Dynamics of alternative modes of RNA replication for positive-sense RNA viruses.

Abstract

We propose and study nonlinear mathematical models describing the intracellular time dynamics of viral RNA accumulation for positive-sense single-stranded RNA viruses. Our models consider different replication modes ranging between two extremes represented by the geometric replication (GR) and the linear stamping machine replication (SMR). We first analyse a model that quantitatively reproduced experimental data for the accumulation dynamics of both polarities of turnip mosaic potyvirus RNAs. We identify a non-degenerate transcritical bifurcation governing the extinction of both strands depending on three key parameters: the mode of replication (α), the replication rate (r) and the degradation rate (δ) of viral strands. Our results indicate that the bifurcation associated with α generically takes place when the replication mode is closer to the SMR, thus suggesting that GR may provide viral strands with an increased robustness against degradation. This transcritical bifurcation, which is responsible for the switching from an active to an absorbing regime, suggests a smooth (i.e. second-order), absorbing-state phase transition. Finally, we also analyse a simplified model that only incorporates asymmetry in replication tied to differential replication modes.

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doi: 10.1098/rsif.2011.0471
, 768-776 first published online 7 September 20119 2012 J. R. Soc. Interface
Josep Sardanyés, Fernando Martínez, José-Antonio Daròs and Santiago F. Elena
positive-sense RNA viruses
Dynamics of alternative modes of RNA replication for


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Dynamics of alternative modes of
RNA replication for positive-sense
RNA viruses
Josep Sardanye ´s1,*, Fernando Martı ´nez1,
Jose ´-Antonio Daro `s1and Santiago F. Elena1,2
1Instituto de Biologı ´a Molecular y Celular de Plantas, Consejo Superior de Investigaciones
Cientı ´ficas-Universitat Polite `cnica de Vale `ncia, Ingeniero Fausto Elio s/n,
46022 Vale `ncia, Spain
2Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
We propose and study nonlinear mathematical models describing the intracellular time
dynamics of viral RNA accumulation for positive-sense single-stranded RNA viruses. Our
models consider different replication modes ranging between two extremes represented by the
geometric replication (GR) and the linear stamping machine replication (SMR). We first ana-
lyse a model that quantitatively reproduced experimental data for the accumulation dynamics
of both polarities of turnip mosaic potyvirus RNAs. We identify a non-degenerate transcritical
bifurcation governing the extinction of both strands depending on three key parameters: the
mode of replication (a), the replication rate (r) and the degradation rate (d) of viral strands.
Ourresultsindicatethatthebifurcationassociatedwithagenericallytakesplacewhentherepli-
cation mode is closer to the SMR, thus suggesting that GR may provide viral strands with an
increased robustness against degradation. This transcritical bifurcation, which is responsible
for the switching from an active to an absorbing regime, suggests a smooth (i.e. second-
order), absorbing-state phase transition. Finally, we also analyse a simplified model that only
incorporates asymmetry in replication tied to differential replication modes.
Keywords: complex systems; intracellular viral dynamics; nonlinear dynamics;
replication mode; RNA viruses; systems biology
1. INTRODUCTION
A multitude of theoretical and computational models
considering different levels of detail and complexity
have been proposed to better understand the dynamics
of viral populations. Broadly speaking, such models can
be divided into structured and unstructured. On the
one hand, the structured models consider a high
degree of detail in the interactions as well as in the pro-
cesses governing replication and infection of viruses and
have been applied to cases such as the bacteriophage T7
[1,2], human immunodeficiency virus type 1 [3], sub-
genomic hepatitis C virus [4], influenza A virus [5]
and vesicular stomatitis virus [6]. This type of model
often entails high dimensionality and a large number
of parameters, which make analytical calculations and
the characterization of bifurcations a difficult task. On
the other hand, unstructured models can also be used
to study viral dynamics in a more abstract way. Con-
trary to structured models, such models only consider
the main processes linked to virus replication and/or
infection, ignoring the details of the interactions, the
cellular compartments where they take place and the
different macromolecules participating in the entire
virus infectious cycle. Many examples of unstructured
models for viruses are found in the literature (see
[7–10] and references therein). This type of model,
although carrying more assumptions than the struc-
turedmodels,oftenenables
characterization of the equilibrium points and their
stability, thus providing clearer and valuable infor-
mation about the role of the parameters in the overall
dynamics of the system.
RNA viruses are obligate cellular parasites infecting
bacteria, fungi, plants and animals. They are character-
ized by large populations, short generation times and
high error rates [11–14]. The reproductive cycle of
RNA viruses consists of several processes crucial for
the success in replication and spreading of the virus
[15]. Viruses can enter into the cell as complete par-
ticles, as ribonucleoprotein complexes or even as
naked nucleic acid molecules. For the particular case
of positive-sense single-stranded RNA (ssRNA) viruses
such as those belonging to the picorna-like group
(studied in this article), the next step is viral uncoating,
by which the genome becomes available for translation,
genome replication and the production of new viral
particles. After infection, the genomic RNA is trans-
lated by the host ribosomes giving rise to the viral
polyprotein, which is self-processed giving place to
detailedanalytical
*Author for correspondence (josep.sardanes@upf.edu).
J. R. Soc. Interface (2012) 9, 768–776
doi:10.1098/rsif.2011.0471
Published online 7 September 2011
Received 15 July 2011
Accepted 15 August 2011
768
This journal is q 2011 The Royal Society
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Page 3

non-structural and structural proteins. The non-
structural proteins give rise to the enzymes needed to
replicate the genomes, a process conducted by an
RNA-dependent RNA polymerase. After some rounds
of amplification of the viral genomes, the structural pro-
teins pack the new genomes forming the virions, which
eventually will infect other available host cells.
A key but poorly understood parameter during intra-
cellular amplification of viral genomes is the mode of
genome replication. Few studies have explored the effect
of the mode of replication on the population dynamics
of viral genomes from a dynamical point of view [16,17],
and even fewer experimental studies have investigated
the dynamics of viral amplification quantitatively [18].
Nevertheless, the few available data suggest different
models of replication for different viruses such as the
stamping machine replication (SMR) mode. For SMR,
and considering an infecting virus with a positive-sense
RNA genome, the progeny of strands will be synthesized
from negative strands complementary to the infecting
positive-sense genome, and thus the expected fraction
of mutant genomes produced per infected cell follows
1 2 e2m, m being the genomic mutation rate. In this
case, the distribution of mutants per infected cell before
the action of selection follows a Poisson distribution.
Such a distribution was identified for bacteriophage
fX174 [19]. Another suggested mode of replication,
opposed to the SMR, is the geometric replication (GR)
mode. For GR, both positive- and negative-sense RNA
strands are used as templates for viral amplification.
For this mode of replication, the expected fraction of
mutant genomes produced per infected cell depends
on the number of replication cycles, n, according to the
expression 1 2 e2nm. The resulting distribution of
mutants then follows the Luria–Delbru ¨ck distribution.
Deviations from the Poisson distribution were found for
the phage T2 [20], thus suggesting that such a virus repli-
cates according to the GR model. Intermediate modes of
replication may also exist, as described for bacteriophage
F6 [21]. In this case, the distribution of mutants slightly
deviated from the Poisson distribution, thus suggesting
thatthereplicationwasmainlyachievedbyanSMRstrat-
egy plus a small contribution of GR.
A direct experimental evaluation of the mode of
replication for a eukaryotic RNA virus has only been
provided very recently. Martı ´nez et al. [18] monitored
and quantified the accumulation of both RNA polarities
of turnip mosaic virus (TuMV) infecting protoplasts of
the plant Nicotiana benthamiana. There, we developed
a simple dynamical system describing the production of
negative- and positive-sense RNA strands including con-
stitutive transcription of positive strands from a plasmid
and interference of positive strands on the synthesis of
negative ones (see §2). Such a model was used to fit the
experimental data (figure 1a). In the present study, we
first analyse a simplified version of this model that may
be useful to better understand the within-cell dynamics
of positive-sense RNA viruses. We analyse in detail the
dynamics of the model, studying the equilibrium points
and their stability depending on the key parameters
governing the intracellular dynamics of viral RNA
replication, paying special attention to the role of the
replication mode. In short, we characterize a non-
degenerate transcritical bifurcation governing the shift
from an active phase, where both viral strands coexist,
to an absorbing state, for which both sequences become
extinct. The critical expression responsible for the bifur-
cation depends on the asymmetry of replication (i.e. the
replication mode), on the replication rate as well as on
RNA degradation rates. Our results suggest that SMR
is more sensitive to the bifurcation and thus GR confers
viral genomes with a dynamical advantage. We finally
analyse a simpler model that only considers the mode of
replication, also studying its equilibria and stability
properties.
2. MATHEMATICAL MODEL
This section introduces a mathematical model that we
recentlyusedtofitexperimentaldataonthe
02040
60
10–4
10–2
100
(a)(b)
α α r
α α r
transcription
φ φ
δ δ
δ δ
φ φ
r
translation
replication
RdRp
translation
concentration of strands
replication dynamics of viral RNA
... AUGCAGCUAGCGAUGCUAGUACGAUC...
(+) RNA: genomic transcripts
... AUGCAGCUAGCGAUGCUAGUACGAUC...
δ δ
replication
hours
... UACGUCGAUCGCUACGAUCAUGCUAG...
(−) RNA: antigenomic strands
τ τ
(+) RNA: genomic strands
(+)
(−)
pTuMV
13 254 bp
t Nos
P 35S
TuMV
Figure 1. (a) Intracellular dynamics of viral RNA accumulation for turnip mosaic virus (TuMV). The circles and the triangles
correspond, respectively, to normalized experimental data for positive (þ) and negative (2) sense strands (see [18]). Solid
and dashed trajectories show the fitting obtained numerically from equations (2.1) and (2.2) using a ¼ 0.0642 (see [18] for
the other parameter values). (b) Schematic of the system under study. In the experiments, a plasmid was used to produce a
viral transcript responsible for initiating the process of viral replication, which starts once the RNA-dependent RNA-polymerase
(RdRp) is translated from the RNA transcript. The grey box displays a schematic of the processes analysed in this study using
equations (3.1) and (3.2). Our model considers strand replication at a rate r . 0. To include the replication mode, we use par-
ameter a [ (0, 1]. When a ¼ 1, the replication is geometric because it is equal for both types of strands. If a ! 0, production of
strands is mainly from (2) to (þ) strands in the stamping machine mode. We also include degradation of the strands pro-
portional to the parameter d . 0.
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J. Sardanye ´s et al.
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accumulation for TuMV genomic positive and antige-
nomic negative RNAs [18]. The model describes the
time dynamics of well-mixed populations of positive
and negative strands during intracellular viral amplifi-
cation considering key parameters such as different
replication modes and degradation of viral strands in
the limit of infinite diffusion (figure 1b). The model is
given by the next couple of nonlinear differential
equations:
dp
dt¼ ðt þ rmÞFðp;mÞ ? dpp
ð2:1Þ
and
dm
dt¼
ar
1 þ cppFðp;mÞ ? dmm:
ð2:2Þ
The state variables p and m denote, respectively, the
concentration of positive (p: plus) and negative (m:
minus) strands. Note that for simplicity, we obviate
the intermediates of replication given by double-
stranded RNAs as well as the explicit consideration of
mutation. We consider that the replication of the viral
RNA is constrained by a logistic-like function, given by
Fðp;mÞ ¼ 1 ?ðp þ mÞ
K
;
assuming finite resources. Here, K is the cellular carry-
ing capacity. The parameter r . 0 corresponds to the
replication rate of the strands, which is assumed to be
symmetric. However, we introduce the parameter a
(with 0 , a ? 1) in equation (2.2) to model all the scen-
arios of asymmetric replication between the GR mode
(with a ¼ 1, i.e. both strands replicate at the same
rate) and the SMR mode (with a p 0). For the latter
case, the replication proceeds mainly from negative to
positive strands. Finally, the model also considers,
respectively, that positive- and negative-sense strands
are degraded at rates dp. 0 and dm. 0.
Equations (2.1) and (2.2) provide a simple scheme to
study the dynamics of the strands during intracellular
viral replication considering asymmetries in the replica-
tion rates owing to different replication modes. As we
mentioned in §1, this model was successfully applied
to reproduce experimental data obtained during the
amplification phase of TuMV in N. benthamiana proto-
plasts [18] ( figure 1a). In that work, we estimated the
key parameters in the replication of this virus, especially
the mode of replication. The parameter t was intro-
duced because in the experiments viral infection was
initiated by the constitutive transcription of positive
strands from an infectious cDNA containing the viral
genome (figure 1). Moreover, the term 1/(1þcp) was
introduced in equation (2.2) to account for possible
interference of positive strands in the synthesis of nega-
tive strands that could result in differences between the
two RNA accumulation curves. The inference of par-
ameter c (which measures the negative effect of
accumulating positive strands on the synthesis of the
negative ones) indicated that no major interference
existed and thus that the differences in accumulation
observed between the two strands emerged mainly
from the different replication modes [18].
Given that interference was not significant in the
TuMV system, in the following, we will not consider it
(i.e. c ¼ 0). Moreover, the strand initiating the process
of viral replication for positive-sense ssRNA viruses in
natural infections is the positive genome of the virus.
Henceforth, in our analyses, we will set t ¼ 0, and the
biologically meaningfulinitial
p(0) . 0 and m(0) ¼ 0.
conditionswillbe
3. RESULTS
3.1. Analyses with degradation of strands
In this section, we provide analytical and numerical
results to characterize the dynamics of equations (2.1)
and(2.2)(with
c ¼ 0,
We note that setting dp¼ dm; d enables clearer
identification of the effect of the mode of replication
on the population dynamics of the two strands, which
is the central question we sought to address in this
study. Hence, the dynamical system we will explore is
given by
?
and
?
The concentration variables or population numbers
span the two-dimensional open space:
t ¼ 0 and
dm¼ dp; d).
dp
dt¼ rm 1 ?p þ m
K
?
? dp
ð3:1Þ
dm
dt¼ arp 1 ?p þ m
K
?
? dm:
ð3:2Þ
R2: fp;m;?1 , p;m , 1g
and only part of which is physically meaningful:
G [ R2;G :¼ fp;m [ Rþ: 0 ? p;m ? 1g:
According to the logistic-like constraint (hereafter
assuming K ¼ 1), the biologically meaningful equili-
brium points are in the triangular phase plane,
GK[ G, with
GK:¼ fp;m [ Rþ: 0 ? p þ m ? K ¼ 1g:
It can be shown that the system of equations (3.1)
and (3.2) has three fixed points or equilibria, which
are calculated from dp/dt ¼ 0 and dm/dt ¼ 0. The
first equilibrium is trivial, given by P*1¼ (0,0), which
involves, if stable, the extinction of both types of
strands. The other two fixed points, denoted as P*+¼
(p*+, m*+), are given by
p?
+¼?raðr þ dÞ + g
ba
;
with
g ¼ rðra þ dÞ
ffiffiffi
a
p
and
b ¼ r2ða ? 1Þ
and
m?
+¼ +r
ffiffiffi
a
p
+ d
p :
r + r
ffiffiffi
a
The next step is to study the stability of these equili-
brium points, which will depend on the model
770
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parameters. To do so, we perform linear stability analy-
sis, evaluating the Jacobian matrix at each of the
equilibrium points of the system. The general form of
the Jacobian matrix for the dynamical system under
study is given by
?
The equilibria can be categorized as stable or
unstable depending on the sign of the resulting eigen-
values. For a two-dimensional dynamical systems such
as the present system, two negative eigenvalues mean
that the fixed point is stable. However, if one or both
eigenvalues are positive, then such an equilibrium will
be unstable, and initial conditions in the immediate
vicinity of this fixed point will move away from this
equilibrium. The general expressions of the eigenvalues,
obtained from det(J 2 lI) ¼ 0, take the form
l+¼1
2
with
J ¼
?rm ? d
arð1 ? 2p ? mÞ
rð1 ? p ? 2mÞ
?arp ? d
?
:
rð?m ? pa +
ffiffiffiffi
v
p
Þ ? 2d
??;
v ¼ m2þ a½4 þ pðpa þ 8p þ 18m ? 12Þ
? 4mð3 ? 2mÞ?:
The corresponding eigenvectors are
?
Wecan study the
equations (3.1) and (3.2), especially for those scenarios
of extinction and persistence of both strands. To do so,
and for simplicity, we first study the stability of the tri-
vial fixed point, P1
while the stability of the other equilibria is numerically
studied (all numerical results shown in this study are
obtained using the fourth-order Runge–Kutta method
with a time stepsize Dt ¼ 1022).
It can be shown that the two eigenvalues evaluated
at the trivial equilibrium point (i.e. computed from
det(J(0) 2 lI) ¼ 0) are l+¼ ?d + r
eigenvectors are n+¼ f+ð
always negative because of the positivity of all par-
ameters. Hence, the stability of the equilibrium point
P1
we can compute the critical value of the mode of repli-
cation, ac, associated with the changes in the stability
of the extinction equilibrium given by P1
value is given by
? ?2
From the previous calculations, it follows that if a ,
acthen lþ, 0, and the origin will be stable because
l+, 0, and both strands will become extinct. However,
if a . acthen lþ. 0, and the origin will be a saddle
and the flows will be attracted by the equilibrium
point Pþ
shows that when the replication mode gets closer to
the SMR (a ! 0), the asymptotic concentration of
n+¼ ??m þ pa +
ffiffiffiffi
v
p
2aðm þ 2p ? 1Þ;1
qualitative
?
behaviour
:
of
*(0, 0), with linear stability analysis,
ffiffiffi
a
p
, and the
ffiffiffi
a
p
Þ?1;1g. Note that l2is
*entirely depends on lþ. From this second eigenvalue,
*. Such a critical
ac¼
d
r
:
ð3:3Þ
*, which is a stable node (figure 2). Figure 2
positive-sense strands increases much more pronounc-
edly than the concentration of negative-sense strands.
Moreover, when the system is analysed for values of a
closer to 1 (i.e. closer to a purely GR), the equilibria
of both strands get closer. The bifurcation diagram of
figure 2 indicates that the decrease in a below the criti-
cal value given by equation (3.3) results in the
extinction of the two strands (see thin lines in the bifur-
cation diagram of figure 2). The presence of a critical
value of the mode of replication responsible for the
switch between coexistence to extinction allows such a
change to be interpreted as a standard absorbing-state
phase transition. This is a class of non-equilibrium tran-
sition in which the system crosses from an active to an
absorbing phase, by varying the control parameter.
Once the absorbing phase is achieved, the system
remains in such a phase forever, with no possibility to
escape [22] (see the description of the bifurcation
below). For the selected parameters (r ¼ 0.1249 and
d ¼ 0.05), ac¼ 0.1602 .... If a , ac, the populations
of strands cannot self-maintain and becomes extinct.
That is, a decrease in a entailing the approach to the
SMR mode causes the extinction of the whole popu-
lation of viral strands.
Similarly, as we did for the parameter tied to the
mode of replication, we can also obtain the critical
values of replication rate as well as of degradation
rate responsible for the extinction of the strands. The
critical values for these two parameters are given,
respectively, by
rc¼ da?1=2
ð3:4Þ
and
dc¼ r
ffiffiffi
a
p
:
ð3:5Þ
If r . rc: lþ. 0, and the trivial equilibrium point
will be a saddle point (recall that l2, 0). If r , rc:
lþ, 0, and then the origin will be stable because
both eigenvalues are negative. For the degradation par-
ameter, the trivial fixed point will be stable if d . dc.
However, if d , dc, lþ. 0 and then the trivial equili-
brium will be a saddle. Under this scenario, the two
RNA strands will achieve a non-trivial steady state.
Such steady state is given by the fixed point Pþ
ally, the third fixed point, P2
plane G: the second coordinate of the equilibrium
point P2
tration of the negative-sense strands, is always negative
under the biologically meaningful ranges of the par-
ameters. Note that under the ranges 0 , a , 1 and
r
a
, r, the denominator of the coordinate m*2will
be always positive. Hence m*2will be negative always.
The dynamics of equations (3.1) and (3.2) are illus-
trated in figures 2 and 3 by means of the solutions
obtained numerically. Some illustrative examples of
the dynamics arising using differential values of a, para-
metrizing differential replication modes, are shown in
figure 2. As previously mentioned, the equilibrium con-
centration between both positive and negative strands
largely differs when a is near 0, and thus replication
proceeds via SMR. As a grows towards 1, the concen-
tration of both strands become more similar, being
*. Actu-
*, is outside the phase
*, which corresponds to the equilibrium concen-
ffiffiffi
p
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equal for a ¼ 1, which actually correspond to the pure
GR. In the analyses shown in figure 2, the degradation
rate of the strands is below the critical value, and the
equilibrium points of GKare an unstable fixed point
(the origin) and the stable node Pþ
shown in figure 2a–d display the effect of increasing a:
the stable node travels towards symmetric equilibrium
values for both strands. We note that the dyna-
mics associated with the meaningful equilibria for
equations (3.1) and (3.2) (given by the origin and Pþ
are independent of the initial conditions. It means that
fromanyarbitraryinitialcondition,theflowswillachieve
a single attractor depending on whether the system is in
the survival or in the extinction scenario.
*. The phase portraits
*)
Figure 3a shows the equilibrium concentration of
both types of strands in the parameter space (a, d).
In agreement with the results reported above, two
different scenarios are found: (i) survival and (ii) extinc-
tion of strands. Scenario (i) occurs for those values of
d , dc, and the dynamics asymptotes towards the
non-trivial steady state are given by Pþ
cussed above, for a given value of d and r, the system
will become extinct as a is decreased and the replication
gets closer to the SMR mode. An interesting result
shown in figure 3a is that GR confers a higher resistance
to degradation. This is due to the fact that all pro-
duced strands are used as templates for further
replication. On the contrary, for the SMR mode, only
*. As we dis-
0.5
1.0
0
0.5
p
1.0
0.5
1.0
0
0.5
p
1.0
0.2 0.4
0.6
0.81.0
0
0.2
0.4
0.6
m
m
(a)(b)
(c)(d)
equilibrium concentration
(a)
α α
(+)
(c)
(b)
(d)
(−)
(+)
(−)
Figure 2. (Top panel) Bifurcation diagram for equations (3.1) and (3.2) showing the equilibrium concentration for (þ) (p, solid
lines) and (2) (m, dashed lines) strands using a as control parameter with d ¼ 0.05 and r ¼ 0.75 (thick lines) and r ¼ 0.1249
(thin lines) (for all data points we use p(0) ¼ 0.1 and m(0) ¼ 0 as initial conditions). Below we show four phase portraits in G
for several values of a indicated with the arrows in the bifurcation diagram: (a) a ¼ 0.06, (b) a ¼ 0.3, (c) a ¼ 0.5 and (d)
a ¼ 0.8 (using r ¼ 0.75 and d ¼ 0.05). In all the phase portraits, the origin is unstable and the fixed point Pþ
node. The arrows indicate the direction of the flows.
*is a stable
772
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a small fraction of negative-sense strands are used as
templates and thus the population is much more sensi-
tive to degradation.
The results from linear stability analysis and from
the bifurcation diagram shown in figures 2 and 3b
suggest that the transition from survival to extinction
of strands associated with the critical parameters pre-
viously characterized is governed by a transcritical
bifurcation. The transcritical bifurcation occurs when
two equilibrium points collide and interchange their
stability [23]. This type of bifurcation suggests the pres-
ence of a smooth, absorbing-state phase transition
because the order parameter (i.e. equilibrium values of
both types of strands) decreases in a continuous way
as the control parameter is driven towards its bifur-
cation value, and no sharp transitions occur as one
might found for saddle-node bifurcations tied to first-
order phase transitions. For our system, it can be
shown that
P?
þjac¼ðd=rÞ2 ¼ ðp?
þ¼ 0;m?
þ¼ 0Þ:
ð3:6Þ
Hence, as the critical value of the parameter tied to
the replication mode, both fixed points P1
lide.Thestabilityofthe
numerically investigated by evaluating the sign of the
two eigenvalues from
*and Pþ
point
*col-
was
fixed
Pþ
*
l+ðP?
þÞ ¼1
2
r ?m?
þ? p?
þa +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
vjðp?
þ;m?
þÞ
q
??
? 2d
hi
;
with
vjðp?
þ;m?
þÞ¼ m?2
þþ a½4 þ p?
? 4m?
þðp?
þÞ?:
þa þ 8p?
þþ 18m?
þ? 12Þ
þð3 ? 2m?
Numerical computations of the eigenvalues for the
fixed point Pþ
in the bifurcation diagram of figure 2 were performed,
*using the same parameter range shown
obtaining (results not shown)
l?ðP?
and lþ(Pþ
values in the parameter range studied in figure 3b (i.e.
0 ? d ? 0.06) also indicated that l2(Pþ
negative and that
þÞ , 0;
*)|a .ac, 0. Moreover, the study of the eigen-
lþðP?
þÞja,ac. 0;
lþðP?
þÞja¼ac¼ 0;
*) was always
lþðP?
lþðP?
þÞjd,dc, 0;
þÞjd.dc. 0
lþðP?
þÞjd¼dc¼ 0;
(results not shown). The two insets in figure 3b show,
respectively, the phase portraits for the scenarios of
survival and extinction of the strands (we show the
flows in the phase plane G). The origin with d , dcis
an unstable equilibrium (indicated with a white
circle), and the non-trivial equilibrium Pþ
node (indicated with a black circle). Once the degra-
dationrateovercomes
equilibrium Pþ
P1
All the previous analyses indicate that the fixed
points P1
collide at the critical parameter
values and that one eigenvalue (i.e. lþ) interchanges
its stability. Under such conditions, the transcritical
bifurcation is called non-degenerate. The same type of
transition occurs as the other two parameters (i.e. r
and d) cross their critical values given by equations (3.4)
and (3.5). From equation (3.6), we also obtain
*is a stable
thecriticalvalue,the
*goes outside the phase plane G, and
*becomes stable.
*and Pþ
*
P?
þjr¼rc¼ ð0;0Þ
and
Pþ?jd¼dc¼ ð0;0Þ:
3.2. Analyses without degradation of strands
Since we are mainly interested in the dependence
between the viral RNA dynamics and the asymmetry
in the mode of replication, we further simplify the
model by only keeping parameter a, and assuming
r ¼ 1 and d ¼ 0. The parameters estimated in Martı ´nez
0
0.02
0.040.06
0.5
1.0
0
0.5
p
1.0
0.5
1.0
0.5
p
1.0
0
0.5
1.0
δ δ
equilibrium concentration
(+)
strands extinction
m
equilibrium
concentration
α α
SMR
GR
1.0
0.5
0
0
0.04
0.08
0.12
1.0
0.5
δ δ
(a)
(b)
(−)
m
Figure 3. (a)Equilibriumconcentrationofpositive-sense(griddedsurface)andnegative-sense(flatsurface)strandsusinga(with0 ,
a ? 1)anddascontrolparameterswithr ¼ 0.1249.Initialconditions:p(0) ¼ 0.15andm(0) ¼ 0.(b)Bifurcationdiagramnumerically
obtainedformequations(3.1)and(3.2),usingtheequilibriumconcentrationofbothstrandsasorderparameters(y-axis:thicklinefor
positiveandthinlinefornegativestrands)anddascontrolparameter(x-axis).Weusethevaluesr ¼ 0.1249anda ¼ 0.0642(asinthe
previous figure). Small arrows indicate the stability of the trivial equilibrium point. The vertical dashed line shows the critical degra-
dation value, dc, given by equation (3.5) (under the selected parameter values, dc? 0.03164...). The insets show the phase portraits
for twodifferent valuesofdegradationrate(indicatedwiththickdashed arrows). For thecase d , dc,weuse d ¼ 0.022; for theextinc-
tion scenario, we use d ¼ 0.037 . dc. Henceforth, stable and unstable equilibria in the phase portraits will be displayed, respectively,
with black and white solid circles.
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et al. [18] revealed that the degradation rates of the viral
RNAs were approximately between one and two orders
of magnitude lower than the replication rate. Hence, the
assumption d ¼ 0 is a first good approach to study the
simplest model of viral strand amplification under
differential replication modes. The model is then simpli-
fied to the following couple of dynamic equations (also
setting K ¼ 1):
dp
dt¼ mð1 ? p ? mÞð3:7Þ
and
dm
dt¼ apð1 ? p ? mÞ:
ð3:8Þ
The dynamical system above has four fixed points
given by
P?
a¼ ð0;0Þ;
P?
b¼ ð0;1Þ;
P?
c¼ ð1;0Þ
and
P?
d¼ ð1 ? m?;m?Þ:
Note that all the points of the form (1 2 m*, m*),
that is, p* ¼ 1 2 m* (which is the diagonal border in
GK), are equilibrium points. The Jacobian matrix for
this reduced model is given by
?
The eigenvalues of matrix (3.9) have the general
form
J ¼
?m
1 ? p ? 2m
?ap
að1 ? 2p ? mÞ
?
:
ð3:9Þ
l+¼1
2
?m ? pa +
ffiffiffiffi
v
p
??
(here also with v ¼ m2þ a [4 þ p(pa þ 8p þ 18m 2
12) 2 4m(3 2 2m)]). The general expressions for the
respective eigenvectors are
?
The eigenvalues evaluated at the trivial fixed point,
obtained from det(J(0) 2 lI) ¼ 0, are lðaÞ
and the eigenvectors are nðaÞ
this equilibrium point will always be a saddle because,
under the biologically meaningful range (i.e. 0 , a? 1),
lþ
given byequations (3.1) and(3.2), the trivialequilibrium
is always unstable because no degradation is considered.
We note that the fixed points Pb
are the extremes of the equilibria contained in
P?
from det(J(Pb
with eigenvectors n1
Moreover, the eigenvalues and the corresponding eigen-
vectors for the fixed point Pc
equilibria P?
and l2
n1
libria, Pb,c
and P?
subspace.
n+¼??m þ pa +
ffiffiffiffi
v
p
2ðm þ 2p ? 1Þa;1
?
:
+¼ +
Þ?1;1g. Then,
ffiffiffi
a
p
;
+¼ f+ð
ffiffiffi
a
p
(a). 0 and l2
(a), 0. As a difference from the model
*¼ (0, 1) and Pc
*¼ (1, 0)
*, computed
(b)¼ 0,
(b)¼ f 2 1, 1g.
d¼ ð1 ? m?;m?Þ. The eigenvalues of Pb
*) 2 lI) ¼ 0, are l1
(b)¼ f1, 0g and n2
(b)¼ 2 1 and l2
*¼ (1, 0) and the line of
dare equal, and are given by l1
(c)¼ 2 a, respectively, and with eigenvectors
(c)¼ f 2 1, 1g and n2
*
d, the flows are attracted in one
(c)¼ 0
(b)¼ f 0, 1g. Note that for the equi-
The previous results on linear stability analysis are
illustrated in figure 4. The dependence of the equilibrium
concentrationsofthepositive-andnegative-sensestrands
is shown at increasing values of a. Here, as a difference of
the previous model analysed, such concentrations are
dependent on the initial conditions, because of the invar-
iant line of infinite point attractors given by Pd? (see the
phase portraits displayed in figure 4a–d). The highest
difference between the equilibrium concentration of
both strands is found for values of a close to 0, where
replication proceeds closer to the stamping machine
mode and there is greater production of positive-sense
strands. As a grows towards 1, approaching the
geometric mode of replication, the two equilibria
approach each other. If the initial condition of positive-
sense strands is low (as one might expect during viral
infection) and m(0) ¼ 0, the equilibrium concentrations
become very close for GR. However, if the initial con-
dition for positive-sense strands is increased, the
equilibrium concentrations for both strands are not so
symmetric as one might expect considering a ! 1
(figure 4).
4. CONCLUSIONS
Whether the choice of the mode of replication might
confer viruses with increased robustness against the
accumulation of deleterious mutations has been a
recent subject of debate and some studies have theoreti-
cally handled questions related to this hypothesis
[17,18]. Experimental studies giving clues about the
model of replication for viruses, mainly obtained from
the analysis of mutant distributions, were performed
many years ago. For instance, the studies of Luria [20]
and Denhardt et al. [19] already suggested different
models of replication for different viruses. Some other
investigations have shown that the replication strategy
in viruses as different as TuMV [18], F6 [21] and
fX174 [19] is closer to the SMR model. These findings
suggest that selection may have favoured this replica-
tive scheme operating on independent viral lineages,
perhaps as a way of reducing the population mutational
load, thus increasing mutational robustness. Together
with the mode of replication, several mechanisms have
been proposed to contribute to the robustness of RNA
virus populations [24]. Robustness, defined as the con-
stancyofthephenotype
perturbations, has been proposed to arise owing to
mechanisms like trans-complementation or neutrality,
intrinsic to virus replication, as well as due to other
extrinsicmechanismsexploiting
mechanisms like the heat-shock chaperones [25].
In this present study, we have analysed the dynamics
of simple models describing the intracellular ampli-
fication dynamics of viral RNA taking into account
differences in the mode of replication. We first studied
a simple model obtained from a dynamical system and
successfully used this to reproduce experimental data
on the accumulation of positive- and negative-sense
strands during the amplification phase of TuMV [18].
The conclusions of our previous study were that, for
TuMV, the model of replication occurred through a
undermutationsor
cellularbuffering
774
Alternative modes of RNA replication
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mixed strategy but approximately 90 per cent of the
genomes were produced via SMR. Here, we studied
the equilibria and stability properties for such a model
describing intracellular dynamics of positive-sense
ssRNA viruses. We reported a non-degenerate transcri-
tical bifurcation responsible for the extinction of the
viral genomes. Such a bifurcation separates two differ-
ent regimes, given by the coexistence of both types of
strands (i.e. condition found in the experimental data)
and by the extinction of strands corresponding to an
absorbing state. The transition between such phases is
governed by a smooth change, suggesting the presence
of a second-order phase transition. Other studies have
reported this type of transition for quasi-species dynami-
cal systems modelling RNA viruses considering mutation
[26,27].Moreover,recentinvestigationshavealso
characterized transcritical bifurcations for quasi-species
models considering both mutation and complementation
phenomena [28].
A second model only considering the mode of repli-
cation was also investigated. This model indicated
that when degradation rates of the viral RNAs are
not considered, there exists an invariant line of attrac-
tors resulting in the coexistence of both strands.
Under these dynamics, the equilibrium values of the
viral RNAs depend on the initial conditions, as a differ-
ence from the first analysed model, for which the
asymptotic dynamics was independent of the initial
conditions.
The results reported with the model considering viral
RNA degradation reveal that as replication gets closer
to the SMR model, the viral strands can suffer the
0
0.5
1.0(a)(b)
(c)(d)
0.5
p
1.0
0
0.5
1.0
0
0.5
p
1.0
0.20.4
0.6
0.81.00
0.2
0.4
0.6
0.8
1.0
m
m
equilibrium concentration
α α
(a)
(b)
(d)
(+)
(c)
(−)
Figure 4. Same as in figure 3 for the simplified model given by equations (3.7) and (3.8). The upper panel shows the two equilibria
for positive (thick lines) and negative (thin lines) strands using three different initial conditions for the positive-sense strands (all
of them with m(0) ¼ 0): p(0) ¼ 0.1 (solid lines), p(0) ¼ 0.5 (dotted lines) and p(0) ¼ 0.7 (dashed lines). Phase portraits for the
same values of a analysed in figure 3, also indicated with arrows in the bifurcation diagram: (a) a ¼ 0.06, (b) a ¼ 0.3, (c) a ¼ 0.5
and (d) a ¼ 0.8. The diagonal displayed with the thick dotted line corresponds to the line of equilibria, P?
d¼ ð1 ? m?;m?Þ.
Alternative modes of RNA replication
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bifurcation and become extinct because of increa-
sed degradation rates or decreased replication rates.
Although SMR might confer RNA viruses with muta-
tional robustness [17], our study suggests that GR
could provide RNA viruses with some other dynamical
advantages. For instance, with an increased resistance
to stop the replication process owing to the degradation
of the negative strands in an environment dominated by
positive-sense strands, or to the degradation of double-
stranded intermediates of replication by the RNA-
silencing machinery [29]. In this sense, one might
expect that RNA viruses may have evolved towards
replication strategies optimizing the interplay between
both mutational and dynamical robustness.
This work was funded by the Human Frontier Science
Program Organization grant RGP12/2008, by the Spanish
Ministerio de Ciencia e Innovacio ´n grants BIO2008-01986
(J.A.D.) and BFU2009-06993 (S.F.E.) and by the Santa Fe
Institute. F.M. is the recipient of a predoctoral fellowship
from Universitat Polite `cnica de Vale `ncia. We also thank the
hospitalityand support of
Theoretical Physics (University of California at Santa
Barbara), where part of this work was developed (grant
NSF PHY05-51164).
theKavliInstitutefor
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