Download Levels of selection in positive-strand virus

Levels of selection in positive-strand virus dynamics
D. C. KRAKAUER* & N. L. KOMAROVA *Santa Fe Institute, Santa Fe, NM, USA
Institute for Advanced Study, Princeton, NJ, USA
Keywords:
Abstract
coevolution;
levels of selection;
RNA virus;
robustness;
translation kinetics virulence.
Conflicting selection pressures occurring over the life cycle of an organism
constitute serious challenges to the robustness of replication. Viruses present a
credible model system for analysing problems that arise through evolutionary
conflicts of interest. We present a multi-level selection model for the life cycle
of positive-strand RNA viruses. The model combines within-cell replication
kinetics and protein synthesis, and between-cell population dynamics of virion
production and transmission. We show how these two levels of within-host
selection interact to produce tradeoffs in the life history strategy of a virus
without consideration of host mortality. We find that viruses evolve towards
intermediate rather than maximum encapsidation rates. This can be interpreted as selection for intermediate virulence through cellular persistence. We
characterize a theoretical persistence threshold arising from the trade-off
between genome replication and genetic translation within the cell. We
present counter-intuitive relationships whereby increasing genome decay
rates and rates of encapsidation lead to increases in the abundance of virusencoded proteins. Data from poliovirus suggest that viruses might be unable to
resolve the vertical conflicts of interests among different levels of selection.
Introduction
Levels of selection problems for a virus
Viruses are infectious, obligate intracellular parasites.
Within a target host cell, the viral genome is replicated
and directs the synthesis of essential virion components
by making use of host cellular pathways. Progeny virions
are constructed from newly synthesized virion components, and constitute vehicles for the transmission of the
viral genome into further target cells.
The replication of a virus genome depends upon
successful completion of a within-cell phase and a
between-cell phase. Those genes encoding for proteins
and thereby strategies effective within the cell are often
distinct from those influencing the biology of cellular
transmission, and with higher levels, including the
infection of new hosts. In many cases, including those
we consider in this paper, these strategies can come into
Correspondence: David C. Krakauer, Santa Fe Institute, Hyde Park Road,
Santa Fe, NM 87501, USA. e-mail: [email protected]
64
direct conflict. Strategies effective at ensuring effective
replication within the cell lead to reduced infection of
susceptible cells. This constitutes a canonical levels of
selection problem (Williams, 1992; Keller, 1999) in
which replication rates at distinct levels of organization
become negatively correlated. The question that arises is,
how viruses if at all resolve this conflict and how it relates
to the design of the virus life cycle.
The virus life cycle
Viruses are translational parasites. All RNA of viral origin
must be translated by host-cell protein synthesis pathways. Either this is achieved directly as in the case of
positive-strand viruses, where the genome can serve
immediately as mRNA template, or some additional
means of synthesizing mRNAs from the negative-strand
template must be employed. The virus typically piggybacks on normal cellular processes and as these are finely
regulated and highly compartmentalized, the virus must
tune its own replication mechanisms and parameters to
exploit those of the host. The essential steps in the
J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD
Virus genome evolution
replication cycle are the formation of mRNA (in DNA
viruses and negative strand RNA viruses), the synthesis
of viral proteins, the replication of viral genomes, the
encapsidation of viral genomes by viral proteins and the
infection of further susceptible cells.
RNA synthesis (replication) is catalysed by virusencoded RNA-dependent RNA polymerase. In some
cases, accessory viral proteins or host-derived factors
are also required to direct the RNA template or prime the
polymerase. The new RNA is synthesized by templatedirected stepwise incorporation of ribonucleotides into
the 3¢-OH end (3¢ to 5¢ direction) of the growing
negative-strand RNA chain, which extends in the 5¢ to
3¢ direction. Genome synthesis occurs mainly in the
cytoplasm, although some viruses synthesize new RNA
when bound to cellular membranes.
The synthesis of viral proteins during translation
proceeds in the 5¢ to 3¢ direction in which the resulting
protein is synthesized from the amino to the carboxy
terminus. Unlike replication, which is largely under the
control of virus-encoded proteins, translation depends
almost entirely on host factors. An important implication
of these observations is that genome replication and
genome expression are mutually exclusive. With replication moving along the positive strand in the 3¢ to 5¢
direction and translation moving 5¢ to 3¢, ribosomes must
be cleared from the viral RNA before serving as a
template for negative-strand RNA synthesis. Replication
appears to be regulated according to the natural clearance rate of translating ribosomes (Gamarnik & Andino,
1998; Barton et al., 1999). A kinetic exploration of the
intracellular implications of this trade-off is provided by
Eigen et al. (1991).
Virion production requires that the protein and nucleic
acid components produced during replication and synthesis are assembled and sorted within the cell for
eventual export. Once at the assembly site, most viruses
are capable of self-assembly and packaging of the virion
components and genome. The release of the virions from
the infected cells varies from non-cytolytic budding,
whereby enveloped viruses leave the cell intact during
release, to lysis where the cell is destroyed spilling the
virions into the extracellular spaces. Once released and
mature, these virions travel within the host initiating
new infections in susceptible cells for which they express
a tropism.
We have chosen to focus on the single-stranded,
positive-sense RNA viruses, as these viruses possess the
simplest of designs. This is because the single RNA
genome performs at least three functions: (i) translation
into proteins, (ii) replication and (iii) encapsidation by
virus-encoded protein for egress from the cell. There is
relatively little in the way of regulation, and hence
evolutionary tradeoffs can be derived directly from
replication kinetics. In order to provide a more concrete
understanding of the RNA virus life cycle, we summarize
below the life cycle of the poliovirus.
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65
Poliovirus: a case study
The poliovirus is a positive, single-stranded RNA virus
(Koch & Koch, 1985). The genome comprises around
7500 nucleotides and encodes between 5 and 10 averagesized proteins. The genome is directly infectious and
contains a single major initiation site for translation. The
entire coding sequence is translated into a 2200 amino
acid polyprotein post-translationally modified into individual protein products. Approximately one-third of the
genome is used to encode a capsid protein required as
protective packaging for the genome during transmission. The capsid is an icosahedral lattice structure of
60 asymmetrical structural protein units. The life cycle
of the virus can be broken down into a number of
sequential steps. The first step adsorbtion requires specific
host-cell receptor binding. This is followed by entry of the
virus genome and structural proteins into the host cell.
During uncoating the genome is released from the
structural proteins. The virus genome is then translated
by host factors, leading to the synthesis of virus proteins
detectable at around 1.5 h post-adsorbtion. This interval
represents the eclipse period. The total amount of virus
RNA produced then increases for 2.5 h, rising to an
asymptotic level maintained for about 1 h before declining around the time of cell death. During the phase of
maximum genome copy number, up to 6000 full
genomic RNAs are synthesized per minute. This value
represents the number of particles produced per minute
at equilibrium densities. The progeny plus-strand RNA
molecules have one of three fates: (1) function as
templates for replication (through a negative-strand
intermediate), (2) serve as mRNA for translation or (3)
associate with capsid proteins in the creation of new
progeny virions. The dynamics of replication and translation give rise to a temporal sequence of each of these
fates. Early in infection, when virus protein level is low
there is a rapid increase in RNA genome copies (fate 1).
As the copy number increases a large fraction becomes
engaged with ribosomes in protein translation, depleting
the store of replicating sequences. This results in a linear
phase of genome replication. As translation proceeds,
increasing numbers of structural proteins become available, including capsid proteins, and lead to the encapsidation of RNA progeny strands (fate 3). From the
population of liberated virions, about 50% penetrate
susceptible cells, whereas only 1% leads to a complete
virus replication cycle and the establishment of productive infection.
Objectives
This paper is organized as follows. In the section on ‘The
intracellular level of selection’, we describe the intracellular viral dynamics and find two equilibria, one corresponding to a productively infected cell and the other to
extinction of virus. We further show that the rate of
66
D. C. KRAKAUER AND N. L. KOMAROVA
encapsidation must be bounded from above in order for
the virus to persist within the cell. In the section on ‘The
extracellular level of selection’, we present the intercellular dynamics and find that increasing the rate of
encapsidation is evolutionarily advantageous at this
higher level. The optimal encapsidation rate, derived
from considering both levels, can be found analytically in
terms of the parameters of the system. In the ‘Discussion’
section, our results in relation to empirical studies, where
available, are discussed along with the horizontal conflicts of interest among distantly related virus lineages
and vertical conflicts of interest among levels of organization.
The intracellular level of selection
The following model aims to make explicit the dynamical
consequences of conserved features of positive strand
RNA virus life cycles, identifying parameters and states
that contribute to intracellular behaviour. This simple
model will enable us to construct a bottom-up model of
extracellular dynamics in the following sections. A more
detailed exploration of the intracellular phase is provided
by Eigen et al. (1991). We are not so much interested in
the ‘Phage clock’ – describing the apparently co-ordinated series of events during virion production – but the
critical persistence thresholds of infection, both intracellular and extracellular. In this section, we describe (1) the
steps essential to the persistence or extinction of virus
genomes within the cell, (2) identify rate-limiting reactions during virion production, (3) characterize parameter dependencies for virus export, and (4) assess the
impact of selection pressures acting at the level of
mature, infectious virions on within-cell genome replication and protein synthesis pathways.
Throughout we shall adopt the following notation:
• m for the number of mRNA strands;
• R for the number of ribosomes;
• p for the abundance of virus protein;
• y for [mp], the mRNA–virus protein complex or virion;
• z for [mR], the mRNA–ribosome or translation
complex.
Intracellular dynamics
We shall assume that a constant fraction 0 £ b £ 1 of the
total virus-encoded protein synthesized is RNA-dependent RNA polymerase. We can then write down the
following reactions as irreversible kinetic relations describing a part of the infectious life cycle of an RNA virus:
k1
m þ bp ! y;
k2
y ! 2m þ bp;
k3
m þ R ! z;
ð1Þ
ð2Þ
ð3Þ
k4
z ! m þ R þ p;
k5
m þ p ! out:
ð4Þ
ð5Þ
The positive-strand RNA genome serves directly as
mRNA – binding with polymerase to form a complex y at
a rate k1 The polymerase replicates the genome and is
released out of complex where it is ready to participate in
further reactions at a rate k2. The genome is also bound to
ribosomes to form the complex z within which virus
proteins are synthesized at a rate k4. RNA present in the
translation complex is not available for replication by
ribosomes. The new genomes and proteins are then
packaged and egress from the cell at a rate k5. The ki are
the reaction rates for each process, and by assuming mass
action, we can write these reactions down as the
following system of differential equations:
m_ ¼ k1 bmp þ 2k2 y k3 mR þ k4 z k5 mp dm m;
ð6Þ
R_ ¼ k3 mR þ k4 z dR R;
ð7Þ
p_ ¼ k1 bmp þ k2 by þ k4 z k5 mp dp p;
ð8Þ
y_ ¼ k1 bmp k2 y dy y;
ð9Þ
z_ ¼ k3 mR k4 z dz z;
ð10Þ
where we have additionally included death rates (di for
each of the components). We shall assume that
dR ¼ dz ¼ 0:
ð11Þ
We make this assumption because the host-derived
factors are constitutively expressed within the infected
cell to serve ongoing cellular processes.
Note that under this assumption, R_ þ z_ ¼ 0 (this
follows from adding equations (7) and (10)). In other
words, we have
RðtÞ þ zðtÞ ¼ Rð0Þ þ zð0Þ C init :
ð12Þ
Upon infection, p(0) ¼ y(0) ¼ z(0) ¼ 0. This is
because the viral proteins have not yet been synthesized.
The initial infectious dose of the virus determines m(0),
whereas R(0) is the homeostatically regulated abundance
of ribosomes within the cell and is always greater than
zero.
Once a cell is infected, genome replication and protein
production follows. This leads to the establishment of a
productively infected cell. At a certain point (in poliovirus 4 h after adsorbtion), the cell can be thought of as
reaching a dynamical equilibrium. This equilibrium does
not imply that components are no longer being produced
or destroyed in some defined sequence, but rather that
the time-averaged abundance of each component has
reached an approximately constant turnover. Where we
have described only five creative kinetic parameters, in
reality there are many more. This is because each
J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD
Virus genome evolution
reaction that we treat as a single step, comprises a
number of steps in nature. For example, the parameter
k5 which we term the rate of encapsidation, involves the
assembly of the virion, transport within the cell, fusion
with the cell membrane and release from the cell
surface. Our parameters are best thought of as ratelimiting steps of these events. For this reason, determining the precise value of the parameters from published
studies becomes very difficult. We therefore speak of
parameters in terms of their relative values, and only
discuss those results where absolute values are not
required.
Viral equilibria
System (6)–(10) under condition (11) has two equilibria.
One of them is given by
m ¼ p ¼ y ¼ z ¼ 0;
R ¼ R~ 0;
ð13Þ
where R~ is an arbitrary constant. In terms of the initial
condition of the system, we have R~ ¼ C init (this is a
consequence of (12)). Local stability analysis tells us that
the solution (13) is always (neutrally) stable.
The other equilibrium of the system is given by the
following expressions:
Re ¼ dm ½dp ðdy þ k2 Þ þ me ðbk1 k2 ð2 bÞ QÞ
=ðk3 me QÞ;
ð14Þ
pe ¼ dm ðdy þ k2 Þ=Q;
ye ¼ bdm k1 me =Q;
ze ¼ Re k3 me =k4 ;
where Q ¼ bk1(k2 ) dy) ) (k2 + dy)k5 and me > 0 is an
arbitrary constant. The existence condition can be
written as
Q ¼ bk1 ðk2 dy Þ ðk2 þ dy Þk5 > 0;
ð16Þ
dp ðdy þ k2 Þ þ me ðbk1 k2 ð2 bÞ QÞ > 0:
ð17Þ
If we assume that
dy ¼ 0;
ð18Þ
all the expressions simplify greatly. Instead of (14) and
(15), we have
Re ¼
dm ½dp þ me ðbk1 ð1 bÞ þ k5 Þ
;
k3 me ðbk1 k5 Þ
ð19Þ
dm
;
bk1 k5
ð20Þ
bdm k1 me
;
k2 ðbk1 k5 Þ
ð21Þ
pe ¼
ye ¼
ze ¼ Re k3 me =k4 :
Solution (19)–(22) exists if
k5 < bk1 :
ð22Þ
J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD
ð23Þ
This condition highlights an essential instability in the
positive-strand RNA virus cellular kinetics. It states that
productive infection can only be achieved when the rate
of encapsidation lies below the rate of polymerase–
genome complex formation. The intuitive explanation
for this requirement is that when the rate of encapsidation becomes too large, replication is unable to preserve
genomes within the cell as they are being removed more
quickly than they are being produced. There is a need for
fine-tuning of parameters to ensure the continued
production of virus.
The precise position of the equilibrium depends on the
virus-determined initial infectious dose, and the hostdetermined abundance of ribosomes. The value me is
related to these initial conditions and is found from
Re þ ze ¼ C init ;
ð24Þ
and equals
me ¼
b þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2 4ac
;
2a
ð25Þ
where a ¼ dmk3[bk1(1 ) b) + k5] > 0, b ¼ dm[dpk3 +
k4(bk1(1 ) b) + k5)] ) k3k4(k1b ) k5)Cinit and c ¼ dmdpk4
> 0.
Let us denote
D > Dc ð15Þ
67
dm ½dp k3 þ k4 k1 bð2 bÞ
:
k4 ½k3 C init þ dm ð26Þ
A simple analysis of formula (25) shows that in order
for the solution, me, to exist and be positive, we need to
satisfy
k5 < bk1 Dc :
ð27Þ
If this condition holds, the system relaxes to the virus
equilibrium [(19)–(22)]. If condition (27) is violated, the
system relaxes to the first, trivial fixed point (13), where
no virus is present. Figure 1 illustrates this point. We
simulated the system of equations with the same initial
conditions for the case where k5 satisfied condition (27)
(plot (a)) and also for the case where k5 violated
condition (27), but satisfied condition (23) (plot (b)). In
the former case, we have a stable non-trivial virus
equilibrium whereas in the latter case the virus disappears from the cell as time progresses. It is important to
note that when the system converges to the nonproductive fixed point, virus RNA decays exponentially
from the initial infecting dose. There is no transient burst
in production that might allow the virus to persist at the
population level.
This result demonstrates that for reasonable initial
conditions, the critical threshold relating encapsidation
and genome replication, given by inequality (23), is
never reached. This is because the dynamical system
68
D. C. KRAKAUER AND N. L. KOMAROVA
(a)
2.0
y
1.5
1.0
R
p
z
m
0.5
0.0
0
100
t
200
300
(b)
2.0
1.5
R
1.0
0.0
0
m,y,z
p
0.5
100
200
bution to the production/decay of mRNA from processes
(3) and (4). This is hardly surprising because during these
processes, the mRNA is conserved. Now, if we look at the
first set of brackets in (28) and use equation (9), we can
see that the corresponding contribution is non-zero and
is equal to bk1pm. This is a manifestation of the fact that
during processes (1) and (2), mRNA is produced, and the
rate of production equals bk1p. Now, we can balance the
three contributions to the change of m: production, virus
egress from the cell and death, given by the three terms:
ðbk2 p k5 p dm Þm ¼ 0:
t 300
Fig. 1 The time-evolution of the variables of the system, m,R,p,y
and z, in the two cases: (a) k5 ¼ 1.3 < bk1 ) Dc and (b) k5 ¼ 3,
bk1 ) Dc < k5 < bk1 (below). In the former case, the system relaxes
to the virus equilibrium (19)–(22), the calculated values for the
equilibrium are marked by horizontal dotted lines. In the latter case,
the system relaxes to the trivial fixed point (13), where no virus is
present. The other parameters are chosen to be k1 ¼ 5, k2 ¼ 0.2,
k3 ¼ 5, k4 ¼ 0.2, b ¼ 0.8, dm ¼ 1 and Cinit ¼ 1.
imposes an additional accessibility restriction on the
parameter values, in effect ensuring that whenever a
productive equilibrium is observed, it is certain to lie
some distance away from the critical threshold. In the
vicinity of the threshold, the virus population evolves
towards extinction.
Parameter dependence of the virus equilibrium
Let us examine the form of the virus equilibrium [(19)–
(22)]. In particular, formula (20) strikes us as counterintuitive at first glance. Why should the amount of
protein produced grow with the death rate for the
mRNA,dm? Surely reducing the half-life of the mRNA
would lead to a reduction in its protein product.
Furthermore, why does p grow as the rate of its removal
from the cell, k5 increases? We might expect that the
amount of protein within the cell decline with increasing
rates of encapsidation. These counterintuitive results can
be appreciated by examining more closely the implications of equilibrium on the dynamical system.
Expression (20) can be interpreted in the following
intuitive way. Let us recall the equation for the dynamics
of m:
m_ ¼ ðk1 bmp þ 2k2 yÞ þ ðk3 mR þ k4 zÞ k5 mp dm m:
ð28Þ
The two terms in brackets on the right-hand side
correspond to processes (1) and (2), the second brackets
contain the contribution of the processes (3) and (4), the
term –k5mp is process (5) and the last term is the decay of
mRNA. Let us start from the second set of brackets in
equation (28). From (10) it follows that at equilibrium,
)k3mR + k4z ¼ 0, which means that there is no contri-
ð29Þ
Formula (20), in which increasing the decay of mRNA
or increasing the rate of encapsidation leads to an increase
in the abundance of protein, follows immediately. To
ensure equilibrium, an increase in the decay of mRNA or
an increase in the rate of encapsidation must be matched
by an increase in the replication rate of the virus genome.
At equilibrium, these conditions are required to prevent
selection of the trivial, zero equilibrium for the virus.
Dynamics of a simplified system
System of equations (6)–(10) is non-linear, and the
general analytical solution cannot be obtained. So far
we have only found the equilibrium points of the system.
It is however possible to characterize the dynamics by
recognizing important properties of virus biology. We
consider an approximation to system (6)–(10) under the
assumption that some of the processes have a much faster
time scale. In particular, complex formation is much
faster than either genome replication or protein synthesis.
Another assumption we are making is that the system is
far from threshold, i.e. k5 bk1. The advantage is that
the system now becomes linear, and we obtain analytical
expressions for all the variables as a function of time.
We assume that
k1 ; k3 k2 ; k4 ; k5 :
ð30Þ
From equations (19)–(22), it follows that R 1/k3 and
p 1/k1, so it is convenient to re-scale the variables
as R ¼ R¢/k3, p ¼ p¢/k1. Substituting this into system
(6)–(10) and neglecting small terms, we can see that
R¢ ¼ k4z/m, p¢ ¼ (k4z + k2by)/(bm) and there are only
three differential equations left:
m_ ¼ ð2 bÞk2 y k4 z dm m;
ð31Þ
y_ ¼ k4 z þ k2 yðb 1Þ;
ð32Þ
z_ ¼ 0;
ð33Þ
with initial conditions m(0), y(0) and z(0). Note that the
initial conditions are not the same as the ones in the full
system; effectively, they correspond to the values of m, y
and z after a short initial stage of the dynamics where
quick, of the order of t 1/k1 or 1/k3, readjustment
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Virus genome evolution
takes place. In particular, it can be shown that the
correct initial condition for z is z(0) ¼ Cinit. System
(31)–(33) is linear, and its eigenvalues are 0, )dm and
)k2(1)b). The solution of system (31)–(33) can be
written as follows:
0 1
0 1
0 k4 1
m
1
dm ð1bÞ
B C
B C
B
C
@ y A ¼ A1 @ k4 A þ A2 @ 0 Aedm t
k2 ð1bÞ
z
0
B
þ A3 @
1
ð2bÞk2
dm ð1bÞk2
1
1
0
C ð1bÞk2 t
Ae
ð34Þ
0
where the constants are expressed in terms of the initial
conditions for m, y and z:
A1 ¼ zð0Þ;
A2 ¼ mð0Þ þ
ðk2 þ dm Þk4 zð0Þ dm k2 ð2 bÞyð0Þ
;
dm ðdm ð1 bÞk2 Þ
A3 ¼ yð0Þ k4 zð0Þ
:
k2 ð1 bÞ
The stable fixed point of this system is given by
1
0
0 1
k4
me
dm ð1bÞ
C
@ ye A ¼ C init B
@ k4 A;
k2 ð1bÞ
ze
1
ð35Þ
ð36Þ
ð37Þ
proceeds at a much faster rate than between cell
transmission.
Dynamics
Let S be the abundance of susceptible cells, I the
abundance of infected cells and V the amount of free
virus. We have the following standard system:
S_ ¼ r cVS d1 S;
ð39Þ
I_ ¼ cVS d2 I;
ð40Þ
V_ ¼ dI d3 V ;
ð41Þ
where r determines the rate at which susceptible cells are
generated from source tissues, c, the efficiency of infection when viruses meet susceptible cells, di are decay
constants and d ¼ k5m(t)p(t) is the rate at which the virus
is liberated from infected cells as described in the
previous within-cell model. We assume that the solution
m(t)p(t) relaxes very quickly to its equilibrium value, so
we can replace it with
d ¼ k5 me pe ;
ð38Þ
and Re ¼ dm(1)b)/k3, pe ¼ dm/(k1b).
One sees from the solution that the dominant
factor in determining the time required for the
system to reach equilibrium is the smaller of the
quantities (1 ) b)k2 and dm. In particular, the smaller
the rate at which new genomes are replicated, k2 the
longer it takes for the system to converge to its
equilibrium.
The extracellular level of selection
The previous results all relate to the within-cell phase of
virus infection. However, a virus must leave a cell and
infect new susceptible cells to ensure that the life cycle is
re-established in further target cells. Parameters critical
for between-cell dynamics need not be the same as those
within the cell: the biological basis for cell tropism and
infectivity (population level parameters) are different
from those describing replication and synthesis (cellular
parameters). Both levels of replication need to satisfied.
In this section, We explore how virus population
dynamics feeds-back to influence the selection of parameters determining cellular kinetics. An important assumption is that the time scale over which the population of
susceptible cells is infected is greater than the time
required for an individual cell to reach its maximum rate
of virion production. In other words, within-cell kinetics
J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD
69
ð42Þ
where me and pe are the equilibrium values, and treat d as
a constant within the system. In this way, we have
combined two time scales of infection, the rapid time
scale of within-cell kinetics and the relatively slow time
scale of cellular infection.
An equilibrium of system (39)–(41) is given by S ¼ rd1,
I ¼ V ¼ 0. This point is stable if R0 < 1, with R0 ¼ cdr/
(d1d2d3). The value R0 is the familiar basic reproductive
ratio. For R0 > 1, the virus is able to invade the
population of susceptible cells and we have another
stable fixed point:
S ¼ d2 d3 =ðcdÞ;
ð43Þ
I ¼ r=d2 ð1 1=R0 Þ;
ð44Þ
V ¼ dr=ðd2 d3 Þð1 1=R0 Þ:
ð45Þ
Increasing the rate of production of susceptible cells or
increasing the value of d always leads to an increase in the
equilibrium abundance of virus. Now, let us assume that
there are two types of viruses, a wild-type V and a rare
mutant V*, which are liberated from infected cells at the
corresponding rates d and d*.The number of cells infected
by each kind of virus is denoted by I and I*. We have
S_ ¼ r cðV þ V ÞS d1 S;
ð46Þ
I_ ¼ cVS d2 I;
ð47Þ
I_ ¼ cV S d2 I;
ð48Þ
V_ ¼ dI d3 V ;
ð49Þ
V_ ¼ d I d3 V :
ð50Þ
70
D. C. KRAKAUER AND N. L. KOMAROVA
An equilibrium of this system is given by equations (43)–
(45), and I* ¼ V* ¼ 0. This solution is stable as long as
R0 > 1 and d > d*. This means that the virus for which
the quantity k5m0p0 is maximized is stable with respect to
invasion of any other strain of virus differing in the value
of the population level parameter d.
An evolutionary stable encapsidation rate
We know that population dynamics favours a steady
increase in the rate of egress of virus from infected cells.
However, we need to consider the impact of this
population-level selection on the stability of the withincell virus dynamics. We can use the results of the section
on ‘The intracellular level of selection’ to find the
conditions under which the quantity k5m0p0 is maximized. Let us assume that
dp ¼ 0:
ð51Þ
With this simplification, which amounts to assuming
that the rate of decay of virus-encoded proteins is very
slow, all the expressions become much more concise and
it is easier to carry out the analysis. We stress, however,
that very similar behaviour is observed for nonzero dp.
We find after some calculation that the evolutionarily
stable strategy for a virus is to have
k5 ¼ bk1 Dopt ;
ð52Þ
where the optimal value of D is given by
Dopt ¼
1þ
ð2 bÞbk1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
ð1 bÞk3 C init =dm
ð53Þ
Figure 2 illustrates this result: the quantity bk1)Dopt
optimizes the curve k5m0p0 as a function of k5. We also
present the result for the case dp > 0, which is qualitatively similar.
From formula (53), we can see that increasing the
decay rate of virus-encoded proteins reduces the maximum value that k5m0p0 can attain. Increasing the rate of
polymerase–genome complex formation (k1), reducing
the rate of ribosome–genome complex formation (k3) or
increasing the decay rate of the virus genome (dm) all
cause the optimum value of the parameter k5 to be
reduced. In other words, these all favour a reduction in
the rate of encapsidation of genomes by virus-encoded
proteins for egress from the infected cell.
tion. A trade-off between these two within host levels of
selection is established, limiting virus replication rates,
and hence limiting virulence in the absence of host
mortality.
Viral genomic architectures and life cycles are highly
diverse. General classifications of viruses are often
composites of these two features (Wildly, 1971). The
first feature – architecture – relates to the organization
of the virus genes and associated proteins, whereas the
second feature – life cycle – is a dynamical property
defined at the level of populations of infected cells.
Evolutionary approaches to biology endeavour to
explain the relationship between organization and
dynamics. Frequently, even within evolutionary studies,
these are treated as functionally independent. This is
often for reasons of tractability. Thus, evolutionary
models will assign payoffs or fitness values to onedimensional entities, variously described as genes,
genomes, phenotypes or populations, without considering their detailed architecture and concentrate on their
dynamics and long-term distributions. This is problematic, not least because the bulk of experimental research
in biology is largely concerned with organization and
tends to neglect dynamics. For example, virologists
interested in virulence study virulence genes and their
expression and consequences within the cell, whereas
theorists tend to treat virulence as synonymous with high
replication rates from infected cells.
The world of viruses, as a result of their relative
simplicity, provide a credible model system in which
these two features can be integrated. We can treat viruses
0.12
k 5 m e pe
b k1 – ∆ opt
dp = 0
0.1
b k1
0.08
0.06
dp > 0
b k 1– ∆ c
0.04
0.02
Discussion
We develop a model of virus replication dynamics in
which the within-cell kinetics and between-cell population dynamics are both described explicitly. We
characterize a bottleneck that arises as a result of
incompatibilities between virus genome replication and
virus protein synthesis. This bottleneck limits the maximum rate of virion production by infected cells, and
opposes selection for maximum rates of cellular infec-
1
2
3
k5
4
Fig. 2 The quantity d ¼ k5m0p0 as a function of k5 for the two cases:
(i) dp ¼ 0 and (ii) dp ¼ 0.01. The other parameters are as in Fig. 1.
The vertical lines indicate the important values of k5: the threshold
value, bk1, which is never reached (plot b), the existence
condition for the virus equilibrium, bk1)Dc (for dp ¼ 0, equation
(26)) and the selected value, bk1)Dopt (for dp ¼ 0, equation (53)).
J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD
Virus genome evolution
as a microcosm of the problems of evolution, and ‘carve’
up the virus world into groups according to combined
organizational and dynamical features. In this paper, we
have presented some speculations on one of these
groups, the single-stranded, positive-sense, RNA viruses.
In our approach, organization is treated using a kinetic
model (developmental model) of replication, and this is
then related to mutation and selection at the population
level through a separation of time scales: the organization emerges quickly, whereas the feedback from the
population dynamics occurs more slowly. This multilevel selection approach highlights important constraints,
or life history tradeoffs, that arise during the course of
evolution. We describe some of these below.
(1) We derive a critical persistence threshold representing a balance between replicating virus genomes
within the cell, and synthesizing proteins in order to
export virions from the cell. If the rate of export is too
high, all genomes are removed from the cell before they
have an opportunity to replicate.
The replication of the positive-sense RNA hepatitis A
virus (HAV) in BS-C-1 cells was examined under singlecycle growth conditions using strand-specific probes for
detection of viral RNA species (Anderson et al., 1988).
The results of this study suggest that encapsidation of
positive-strand HAV RNA inhibits transcription at all
times during the growth cycle, thereby reducing the pool
of replicating RNA and the final yield of infectious HAV.
In other words, encapsidation threatens to exhaust the
store of RNA genomes within the cell. It seems that one
way around this for the virus is to ensure that genomes
be replicated before they are encapsidated. This strategy
is found for flavivirus RNA genomes (Khromykh et al.,
2001). Another solution is to divide the RNA into satellite
RNA destined for replication and normal RNA destined
for encapsidation. This is the solution favoured by velvet
tobacco mottle virus (Hanada & Francki, 1989).
It has even been suggested that persistent infection
with enteroviruses in the central nervous system could
depend on defective transcription, thereby biasing kinetics towards replication and away from protein synthesis
and encapsidation. Evidence from coxsackie virus in
human skeletal muscle cells does not support this
position (Gow et al., 1997).
(2) The model tells us that increasing the rate of decay
of the virus genome (mRNA), or increasing the export of
protein from the cell, both lead to an increase in the total
amount of protein within the cell at equilibrium.
For most viruses, the half-life of mRNAs varies
according to the quantity of protein required within the
cell. Long-lived RNAs generally encode structural proteins that are required most of the time, whereas shortlived RNAs are often associated with regulatory proteins
that are only required transiently. For any RNA that is
required constantly, the decay rate must not exceed the
production rate; otherwise, the RNA will disappear.
Hence at equilibrium, any increase in the rate of decay
J. EVOL. BIOL. 16 (2003) 64–73 ª 2003 BLACKWELL PUBLISHING LTD
71
must be matched by an increase in production. The
immediate outcome of this requirement is that increasing
the decay rate can in some instances lead to an increase
in the abundance of RNA.
Herpes simplex virus carries within its virion a host
shutoff protein (Vhs) that is delivered into cells prior to
virus gene expression. Herpes also encodes the protein
within its genome for synthesis following late gene
expression. The effect of the Vhs protein is to reduce the
stability of mRNA in infected cells (Zelus et al., 1996). The
protein does not discriminate between host-derived and
virus-derived mRNA and leads to an accelerated decay of
both. The standard explanation for Vhs is that the protein
reduces competition between virus- and host-derived
mRNAs for access to translational machinery, producing
a net benefit for the virus. We demonstrate, however,
that such a strategy can increase genomic RNA without
invoking host factors.
An opposite finding is observed in the papilloma
viruses, in which transformed cell lines producing benign
tumours express unstable RNAs rich in AU nucleotides.
In malignant, cervical carcinoma cells, as a consequence
of integration into the host genome, the E6 and E7 gene
mRNAs lack the AU rich sequences, and are consequently more stable (Jeon & Lambert, 1995). This
increased stability is associated with a higher concentration of tumour-specific proteins.
These two cases serve to illustrate that RNA decay rates
are in themselves insufficient to determine protein
abundance. Estimates of proteins require knowledge of
the full chemical reaction scheme and decay rates for
proteins.
(3) The fully analytical solution to our system (under
certain assumptions) shows that the rate of synthesis of
new viral genomes from the polymerase complex is rate
limiting and determines the time to reach equilibrium:
the higher the replication rate the more quickly a fixed
point is reached.
The virus life cycle is made up from a number of
consecutive steps. Although the cleavage of polyproteins
and the subsequent assembly of capsids can be protracted, they are rarely rate-limiting. The rate-limiting step is
most often associated with the production of new protein
(k4) (Borovec & Anderson, 1993) or the replication of the
virus genome (k2) (Anderson et al., 1988). Reduced rates
of protein synthesis by polymerase can be sufficiently
important to form the basis of restricted host ranges
(Lemm et al., 1990).
(4) We showed that evolution favours those viruses
that maximize their rate of egress from infected cells. We
then used this result to determine the evolutionary
optimum value of the virus-determined rate parameter
k5 – the rate of virus encapsidation and export from the
infected cell. We found that k5 is bounded from below
and from above by the intracellular and intercellular
selection forces, and that an intermediate optimum
exists. This optimum is greater for lower rates of genome
72
D. C. KRAKAUER AND N. L. KOMAROVA
decay and lower rates of polymerase binding. Thus,
population level selection feeds-back onto parameters
influencing within-cell kinetics.
(5) Instabilities arising through competing selection
pressures at the intracellular and extracellular levels give
rise to critical points in virus persistence, limiting virus
virulence.
There has been much discussion on the evolution of
virulence (Frank, 1996; see also the volume by Diekman
et al., 2002). In formal treatments of this problem
virulence is modelled as increasing rates of virion
production (Nowak & May 1994). Competition among
virus strains within a single infected host is thought to
produce the most productive strain capable of effective
transmission between hosts. Thus host mortality and
transmission opportunity place a ceiling on the virulence
(e.g. Bonhoeffer & Nowak, 1994; Haraguchi & Sasaki,
2000). We have shown that an explicit coupling of the
intracellular life cycle with the population dynamics
leads to an additional kinetic restriction on the evolution
of virulence: viruses that emerge too quickly from within
the cell are incapable of persistent infection. This suggests
that population selection increasing the encapsidation
rate of viral genomes might effectively rid cells of
infection. Evolutionary approaches to virulence could
benefit from considering in more detail the mechanisms
of virus replication within cells, as increasing fitness as
measured at the cellular level, can correspond to the
reduction of long-term fitness within cells.
(6) Competition among the levels of selection organizes the virus life cycle.
As a result of the distinct adaptations required for
within-cell virus replication and protein production, and
between-cell tropism and transmission, virus genomes
come under selection for the expression of structurally
independent traits. However, these traits are rarely
functionally independent, as the previous discussion
has illustrated. This suggests that selection for efficient
kinetics will often oppose selection for efficient cellular
population dynamics. This evolutionary conflict of interest between two levels of selection is played out in the
structure of the virus life cycle. The poliovirus example
presented in the Introduction suggests that this conflict is
not always resolved. Whereas about 50% of polioviruses
penetrate susceptible cells, only 1% establishes a productive infection. Traits valuable at the within-cell level
do not perform as well as traits required at the level of the
population of cells. Such a large reduction in viability
might only be sustainable because of the large number of
virus particles generated, reflecting the relationship
between genomic stability and population size (Krakauer
& Plotkin, 2002).
A further dimension of the problem not addressed in
the model is the consequence of heterogeneity in the
genetic structure of the virus population. Thus, genetic
conflicts arise both vertically among levels of organization and horizontally among more distantly related virus
lineages. The study of Turner & Chao (1999) on the
within-cell competition dynamics of the /6 phage shows
how within-cell kinetic competition can lead to a mean
reduction of virus fitness defined at the level of cellular
propagation. As the within-cell phase becomes more
intense, the ability to perform at the cellular level is
increasingly compromised. This study also suggests how
promoting competition at a suitable level of selection
might be exploited towards therapeutic advantage, in
which cellular pathology is promoted, in order to
minimize further infection (Krakauer, 2002).
Acknowledgments
DCK thanks the John D. and Catherine T. MacArthur
foundation, the National Science Foundation, and the US
Department of Energy. NLK thanks the Alfred P. Sban
Foundation, the Ambrose Monnell Foundation and the
National Science Foundation. Thanks also to the anonymous referees for their constructive comments.
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Received 15 August 2002; accepted 30 August 2002