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A dynamic model for in vivo virus replication
J. theor. Biol. (1981) 90,265-281
A Dynamic
Model for in uivo Virus Replication?
JOHN E. MACCARTHY
AND JOHN J. KOZAK
Program in Biophysics and Biochemistry, and Radiation University of Notre Dame, Notre Dame, Indiana 46556, U.S.A. (Received
30 January
1980, and in revisedform
Laboratory,
23 December
1980)
In this paper we present a dynamic model of in viva virus replication. Kinetic equations are formulated to describe the overall process of replication and then analyzed using a “synergetic” approach. First the importanceof a rate-limiting substrateis taken explicitly into account, and secondly the coupling between the processesconsidered (translation, replication and assembly)is strictly preserved;the analysisitself is carried out in the linear regime. We treat the problems of defective-particle infections, standard-virus infections, inhibition of cellular synthesis, and the caseof co-infected cells. The various parametersof the model (initial
cellular concentrations, rate constants) are specified using existing experimental data and the full (numerical)consequences of the mode1are explored in detail and compareddirectly with experimentsby Baltimore, Cole and others on viral systems.Quite surprisingly, the simple model developed in this paper is able to account qualitatively, and occasionally quantitatively, for the behavior observedexperimentally for each of the problemscited above. 1. Introduction The replication of poliovirus in Hela cells has been studied extensively in recent years and is one of the best understood systems in virology. In this paper we present a relatively simple dynamical mode1 whose elaboration allows us to organize, within a single theoretical framework, a variety of experimental data on replication of standard (S-) poliovirus (Baltimore, 1969; Baltimore, Girard & Darnell, 1966; Darnell & Levintow, 1960; Cole & Baltimore, 1973a,b), replication of polio defective interfering (DI-) particles (Cole & Baltimore, 1973a,b; Cole, 1975), inhibition of cellular synthetic processes (Holland & Peterson, 1964; Holland, 1964; Contreras et al., 1973), and the effects of co-infection of cells by both S-virus and DI-particles on the yields of each (Cole & Baltimore, 1973; Cole, 1975). In t Theresearch described herein was supported in part by the Office of Basic Energy Sciences of the Department of En_ergy. This is Document No. NDRL-2074 from the Notre Dame Radiation Laboratory. 265
0022-5193/81/100265+17$02.00/0
@ 1981 Academic Press Inc. (London) Ltd.
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dealing with these problems, our main concern will be the dynamic aspects of replication. As such, the work may be distinguished from the earlier contributions of Lauffer (1975) and Casper (1963) on the thermodynamics of virus assembly. And, although Butterworth & Rueckert i1972) developed a kinetic model for EMC protein cleavage, to our knowledge there has been presented no comprehensive model for the overall processof viral infection in viva. The spirit of our approach is similar to that taken by the Gijttingen school [Eigen (1971) and Eigen & Schuster (1977a, 1978h,c)] in the theory of competing biological speciesor that of the Bruxelles school [Babloyantz & Sanglier, (1972)] in their work on the regulation of the lac operon, in that the co-operative aspects of the problem are recognized and stressed. The mathematical techniques for dealing with such problems are widely known, and have proved successful in addressing problems in a variety of different fields (Haken, 1977; Nicolis & Prigogine, 1978). Although the present model is admittedly naive, it does have the advantage that the co-operative features of virus replication can be traced using simple, analytic arguments; moreover, it will be seen that the model is surprisingly successful in accounting for many of the characteristic trends observed in the specific systems rited above. At the very least, it is hoped that the present study may stimulate the development of more sophisticated mathematical models for this important problem. In section 2 we focus on the problems of DI-particle infection and S-virus infection. Then, in section 3 we take up inhibition of cellular synthesis and the problem of co-infected cells. In section 4 the limitations of the model are discussed. The final section is given over to a statement of the principal conclusions that may be drawn from this work. 2. The Model The goal of this section is to develop the simplest possible dynamic model for the post-infection processesof virus replication. Each process is defined by a characteristic class of substrates and products. In the spirit of Eigen (1971) and Babolyantz & Sanglier (1972), a classof substrates or products will be represented by a single substrate or product respectively, -and complex biochemical polymerizations will be idealized as a single step reaction of first order in each of the reactants. What is proposed then, is that the abstract dynamic structure of the real viral replication process can be modeled by the simplest idealized chemical system of identical dynamic structure. It will be seen that the analysis of this abstract system predicts kinetic behavior that is in qualitative agreement with experimental data on the real system.
MODEL
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REPLICATION
Experimentally poliovirus replication is known to involve three processes once a cell is infected: translation, replication and assembly (Baltimore, 1969). The process will be idealized as follows. (1) Translation. R (viral RNA) and N (cellular nucleotides) act as catalysts? for the irreversible reaction that changes A (cellular amino acids) into P (viral protein). A+R+Nfi:R+N+P. (2) Repiication. R acts autocatalytically
in an irreversible
reaction with N
as its substrate. R+N32R. (3) Assembly. The terminating irreversible reaction in which P and R act as substrates for the production of V (virions).
R+P>V. It should be noted that all the above reactions are irreversible. In the first two cases it is because phosphate bonds are broken, while in the last case it is because the protein cleaves. The kinetic equations that follow from this model are: /i = -klRNA, ni= -k*RN, P= klRNA-k3RP,
(1)
ti = kzRN-k3RP, 3= kjRP. The dynamic structure of this idealized system is quite similar to that of the simplest general system in which a single molecule (R) acts simultaneously in separate reactions as a catalyst, an autocatalyst, and a substrate. Such a system would be represented as follows:
R+X + V. t Literally catalyst.
as a template
and a catalyst
respectively;
however,
a template
acts abstractly
as a
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The model proposed above is a variation on this scheme, in which N is also needed as a catalyst for the first step and in which X is just the product (PI of the first reaction. If now we consider a system containing substrates A and N (see Fig. l), no reaction will occur until the catalytic molecule R is added. Then, at least some of the reactions will go until both N and R (or P) are entirely consumed. Since N is also required for production of R and P, it must be the limiting substrate. These results may be obtained from a steady-state analysis of the above set of kinetic equations. A schematic representation of the biologically relevant case, in which it is P that is depleted rather than R is shown in Fig. 1. (Note that it must be P that is depleted rather than R since 50% of the progeny RNA remains unencapsidated at the termination of the replicative process.)
Infection
t=o
FIG. 1. A schematic representation initial cellular pool of amino acids respectively. A, N. R, P, and V are RNAs, viral proteins, and viruses in the final state of the cell when virus
Repllcatlon o
Terminal~on f = 1,
of the replication process. A,), N,,. and R,, represent the and nucleotides and the number of infecting viral RNAs. respectively the number of amino acids, nucleotides, viral the cell at some time t (0 < r < rf). A,, Rf, and V, represents replication is complete at time tr.
Equations (1) do not admit analytic solutions; however, when rate constants and initial concentrations are provided (see section 4 for a discussion of how these parameters are determined), numerical calculations may be performed to determine the evolution of the system. Figures 2 and 3 compare the results of such numerical calculations with experimental kinetic data from a variety of sources. The qualitative agreement is remarkably good and indicates the wealth of predictive power that lies in considering the abstract dynamic structure of the process. If one could prevent the last (assembly) step from occurring, the equations would admit an analytic solution. Qualitatively, this would simulate the simplest general system in which a molecule (R) acts both as a catalyst and an autocatalyst 4.5 A + P. A N - R,
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2
(h)
3
4
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FIG. 2. Comparison of the experimental (- - - -) and theoretical (-or - - -) evolutions of the rates of synthesis of (a) d-protein (A) and d-RNA (0) given the MO1 = 10; (b) v-protein (a, theoretical curve is hyphenated) and V-RNA (0) given the MOI= 10; (c) d-RNA for different MO1 (40-A, lo-, 2-O) (d) V-RNA for different MO1 (40-a. lo-, 2-O). The following values were adopted for the required initial values and kinetic parameters: Pc=Vn= 0. N,, = 2 x 10’ nucleotide/cell, Aok1 = 6 x 10m6 protein nucleotide-’ RNA-’ h-‘, k: = k2 = 1.2x lo-’ nucleotide-’ h-l, k3 = 5.3 x 10m6 virus protein-’ RNA-’ h-’ (ky = 0). All data is normalized relative to maximum value for the variable in question.
provided we require that N is present as a co-catalyst in the first reaction. Serendipitously, there is an important analogue in virology corresponding to this case, viz. defective interferring (DI-) particles. DI particles are defective in that the protein they produce is unable to form capsids, hence the encapsidation process does not occur. Mathematically this case corresponds to setting k3 = 0. The analytic solutions that are obtained in this case are: A(t) = Ad(f), N(t)=NO+Ro[l-aI(r P(r) = PO+ Ao[ 1 - A(t)], R(r) = Roar(t),
(2)
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270 where
N =N,,+R,i r(t) = (NC, e “k”+
R,,) ’ R$JJ(t)(l
Time
-ecrk,‘)
(h)
Time
(hi
Cc) 100 _.....................,....,,~,~,~ 80s 4.I 60> ? ;,
40-
E 20 -
4
6 Time
8
Ch)
FIG. 3. Comparison of experimental (----I and theoretical (or -~ -) rates of synthesis in S-virus infected cells. (a) Early stages of V-RNA synthesis, MOI = 30 (Baltimore. 1966). (b) Early stages of v-assembly, MO1 = 20 (Cole & Baltimore, 1973). (c) Total v-protein production (A experiment, theory), virus assembly (0 experiment, - - - theory), and nucleotide depletion with MO1 = 40 (. theory). The experimental points are the averages of two trials at an MOI of 40 (DarnelI& Levintow, 1960).
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Since experimentally it is the rates of protein and RNA synthesis that are measured, to compare the model with experiment we must solve for P and k. P = kl RNA = k1RoNoAoa2 e-ak2’I(f)2A(f), k = k2 RN = k2RoNocr2 edak211’( t)‘.
(3)
In Figs 2a and b we compare theoretical with experimental kinetic data on the system and find excellent qualitative agreement between the two sets of results. In particular, the experimentally observed shift in peak maxima with a change in MO1 is predicted in a quantitative fashion. 3. Further Applications In this section we describe how the ideas of the previous section may be applied to systems of greater complexity. Let us consider first the case of a cell co-infected with both S-virus and DI-particles. Since both the S-RNA and the DI-RNA compete for N and A, and since S-protein encapsidates both S- and DI-RNA (thus S- and DI-RNA compete for S-protein), modeling this problem is more complex than constructing a simple superposition of the two first-order models discussed previously. The kinetic equations become: A=-(k;R, I+ = -(k;
+ kyR&NA R, + kFR&N
p,= k;R,NA-(k;R,+k:,R,)P, &, = kfRDNA (4)
fi, = k;R,N-kjR,P,q RD = k?RDN-
D
ks RDP,
$‘= k;R,P, D
D = k3 RDP,
where the sub(super)script D indicates a quantity associated with DIparticles and the sub(super)script s indicates one associated with S-viruses. It should be noted that the processes of S-virus and DI-particle replication are coupled in three ways: competition for A, N and P,. For simplicity consider the case in which k; = k? (no preferential replication) and ki = ky (no preferential encapsidation). It can then be shown
272
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that for any time t,
where R% and Ry are the number of infecting DI- and S-RNAs, respectively. The fraction of the final-particle yield that are S-virus particles follows immediately; it is R:’ Vf (h) V,cDi=Ri’ In S-virus infections the same number of progeny virions (-2 x 105) are always produced regardless of the MOI. If we set this number equal to V* and assume the cellular amino acid pool is sufficiently large that it is not significantly depleted (i.e. A = A,, = a constant), the following expression relating co-infection yields to yields from cells infected with S-virus only may be obtained: V,+D, R: 17) -$-=-r--T R,+RD
from which,
in conjunction
with equation
(6), it immediately
follows
that:
In Fig. 4 we compare the results of our theory with experimental data on the system and find excellent quantitative agreement between the two sets of results. The deviation from theory is a consequence of enrichment, and will be discussed in the next section. As a final application of the approach taken in this paper, consider the effect of S-virus infection on an idealized scheme for the cellular metabolic processes translation, transcription and replication. Following the program described in the previous section we consider: (1) Translation. R, (cellular m-RNA) and N act as catalysts for the reaction that changes A into P, (a cellular protein). R,+N+Az (2) Transcription. D, (cellular DNA) production of R, from N.
R+N+P, acts as a template or catalyst for the
D, +N%RR,.+D, (3) Replication.
D,. acts autocatalytically
with N as its substrate
D, + N 2 2D,..
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standard
VIRUS
virus
REPLICATION
273
in ~noculum
FIG. 4. Interference of standard virus growth by DI particle as a function of their relative input multiplicities. Here, o denotes total yield of physical particles from cells infected by different relative multiplicities of standard and DI particles; x denotes percentage of standard virions among the progeny; A denotes the yield of plaque-forming virus (percentage of control). The data is taken from Cole & Baltimore (1973b). The lines are the theoretical predictions.
The resulting kinetic equations are: PC = kfR,NA, 6, = k;D,N, d, = k;D,N. Coupling these equations to those developed in the last section for virus replication and solving for the rates of cellular synthesis in an infected cell relative to those for an uninfected cell, one can generate theoretical curves that may be compared with experimental data. This is done in Fig. 5. Again the results agree qualitatively with the experimental results. One notes that D,, as it appears on the right-hand side of the rate equations, is a constant over the time scales of interest. Such agreement is especially surprising in view of the currently accepted explanations for inhibition of cellular synthetic process. Our model suggests that the depletion of the nucleotide pool may be more important than the action of viral inhibitor proteins in the inhibition of cellular processes.
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FIG. 5. Comparison of the experimental and theoretical rates of cellular synthesisof c-DNA [A-experiment (Holland & Peterson, 1964). --theory, MO1 = 1001, c-RNA [O-experiment (Contreras et a/., 1973), ----theory, MOI=300], and c-protein [O-experiment, __theory, MO1 = 1001 in infected cell.
4. Discussion
and Limitations
of the Model
Let us begin with a discussion of the parameters used in this model. At first sight it would appear that to carry out the calculations described in section 2? one must supply the initial concentration of each system component as well as each of the rate constants. However, a number of these parameters may be quickly eliminated (or finessed) by appropriate assumptions based on a knowledge of the system. For example, it is obvious that V,, and P,, must be set to zero since an uninfected cell would not contain virus or viral protein. R. is just the multiplicity of infection (MOI) and No may be determined from the fact that the final yield of viral RNA is -2 x 10’ regardless of MO1 (since in our idealized model each R arises from only one N, No = 2 x 10’). The species A is assumed to be present in great excess and is taken to be a constant, A”. Hence its evolution equation may be eliminated. It is known that the number of replication intermediates doubles every 20 min; this would correspond to kzN==2.l/h. In the early stages of evolution, N changes slowly and hence may be set equal to N,,. From data on V-RNA accumulation klNo=3.1/h. So that the theoretical peak maximum for S-RNA replication at an MO1 of 10 corresponds to the experimental curve, in this paper we set kZN0=2*4/h. It should be noted that although the specific value chosen for k2 was the result of a calibration, the range of values was not.
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To summarize, at this point one has specified all the parameters needed to predict the behavior of the DI system as displayed in Fig. 2. The constant kl is finessed by calibrating p(t) to p,,,,, (the maximum value of P). The ability of the model to predict the proper qualitufiue behavior is independent of the value chosen for kZ and is a direct consequence of the underlying dynamic structure of the system. To achieve quantitative agreement for the peak time, only one parameter, kz, had to be fitted. Furthermore, as seen above, an approximate value for this parameter can be determined from experiment. Once this single parameter is determined from the peak time at one MOI, the shift in peak time with a change in MO1 is a necessary consequence of the dynamics. The S-virus system requires the specification of kl and k3. A rough, order-of-magnitude argument can be made to specify the value of kl. Since only -50% of the V-RNA produced is encapsidated by the time N is depleted, only about half as much P has been produced as R. Hence, kiAoNo-$kzNo=l*2/h. Therefore, k1A0=6*0 x 1O-6 proteins/nucleotide/RNA/h. An estimate of k3 follows from the work of Darnell & Levintow (1960) in which the rate of virus assembly is roughly 20%/h when -38% of the v-protein and hence about half the total V-RNA is present in the free form. Thus $‘- O-2 x 105/virus/h and, from the previously determined constants, k3 z 5.3 x 10e6 virus/protein/RNA/h. Once again the qualirutive features of the system are independent of these constants, while quuntitutioe agreement for alJ MO1 may be achieved by calibrating the curves at one MOI. In section 3 the prediction of the yield of co-infected cells does not depend in any way on calibration or curve fitting. For the effect of virus replication on cellular synthetic processes, specific knowledge of the rate constants is bypassed by taking the ratio of the rates of synthesis in infected cells to those in uninfected cells. It was also assumed that the drain on the cellular N and A pools due to cellular processes was so much less than that due to viral processes that they could be ignored in the expressions for # and A. Hence these results are also independent of any calibration other than that involved to determine kZ. Since experiments were done at different MOIs, it was necessary to carry out the calculations with correspondingly different values of Ro. To conclude this section, let us review critically the approximations made in developing our model and suggest ways in which the model may be modified to account for the complexity of the real system. In our model one specifies one amino acid, nucleotide, or protein to make a protein, RNA, or capsid, respectively; in an actual system -6000 nucleotides make up the RNA, -2000 amino acids make up the initially translated protein, and 60
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capsid proteins must combine to form the capsid. If a simple steady state polymerization scheme is assumed, the only change required of the model is to multiply the depletion term in each rate equation by the number of units used to make the product (e.g. &r = 6000 kz RN and hence N,, becomes -1.2 x 10’). However, the polymerizations are not simple. Translation and replication rely on complex, self-regulatory cellular pathways whose mathematical description lies beyond the scope of this paper. It is possible, however, that the polymerization of the capsomers to form a capsid could be treated in a way analogous to Casper’s treatment t 1963) of the assembly of the TMV helical aggregates. As a second point, there are some 20 different amino acids (or more, if one considers post-translational modifications) and four nucleotides rather than the one of each species considered in our model. However, in the absence of a more detailed description of the translation and replicative processes, we have assumed that one of the amino acids and one of the nucleotides will be present in a lower concentration than the others of its type and that the rate of polymerization will be determined by the rate at which that amino acid or nucleotide is provided to the forming polymer. At least in the case of nucleotides, it is known that there are cellular regulatory pathways that tend to maintain constant, the relative amounts of the four nucleotides. In translation, N provides the energy to make the peptide bonds and to initiate translocation by the irreversible hydrolysis of GTP to GDP and P,. Thus it acts as a substrate rather than the catalyst indicated by the model. However if one assumes that it is rapidly rephosphorylated by ATP (which is in turn rephosphorylated by the electron transport system 1,N would appear to act as a catalyst on the time scale of interest. The assumption of non-renewable initial concentrations of A and N is, of course, unrealistic. A cell has homeostatic mechanisms that maintain the concentations of A and N within some acceptable range. Thus one might want to add a [+k,(A-Ao)N] and a [+k,,(N-NoIN] source term to the rate equations for A and N, respectively, to account for homeostasis [the multiplicative N appearing in the above expressions reflects the fact that energy is required to produce A and N directly or indirectly in the form of ATP (N)]. Our calculations show that such corrections do not change significantly the results reported above. The function of viral proteins is, of course, a much more complicated one than simply to provide capsid precursors. Although the whole genome is probably translated as a single polypeptide, it quickly undergoes a series of cleavages (that are still not completely mapped) to form the tripeptide capsomer as well as a polymerase, a protease, and probably at least one protein responsible for inhibition of cellular RNA and protein synthesis.
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Cellular enzymes are responsible for the cleavage of the initial protein. See, for example, the contributions of Shih et al. (1979). Subsequent cleavages of the resulting capsomer precursor depend on the presence of viral protease (which is probably a component of the polymerase precursor). Subsequent cleavage of the noncapsid proteins may be due to either cellular or viral proteases. In a seminal paper by Butterworth & Rueckert (1972), a first step is taken towards a mathematical description of the cleavage process and their theoretical results compare favourably with kinetic data. It is possible that their approach could be incorporated into the present theory. Inhibition of cellular protein synthesis is thought to be caused by a viral inhibitor protein that binds to cellular ribosomes causing the breakdown of host-cell polyribosomes (Baltimore, 1969). More recently it has been shown that the inhibitor somehow causes the inactivity of eIF-4B, a cellular translation initiation factor which binds to the S-terminal cap of m-RNA (Rose ef al., 1978; Trachsel ef al., 1980).The inhibition of host-cell RNA synthesis is believed due to a viral inhibitor protein acting directly on the cellular polymerase. Inhibition of DNA synthesis.is thought to be a secondary result of one of the other two types of inhibition (Baltimore, 1969). Although the model may be modified to take into account inhibitor proteins, theory suggests that their effect may be treated as a perturbation to the process of nucleotide depletion. In addition one might want to consider competition between cellular m-RNA and viral RNA for free ribosomes. A more realistic model would also have to take into account the fact that the virus produces its own replicase and that replication and translation are confined to the smooth and rough endoplasmic reticulum, respectively. Such a model would involve reaction-diffusion and saturation kinetics. In the context of this more detailed theory it would be necessary to consider more closely the role of VPg, a viral protein whose capping of the 5’-terminal end of viral RNA may initiate RNA synthesis and whose absence or presence may determine whether the RNA functions in translation, replication, or encapsidation (Rothberg et al., 1978; Nomoto ef al., 1977). Furthermore, there is not one type of RNA that acts as an autocatalyst, but rather two types, plus and minus, each of which acts as a catalyst (or equivalently as templates) for the production of the other, thus forming a catalytic cycle. Although it only takes 45-60 s for a plus strand to be produced on a minus template, it takes about 20 min for the number of minus strands to double. Eigen has pointed out that such a catalytic cycle can be represented as a single autocatalyst. In the models presented in sections 2 and 3, the rates of DI- and S-RNA replication encapsidation were considered to be equal. Although this is not
278
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strictly true, the fact that the rates of RNA synthesis peak at about the same time in cells infected at any given MO1 with one or the other. as well as the fact that ratio of progeny S-virus to total progeny is about the same as the ratio of infecting virus to the total number of infecting particles in coinfected cells, indicates that the rates are approximately equal. On the other hand, experiments by Cole & Baltimore, (1973b,c) indicate that DI-RNA probably enjoys the dual advantages of preferential replication and encapsidation, which lead to a S-8% enrichment in DI-particle yield relative to S-virus. The mode1 presented in section 2 provides a route for determining the relative importance of these two effects and thence to account for the different observed hyperenrichment phenomena. This, as well as the problem of the origin of DI-RNA of specific length after successive passages, will be the subject of a subsequent paper. A more complete model would also consider the competition between DI- and S-RNA for the viral replicase. When considering the interaction between cellular RNA and protein synthetic processes, and virus replicative processes, a number of further effects must be considered. In the same way that cellular nucleotide and amino acid pools are maintained by homeostatic mechanism, the levels of cellular RNA and protein also tend to be maintained within some range. Thus the right hand sides of the expressions for cellular RNA and protein synthesis could be replaced by terms of the form [k>(R;,R,. IN] and [kS (PF,-P‘.)R,NA], respectively. (Note that since D, was constant, this parameter has been incorporated into the new rate constants.) Competition between cell and virus for the cell’s substrate pools could be included by subtracting and above terms from the expressions for the depletion of N and A, respectively. Also neglected in the models of sections 2 and 3 were the rates of degradation of cellular and viral proteins and RNAs by cellular protease and RNases. One might want to add a [-k:‘] or [-k:‘Xi] term to each of the synthesis equations to take into account the degradation of the ith synthetic product. The above review emphasizes the complexity of the processes we have considered, while the nearly quantitative success of the simplistic model of Section II emphasizes the usefulness of coupling reasonable assumptions with a dynamic approach. In this sense (only), we believe that the assumptions utilized in the development of our model are justified. 5. Conclusions Our goal in this paper was to determine the extent to which the qualitative behavior of virus replication could be accounted for by a mathematical
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model. In developing a dynamic model for a physical process, endeavored to find the simplest mathematical structure of physical significance. This was done in section 2 and a steady state analysis of the resulting equations together with the knowledge that about half of the progeny RNA remain unencapsidated led to the conclusion that virus replication is controlled by the depletion of the cellular nucleotide pool, a conclusion that is consistent with the experimental fact that the amount of progeny RNA produced in a cell infected with DI-particles and/or S-virus is the same regardless of the MOIs. The controlling reaction is the autocatalytic replicative step which depletes the nucleotide pool. Consideration of this step alone is capable of predicting the observed exponential and linear phases of V-RNA synthesis as well as the cessation of synthesis and the shift in the peak of the rate of synthesis with increasing MOI. Interference by DI particles is a natural consequence of competition for the limited resource N, as are the effects of varying the relative MOIs. The model also predicts the decreased interference when cells initially infected with S-virus are later superinfected with DI particles and, in addition, the dramatic increase in interference when DI-particle infection preceeds an S-virus superinfection. Inhibition of cellular synthetic processes, as well as the shift of the virus-synthesis curves to the left with increasing MOI, also follows from the depletion of nucleotides. Such a depletion may also explain the timing of cell lysis. Since cell membranes require continued maintenance to retain their integrity and since such maintenance requires energy, the depletion of nucleotides would inhibit maintenance and it is reasonable to suppose that deterioration (lysis) of the membrane would follow. These qualitative results are a consequence of the mathematical structure and do not depend in any way on a calibration (or curve fitting). A single-parameter calibration does, however, bring the theory into first-order quantitative agreement with experiment with respect to the approximate duration of the exponential and linear phases, when they occur, and how their phases shift with MOI. That all the above results should apply to DI-particles as well as to S-virus is a consequence of the fact that both replication processes have the same fundamental RNA replicative step at their foundations. By considering the further refinements of the model described in the last section, a better quantitative agreement with experiment might be realized, as well as a more fundamental reason for choosing Aok = 6.0~ lo-” proteins/nucleotide/RNA/h and k3 = 5.3 x 10e6 virus/protein/RNA/h. Moreover, by carrying out the proposed modifications, a more realistic model would result, and other experimentally observed behavior could be understood within the framework of the theory.
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The production of protein capable of encapsidating the RNA constitutes a perturbation to the fundamental process of autocatalysis. An eclipse period, a phase of exponential growth followed by a linear accumulation of virus particles lagging behind RNA and protein accumulation, as well as the tinal cessation of encapsidation follow naturally from theory. The eclipse period, for example, is due to the second-order nature of the encapsidation step and the low early concentration of progeny RNAs and top components. By fitting the parameter k,A,, and kj to data on protein synthesis subject to the constraint that only half the progeny RNA is finally encapsidated. the approximate durations of each of the phases was predicted was well as the approximate lag time between protein (and RNA) accumulation and accumulation of progeny virus. Alternatively. an approximate value for k, A,, may be obtained by an order-of-magnitude calculation and used to determine kz on the basis of the observed lag in the accumulation of progeny virus relative to total viral protein and RNA accumulation. That the values of k,A,) and k3 are of the same order of magnitude regardless of the method or experiment used in their calculation indicates the consistancy of the theory. The theory predicts a shift to the left in the virus accumulation curve with increasing MOI. Theory also predicts that if RNA replication is inhibited temporarily in the linear stage without inhibiting protein synthesis for a period of time, there will be an increase in the percent of progeny RNA encapsidated. In section 3 we saw that both the phenomena of interference and of inhibition of cellular synthetic processes were necessary consequences of competition for cellular nucleotide. The overall aspects of interference (Fig. 4) followed without having to specify any parameters. Using only one. previously-determined parameter, k2, the theoretical curve for the inhibition of cellular synthetic processes approximated the experimentallyobserved curves to first-order. Once again, by considering the modifications suggested in section 4 as a perturbation on the first-order theory, better quantitative agreement with experiment could be obtained. In conclusion, the proposed theory is surprisingly accurate and effective in its ability to account for the experimentally-observed behavior and its ability to organize data from several systems. There is however, one critical prediction upon which the validity of the theory rests that has not yet been studied experimentally: We find that at least one of the nucleotides i.7 depleted significantly by the end of the viral replication process. Once the kz, klA,, and k3 are specified by an appropriate reference system, the simple model proposed in this paper applies not only to poliovirus, but also to any picornavirus (or, more generally to any virus whose replication process is governed by a similar dynamic structure). The
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experimentalist may apply the model with parameters thus specified, to predict the approximate behavior of any other system. It should be noted that steady state infections can be understood if klAo > kz, or if kZ is small relative to the rate at which the cell replaces nucleotides. It may be hoped that the success enjoyed by this dynamic model in describing poliovirus replication will persist when applied to kinetic data on other picornaviruses. If it is, the next step would be to consider viruses with more complex dynamic structure: for exampIe, DNA viruses, RNA viruses with reverse transcriptases, or viruses whose genome spends time integrated with that of the host cell. The authors would like to thank Professor Gary Burleson and MS Maura Glynn for many stimulating and helpful discussions, and for their constant interest in this work. REFERENCES BABLOYANTZ, A. AND SANGLIER,M. (1972). FEBSLerr. 23,364. BALTIMORE,D.,GIRARD,M.& DARNELL,J. E. (1966). Virology,29,179. BALTIMORE,D. (1969). In 77zeBiochemistryo,fViruses, (H. B.Levy,ed.).New York: Marcel Dekker. BUTTERWORTH,B.& RuEcKERT,R.(I~~~). Virology 50,535. CASPAR, D. L. D. (1963). Advs. Protein Chem. 18, 37. COLE,~. N.& BALTIMORE,D.(~~~~~).J. mol. Biol.76,325. COLE,~. N.& BALTIMORE,D. (19736).1. mol. Biol.76,345. COLE,C.N. & BALTIMORE,D. (1973~). J. Virol.12,1414. COLE,C. N. (1975). Prog. med. Vir. 20, 180. CONTRERAS,G.,SUMMERS,D.F.,MAIZEL,J.V.& EHRENFELD,E.(~~~~). Virology 53, 120. DARNELL,J.E. & LEVINTOW,L. (1960). J. biol. Chem.235,70. EIGEN, M. (1971). Nuturwissensh. 58,465. EIGEN,M.& SCHUSTER,~. (1977a),Naturwissen.sh.64,541. EIGEN,M.& SCHUSTER,~. (1978a). Naturwissensh.65,7. EIGEN,M. & SCHUSTER,P. (1978b).Nuturwissensh.65,341. HAKEN, H. (1978). Synergetics, 2nd Ed. Berlin: Springer-Verlag. HOLLAND,J.J. & PETERSON,J. A. (1964). J. mol. Bio[. 8, 556. HOLLAND,J.J. (1964).X. mol. Rio!. 8,574. HUANG, A. S. (1973). Ann. Rev. Mcrobiol. 27, 101. LAUFFER, M. A. (1975). Molecular Biology andBiophysics. Berlin: Springer-Verlag. NICOLIS, G. & PRIGOGINE, I. (1977). Self-Organization in Nonequilibrium Systems. New York: John Wiley. NOMOTO,A.,DETJEN,B.,POZZATTI,R.& WIMMER,E.(~~~~). Nature 268,208. PALMENBERG,A.C.,PALLANSCH,M.A.& RuEcKERT,R.R.(~~~~). J. Viral. 32,770. RosE,J.K.,TRAcHsEL,H.,LEoNG,K.& BALTIMORE,D. (1978). Proc.Natn. Acad. Sci. U.S.A.
73,2732.
ROTHBERG,P.G.,HARRIS,T.J.R.,NOMOTO,A.&WIMMER,E.(~~~~). Sci. U.S.A.
Proc.Natn.Acad.
75,4868.
SHIH,D.S.,SHIH,C.T.,ZIMMERN,D.,RUECKERT,R.R.,&KAESBERG,P. (1979i.J. 30,472. TRACHSEL,H.,SONENBERG,N.,SHATKIN,A.J.,ROSE,J.K.,LEONG,K.,BERGMANN,J. E., GORDON, J. & BALTIMORE, D. (1980). Proc. Natn. Acad. Sci. U.S.A. 77,770.
Viral.
A Dynamic
Model for in uivo Virus Replication?
JOHN E. MACCARTHY
AND JOHN J. KOZAK
Program in Biophysics and Biochemistry, and Radiation University of Notre Dame, Notre Dame, Indiana 46556, U.S.A. (Received
30 January
1980, and in revisedform
Laboratory,
23 December
1980)
In this paper we present a dynamic model of in viva virus replication. Kinetic equations are formulated to describe the overall process of replication and then analyzed using a “synergetic” approach. First the importanceof a rate-limiting substrateis taken explicitly into account, and secondly the coupling between the processesconsidered (translation, replication and assembly)is strictly preserved;the analysisitself is carried out in the linear regime. We treat the problems of defective-particle infections, standard-virus infections, inhibition of cellular synthesis, and the caseof co-infected cells. The various parametersof the model (initial
cellular concentrations, rate constants) are specified using existing experimental data and the full (numerical)consequences of the mode1are explored in detail and compareddirectly with experimentsby Baltimore, Cole and others on viral systems.Quite surprisingly, the simple model developed in this paper is able to account qualitatively, and occasionally quantitatively, for the behavior observedexperimentally for each of the problemscited above. 1. Introduction The replication of poliovirus in Hela cells has been studied extensively in recent years and is one of the best understood systems in virology. In this paper we present a relatively simple dynamical mode1 whose elaboration allows us to organize, within a single theoretical framework, a variety of experimental data on replication of standard (S-) poliovirus (Baltimore, 1969; Baltimore, Girard & Darnell, 1966; Darnell & Levintow, 1960; Cole & Baltimore, 1973a,b), replication of polio defective interfering (DI-) particles (Cole & Baltimore, 1973a,b; Cole, 1975), inhibition of cellular synthetic processes (Holland & Peterson, 1964; Holland, 1964; Contreras et al., 1973), and the effects of co-infection of cells by both S-virus and DI-particles on the yields of each (Cole & Baltimore, 1973; Cole, 1975). In t Theresearch described herein was supported in part by the Office of Basic Energy Sciences of the Department of En_ergy. This is Document No. NDRL-2074 from the Notre Dame Radiation Laboratory. 265
0022-5193/81/100265+17$02.00/0
@ 1981 Academic Press Inc. (London) Ltd.
266
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AND
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KO7AK
dealing with these problems, our main concern will be the dynamic aspects of replication. As such, the work may be distinguished from the earlier contributions of Lauffer (1975) and Casper (1963) on the thermodynamics of virus assembly. And, although Butterworth & Rueckert i1972) developed a kinetic model for EMC protein cleavage, to our knowledge there has been presented no comprehensive model for the overall processof viral infection in viva. The spirit of our approach is similar to that taken by the Gijttingen school [Eigen (1971) and Eigen & Schuster (1977a, 1978h,c)] in the theory of competing biological speciesor that of the Bruxelles school [Babloyantz & Sanglier, (1972)] in their work on the regulation of the lac operon, in that the co-operative aspects of the problem are recognized and stressed. The mathematical techniques for dealing with such problems are widely known, and have proved successful in addressing problems in a variety of different fields (Haken, 1977; Nicolis & Prigogine, 1978). Although the present model is admittedly naive, it does have the advantage that the co-operative features of virus replication can be traced using simple, analytic arguments; moreover, it will be seen that the model is surprisingly successful in accounting for many of the characteristic trends observed in the specific systems rited above. At the very least, it is hoped that the present study may stimulate the development of more sophisticated mathematical models for this important problem. In section 2 we focus on the problems of DI-particle infection and S-virus infection. Then, in section 3 we take up inhibition of cellular synthesis and the problem of co-infected cells. In section 4 the limitations of the model are discussed. The final section is given over to a statement of the principal conclusions that may be drawn from this work. 2. The Model The goal of this section is to develop the simplest possible dynamic model for the post-infection processesof virus replication. Each process is defined by a characteristic class of substrates and products. In the spirit of Eigen (1971) and Babolyantz & Sanglier (1972), a classof substrates or products will be represented by a single substrate or product respectively, -and complex biochemical polymerizations will be idealized as a single step reaction of first order in each of the reactants. What is proposed then, is that the abstract dynamic structure of the real viral replication process can be modeled by the simplest idealized chemical system of identical dynamic structure. It will be seen that the analysis of this abstract system predicts kinetic behavior that is in qualitative agreement with experimental data on the real system.
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Experimentally poliovirus replication is known to involve three processes once a cell is infected: translation, replication and assembly (Baltimore, 1969). The process will be idealized as follows. (1) Translation. R (viral RNA) and N (cellular nucleotides) act as catalysts? for the irreversible reaction that changes A (cellular amino acids) into P (viral protein). A+R+Nfi:R+N+P. (2) Repiication. R acts autocatalytically
in an irreversible
reaction with N
as its substrate. R+N32R. (3) Assembly. The terminating irreversible reaction in which P and R act as substrates for the production of V (virions).
R+P>V. It should be noted that all the above reactions are irreversible. In the first two cases it is because phosphate bonds are broken, while in the last case it is because the protein cleaves. The kinetic equations that follow from this model are: /i = -klRNA, ni= -k*RN, P= klRNA-k3RP,
(1)
ti = kzRN-k3RP, 3= kjRP. The dynamic structure of this idealized system is quite similar to that of the simplest general system in which a single molecule (R) acts simultaneously in separate reactions as a catalyst, an autocatalyst, and a substrate. Such a system would be represented as follows:
R+X + V. t Literally catalyst.
as a template
and a catalyst
respectively;
however,
a template
acts abstractly
as a
268
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.&ND
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The model proposed above is a variation on this scheme, in which N is also needed as a catalyst for the first step and in which X is just the product (PI of the first reaction. If now we consider a system containing substrates A and N (see Fig. l), no reaction will occur until the catalytic molecule R is added. Then, at least some of the reactions will go until both N and R (or P) are entirely consumed. Since N is also required for production of R and P, it must be the limiting substrate. These results may be obtained from a steady-state analysis of the above set of kinetic equations. A schematic representation of the biologically relevant case, in which it is P that is depleted rather than R is shown in Fig. 1. (Note that it must be P that is depleted rather than R since 50% of the progeny RNA remains unencapsidated at the termination of the replicative process.)
Infection
t=o
FIG. 1. A schematic representation initial cellular pool of amino acids respectively. A, N. R, P, and V are RNAs, viral proteins, and viruses in the final state of the cell when virus
Repllcatlon o
Terminal~on f = 1,
of the replication process. A,), N,,. and R,, represent the and nucleotides and the number of infecting viral RNAs. respectively the number of amino acids, nucleotides, viral the cell at some time t (0 < r < rf). A,, Rf, and V, represents replication is complete at time tr.
Equations (1) do not admit analytic solutions; however, when rate constants and initial concentrations are provided (see section 4 for a discussion of how these parameters are determined), numerical calculations may be performed to determine the evolution of the system. Figures 2 and 3 compare the results of such numerical calculations with experimental kinetic data from a variety of sources. The qualitative agreement is remarkably good and indicates the wealth of predictive power that lies in considering the abstract dynamic structure of the process. If one could prevent the last (assembly) step from occurring, the equations would admit an analytic solution. Qualitatively, this would simulate the simplest general system in which a molecule (R) acts both as a catalyst and an autocatalyst 4.5 A + P. A N - R,
MODEL
2
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5 Tize
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2
(h)
3
4
5
6
Ttme (hl
FIG. 2. Comparison of the experimental (- - - -) and theoretical (-or - - -) evolutions of the rates of synthesis of (a) d-protein (A) and d-RNA (0) given the MO1 = 10; (b) v-protein (a, theoretical curve is hyphenated) and V-RNA (0) given the MOI= 10; (c) d-RNA for different MO1 (40-A, lo-, 2-O) (d) V-RNA for different MO1 (40-a. lo-, 2-O). The following values were adopted for the required initial values and kinetic parameters: Pc=Vn= 0. N,, = 2 x 10’ nucleotide/cell, Aok1 = 6 x 10m6 protein nucleotide-’ RNA-’ h-‘, k: = k2 = 1.2x lo-’ nucleotide-’ h-l, k3 = 5.3 x 10m6 virus protein-’ RNA-’ h-’ (ky = 0). All data is normalized relative to maximum value for the variable in question.
provided we require that N is present as a co-catalyst in the first reaction. Serendipitously, there is an important analogue in virology corresponding to this case, viz. defective interferring (DI-) particles. DI particles are defective in that the protein they produce is unable to form capsids, hence the encapsidation process does not occur. Mathematically this case corresponds to setting k3 = 0. The analytic solutions that are obtained in this case are: A(t) = Ad(f), N(t)=NO+Ro[l-aI(r P(r) = PO+ Ao[ 1 - A(t)], R(r) = Roar(t),
(2)
.I. t-. Maci‘AK’I‘H\/\NI) I I KO%Ah
270 where
N =N,,+R,i r(t) = (NC, e “k”+
R,,) ’ R$JJ(t)(l
Time
-ecrk,‘)
(h)
Time
(hi
Cc) 100 _.....................,....,,~,~,~ 80s 4.I 60> ? ;,
40-
E 20 -
4
6 Time
8
Ch)
FIG. 3. Comparison of experimental (----I and theoretical (or -~ -) rates of synthesis in S-virus infected cells. (a) Early stages of V-RNA synthesis, MOI = 30 (Baltimore. 1966). (b) Early stages of v-assembly, MO1 = 20 (Cole & Baltimore, 1973). (c) Total v-protein production (A experiment, theory), virus assembly (0 experiment, - - - theory), and nucleotide depletion with MO1 = 40 (. theory). The experimental points are the averages of two trials at an MOI of 40 (DarnelI& Levintow, 1960).
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Since experimentally it is the rates of protein and RNA synthesis that are measured, to compare the model with experiment we must solve for P and k. P = kl RNA = k1RoNoAoa2 e-ak2’I(f)2A(f), k = k2 RN = k2RoNocr2 edak211’( t)‘.
(3)
In Figs 2a and b we compare theoretical with experimental kinetic data on the system and find excellent qualitative agreement between the two sets of results. In particular, the experimentally observed shift in peak maxima with a change in MO1 is predicted in a quantitative fashion. 3. Further Applications In this section we describe how the ideas of the previous section may be applied to systems of greater complexity. Let us consider first the case of a cell co-infected with both S-virus and DI-particles. Since both the S-RNA and the DI-RNA compete for N and A, and since S-protein encapsidates both S- and DI-RNA (thus S- and DI-RNA compete for S-protein), modeling this problem is more complex than constructing a simple superposition of the two first-order models discussed previously. The kinetic equations become: A=-(k;R, I+ = -(k;
+ kyR&NA R, + kFR&N
p,= k;R,NA-(k;R,+k:,R,)P, &, = kfRDNA (4)
fi, = k;R,N-kjR,P,q RD = k?RDN-
D
ks RDP,
$‘= k;R,P, D
D = k3 RDP,
where the sub(super)script D indicates a quantity associated with DIparticles and the sub(super)script s indicates one associated with S-viruses. It should be noted that the processes of S-virus and DI-particle replication are coupled in three ways: competition for A, N and P,. For simplicity consider the case in which k; = k? (no preferential replication) and ki = ky (no preferential encapsidation). It can then be shown
272
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that for any time t,
where R% and Ry are the number of infecting DI- and S-RNAs, respectively. The fraction of the final-particle yield that are S-virus particles follows immediately; it is R:’ Vf (h) V,cDi=Ri’ In S-virus infections the same number of progeny virions (-2 x 105) are always produced regardless of the MOI. If we set this number equal to V* and assume the cellular amino acid pool is sufficiently large that it is not significantly depleted (i.e. A = A,, = a constant), the following expression relating co-infection yields to yields from cells infected with S-virus only may be obtained: V,+D, R: 17) -$-=-r--T R,+RD
from which,
in conjunction
with equation
(6), it immediately
follows
that:
In Fig. 4 we compare the results of our theory with experimental data on the system and find excellent quantitative agreement between the two sets of results. The deviation from theory is a consequence of enrichment, and will be discussed in the next section. As a final application of the approach taken in this paper, consider the effect of S-virus infection on an idealized scheme for the cellular metabolic processes translation, transcription and replication. Following the program described in the previous section we consider: (1) Translation. R, (cellular m-RNA) and N act as catalysts for the reaction that changes A into P, (a cellular protein). R,+N+Az (2) Transcription. D, (cellular DNA) production of R, from N.
R+N+P, acts as a template or catalyst for the
D, +N%RR,.+D, (3) Replication.
D,. acts autocatalytically
with N as its substrate
D, + N 2 2D,..
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VIRUS
virus
REPLICATION
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in ~noculum
FIG. 4. Interference of standard virus growth by DI particle as a function of their relative input multiplicities. Here, o denotes total yield of physical particles from cells infected by different relative multiplicities of standard and DI particles; x denotes percentage of standard virions among the progeny; A denotes the yield of plaque-forming virus (percentage of control). The data is taken from Cole & Baltimore (1973b). The lines are the theoretical predictions.
The resulting kinetic equations are: PC = kfR,NA, 6, = k;D,N, d, = k;D,N. Coupling these equations to those developed in the last section for virus replication and solving for the rates of cellular synthesis in an infected cell relative to those for an uninfected cell, one can generate theoretical curves that may be compared with experimental data. This is done in Fig. 5. Again the results agree qualitatively with the experimental results. One notes that D,, as it appears on the right-hand side of the rate equations, is a constant over the time scales of interest. Such agreement is especially surprising in view of the currently accepted explanations for inhibition of cellular synthetic process. Our model suggests that the depletion of the nucleotide pool may be more important than the action of viral inhibitor proteins in the inhibition of cellular processes.
274
.I F MacC’AKl’H\
ANI> .I I
lime
hO/,\h
ihl
FIG. 5. Comparison of the experimental and theoretical rates of cellular synthesisof c-DNA [A-experiment (Holland & Peterson, 1964). --theory, MO1 = 1001, c-RNA [O-experiment (Contreras et a/., 1973), ----theory, MOI=300], and c-protein [O-experiment, __theory, MO1 = 1001 in infected cell.
4. Discussion
and Limitations
of the Model
Let us begin with a discussion of the parameters used in this model. At first sight it would appear that to carry out the calculations described in section 2? one must supply the initial concentration of each system component as well as each of the rate constants. However, a number of these parameters may be quickly eliminated (or finessed) by appropriate assumptions based on a knowledge of the system. For example, it is obvious that V,, and P,, must be set to zero since an uninfected cell would not contain virus or viral protein. R. is just the multiplicity of infection (MOI) and No may be determined from the fact that the final yield of viral RNA is -2 x 10’ regardless of MO1 (since in our idealized model each R arises from only one N, No = 2 x 10’). The species A is assumed to be present in great excess and is taken to be a constant, A”. Hence its evolution equation may be eliminated. It is known that the number of replication intermediates doubles every 20 min; this would correspond to kzN==2.l/h. In the early stages of evolution, N changes slowly and hence may be set equal to N,,. From data on V-RNA accumulation klNo=3.1/h. So that the theoretical peak maximum for S-RNA replication at an MO1 of 10 corresponds to the experimental curve, in this paper we set kZN0=2*4/h. It should be noted that although the specific value chosen for k2 was the result of a calibration, the range of values was not.
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To summarize, at this point one has specified all the parameters needed to predict the behavior of the DI system as displayed in Fig. 2. The constant kl is finessed by calibrating p(t) to p,,,,, (the maximum value of P). The ability of the model to predict the proper qualitufiue behavior is independent of the value chosen for kZ and is a direct consequence of the underlying dynamic structure of the system. To achieve quantitative agreement for the peak time, only one parameter, kz, had to be fitted. Furthermore, as seen above, an approximate value for this parameter can be determined from experiment. Once this single parameter is determined from the peak time at one MOI, the shift in peak time with a change in MO1 is a necessary consequence of the dynamics. The S-virus system requires the specification of kl and k3. A rough, order-of-magnitude argument can be made to specify the value of kl. Since only -50% of the V-RNA produced is encapsidated by the time N is depleted, only about half as much P has been produced as R. Hence, kiAoNo-$kzNo=l*2/h. Therefore, k1A0=6*0 x 1O-6 proteins/nucleotide/RNA/h. An estimate of k3 follows from the work of Darnell & Levintow (1960) in which the rate of virus assembly is roughly 20%/h when -38% of the v-protein and hence about half the total V-RNA is present in the free form. Thus $‘- O-2 x 105/virus/h and, from the previously determined constants, k3 z 5.3 x 10e6 virus/protein/RNA/h. Once again the qualirutive features of the system are independent of these constants, while quuntitutioe agreement for alJ MO1 may be achieved by calibrating the curves at one MOI. In section 3 the prediction of the yield of co-infected cells does not depend in any way on calibration or curve fitting. For the effect of virus replication on cellular synthetic processes, specific knowledge of the rate constants is bypassed by taking the ratio of the rates of synthesis in infected cells to those in uninfected cells. It was also assumed that the drain on the cellular N and A pools due to cellular processes was so much less than that due to viral processes that they could be ignored in the expressions for # and A. Hence these results are also independent of any calibration other than that involved to determine kZ. Since experiments were done at different MOIs, it was necessary to carry out the calculations with correspondingly different values of Ro. To conclude this section, let us review critically the approximations made in developing our model and suggest ways in which the model may be modified to account for the complexity of the real system. In our model one specifies one amino acid, nucleotide, or protein to make a protein, RNA, or capsid, respectively; in an actual system -6000 nucleotides make up the RNA, -2000 amino acids make up the initially translated protein, and 60
276
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KO%AK
capsid proteins must combine to form the capsid. If a simple steady state polymerization scheme is assumed, the only change required of the model is to multiply the depletion term in each rate equation by the number of units used to make the product (e.g. &r = 6000 kz RN and hence N,, becomes -1.2 x 10’). However, the polymerizations are not simple. Translation and replication rely on complex, self-regulatory cellular pathways whose mathematical description lies beyond the scope of this paper. It is possible, however, that the polymerization of the capsomers to form a capsid could be treated in a way analogous to Casper’s treatment t 1963) of the assembly of the TMV helical aggregates. As a second point, there are some 20 different amino acids (or more, if one considers post-translational modifications) and four nucleotides rather than the one of each species considered in our model. However, in the absence of a more detailed description of the translation and replicative processes, we have assumed that one of the amino acids and one of the nucleotides will be present in a lower concentration than the others of its type and that the rate of polymerization will be determined by the rate at which that amino acid or nucleotide is provided to the forming polymer. At least in the case of nucleotides, it is known that there are cellular regulatory pathways that tend to maintain constant, the relative amounts of the four nucleotides. In translation, N provides the energy to make the peptide bonds and to initiate translocation by the irreversible hydrolysis of GTP to GDP and P,. Thus it acts as a substrate rather than the catalyst indicated by the model. However if one assumes that it is rapidly rephosphorylated by ATP (which is in turn rephosphorylated by the electron transport system 1,N would appear to act as a catalyst on the time scale of interest. The assumption of non-renewable initial concentrations of A and N is, of course, unrealistic. A cell has homeostatic mechanisms that maintain the concentations of A and N within some acceptable range. Thus one might want to add a [+k,(A-Ao)N] and a [+k,,(N-NoIN] source term to the rate equations for A and N, respectively, to account for homeostasis [the multiplicative N appearing in the above expressions reflects the fact that energy is required to produce A and N directly or indirectly in the form of ATP (N)]. Our calculations show that such corrections do not change significantly the results reported above. The function of viral proteins is, of course, a much more complicated one than simply to provide capsid precursors. Although the whole genome is probably translated as a single polypeptide, it quickly undergoes a series of cleavages (that are still not completely mapped) to form the tripeptide capsomer as well as a polymerase, a protease, and probably at least one protein responsible for inhibition of cellular RNA and protein synthesis.
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Cellular enzymes are responsible for the cleavage of the initial protein. See, for example, the contributions of Shih et al. (1979). Subsequent cleavages of the resulting capsomer precursor depend on the presence of viral protease (which is probably a component of the polymerase precursor). Subsequent cleavage of the noncapsid proteins may be due to either cellular or viral proteases. In a seminal paper by Butterworth & Rueckert (1972), a first step is taken towards a mathematical description of the cleavage process and their theoretical results compare favourably with kinetic data. It is possible that their approach could be incorporated into the present theory. Inhibition of cellular protein synthesis is thought to be caused by a viral inhibitor protein that binds to cellular ribosomes causing the breakdown of host-cell polyribosomes (Baltimore, 1969). More recently it has been shown that the inhibitor somehow causes the inactivity of eIF-4B, a cellular translation initiation factor which binds to the S-terminal cap of m-RNA (Rose ef al., 1978; Trachsel ef al., 1980).The inhibition of host-cell RNA synthesis is believed due to a viral inhibitor protein acting directly on the cellular polymerase. Inhibition of DNA synthesis.is thought to be a secondary result of one of the other two types of inhibition (Baltimore, 1969). Although the model may be modified to take into account inhibitor proteins, theory suggests that their effect may be treated as a perturbation to the process of nucleotide depletion. In addition one might want to consider competition between cellular m-RNA and viral RNA for free ribosomes. A more realistic model would also have to take into account the fact that the virus produces its own replicase and that replication and translation are confined to the smooth and rough endoplasmic reticulum, respectively. Such a model would involve reaction-diffusion and saturation kinetics. In the context of this more detailed theory it would be necessary to consider more closely the role of VPg, a viral protein whose capping of the 5’-terminal end of viral RNA may initiate RNA synthesis and whose absence or presence may determine whether the RNA functions in translation, replication, or encapsidation (Rothberg et al., 1978; Nomoto ef al., 1977). Furthermore, there is not one type of RNA that acts as an autocatalyst, but rather two types, plus and minus, each of which acts as a catalyst (or equivalently as templates) for the production of the other, thus forming a catalytic cycle. Although it only takes 45-60 s for a plus strand to be produced on a minus template, it takes about 20 min for the number of minus strands to double. Eigen has pointed out that such a catalytic cycle can be represented as a single autocatalyst. In the models presented in sections 2 and 3, the rates of DI- and S-RNA replication encapsidation were considered to be equal. Although this is not
278
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strictly true, the fact that the rates of RNA synthesis peak at about the same time in cells infected at any given MO1 with one or the other. as well as the fact that ratio of progeny S-virus to total progeny is about the same as the ratio of infecting virus to the total number of infecting particles in coinfected cells, indicates that the rates are approximately equal. On the other hand, experiments by Cole & Baltimore, (1973b,c) indicate that DI-RNA probably enjoys the dual advantages of preferential replication and encapsidation, which lead to a S-8% enrichment in DI-particle yield relative to S-virus. The mode1 presented in section 2 provides a route for determining the relative importance of these two effects and thence to account for the different observed hyperenrichment phenomena. This, as well as the problem of the origin of DI-RNA of specific length after successive passages, will be the subject of a subsequent paper. A more complete model would also consider the competition between DI- and S-RNA for the viral replicase. When considering the interaction between cellular RNA and protein synthetic processes, and virus replicative processes, a number of further effects must be considered. In the same way that cellular nucleotide and amino acid pools are maintained by homeostatic mechanism, the levels of cellular RNA and protein also tend to be maintained within some range. Thus the right hand sides of the expressions for cellular RNA and protein synthesis could be replaced by terms of the form [k>(R;,R,. IN] and [kS (PF,-P‘.)R,NA], respectively. (Note that since D, was constant, this parameter has been incorporated into the new rate constants.) Competition between cell and virus for the cell’s substrate pools could be included by subtracting and above terms from the expressions for the depletion of N and A, respectively. Also neglected in the models of sections 2 and 3 were the rates of degradation of cellular and viral proteins and RNAs by cellular protease and RNases. One might want to add a [-k:‘] or [-k:‘Xi] term to each of the synthesis equations to take into account the degradation of the ith synthetic product. The above review emphasizes the complexity of the processes we have considered, while the nearly quantitative success of the simplistic model of Section II emphasizes the usefulness of coupling reasonable assumptions with a dynamic approach. In this sense (only), we believe that the assumptions utilized in the development of our model are justified. 5. Conclusions Our goal in this paper was to determine the extent to which the qualitative behavior of virus replication could be accounted for by a mathematical
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model. In developing a dynamic model for a physical process, endeavored to find the simplest mathematical structure of physical significance. This was done in section 2 and a steady state analysis of the resulting equations together with the knowledge that about half of the progeny RNA remain unencapsidated led to the conclusion that virus replication is controlled by the depletion of the cellular nucleotide pool, a conclusion that is consistent with the experimental fact that the amount of progeny RNA produced in a cell infected with DI-particles and/or S-virus is the same regardless of the MOIs. The controlling reaction is the autocatalytic replicative step which depletes the nucleotide pool. Consideration of this step alone is capable of predicting the observed exponential and linear phases of V-RNA synthesis as well as the cessation of synthesis and the shift in the peak of the rate of synthesis with increasing MOI. Interference by DI particles is a natural consequence of competition for the limited resource N, as are the effects of varying the relative MOIs. The model also predicts the decreased interference when cells initially infected with S-virus are later superinfected with DI particles and, in addition, the dramatic increase in interference when DI-particle infection preceeds an S-virus superinfection. Inhibition of cellular synthetic processes, as well as the shift of the virus-synthesis curves to the left with increasing MOI, also follows from the depletion of nucleotides. Such a depletion may also explain the timing of cell lysis. Since cell membranes require continued maintenance to retain their integrity and since such maintenance requires energy, the depletion of nucleotides would inhibit maintenance and it is reasonable to suppose that deterioration (lysis) of the membrane would follow. These qualitative results are a consequence of the mathematical structure and do not depend in any way on a calibration (or curve fitting). A single-parameter calibration does, however, bring the theory into first-order quantitative agreement with experiment with respect to the approximate duration of the exponential and linear phases, when they occur, and how their phases shift with MOI. That all the above results should apply to DI-particles as well as to S-virus is a consequence of the fact that both replication processes have the same fundamental RNA replicative step at their foundations. By considering the further refinements of the model described in the last section, a better quantitative agreement with experiment might be realized, as well as a more fundamental reason for choosing Aok = 6.0~ lo-” proteins/nucleotide/RNA/h and k3 = 5.3 x 10e6 virus/protein/RNA/h. Moreover, by carrying out the proposed modifications, a more realistic model would result, and other experimentally observed behavior could be understood within the framework of the theory.
280
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The production of protein capable of encapsidating the RNA constitutes a perturbation to the fundamental process of autocatalysis. An eclipse period, a phase of exponential growth followed by a linear accumulation of virus particles lagging behind RNA and protein accumulation, as well as the tinal cessation of encapsidation follow naturally from theory. The eclipse period, for example, is due to the second-order nature of the encapsidation step and the low early concentration of progeny RNAs and top components. By fitting the parameter k,A,, and kj to data on protein synthesis subject to the constraint that only half the progeny RNA is finally encapsidated. the approximate durations of each of the phases was predicted was well as the approximate lag time between protein (and RNA) accumulation and accumulation of progeny virus. Alternatively. an approximate value for k, A,, may be obtained by an order-of-magnitude calculation and used to determine kz on the basis of the observed lag in the accumulation of progeny virus relative to total viral protein and RNA accumulation. That the values of k,A,) and k3 are of the same order of magnitude regardless of the method or experiment used in their calculation indicates the consistancy of the theory. The theory predicts a shift to the left in the virus accumulation curve with increasing MOI. Theory also predicts that if RNA replication is inhibited temporarily in the linear stage without inhibiting protein synthesis for a period of time, there will be an increase in the percent of progeny RNA encapsidated. In section 3 we saw that both the phenomena of interference and of inhibition of cellular synthetic processes were necessary consequences of competition for cellular nucleotide. The overall aspects of interference (Fig. 4) followed without having to specify any parameters. Using only one. previously-determined parameter, k2, the theoretical curve for the inhibition of cellular synthetic processes approximated the experimentallyobserved curves to first-order. Once again, by considering the modifications suggested in section 4 as a perturbation on the first-order theory, better quantitative agreement with experiment could be obtained. In conclusion, the proposed theory is surprisingly accurate and effective in its ability to account for the experimentally-observed behavior and its ability to organize data from several systems. There is however, one critical prediction upon which the validity of the theory rests that has not yet been studied experimentally: We find that at least one of the nucleotides i.7 depleted significantly by the end of the viral replication process. Once the kz, klA,, and k3 are specified by an appropriate reference system, the simple model proposed in this paper applies not only to poliovirus, but also to any picornavirus (or, more generally to any virus whose replication process is governed by a similar dynamic structure). The
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experimentalist may apply the model with parameters thus specified, to predict the approximate behavior of any other system. It should be noted that steady state infections can be understood if klAo > kz, or if kZ is small relative to the rate at which the cell replaces nucleotides. It may be hoped that the success enjoyed by this dynamic model in describing poliovirus replication will persist when applied to kinetic data on other picornaviruses. If it is, the next step would be to consider viruses with more complex dynamic structure: for exampIe, DNA viruses, RNA viruses with reverse transcriptases, or viruses whose genome spends time integrated with that of the host cell. The authors would like to thank Professor Gary Burleson and MS Maura Glynn for many stimulating and helpful discussions, and for their constant interest in this work. REFERENCES BABLOYANTZ, A. AND SANGLIER,M. (1972). FEBSLerr. 23,364. BALTIMORE,D.,GIRARD,M.& DARNELL,J. E. (1966). Virology,29,179. BALTIMORE,D. (1969). In 77zeBiochemistryo,fViruses, (H. B.Levy,ed.).New York: Marcel Dekker. BUTTERWORTH,B.& RuEcKERT,R.(I~~~). Virology 50,535. CASPAR, D. L. D. (1963). Advs. Protein Chem. 18, 37. COLE,~. N.& BALTIMORE,D.(~~~~~).J. mol. Biol.76,325. COLE,~. N.& BALTIMORE,D. (19736).1. mol. Biol.76,345. COLE,C.N. & BALTIMORE,D. (1973~). J. Virol.12,1414. COLE,C. N. (1975). Prog. med. Vir. 20, 180. CONTRERAS,G.,SUMMERS,D.F.,MAIZEL,J.V.& EHRENFELD,E.(~~~~). Virology 53, 120. DARNELL,J.E. & LEVINTOW,L. (1960). J. biol. Chem.235,70. EIGEN, M. (1971). Nuturwissensh. 58,465. EIGEN,M.& SCHUSTER,~. (1977a),Naturwissen.sh.64,541. EIGEN,M.& SCHUSTER,~. (1978a). Naturwissensh.65,7. EIGEN,M. & SCHUSTER,P. (1978b).Nuturwissensh.65,341. HAKEN, H. (1978). Synergetics, 2nd Ed. Berlin: Springer-Verlag. HOLLAND,J.J. & PETERSON,J. A. (1964). J. mol. Bio[. 8, 556. HOLLAND,J.J. (1964).X. mol. Rio!. 8,574. HUANG, A. S. (1973). Ann. Rev. Mcrobiol. 27, 101. LAUFFER, M. A. (1975). Molecular Biology andBiophysics. Berlin: Springer-Verlag. NICOLIS, G. & PRIGOGINE, I. (1977). Self-Organization in Nonequilibrium Systems. New York: John Wiley. NOMOTO,A.,DETJEN,B.,POZZATTI,R.& WIMMER,E.(~~~~). Nature 268,208. PALMENBERG,A.C.,PALLANSCH,M.A.& RuEcKERT,R.R.(~~~~). J. Viral. 32,770. RosE,J.K.,TRAcHsEL,H.,LEoNG,K.& BALTIMORE,D. (1978). Proc.Natn. Acad. Sci. U.S.A.
73,2732.
ROTHBERG,P.G.,HARRIS,T.J.R.,NOMOTO,A.&WIMMER,E.(~~~~). Sci. U.S.A.
Proc.Natn.Acad.
75,4868.
SHIH,D.S.,SHIH,C.T.,ZIMMERN,D.,RUECKERT,R.R.,&KAESBERG,P. (1979i.J. 30,472. TRACHSEL,H.,SONENBERG,N.,SHATKIN,A.J.,ROSE,J.K.,LEONG,K.,BERGMANN,J. E., GORDON, J. & BALTIMORE, D. (1980). Proc. Natn. Acad. Sci. U.S.A. 77,770.
Viral.