- Email: [email protected]
Stochastic processes in messenger RNA turnover
J. theor. Biol. (1974) 48, 161-171
Stochastic Processes in Messenger RNA. Turnover HAYWOOD BLUM Drexel University, Philadelphia, Penlzsylvania 19104, U.S.A. (Received 9 November 1973)
It has been proposed that mRNA stability in Escherichia coIi is enhanced by association with ribosomes and that failure of ribosome initiation into polysomes results in message inactivation. This hypothesis is examined with the aid of a simple steady queuing model from which mRNA lifetimes and other cell parameters may be calculated. Agreement with experimentally determined lifetimes is good. 1. Introduction The mechanisms by which information stored in nucleic acid is transcribed and translated into protein are very complex, involving many biochemical steps. Each step is a chemical reaction whose rate can be described by a deterministic set of equations when the concentration of reactants is high enough so that fluctuations in concentration are negligible. This is usually the case in biological reactions. However, when we deal with the small number of macromolecules, enzyme complexes and ribosomes involved in protein synthesis it becomes important to consider the fluctuations in some detail. In most instances fluctuations in reaction rates will lead to trivial consequences such as the loss in synchrony in the phases of the cell cycle of cells grown in culture. Qther cases, such as the one considered here, in which timing of reactions is critical, can lead to choices of alternate biological pathways. The transcription, translation, inactivation and degradation of certain mRNAs in Escherichia coli has been experimentally examined in some detail. Most of the recent experimental information has been summarized by Apirion (1973). He proposes with others (Singh, 1973) that the critical step for mRNA functional inactivation occurs at the ribosome initiation site. Unless a ribosome is available for initiation (and is also able to actually accomplish initiation) a signal for inactivation is given which prevents any new ribosome from initiating thereafter. The mRNA is thus gradually emptied of ribosomes which are already in the translation process as they reach the 3’ terminus. Chemical degradation of the mRNA is a separate T.B.
161
11
162
H.
BLUM
process from inactivation with a different characteristic lifetime (Blundell, Craig & Kennell, 1972). This hypothesis for inactivation supposes that the very important and necessary elimination of used mRNA is essentially unregulated in E. coli depending only on a random fluctuation either in local concentration of ribosomes at the 5’ end of mRNA or on the initiation process. In this article we examine the possibility that fluctuations in the ribosome concentration at the initiation site can lead to the observed lifetimes for mRNA. We also explore other consequences of this picture. Previous applications of stochastic models to protein synthesis have dealt with stable mRNA (Gordon, 1969; MacDonald & Gibbs, 1969). Other protein synthesis models (Singh, 1969) have assumed a lifetime for mRNA but have not analyzed the mechanisms. 2. Queuing Model The stochastic model used to describe the hypothesis outlined above is a simple steady state single queue. The mRNA is considered to be the servicing station with ribosomes as the customers. Newly made ribosomes or ribosomal subunits plus recycled ribosomes released from completed mRNA form a pool. We assume that pooled ribosomes randomly queue up to all uninactivated mRNA so that arrivals will follow the Poisson distribution with average rate 1. Other models are possible and even likely. If ribosomes remained pooled until they initiated on a mRNA we would have a set of parallel queues with much greater efficiency. It is unlikely that a single pool would suffice since the mRNAs are spatially separated with the cell. Finite ribosomal diffusion rates prevent a random selection from a single pool, However, subsets, that is subpools feeding a limited number of mRNAs, are not out of the question. The choice of the most realistic model is partly an experimental problem. Once attached, ribosomes are serviced at an average rate p. The distribution in the rate of servicing, while affecting the average queue length, does not perturb the probability P(O), that no ribosomes will be available for initiation. This probability is given by (Morse, 1958) P(0) = 1 --I/p. (1) Since inactivation is assumed to occur when no ribosomes are available for initiation, the mRNA inactivation halflife z is related to P(0); in fact, In 2/r = @(O), (2) where s is a number slightly greater than unity. This takes into account some inactivations occurring even in the presence of ribosomes. It is consistent with the idea of a chemical inactivation with the attachment site competed for by
STOCHASTIC
PROCESSES
IN
MRNA
163
TURNOVER
the ribosome and the inactivator. The ribosome attachment rate is much greater than the inactivator attachment rate. Another way of putting this argument is that ribosomes do not attach with 100% efficiency even when they are available, leaving the attachment site susceptible to inactivation. Combining equations (1) and (2), In 2jz = ps(l - ;l/p). The main job left is to calculate ,J and ,u. This is done in the following sections. The queue size will depend on the specifics of the servicing. Each step in the translation process must be accomplished sequentially. Queuing is only possible for the ribosome initiation step. If each step in translocation takes the same time on the average, the Erlang distribution would be an appropriate description of the servicing. This distribution tends toward a uniform rate of servicing for many steps. The average queue length(n) for regular servicing is
-n//L)-‘.
This queue length is approximately one-half the length expected for a situation where a single step determines the average step rate ,u. TABLE
Parameters
1
of the queuing model Queue
pm 1 -UP 0.800 0.900 0.950 0.960 o-970 0.980 0.990
0.200 0~100 0.050 0.040 0.030 0.020 0.010
Exponential service (Viu>“(l - 44 - 1
18.05 23.04 31.36 48.02 98.01
(~/,a1
length Regular - x4
service - Yl -
VW
1.92 4.45 9.48 11.98 16.15 24.49 49.50
3. mRNA Populations In order to determine the average arrival and servicing rates used in the queuing model it is necessary to know the distribution in the mRNA population in the various phases of its life cycle. The processes involved are illustrated in Fig. 1, in which we consider a mRNA whose completed length is L nucleotides. Transcription begins after RNA polymerase attaches to DNA (not shown). As the polymerase steps
164
H. BLUM
(a)
Queues
Messages
“,“;I$
I
000
2 3
0
4 5
(b) 01 2
000
3
0
4 5
(cl .I 2 3 4 5
FIG. 1. Illustration of some of the various processes involved in the mRNA life cycle. Open circles represent ribosomes, triangles represent RNA polymerase, squares are inactivation mechanism preventing new ribosome initiation. mRNA is represented by the horizontal line. (a) Shows five mRNAs just before a step. Message 1 is completed and uninactivated, 2 is nascent and uninactivated, 3 and 5 are nascent and inactivated, 4 is initiated. (b) After a step but before ribosomes in queues attach. Message 1 releases a ribosome to the pool, 5 is now complete. (c) After initiation of ribosomes. Message‘1 is now inactivated, 4 is nascent and uninactivated.
along, a ribosome attachment site is formed on the nascent mRNA. This occurs at an average rate ,u so that the ribosome step time is p-r. This is the time for an attachment site to be completed and is, of course, somewhat longer than a single nucleotide attachment time. At this moment a ribosome can attach if it is available, creating a nascent uninactivated mRNA, or the message is still-born. After each step time this choice must be made anew. If the message is inactivated after one or more ribosomes are attached, it is considered to be nascent and inactivated. The nascent messages continue to
STOCHASTIC
PROCESSES
IN mRNA
TURNOVER
165
grow until they are completed. At this time ribosomes having completed the translation cycle are free to leave the message and rejoin the pool. The amino acid chain elongation rate for the protein coded by this message is R and the nucleotide chain elongation iate is 3R. The time to transcribe this message, tT, is t, = L;/3R. (5) Figure 2 presents the schematic life cycle of a mRNA. An uninactivated mRNA is one upon which ribosomes are able to initiate and translate the codons into polypeptides. Inactivated mRNA still contains translating
FIG.
2. Phases of mRNA life cycle and transition rates. Symbols are defined in the text.
ribosomes but does not initiate new ones. Nascent mRNA contains ribosomes but has not been completely transcribed from its DNA hybrid. Nascent mRNA does not, therefore, release ribosomes whereas completed mRNA releases ribosomes as they reach the 3’ termination position. As ribosomes leave inactivated mRNA, it gradually empties. Chemical degradation can begin with inactivation but it is not directly coupled to inactivation since it has its own halflife (Kennel1 & Bicknell, 1973). The inactivation rate is given by dN,/dt = kN, (6) where Nu is the number of uninactivated mRNA and k = In 2/r. Initiated mRNA is converted to nascent uninactivated mRNA by picking up a ribosome. After a time tT has elapsed, the mRNA is completed. During
166
H.
BLUM
this time a fraction f, is inactivated. Of the nascent uninactivated mRNA initiated at time (t- t& only a fractionfu survive uninactivated at time t. f”t-.h = 1. (7) From equation (6) it is clear that fu = exp (- kt,). (8) We consider two cases below. First, a no-growth situation (or alternatively the case where all populations are per genome equivalent) and second, exponential growth. The second case, in which the cell doubling time is explicitly treated, includes the first. Nevertheless, we prefer to do the calculation in stages. 4. No-growth Case For no-growth, we assume that the mRNA population in each of the five categories remains constant and the rate at which mRNA is initiated is constant. The rate equations are [dN(t)/dt]i,itiated = v = constant,
dN.dt)ldt = 0 = f”CdN(l>jdtlinitiatea
-N&t) *k-f”CdN(t-t*)/dilinitiated *A, dNdt)/‘dt = 0 = kNdt)-f,f,lIdN(ttdldtlinitiated, dNdt)ldt = 0 = fvf”lIdN(t- t*)idtlinitiated-kWCU(t),
(9)
dNdt)/dt = 0 =f,fI~d~i(t-tT)ldt]initiated-kNu(t-tT)+kNc"(t), where NW, NM, NW, N,-, are the numbers of mRNA which are nascent uninactivated, nascent inactivated, complete uninactivated and complete inactivated, respectively. f, is the fraction of initiated mRNA which successfully attaches a ribosome. From the preceding discussion, it is clear that f, = exp C-k/14. (10) In the case we are presently considering,
CdN(t-tT)/dtlinitiated= [dN(t)/‘dtlinitiated-
(11)
Using equations (9) and (11) we find
NNU(~)= f&k-'v N,,(t) = f,fuk-'v
(12)
and N, = N,,+N,,
= vf,k-?
(13) To find NC, and NNI we use the following arguments. From equations (6) and (13), the number of mRNAs entering inactivation/time is dNjdt = vJ;,. All those entering in the time interval (t, t + tT) will still be in this category at
STOCHASTIC
PROCESSES
time t + t,. Other inactivated completely emptied at ti-t,. mRNA, N,, is given by
IN
mRNA
167
TURNOVER
mRNA entering inactivation Therefore, the total number
earlier will be of inactivated
rftT
N, =
vf, dt = vf, tT. (14) * Similarly for nascent mRNA. The rate at which nascent mRNA appears is dN/dt = vfV. mRNA initiated before time t will already be completed at t + t,. mRNA initiated in the interval (t, t + fr) will still be nascent at t+ t,. Therefore the total nascent mRNA, iVN, is NN = NNU + N,, = vf, tT. (15) From equations (12), (14), and (15), NN, = vfVh-fi~-l), N,,
j
(16)
= vf,fIk-‘3
and NC = N,,+N,,
= vf,k-’
(17)
and N = NNu+NCU+NNI+NC,
= vfV(tT+k -‘).
(18)
The above expressions for the mRNA populations are not completely accurate because equation (6) and the rate equations assume that inactivation can occur in a continuous fashion instead of a discrete inactivation at each step. If we assume inactivation occurs only during a step, the expressions become NNU N, Nc NC,
= = = =
vf,KIU -f,fuNl-fvl-‘3 vf, tT, vf,FIW-fX1, vf,f&-‘(1 -fJ-‘3
NU
= vfW1(l+fu+.Lfu>(l
(19) -fd-‘.
Equations (19) reduce to the previous expressions in the appropriate k/p 4 1.
limit,
5. Exponential Growth Case The rate equations are the same as in section 4 except that dNildt = Ni k, (20) instead of zero, where i represents the various categories of mRNA and kD is the celi doubfing rate; that is, k,, = In 2/r,, where ru is the time for one cell division. Additionally, V(t)
=
V(0)
eXp
(k, t)
(21)
168
H. BLUM
and
CdNtt- tT)ldtlinitiated = fD[dN(t)ldtlinitiated,
where
fD = exp t - kD f4. Following the arguments of the previous section we find, N,,(t) Nmtt) N,(t) NI(t)
= v(O.LfufXk + kd- ‘, = vtOf,U -f-A,>@ + kd- ‘, = N,,(t) + Niw(O = v(Of,(k + kd- l, t+tr = l kN,(t) dt = v(t)iY(j; i - l)(l+ k,/k)- ‘k&
Nidt)
= N:~(t) + N&t)
(22)
(23)
(24)
= vtOfvtf,- ’ - l)kii ‘,
1 - W, ’ - (1 -fuh>(k + kd- II, NM(~) = vtMKf; NC,(~) = v(Of,tk+ kd- ‘(2 -hi’ ’ -fu&>, N,(t) = N,,(t) + N,,(t) = v(Ofy@ -fi ‘>tk + M- ’ N(t) = v(tl.Ltk+ kd- ‘L-1+tf; ’ - ~WkJl. In order to evaluate the predictions of these equations we have taken typical values for the various cell parameters from the literature to substitute into equations (24). Table 2 and Table 3 contain experimental and calculated values of these parameters which are used in equations (24) to arrive at the TABLE 2
Typical valuesfor cell parameters Symbol
Definition
Value - _.--
M N R F L
Number of ribosomes Number of mRNA Amino acid chain elongation rate Fraction of ribosomes in polysomes mRNA length in nucleotides
30,OOop 7502
16 set-lg o-59
9Wl
7 Watson (1970). $ Assuming 20 ribosomesjmessage and 15,ooO ribosomes in polysomes. 0 Harvey (1973) for rD = 60 min. //This mRNA would code for a polypeptide of 300 residues with a molecular weight of approximately 36,000.
results listed in Table 4. Assuming three possible mRNA haltlives we see that most of the mRNA tends to be complete and uninactivated but that there are appreciable amounts of both nascent uninactivated and complete inactivated mRNA.
STOCHASTIC
PROCESSES
IN mRNA
TURNOVER
169
TABLE 3 Cell growth and transcription parameters Symbol k ;: P
Definition Transcription In 2/t, exp (-kdd MF1t-J’
Value? 18.75 see 1.925 x low4 see-I$ 0.9964 1.067 set- 1
time = L/3R
t Using values given in Table 2. $ Assuming a 60 min doubling time. TABLE 4
mRNA populations Symbol
Definition
z (set) mRNA halflife k (~7 - ‘) In 2/s ” exp (-kfd
;
Y (set-I) h&/N Ncr/N N&N NNI/N
1-fv ew (--k/.4 Equation Equation Equation Equation Equation
(24) (24) (24) (24) (24)
Value? 30 0.02310 O-6484 0.3516 0.9786 12.454 0.4507 0.2444 O-2469 0.0581
60 0.01155 0.8052 0.1948 0.9892 7.317 O-6594 0.1595 O-1625 0.0186
120 0.005776 04974 0.1026 0.9946 4.053 O-8052 0.0920 0.0953 0.0056
t Using values given in Tables 2 and 3.
6. Message Lifetime The average arrival rate, I, for ribosomes at any uninactivated message is derived from the average rate at which ribosomes are released from completed mRNA plus the rate at which new ribosomes are manufactured. The latter rate, dM/dt, is dM/dt = k,M(?) (25) where M(t) is the number of ribosomes present in the cell at time t. Studying Fig. 1, we see that the rate at which ribosomes arrive at the pool is [,uN,+dM/dt]
and theseribosomes are shared by [N,+ v/p] ribosome initia-
tion sites so that (26)
170
H.
BLUM
From equations (1) and (2) P(0) = 1--n/p = 1 -
NC+/?
dM/dt
NU + VIP Let x = k/p. Then from equations (24), (25) and (27)
= k/p.
(27)
’
(28)
x_ = 1 _ vf,(2-f,-%k+k,J1+Mk,p-’ s vf,( k + k,) - ’ + v/p
Equation (28) can be rewritten s/,u) = eex[ - x + s(& ’ - l)]
x2 + x(k,,/p - s - Mk, s/v) - (1 + Mk,/v)(k,
where we used the fact thatf,
(29)
= emx.
(A)
NO-GROWTH
LIMIT
For kD + 0, equation (29) reduces to x2-sx
= edx(-x),
(30)
with solution for x $ 1 x = [2(s- l)]? (31) This interesting result in the no-growth limit says that if ribosome attachment efficiencies are perfect (s = l), then mRNA lifetimes will be infinite (x = 0). This would result in v = 0 also so that with no mRNA being made the total number will remain constant (and finite). From Table 2, we can calculate the average queue length(n). (n)
= (1 -F)M/N
= 20.
(32) Consulting Table 1, choosing regular servicing, for (n) = 20, I/p N 0.975. For this value of n/,u, using equations (3) and (31) we have for s s = [l -(l-2(1 -~/j.Q2)“2](1 +,u)-” = 1.00031264 (33) which results in a prediction for z from equation (3) of z = 25.98 sec. We see that the ribosome attachment efficiency is, in fact, very high, since s is only slightly greater than one. (B)
Returning
--s[l+Mk,v-’
x(l+k,p-+
EXPONENTIAL
GROWTH
LIMIT
to equation (29), since x < 1, we find
-l>
= 4a?-~> +kD,u-l(l-i-Mkg-l)]
(34)
and with equation (3), s=
(l-l/~)(l+kD~-l)-(f~l-l)-k,~-l(l+Mk,v-l)
(l-/Z/,u)[l+Mknv-l-(f,-‘-l)]
*
(35)
STOCHASTIC
PROCESSES
IN
mRNA
TURNOVER
171
With the values given in Tables 2 and 3 this reduces to s z (l+Mk~v-l)-l
(36)
and x = (l-~//L)(lfMknv-l)-l
(37) with consistent solution [using equation (24)] r = 42 see, v = 9.6 set-‘, s = 0.62. This result is somewhat surprising since s is less than unity; that is, even if no ribosomes are available for initiation, inactivation may not occur. This suggests that the number of inactivating enzymes is lagging behind the production of mRNA. This could be a possible regulatory device. 7. Discussion
The predicted halflives in section 6 are for an average mRNA. Agreement with experiment (Blundel, Craig & Kennell, 1972) is good. If without changing any of the average cell parameters, we have a message with higher p, equation (26) for 1 remains unchanged so that P(0) increases; that is, the lifetime would shorten. If, as reported (Harvey, 1973), ribosomal protein mRNA has a significantly reduced translation rate, its lifetime should be much higher than for other mRNA. Our results suggest that mRNA turnover can be understood on the basis of random inactivation of message whose ribosome initiation sites are empty due to fluctuations in the local ribosome concentration. However, some regulation may occur through inactivation enzyme concentration control. The steady state model we have employed is not, in its present form, able to handle transient responses to changes in growth conditions or inducers. This defect can be corrected and will be the subject of another article. This work was supported in part by NIH Special Research Fellowship GM54114-01.
1 FQ3
REFERENCES APIRION, D. (1973). Molec. gen. Genet. 122, 313. BLUNDELL, M., CRAIG, E. & KENNELL, D. (1972).Nature New Biol. 238,46. GORDON, R. (1969). J. theor. Biol. 22, 515. HARVEY, R. J. (1973).J. Bact. 114, 287. KBNNELL. D. & BICKNELL. I. (1973). J. molec. Biol. 74. 21. MACDOI&LD, C. T. & G&s,‘J. H: (1969). Biogolytneis 7, 707. MORSE, P. M. (1958).Queues, Inventories and Maintenance. New York: John Wiley and Sons. Inc. SINGH,.~. N. (1969). J. theor. Biol. 25, 444. SINGH. U. N. (1973).J. theor. Biol. 45. 553. WAXEN, J. B.‘(19fO). Molecular Biology of the Gene, 2nd edn,p. 85. New York: W. A.
Benjamin,Inc.
Stochastic Processes in Messenger RNA. Turnover HAYWOOD BLUM Drexel University, Philadelphia, Penlzsylvania 19104, U.S.A. (Received 9 November 1973)
It has been proposed that mRNA stability in Escherichia coIi is enhanced by association with ribosomes and that failure of ribosome initiation into polysomes results in message inactivation. This hypothesis is examined with the aid of a simple steady queuing model from which mRNA lifetimes and other cell parameters may be calculated. Agreement with experimentally determined lifetimes is good. 1. Introduction The mechanisms by which information stored in nucleic acid is transcribed and translated into protein are very complex, involving many biochemical steps. Each step is a chemical reaction whose rate can be described by a deterministic set of equations when the concentration of reactants is high enough so that fluctuations in concentration are negligible. This is usually the case in biological reactions. However, when we deal with the small number of macromolecules, enzyme complexes and ribosomes involved in protein synthesis it becomes important to consider the fluctuations in some detail. In most instances fluctuations in reaction rates will lead to trivial consequences such as the loss in synchrony in the phases of the cell cycle of cells grown in culture. Qther cases, such as the one considered here, in which timing of reactions is critical, can lead to choices of alternate biological pathways. The transcription, translation, inactivation and degradation of certain mRNAs in Escherichia coli has been experimentally examined in some detail. Most of the recent experimental information has been summarized by Apirion (1973). He proposes with others (Singh, 1973) that the critical step for mRNA functional inactivation occurs at the ribosome initiation site. Unless a ribosome is available for initiation (and is also able to actually accomplish initiation) a signal for inactivation is given which prevents any new ribosome from initiating thereafter. The mRNA is thus gradually emptied of ribosomes which are already in the translation process as they reach the 3’ terminus. Chemical degradation of the mRNA is a separate T.B.
161
11
162
H.
BLUM
process from inactivation with a different characteristic lifetime (Blundell, Craig & Kennell, 1972). This hypothesis for inactivation supposes that the very important and necessary elimination of used mRNA is essentially unregulated in E. coli depending only on a random fluctuation either in local concentration of ribosomes at the 5’ end of mRNA or on the initiation process. In this article we examine the possibility that fluctuations in the ribosome concentration at the initiation site can lead to the observed lifetimes for mRNA. We also explore other consequences of this picture. Previous applications of stochastic models to protein synthesis have dealt with stable mRNA (Gordon, 1969; MacDonald & Gibbs, 1969). Other protein synthesis models (Singh, 1969) have assumed a lifetime for mRNA but have not analyzed the mechanisms. 2. Queuing Model The stochastic model used to describe the hypothesis outlined above is a simple steady state single queue. The mRNA is considered to be the servicing station with ribosomes as the customers. Newly made ribosomes or ribosomal subunits plus recycled ribosomes released from completed mRNA form a pool. We assume that pooled ribosomes randomly queue up to all uninactivated mRNA so that arrivals will follow the Poisson distribution with average rate 1. Other models are possible and even likely. If ribosomes remained pooled until they initiated on a mRNA we would have a set of parallel queues with much greater efficiency. It is unlikely that a single pool would suffice since the mRNAs are spatially separated with the cell. Finite ribosomal diffusion rates prevent a random selection from a single pool, However, subsets, that is subpools feeding a limited number of mRNAs, are not out of the question. The choice of the most realistic model is partly an experimental problem. Once attached, ribosomes are serviced at an average rate p. The distribution in the rate of servicing, while affecting the average queue length, does not perturb the probability P(O), that no ribosomes will be available for initiation. This probability is given by (Morse, 1958) P(0) = 1 --I/p. (1) Since inactivation is assumed to occur when no ribosomes are available for initiation, the mRNA inactivation halflife z is related to P(0); in fact, In 2/r = @(O), (2) where s is a number slightly greater than unity. This takes into account some inactivations occurring even in the presence of ribosomes. It is consistent with the idea of a chemical inactivation with the attachment site competed for by
STOCHASTIC
PROCESSES
IN
MRNA
163
TURNOVER
the ribosome and the inactivator. The ribosome attachment rate is much greater than the inactivator attachment rate. Another way of putting this argument is that ribosomes do not attach with 100% efficiency even when they are available, leaving the attachment site susceptible to inactivation. Combining equations (1) and (2), In 2jz = ps(l - ;l/p). The main job left is to calculate ,J and ,u. This is done in the following sections. The queue size will depend on the specifics of the servicing. Each step in the translation process must be accomplished sequentially. Queuing is only possible for the ribosome initiation step. If each step in translocation takes the same time on the average, the Erlang distribution would be an appropriate description of the servicing. This distribution tends toward a uniform rate of servicing for many steps. The average queue length(n) for regular servicing is
-n//L)-‘.
This queue length is approximately one-half the length expected for a situation where a single step determines the average step rate ,u. TABLE
Parameters
1
of the queuing model Queue
pm 1 -UP 0.800 0.900 0.950 0.960 o-970 0.980 0.990
0.200 0~100 0.050 0.040 0.030 0.020 0.010
Exponential service (Viu>“(l - 44 - 1
18.05 23.04 31.36 48.02 98.01
(~/,a1
length Regular - x4
service - Yl -
VW
1.92 4.45 9.48 11.98 16.15 24.49 49.50
3. mRNA Populations In order to determine the average arrival and servicing rates used in the queuing model it is necessary to know the distribution in the mRNA population in the various phases of its life cycle. The processes involved are illustrated in Fig. 1, in which we consider a mRNA whose completed length is L nucleotides. Transcription begins after RNA polymerase attaches to DNA (not shown). As the polymerase steps
164
H. BLUM
(a)
Queues
Messages
“,“;I$
I
000
2 3
0
4 5
(b) 01 2
000
3
0
4 5
(cl .I 2 3 4 5
FIG. 1. Illustration of some of the various processes involved in the mRNA life cycle. Open circles represent ribosomes, triangles represent RNA polymerase, squares are inactivation mechanism preventing new ribosome initiation. mRNA is represented by the horizontal line. (a) Shows five mRNAs just before a step. Message 1 is completed and uninactivated, 2 is nascent and uninactivated, 3 and 5 are nascent and inactivated, 4 is initiated. (b) After a step but before ribosomes in queues attach. Message 1 releases a ribosome to the pool, 5 is now complete. (c) After initiation of ribosomes. Message‘1 is now inactivated, 4 is nascent and uninactivated.
along, a ribosome attachment site is formed on the nascent mRNA. This occurs at an average rate ,u so that the ribosome step time is p-r. This is the time for an attachment site to be completed and is, of course, somewhat longer than a single nucleotide attachment time. At this moment a ribosome can attach if it is available, creating a nascent uninactivated mRNA, or the message is still-born. After each step time this choice must be made anew. If the message is inactivated after one or more ribosomes are attached, it is considered to be nascent and inactivated. The nascent messages continue to
STOCHASTIC
PROCESSES
IN mRNA
TURNOVER
165
grow until they are completed. At this time ribosomes having completed the translation cycle are free to leave the message and rejoin the pool. The amino acid chain elongation rate for the protein coded by this message is R and the nucleotide chain elongation iate is 3R. The time to transcribe this message, tT, is t, = L;/3R. (5) Figure 2 presents the schematic life cycle of a mRNA. An uninactivated mRNA is one upon which ribosomes are able to initiate and translate the codons into polypeptides. Inactivated mRNA still contains translating
FIG.
2. Phases of mRNA life cycle and transition rates. Symbols are defined in the text.
ribosomes but does not initiate new ones. Nascent mRNA contains ribosomes but has not been completely transcribed from its DNA hybrid. Nascent mRNA does not, therefore, release ribosomes whereas completed mRNA releases ribosomes as they reach the 3’ termination position. As ribosomes leave inactivated mRNA, it gradually empties. Chemical degradation can begin with inactivation but it is not directly coupled to inactivation since it has its own halflife (Kennel1 & Bicknell, 1973). The inactivation rate is given by dN,/dt = kN, (6) where Nu is the number of uninactivated mRNA and k = In 2/r. Initiated mRNA is converted to nascent uninactivated mRNA by picking up a ribosome. After a time tT has elapsed, the mRNA is completed. During
166
H.
BLUM
this time a fraction f, is inactivated. Of the nascent uninactivated mRNA initiated at time (t- t& only a fractionfu survive uninactivated at time t. f”t-.h = 1. (7) From equation (6) it is clear that fu = exp (- kt,). (8) We consider two cases below. First, a no-growth situation (or alternatively the case where all populations are per genome equivalent) and second, exponential growth. The second case, in which the cell doubling time is explicitly treated, includes the first. Nevertheless, we prefer to do the calculation in stages. 4. No-growth Case For no-growth, we assume that the mRNA population in each of the five categories remains constant and the rate at which mRNA is initiated is constant. The rate equations are [dN(t)/dt]i,itiated = v = constant,
dN.dt)ldt = 0 = f”CdN(l>jdtlinitiatea
-N&t) *k-f”CdN(t-t*)/dilinitiated *A, dNdt)/‘dt = 0 = kNdt)-f,f,lIdN(ttdldtlinitiated, dNdt)ldt = 0 = fvf”lIdN(t- t*)idtlinitiated-kWCU(t),
(9)
dNdt)/dt = 0 =f,fI~d~i(t-tT)ldt]initiated-kNu(t-tT)+kNc"(t), where NW, NM, NW, N,-, are the numbers of mRNA which are nascent uninactivated, nascent inactivated, complete uninactivated and complete inactivated, respectively. f, is the fraction of initiated mRNA which successfully attaches a ribosome. From the preceding discussion, it is clear that f, = exp C-k/14. (10) In the case we are presently considering,
CdN(t-tT)/dtlinitiated= [dN(t)/‘dtlinitiated-
(11)
Using equations (9) and (11) we find
NNU(~)= f&k-'v N,,(t) = f,fuk-'v
(12)
and N, = N,,+N,,
= vf,k-?
(13) To find NC, and NNI we use the following arguments. From equations (6) and (13), the number of mRNAs entering inactivation/time is dNjdt = vJ;,. All those entering in the time interval (t, t + tT) will still be in this category at
STOCHASTIC
PROCESSES
time t + t,. Other inactivated completely emptied at ti-t,. mRNA, N,, is given by
IN
mRNA
167
TURNOVER
mRNA entering inactivation Therefore, the total number
earlier will be of inactivated
rftT
N, =
vf, dt = vf, tT. (14) * Similarly for nascent mRNA. The rate at which nascent mRNA appears is dN/dt = vfV. mRNA initiated before time t will already be completed at t + t,. mRNA initiated in the interval (t, t + fr) will still be nascent at t+ t,. Therefore the total nascent mRNA, iVN, is NN = NNU + N,, = vf, tT. (15) From equations (12), (14), and (15), NN, = vfVh-fi~-l), N,,
j
(16)
= vf,fIk-‘3
and NC = N,,+N,,
= vf,k-’
(17)
and N = NNu+NCU+NNI+NC,
= vfV(tT+k -‘).
(18)
The above expressions for the mRNA populations are not completely accurate because equation (6) and the rate equations assume that inactivation can occur in a continuous fashion instead of a discrete inactivation at each step. If we assume inactivation occurs only during a step, the expressions become NNU N, Nc NC,
= = = =
vf,KIU -f,fuNl-fvl-‘3 vf, tT, vf,FIW-fX1, vf,f&-‘(1 -fJ-‘3
NU
= vfW1(l+fu+.Lfu>(l
(19) -fd-‘.
Equations (19) reduce to the previous expressions in the appropriate k/p 4 1.
limit,
5. Exponential Growth Case The rate equations are the same as in section 4 except that dNildt = Ni k, (20) instead of zero, where i represents the various categories of mRNA and kD is the celi doubfing rate; that is, k,, = In 2/r,, where ru is the time for one cell division. Additionally, V(t)
=
V(0)
eXp
(k, t)
(21)
168
H. BLUM
and
CdNtt- tT)ldtlinitiated = fD[dN(t)ldtlinitiated,
where
fD = exp t - kD f4. Following the arguments of the previous section we find, N,,(t) Nmtt) N,(t) NI(t)
= v(O.LfufXk + kd- ‘, = vtOf,U -f-A,>@ + kd- ‘, = N,,(t) + Niw(O = v(Of,(k + kd- l, t+tr = l kN,(t) dt = v(t)iY(j; i - l)(l+ k,/k)- ‘k&
Nidt)
= N:~(t) + N&t)
(22)
(23)
(24)
= vtOfvtf,- ’ - l)kii ‘,
1 - W, ’ - (1 -fuh>(k + kd- II, NM(~) = vtMKf; NC,(~) = v(Of,tk+ kd- ‘(2 -hi’ ’ -fu&>, N,(t) = N,,(t) + N,,(t) = v(Ofy@ -fi ‘>tk + M- ’ N(t) = v(tl.Ltk+ kd- ‘L-1+tf; ’ - ~WkJl. In order to evaluate the predictions of these equations we have taken typical values for the various cell parameters from the literature to substitute into equations (24). Table 2 and Table 3 contain experimental and calculated values of these parameters which are used in equations (24) to arrive at the TABLE 2
Typical valuesfor cell parameters Symbol
Definition
Value - _.--
M N R F L
Number of ribosomes Number of mRNA Amino acid chain elongation rate Fraction of ribosomes in polysomes mRNA length in nucleotides
30,OOop 7502
16 set-lg o-59
9Wl
7 Watson (1970). $ Assuming 20 ribosomesjmessage and 15,ooO ribosomes in polysomes. 0 Harvey (1973) for rD = 60 min. //This mRNA would code for a polypeptide of 300 residues with a molecular weight of approximately 36,000.
results listed in Table 4. Assuming three possible mRNA haltlives we see that most of the mRNA tends to be complete and uninactivated but that there are appreciable amounts of both nascent uninactivated and complete inactivated mRNA.
STOCHASTIC
PROCESSES
IN mRNA
TURNOVER
169
TABLE 3 Cell growth and transcription parameters Symbol k ;: P
Definition Transcription In 2/t, exp (-kdd MF1t-J’
Value? 18.75 see 1.925 x low4 see-I$ 0.9964 1.067 set- 1
time = L/3R
t Using values given in Table 2. $ Assuming a 60 min doubling time. TABLE 4
mRNA populations Symbol
Definition
z (set) mRNA halflife k (~7 - ‘) In 2/s ” exp (-kfd
;
Y (set-I) h&/N Ncr/N N&N NNI/N
1-fv ew (--k/.4 Equation Equation Equation Equation Equation
(24) (24) (24) (24) (24)
Value? 30 0.02310 O-6484 0.3516 0.9786 12.454 0.4507 0.2444 O-2469 0.0581
60 0.01155 0.8052 0.1948 0.9892 7.317 O-6594 0.1595 O-1625 0.0186
120 0.005776 04974 0.1026 0.9946 4.053 O-8052 0.0920 0.0953 0.0056
t Using values given in Tables 2 and 3.
6. Message Lifetime The average arrival rate, I, for ribosomes at any uninactivated message is derived from the average rate at which ribosomes are released from completed mRNA plus the rate at which new ribosomes are manufactured. The latter rate, dM/dt, is dM/dt = k,M(?) (25) where M(t) is the number of ribosomes present in the cell at time t. Studying Fig. 1, we see that the rate at which ribosomes arrive at the pool is [,uN,+dM/dt]
and theseribosomes are shared by [N,+ v/p] ribosome initia-
tion sites so that (26)
170
H.
BLUM
From equations (1) and (2) P(0) = 1--n/p = 1 -
NC+/?
dM/dt
NU + VIP Let x = k/p. Then from equations (24), (25) and (27)
= k/p.
(27)
’
(28)
x_ = 1 _ vf,(2-f,-%k+k,J1+Mk,p-’ s vf,( k + k,) - ’ + v/p
Equation (28) can be rewritten s/,u) = eex[ - x + s(& ’ - l)]
x2 + x(k,,/p - s - Mk, s/v) - (1 + Mk,/v)(k,
where we used the fact thatf,
(29)
= emx.
(A)
NO-GROWTH
LIMIT
For kD + 0, equation (29) reduces to x2-sx
= edx(-x),
(30)
with solution for x $ 1 x = [2(s- l)]? (31) This interesting result in the no-growth limit says that if ribosome attachment efficiencies are perfect (s = l), then mRNA lifetimes will be infinite (x = 0). This would result in v = 0 also so that with no mRNA being made the total number will remain constant (and finite). From Table 2, we can calculate the average queue length(n). (n)
= (1 -F)M/N
= 20.
(32) Consulting Table 1, choosing regular servicing, for (n) = 20, I/p N 0.975. For this value of n/,u, using equations (3) and (31) we have for s s = [l -(l-2(1 -~/j.Q2)“2](1 +,u)-” = 1.00031264 (33) which results in a prediction for z from equation (3) of z = 25.98 sec. We see that the ribosome attachment efficiency is, in fact, very high, since s is only slightly greater than one. (B)
Returning
--s[l+Mk,v-’
x(l+k,p-+
EXPONENTIAL
GROWTH
LIMIT
to equation (29), since x < 1, we find
-
= 4a?-~> +kD,u-l(l-i-Mkg-l)]
(34)
and with equation (3), s=
(l-l/~)(l+kD~-l)-(f~l-l)-k,~-l(l+Mk,v-l)
(l-/Z/,u)[l+Mknv-l-(f,-‘-l)]
*
(35)
STOCHASTIC
PROCESSES
IN
mRNA
TURNOVER
171
With the values given in Tables 2 and 3 this reduces to s z (l+Mk~v-l)-l
(36)
and x = (l-~//L)(lfMknv-l)-l
(37) with consistent solution [using equation (24)] r = 42 see, v = 9.6 set-‘, s = 0.62. This result is somewhat surprising since s is less than unity; that is, even if no ribosomes are available for initiation, inactivation may not occur. This suggests that the number of inactivating enzymes is lagging behind the production of mRNA. This could be a possible regulatory device. 7. Discussion
The predicted halflives in section 6 are for an average mRNA. Agreement with experiment (Blundel, Craig & Kennell, 1972) is good. If without changing any of the average cell parameters, we have a message with higher p, equation (26) for 1 remains unchanged so that P(0) increases; that is, the lifetime would shorten. If, as reported (Harvey, 1973), ribosomal protein mRNA has a significantly reduced translation rate, its lifetime should be much higher than for other mRNA. Our results suggest that mRNA turnover can be understood on the basis of random inactivation of message whose ribosome initiation sites are empty due to fluctuations in the local ribosome concentration. However, some regulation may occur through inactivation enzyme concentration control. The steady state model we have employed is not, in its present form, able to handle transient responses to changes in growth conditions or inducers. This defect can be corrected and will be the subject of another article. This work was supported in part by NIH Special Research Fellowship GM54114-01.
1 FQ3
REFERENCES APIRION, D. (1973). Molec. gen. Genet. 122, 313. BLUNDELL, M., CRAIG, E. & KENNELL, D. (1972).Nature New Biol. 238,46. GORDON, R. (1969). J. theor. Biol. 22, 515. HARVEY, R. J. (1973).J. Bact. 114, 287. KBNNELL. D. & BICKNELL. I. (1973). J. molec. Biol. 74. 21. MACDOI&LD, C. T. & G&s,‘J. H: (1969). Biogolytneis 7, 707. MORSE, P. M. (1958).Queues, Inventories and Maintenance. New York: John Wiley and Sons. Inc. SINGH,.~. N. (1969). J. theor. Biol. 25, 444. SINGH. U. N. (1973).J. theor. Biol. 45. 553. WAXEN, J. B.‘(19fO). Molecular Biology of the Gene, 2nd edn,p. 85. New York: W. A.
Benjamin,Inc.