J. Mol. BioZ. 51, 541. Rotman, B. (1961). Proc. Nat. Acad. Sci., Wash. 47, 1981. Rotman, B., Ganesan, A. K. & Guzman, R. (1968). J. Mol. BioZ. 36, 247. Schwartz, T., Craig, E. t Kennell, D. (1970). J. Mol. BioZ. 54, 299. Winslow, R. M. & Lazzarini, R. A. (1969). J. BioZ. Chem. 244, 1128.
A.
L.
KOCH
APPENDIX
Distinction
between Various
Models
for Messenger
Decay
ARTHUR L. KOCH After a single initiation of the transcription of a messenger molecule, a number of ribosomes move down the message, translating it, to produce polypeptide chains. Eventually the messenger becomes inactivated in the sense that there is a last ribosome to successfully translate the message. The life span of the individual message can be defined as the length of time from initiation until the last ribosome completes the synthesis of its polypeptide chain. By this definition, both the delay time until the first polypeptide chain is completed and the life span until the last is finished would be longer for distal messages in a polycistronic message regardless of the type of inactivation process. Assume that ribosomes travel down the message in an evenly spaced fashion, then each initiation results in a delay, a ramp of finished polypeptide production and then a plateau. A series of such curves can be summed in appropriate ways to predict the curve of enzyme induction under various regima of inducer addition and withdrawal, and for various assumed models for messenger inactivation. This summation or integration is most easily carried out for the case in which all messenger molecules have the same delay and life span. The results for the transcription of the tryptophan operon are consistent withafixed delay and the life span model (Morse, Mosteller & Yanofsky, 1969). We spare the reader the tedious algebra, and report that for the step-induction type experiment where the inducer is added and never removed from the culture, the resultant curve also has three phases reflecting the delay, the transient, and the steady-state production of finished polypeptide chains : Z=B
o
Z=B-;(o+@+rt In these expressions, Z is the enzyme level at any time, t; B is the basal level of the uninduced cells; 6 is the delay time as defined in the text; (J, the life span; and r, the rate of steady-state enzyme production. In the steady-state production phase, the average lag time is (r/2) (u + 6)/r = (u + 8)/Z, or the mean of the life span and delay time. Based on this relationship, average life spans of 215 to 250 seconds for batch cultures shown in Table 1 can be computed. As an example, the dashed line in Figure 5 was calculated from the life span computed as the average together with the other paramenters taken for experiment 2 from Table 1. Also shown are the actual data from this experiment. When inducer is removed or prevented from acting during the third or steady-state
13
APPENDIX
0 0
I 60
I 120 Time Lsec)
I 180
FIG. 5. Theoretical curves for step induction. The fixed life-span random inactivation model (solid line) have been fitted to the data Both curves are fitted to the same basal values and to the same production at times from 6 to 20 min shown in Table 1.
phase of induction, same mathematical
I 24(
model (dashed line) and the of experiment 2 of the text. steady-state rate of enzyme
the kinetics of de-induction or run-off can be calculated process. This too leads to three phases as follows:
2 = z,, + r (t - to)
2 = -Go +&
by the
t,
[-2toIJ -
s2 - to2 + 2 (to + u) t - P]
2, = Zto + ;(S + u) = B + rt,
t,+a
This formulation requires enzyme production to continue unaltered for a delay time, 6, after the inducer is removed but so would any model for the inactivation of messenger that assumes that the rate of translation remains constant. I know of no published data where this assumption appears to be satisfied. In the published reports from Magasanik’s laboratory (Nakada & Magasanik, 1964; Kaempfer & Magasanik, 1967a,b), whether initiation is stopped by filtration of the inducer or phage addition, the rate of enzyme production slows well before the delay time is complete. Since a delay time will be expected with any model for message inactivation, it can be concluded that a gentle enough, precise enough method to prevent only initiation is still not available. Still, the phage addition data of Kaempfer & Magasanik (1967a,b) appear to be the best available, so in Figure 6 this model is fitted by a dashed line to data reconstructed from their published data. It should be noted that Kennel1 (1970) has presented evidence that phage infection prevents ribosome association with host messenger RNA, and thus affects translation as well. Of course, the best way to estimate a and its variation would be to study the enzyme formation resulting from initiation events starting over a very brief period of time. The method of pulse induction devised by Kepes (1963), resulting in what
14
5
6
A.
L.
7
8
KOCH
9 IO Time (min)
II
12
15
Wa. 6. Theoretical ourve~ for induoer removal: “run-off kinetios”. Fixedlife-span model (dashed lines) and random inactivation model (solid line) have been fitted to the data of Kaempfer & Magasanik (1967L). In these experiments, inducer was added at time zero and a high mulfiplioity of phage added at 6 min. These curves are fitted to their control rate and we have assumed that the rate of tmnsaription and translation are alower at 30°C (at which the experiment was conducted) than at 37°C. It w&s assumed thet 6 = 120 set, D = 240 aeo, and l/k: = 60 sea.
he called the “elementary wave”, is just this. The way his experiment was carried out was to add a carefully chosen concentration of inducer to a culture, and dilute the culture 40-fold after the 20th second. The inducer level has to be high enough to start induction instantly and low enough so that the dilution instantly blocks further initiation events. If we assume that initiation events take place continuously and exclusively during the 20-second period, then the same summing approach may be used, but in this case it leads to more elaborate equations. The equations are cumbersome and will not be given, but the equations were fitted to Kepes’ published data and are shown in Figure 7 by the dashed line. Both Kepes and Magasanik treat such data by plotting the logarithm of (2, - 2) against time. Such plots frequently give quite straight lines, and the half-times extracted from such plots have been stated to be the half-life of message. The shape of such experimental plots is highly dependent on the accuracy with which 2, is known and its value is sometimes seemingly obtained by a questionable extrapolation; i.e. testing various values until one is found that makes the logarithm plot straight. For the particular case under consideration so far, the true half-time is zero because all messengers have been presumed to have the same duration of life. Thus, it takes no time at all for the number of surviving messenger molecules to drop from l/2 to l/4 the original value. About half of the apparent half-life in this case is simply the result of the enzyme production phase. The remaining portion may either be due to heterogeneity resulting from initiations having taken place after the culture was diluted, but before the internal concentration of inducer falls, or to the clear inadequacy of the fixed life-span model. The point to be stressed,however, is that much of the rise time experimentally measured does not reflect the kinetics of messenger decay but the kinetics of translation.
APPENDIX
15
0.5
c C s ci 9 x 5 :: 5 S s (h
0.4
0.3
0.2
0.1
0
120
240
360
TimeCsec)
FIU. 7. Theoretical ourves for pulse induotion. Fixed life-span model (dashed line) and random inaotivation model (solid line) have been fitted to the data of Kepes (1963,1969). To fit his data at all, a delay time of 60 sea was assumed for the solid line, o was taken to be 190 set and the value of 2, = O-63; for the dashed line l/k = 60 880 and Zm was taken to be 0.56.
It must be made clear that under any model the half-life values from semi-logsrithmic plots are not related directly to the average life or life span becauseexactly the same curves would have been obtained for the fixed life-span model, but simply displaced horizontally, had both a and 6 been increased by the ss,meamount. Such a change in a and 8 would obviously change the life span of the average messenger and would result in an increased rate of synthesis in the cell. The analogy can be made to the case of human mortality. It happens that 95-year old men die with a 50% probability per yesr. Yet is is clearly wrong to say that the half-life of man is 1 year. This is becausehuman survival does not follow random statistics and is not a first-order processstarting from birth. Still, the fit of the exponential model to the date is very good, as can be seen from the new pulse-induction data presented in Figure 4 in the text. In this experiment, there was no ambiguity about extrapolation to 2,. In this case, the inducer ws,s removed by filtration and thus combines the methods used in Kepes’ and M&gas&k’s l&borstories. So, pragmatically, we next chooseto elaborate a model where the number of polypeptide chains produced by messengermoleculesis distributed in an exponential fashion according to eekt. If we assumethat heterogeneity effects due to cell-cell variation or to variation in rates of tmnscript or to experimental artifacts are small, then a rational basis of the Kepes semi-logarithmic plot is to assumethat a messengercan be inactivated with constant probability after each successive ribosome starts to translate the message.This could be mechanistically understood on the adoption of the “killerribosome” hypothesis of David Schlessinger (1970, personal communication-see
16
A. L. KOCH
Kuwano, Schlessinger & Apirion, 1970). Its limitations and other models will be appraised below. On this basis, a messagewould function until a ribosome of a special classwas engaged which would destroy RNA as it translated it. On this basis, many messageswould be inactivated after only a few ribosomeshad translated them, but some would produce an inordinately large number of polypeptide copies. On this model, the number of chains per messagewould approximate to a simple Poisson distribution. The solution to the problems of enzyme production for the three induction regima are shown graphically in Figures 4 to 7 fitted to the experimental data as solid lines. The formulae are : for step induction, Z=B
Ost
for the run-off case the equation given immediately above applies until t = t,, + 6, then,
-’ e-k(&6 z=B+r(to - keta’ for the third case of pulse induction for a duration r, Z=B
o
2
=
B
-
~(6
+
2
=
B
+
,,.,,. _
k)
+
re-+%k
rt
+
f
e-‘t
-
6)k
a
y
As for the previous model, we omit the derivation of these equations, since they are simply repetitious integrations of the polypeptide curves from single initiation events. The derivations will be sent to those requesting them. It can be seen that in all three casesthe second model fits the data somewhat better than the first. Although in principle the elementary wave experiment is most sensitive to the life table of the messengermolecules, valid criticism of all available data for this kind of experiment can be made : one cannot be certain that induction is not abruptly terminated or the rate of protein synthesis is altered by the filtration procedure or the drug used. Similarly, for the run-off experiment, the termination of initiation cannot be consideredeffectively mild and abrupt enough in any single case. The important point is that the step-induction experiment is not subject to the limitations of how quickly or mildly the inducer can be removed and unambiguously established that a fixed life-span model is inconsistent with experiment. Therefore, although models requiring that each copy of messagebe read by a fixed number of ribosomes(time-bomb models)have not been popular for a few years, we believe that
APPENDIX
17
the present data and calculations are the first to exclude them for the case of /3-galactosidase in E. cdi. We add that this question, to our knowledge, is completely unresolved in any eucaryotic system. In fact the equation is given above for the fixed life-span model in the hope that it will be tested in such studies. There is a second circumstance which results in the identical time-course of enzyme production as does the killer-ribosome model. This is the case where an inactivation event can take place anywhere along the length of the message as well as at any time after that portion of the message has been synthesized. The recent finding that material hybridizing as lac messenger has a longer half-life than the entity coding j?-galactosidase synthesis is consistent with this model (Schwartz, Craig & Kennell, 1970). Ribosomes that have gone past the point of scission would proceed to completion, but ribosomes attached at the time of inactivation proximally to the scission point would fail to produce completed functional polypeptide chains. This model appears more plausible since it has recently been shown (Nath & Koch, 1970,1971) that there is a fraction of the cells total peptide synthesis which is made roughly in fixed proportion of the total peptide bond synthesis, that is then degraded with a half-life of less than one hour and produced in amounts that would be expected from the random scission of messenger chains producing incomplete proteins. The rate-constant for degradation of this material is the same as for peptide chains artificially terminated by puromycin. To find the life expectancy for this model, we must compute the combined probability of the message remaining intact while the &h ribosome traverses it but becoming cut before the (n + 1)th ribosome can complete its transit. This can be done easily by reference to Figure 8. In this Figure, time is represented horizontally and position along the message vertically. If chain cuts are completely random, then the chance of inactivation at any point in the field to the right of the solid line depicting messenger synthesis is as likely as at any other point in the field; this way to view the problem
Time
8. Model for random inaativation of messenger. The synthesis of the messenger is indicated by the solid line. Different translation events are designated by the dashed lines. The space-time field can be considered to be divided into little rectangles of duration T/8 and vertical distance along a message of X/s, where T is the time for total translation and X is the entire length of message. For this diagram the rate of transcription of messenger has been assumed equal to that of translation (i.e. the step-time for transcription has been set at 3 times the step-time for translation). FIG.
2
18
A. L. KOCH
was suggestedby Dr George Minty. Let us designate ~1as the probability of inactivation per unit time and per unit distance. Then let us divide the field into little rectangles of duration l/a of the transcription time, T, and of length l/s of the length of the message,X. By making s larger and larger we will be able to consider sufficiently small rectangles so that the chance of two inactivation events is negligible. The chance of an inactivation event is t.~(T/s) (X/s). The chance of surviving in any particular rectangle is (1 - p (T/s) (X/s)). The probability of the first translation event being successfulis this expression raised to a power which is the number of rectangles between the solid and the nearest parallel dashed line. Therefore, the probability of at least one polypeptide being completed is [l p(X/s) (T/s)j@ AT/T, where AT is the time between initiation of transcription and the first event initiating translation’. If the sameinterval exists between the attachment of any successivepair of ribosomes, then the probability of n polypeptide chains being produced, but not the (w + 1)th one is:
This can be recast so that the sameexpressions inside the parenthesis also appear in the exponent:
In the limit as s gets larger and larger, this expression can be simplified by the definition of e as the limit of (1 + H)l’H as H approaches zero. The final expression for P(n) becomes: P(n) = e-PnXAT[l - e-@XAT]. Since the quantity in brackets is constant and nAT is the time after initiation of transcription that the last successfulribosome starts its transit, this model predicts results indistinguishable from the previous model. Both can be written: P(n) = (1 - P)” (P) where P is the probability of any ribosome successfully transversing a previously intact message.The secondmodel has the advantage that no special classof ribosome need be postulated. If this random soissionof the messengerRNA proves to be the correct mode of messengerdecay, then it can easily be elaborated to include those cases where transcription and translation do not take place at the samerate. It is also easy to elaborate the treatment to include regulatory regions which may be transcribed but not translated. Also, the treatment can be expanded to include a separate contribution from ribosomesmoving along a poly-cistronic messageand those attaching at the beginning of a later message.
Alpers, Baker, Boezi, Bremer,
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