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Journal of Virology, April 2000, p. 3566-3571, Vol. 74, No. 8
0022-538X/00/$04.00+0
Copyright © 2000, American Society for Microbiology. All rights reserved.
Diminishing Returns of Population Size in the Rate
of RNA Virus Adaptation
Rosario
Miralles,
Andrés
Moya, and
Santiago F.
Elena*
Institut Cavanilles de
Biodiversitat i Biología Evolutiva and Departament de
Genètica, Universitat de València, 46071 València, Spain
Received 17 September 1999/Accepted 14 January 2000
 |
ABSTRACT |
Whenever an asexual viral population evolves by adapting to new
environmental conditions, beneficial mutations, the ultimate cause of
adaptation, are randomly produced and then fixed in the population. The
larger the population size and the higher the mutation rate, the more
beneficial mutations can be produced per unit time. With the usually
high mutation rate of RNA viruses and in a large enough population,
several beneficial mutations could arise at the same time but in
different genetic backgrounds, and if the virus is asexual, they will
never be brought together through recombination. Thus, the best of
these genotypes must outcompete each other on their way to fixation.
This competition among beneficial mutations has the effect of slowing
the overall rate of adaptation. This phenomenon is known as clonal
interference. Clonal interference predicts a speed limit for adaptation
as the population size increases. In the present report, by varying the size of evolving vesicular stomatitis virus populations, we found evidence clearly demonstrating this speed limit and thus indicating that clonal interference might be an important factor modulating the
rate of adaptation to an in vitro cell system. Several evolutionary and
epidemiological implications of the clonal interference model applied
to RNA viruses are discussed.
| |
INTRODUCTION |
In recent years, increasing
attention has been paid to the study of RNA virus adaptive evolution
from an experimental standpoint. The evolution of biological fitness
(understanding it as a macroscopic characteristic that includes many
components such as speed of replication, efficiency of encapsidation,
ability for cell-to-cell transmission, or resistance to antiviral
factors or defective interfering particles) has been studied in vitro
for virus such as vesicular stomatitis virus (VSV) (4, 9, 10, 21,
24-26), foot-and-mouth disease virus (11, 12), 
6
(2), and X174 (29). Several common features
and conclusions are evident from these studies. For example, the rate
of adaptation decreases with evolutionary time as the evolving
population reaches an adaptive peak which usually represents the best
solution to the problem imposed by the selective environmental
pressures. Another important conclusion is that the rate of adaptation,
as well as the final peak reached, depends on the initial fitness of
the viral clone employed (10). Despite these advances in the
understanding of viral evolutionary adaptation, few experiments have
analyzed the effect of population size on the rate of viral adaptation.
Burch and Chao (2) analyzed the effect of population size on
the fixation of mutations of different magnitudes during the
evolutionary process. A positive correlation was found between
population size and the fitness effect that was fixed. In addition to
this pioneering work, in a recent study (22) we investigated
the effect of population size on the magnitude of the fitness effect
fixed and in the rate of viral adaptation and showed a positive
correlation between viral population size and the fitness effect fixed
as well as a limit imposed by population size to the rate of adaptation.
Recently, Gerrish and Lenski (13) developed a theoretical
framework to explain adaptive evolution in microbial asexual
populations. Experimental support for the model was recently provided
by studies with the bacterium Escherichia coli
(6) and the RNA virus VSV (22), where several of
the predictions made by the model were confirmed. The model is based on
the phenomenon of clonal interference. Briefly, the idea of clonal
interference is as follows. Imagine a beneficial mutation that
spontaneously arose in a genotype. The time required for fixing this
mutation in the population should be long if the population size is
large. However, the frequency of the mutant in the population is low
for a substantial portion of the time (18). During this
period, at which the beneficial mutation is present at low frequency,
there is a certain likelihood, which indeed depends on the mutation
rate, that a second beneficial mutation will arise in a different
genotype. If the population is sexual, both beneficial mutations can be
brought together in forming a new genotype that benefits from their
joint effect. If the population is asexual, the two mutations cannot
recombine and so must compete with one another on their way to
fixation. The result is that only the best one will become fixed,
eliminating the other(s). Even if the first mutation is the best
possible candidate, the time for its fixation is longer due to presence of the second beneficial mutation. Once this beneficial genotype fixes,
secondary beneficial mutations can arise against the new dominant
genetic background. Thus, in asexual populations, beneficial mutations
must fix in a sequential manner.
The main conclusions of the clonal interference model can be summarized
as follows (see reference 13 for a more detailed description of each conclusion). (i) The probability of fixation of a
given beneficial mutation decreases with both population size and
mutation rate. (ii) As population size or mutation rate increases,
adaptive substitutions result in larger fitness increases. (iii) The
rate of adaptation is an increasing but decelerating function of both
population size and mutation rate. (iv) Beneficial mutations that
become transiently common but do not achieve fixation due to
interfering beneficial mutations are relatively abundant. (v) Transient
polymorphisms may give rise to a "leapfrog" effect, where the most
common genotype at a given moment might be less closely related to the
immediately preceding one than with an earlier genotype.
In the present contribution, which must be seen as a complement to our
previously published experiments (22), we want to address
the third of these conclusions, i.e., that the rate of adaptation is an
increasing but decelerating function of population size. To do so, we
will simulate in vitro the evolution of VSV populations of different
population sizes and then measure the rate of adaptation at each
population size. This approach to measuring the effect of population
size in the rate of adaptation is different from and more
straightforward that the one we used in our previous work
(22). Regression analysis of the rate of evolution on
population size will show us if the relationship between the two
parameters is linear or if a maximum rate exists, as predicted by the model.
| |
MATERIALS AND METHODS |
VSV clones.
The first clone employed was a I1
monoclonal antibody (MAb)-sensitive, surrogate wild-type clone. This
wild-type clone was derived from the Mudd-Summers strain of the VSV
Indiana serotype and was replicated for more than two decades in
J. J. Holland's laboratory on BHK-21 cells with either
low-multiplicity passages or plaque-to-plaque clonal propagation to
minimize interference by defective interfering particles. We obtained a
clone from him a decade ago, and to avoid any further genetic change, a
large volume with a high titer (~1010 PFU/ml) was
produced and kept at 
80°C in 1-ml aliquots. This wild-type clone
was used as a common competitor during the competition experiments. The
second viral clone, MARM C (also obtained from Holland's
laboratory), was generated from the wild-type clone and has an Asp259
Ala substitution in the G surface protein. This substitution allows
the mutant to replicate under MAb I1 concentrations that
completely neutralize the wild-type clone (28). All the
evolution experiments were done with MARM C.
Before the evolution experiments, the MARM C and wild-type
clones were plated in the presence of inhibitory concentrations of MAb
I1 to test for the presence or absence of the MAR
phenotype. To start the experiments with a genetically homogeneous
virus, a clone of MARM C was isolated and used to initiate
the evolution experiments.
The fitness of MARM
C relative to the wild-type has been
determined many times under identical or nearly identical conditions
to
those employed here (
4,
7,
8,
15,
20,
21), as
well as 18 times during the course of the present experiment (in
six independent
blocks with three replicates per block). In this
experiment, we
obtained a fitness value not significantly different
from unity (mean
and standard error = 1.04 ± 0.03; Student's
t test = 1.4227;
P = 0.1729). Thus, the mutation
conferring the
MARM phenotype can be considered selectively
neutral.
Cell lines and culture conditions.
Baby hamster kidney cells
(BHK-K) were grown as monolayers in Dulbecco's modified Eagle's
medium (DMEM) containing 5% newborn calf serum and 0.06% proteose
peptone 3. Cells were grown in 25-cm2 plastic flasks
(containing 5 ml of medium) for infections or in 100-cm2
plates (containing 15 ml of medium) for routine maintenance. The cell
density was determined three times during the course of the experiment.
To do so, cells from a 25-cm2 flask were detached (with a
solution containing 0.05 mg of trypsin per ml and 0.2 mg of EDTA per
ml), gently suspended in DMEM, and serially diluted. A convenient
dilution was visually counted using a Neubauer hemocytometer
(0.00625-mm3 grade volume). The cells grew at a density of
(7.7 ± 0.5) × 106 cells/cm2.
The I
1 monoclonal antibody employed in the competition
experiments was produced and characterized by Lefrancois and Lyles
(
17) and VandePol et al. (
28). We propagated
hybridoma cells
in DMEM containing 20% bovine calf serum, 2 µg of
thymidine per
ml, 0.1 µg of glycine per ml, and 14 µg of
hypoxanthine per ml
in large flasks to produce many liters of
high-titer neutralizing
MAb, which was stored at
80°C until
used.
Cell were maintained in incubators at 37°C under a 5%
CO
2 atmosphere.
Effective viral population size.
The effective population
size, Ne for a haploid asexual population under
a batch transfer regimen is given by the expression Ne = N0
(18),
where N0 is the initial population size and is the number of generations of growth per day necessary to reach the
final density. Regardless the different N0
values used in the present experiment, the viral burst from a
25-cm2 flask after complete destruction of the cell
monolayer (1 day of infection in our case) was estimated to be about
3.8 × 1010 PFU. This value depends only on the
available cells for infection, which was a constant throughout the
experiment. Therefore, to obtain different N0
values, it is sufficient to make proper dilutions. We used the three
different N0 values shown in the first column of
Table 1.
The second variable involved in the estimation of
Ne, the number of generations of growth per day,
, has been estimated by
means of equation A1 (see
Appendix). For the
final viral density
and the number of host cells given above, the
number of generations
per day for each initial density is given in the
second column
of Table
1. From these values, it is possible to compute
the
Ne values in our experiment (third column in
Table
1). Obviously,
these values are determined mostly by
N0.
Experimental design.
For each Ne, two
replicates were initiated by infecting flasks with MARM C at
their corresponding N0. After 24 h of
infection, the resulting virus was properly diluted and used to infect
a fresh monolayer. This batch transfer procedure was performed for a
number of consecutive days (given in the fourth column of Table 1).
Virus under each Ne regimen experienced a
different number of generations per day (second column in Table 1);
nonetheless, evolution experiments were stopped when all lines reached
100 generations. Samples were taken periodically from the evolving populations and used to estimate the fitness at different time points.
This basic experiment was performed twice to get statistical power.
Relative-fitness assays.
At the end of each evolution
experiment, the MARM C evolving populations sequentially
isolated were assayed for relative fitness with threefold replication
(15). Relative fitness measures the degree of adaptation of
viral populations to the experimental environment. Each evolving MARM
C population was mixed, in three independent test tubes,
with a known amount of wild-type clone, and the initial ratio for each
replicate mixture, R0, was determined by
performing plaque assays with and without I1 MAb in the
agarose overlay medium. Incorporation of the antibody into the plaque overlay medium (after virus penetration), instead of standard virus
neutralization, avoids the problem of phenotypic mixing and hiding
(i.e., the encapsidation of MARM RNAs within phenotypically wild-type
envelops) (14). Each competition mixture was transferred serially during a sufficient number of passages to obtain good estimates of relative fitness as follows. At each transfer, the resulting virus mixture was diluted by a factor of 104 and
used to initiate the next competition passage by infection of a fresh
cell monolayer. The ratio of MARM C to wild type was determined by plating with and without I1 MAb in the
overlay agarose medium at different transfers. These determinations
gave the proportion MARM C to wild type at transfer
t, Rt. Fitness was defined as W = (Rt/R0)1/t
(3) and obtained by fitting lnW to the time
series data by the least-squares method (27).
Statistical analysis.
All statistical analyses described
were done with the SPSS 8.0.1S for Windows package (23). To
detect the diminishing-returns effect of population size in the rate of
adaptation predicted by the clonal-interference model, we fitted our
estimates of the rate of adaptation obtained for the three different
population sizes to two models. The first is a linear model,
dW/dt = aNe, in which the rate of
adaptation is directly proportional to the effective population size,
i.e., the supply of beneficial mutations. The second model is a
hyperbolic one, dW/dt = aNe/(b + Ne), such that the rate of adaptation shows an upper
limit due to clonal interference (6). The hyperbolic models
was chosen on the basis of its mathematical properties and shape:
faster increases for smaller Ne values, a
decline in the rate of increase with increasing Ne, and the existence of a maximum
dW/dt value. From a statistical point of view, the hyperbola
is also convenient: it uses only two parameters, one more than the
linear model, which allows us to make nested comparisons among models.
In both models, the intercept to the dW/dt value is set to
zero, because it is expected that an asexual population of null size,
that is, with no genetic variability, cannot improve its fitness by
adaptation (6).
| |
RESULTS |
Data description and homogeneity among replicates.
Figure
1 shows the fitness trajectories for all
three effective population sizes and replicates. Although fitness of
the ancestral MARM C clone was measured in six independent
blocks (with three replicates per block), no significant block effect
was detected (nested analysis of variance, F2,3 = 0.1476; P = 0.8687). However, from a statistical point of
view, it is more appropriate to use each particular estimate, obtained
along with the other time point data, in the regression analysis rather
than to use just a single average value.
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FIG. 1.
Fitness trajectories followed by the MARM C
VSV populations for the three different effective population sizes
(indicated in each panel) and replicates. The fitness data have been
log-transformed to obtain a linear regression. Error bars represent
standard errors.
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|
Clearly, under all three different
Ne values,
the fitness significantly increased in an exponential fashion (note the
log
transformation on the
y axis). These exponential
increases in
fitness reflect an improvement in the ability of the virus
to
replicate in the in vitro cell system, in other words, an increase
in viral adaptation. It is a basic principle in evolutionary biology
that adaptation always happens by means of natural selection acting
on
genetic variability (i.e., availability of beneficial mutations).
Therefore, the observed improvements in adaptation are due to
beneficial mutations arising during the experiment. Thus, a requisite
of the model (i.e., the presence of beneficial mutations)
holds.
This result is in accord with previously reported findings for the same
MARM
C clone (
25) after short periods of
evolution.
In experiments involving different VSV clones (
9,
25,
26),
as well as in experiments with the bacterium
E. coli (
18,
19),
trajectories showing two phases have
been observed: fast initial
adaptation followed by a decline in the
rate of adaptation. This
reflects the fact that the speed of evolution
slows as the degree
of adaptation increases as a consequence of a lower
availability
of beneficial mutations for the fine-tuning. (An
alternative explanation,
based on the action of random genetic drift in
large populations,
has been proposed to explain this deceleration
[
26], although
the argument is flawed because it is
based on an incorrect statistical
analysis.)
Despite the clear parallel observed between the trajectories followed
under the three different effective population sizes
(
Ne), an analysis of variance (Table
2) showed a significant
effect associated
with
Ne (
P < 0.0001). However, no
significant
differences were found between the two replicates of the
experiment
(
P = 0.1355). A significant effect of
Ne on the adaptation of
viral populations is
predicted by the clonal-interference model.
However, it is necessary to
test whether the way in which
Ne affects
the
rate of adaptation is exactly that predicted by the model,
i.e., an
increasing but decelerating function. We will gain a
deeper insight
into this question in the next two sections.
Rate of adaptation.
The fitness data shown in Fig. 1 were
fitted to both the log-linear model described in reference
31 and the log-hyperbolic model described in
reference 15. In all six cases, the log-linear model
provides a significantly better fit to the data than did the
log-hyperbolic one. Table 3 shows the
corresponding statistics comparing the fit to both models
(16). This result is not surprising, since we precisely
chose MARM C for this experiment based on the previous
observation that its rate of adaptation was constant during the early
stages of evolution. The derivative of the linear model, dln
W/dt, i.e., its slope, provides an estimate of the rate of
adaptation on a logarithmic scale (last column of Table 3). Note that
the logarithmic transformation has no effect on the conclusions we draw
below.
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TABLE 3.
Fit to alternative log-linear and log-hyperbolic models
of fitness evolution and estimates of the rate of adaptation for each
line in Fig. 1
|
|
Diminishing-returns effect of population size.
The rates of
adaptation calculated above were then regressed with
Ne. As described in Materials and Methods, two
different models were fitted. The first (linear) model implies that the larger the Ne value, the faster evolution takes
place. The second (hyperbolic) model implies the existence of an upper
limit for increasing Ne, as predicted for the
clonal-interference model: the larger the number of beneficial
mutations that coexist at a given time, the greater the competition,
thus slowing adaptation. Figure 2 shows
the fit of both models to the experimental data. The fit to both the
linear (P = 0.0334) and hyperbolic (P = 0.0026) models was significant. Nevertheless, a
partial-F test showed that the fit to the hyperbolic model
significantly increased the goodness of fit, despite the use of an
extra degree of freedom (F1,4 = 25.1180; P = 0.0074). This result clearly confirms the predictions made by
the clonal-interference model.
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FIG. 2.
Rate of adaptation versus effective population size. The
solid line represents the fit to the hyperbolic model that implies the
existence of clonal interference (R2 = 0.5039;
F2,4 = 37.2301; P = 0.0026). The dashed line
represents the fit to a linear model (R2 = 0.5545; F1,5 = 8.4693; P = 0.0334). Note
that the Ne axis has been log-transformed to
separate the data points. As a consequence, both regressions show this
peculiar aspect. Error bars represent standard errors.
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|
| |
DISCUSSION |
Our results clearly confirm the important role played by clonal
interference in the evolution of asexual RNA viruses, as previously demonstrated (22). As expected from their large population
sizes and high genomic mutation rates (more than one per genome and replication round), the number of possible beneficial mutations coexisting at a given time in a population must certainly be large. (The true number of beneficial mutations out of the total number of
mutations produced is still an open question; however, our results
[22] suggest that approximately 1/108
mutations should be beneficial.) This coexistence of beneficial mutations in different genomes implies that they must compete with each
other on their way to fixation. This competition among beneficial
mutations has the effect of slowing the rate of adaptation: the larger
the number of beneficial mutations competing, the smaller the increase
in the rate of adaptation. Acceptance of the clonal-interference model
also allows us to infer some other properties of the adaptive evolution
of RNA viruses. In particular, we discuss three important implications
of the theory for viral evolution: the class of mutations that gets
fixed, the existence of transient polymorphisms, and the reason why a
high-fitness clone can be eliminated from a population of low-fitness genotypes.
The mutation fixed is necessarily the best possible candidate for
the present circumstances.
Regardless of the availability of
beneficial mutations (i.e., the product of beneficial-mutation rate and
population size), beyond a certain population size (usually large), the
adaptive substitutions appear as discrete, rare events. This implies
that a clone harboring a given beneficial mutation that has a low
frequency in the population or is on its way to fixation has a
extremely low probability of undergoing a second beneficial mutation.
This affirmation is based upon our previous estimates of the
beneficial-mutations rate (22), i.e.,
µb = 6.39 × 10
8
beneficial mutations per genome and generation. The likelihood that
such a clone undergoes two beneficial mutations is just
µb2 = 4.08 × 1015; thus, the required population size for the
positive clone (not for the entire population within which it exists)
must be greater than 1/4.08 × 1015 = 2.45 × 1014 to make the second mutational event likely. This
number is far larger than the population sizes reached during our experiments.
As a consequence of the interference among beneficial mutations, the
one that finally gets fixed corresponds to the best possible
alternative, since on its way to fixation it had to outcompete
all
other coexisting beneficial mutations (
13). This fact has
important implications, such as in the dynamics of resistance
to
antiviral drugs: a certain set of mutations conferring resistance
to a
given antiviral does not necessarily represent the entire
set of
possible resistance-conferring mutations but simply a subset
formed by
the better competitors. Therefore, under different circumstances,
a
different set of mutations could be
present.
Common transitory polymorphisms imply viral plasticity.
The
clonal-interference model also has another interesting epidemiological
consequence. Beneficial mutations that become common but do not achieve
fixation due to their interference with the one finally fixed are
abundant. This implies that a high polymorphism is expected. As we
demonstrated in our previous study (22), the larger the
population size, the stronger the effect of clonal interference and
hence the more beneficial mutations competing at a given time. This
increased polymorphism could be important for the adaptiveness of RNA
viruses if the environment fluctuates or rapidly changes, since any of
these suboptimal beneficial mutations, which are destined to vanish in
the existing environment, could potentially the best one in an
alternative environment.
A way to self-protect resident viral population from
outsiders.
de la Torre and Holland (5) previously
observed that a high-fitness VSV clone seeded at low frequency within a
population of lower fitness variants was unexpectedly displaced from
the mixture. It was necessary to introduce the high-fitness clone above
a certain threshold for it to fix (5, 15). This observation, which was difficult to explain from the classical population genetics theory (other than trivial loss by drift), is easily explained if
clonal interference is taken into consideration. Imagine that the
superior clone is initially present at a very low frequency. There is a
high probability that beneficial mutations will arise in the most
common genotype, improving its fitness, interfering with the intruder,
and eventually removing the intruder from the population. However, if
the initial frequency of the fitter clone is high enough, its frequency
in the population will deterministically increase before any
low-fitness variant has a chance to find the appropriate beneficial
mutation. As a consequence, the outsider gets fixed.
| |
APPENDIX |
Number of generations of viral replication in a batch culture.
Let us define Ct as the number of available
cells at the instant t. Similarly, we define
Vt as the number of viral particles produced at
the instant t.
At the beginning of the infection, the number of available cells is
defined as
C0 and the number of viruses
transferred from
the previous day is defined as
V0. At the end of the process of
viral infection
and cell lysis, the density of viruses will be
Vf and no living cells will be present (i.e.,
Cf = 0).
If all the initially seeded viruses efficiently attach to and infect a
cell, the number of available cells at
t = 1 will be
C0 V0. If each infected cell
produces, on average, a progeny
of
R viruses, the number of
virus produced at
t = 1 will be
RV0.
If all these
RV0
viruses efficiently diffuse, as expected in a
liquid culture, and
attach to the remaining
C0 V0 living cells
(
RV0 <
C0 V0), a second round of
infection and lysis will
occur.
At the end of the second round of infection,
RV1 =
R2V0 viruses will be produced, and
therefore,
C0 V0 V1 =
C0 V0 RV0 cells would still be
alive.
It is possible to obtain a general expression for the above recursion
process for any number,
t, of these rounds of infection
and
lysis. The respective densities of virus and number of live
cells are
given by the following set of equations:
We can solve this set of equation with the final conditions
Cf = 0 and
Vf and
obtain an estimate of the total number of cycles
of infection and lysis
that occurred within a culture bottle,
g, to grow the viral
population from
V0 to
Vf:
where
g and
R are the two unknown variables
of the system. Solving for them, we get
and
Remember that this
g does not represents real
generations but instead the number of cell infection cycles. To
determine the
number of generations per infection cycle, we must take
into consideration
the mechanism that VSV uses to replicate. The
infection starts
with a negative-sense RNA, which is copied to a few
intermediate
positive-sense RNAs, which are never encapsidated. These
positive-sense
RNAs are then used as templates to generate the final
negative-sense
offspring (
1). Therefore, according to this
replication model,
we estimated the number of generations per cycle of
cell infection
as 2: the first for generating the positive-sense RNAs
from the
parental negative-sense RNAs and the second for copying them
to
the final negative-sense strains. We define generations in this
way
to take into consideration the fact that there are two moments
during
each cell infection in which mutations can appear: during
the synthesis
of the positive strains and later during the synthesis
of the negative
ones. Whether subsequent positive strains are
synthesized (and from
them fourth-generation negative strains)
is a total mystery. So,
conservatively, we can say that only two
generations of RNA replication
per cell occurred. Thus, the total
number of RNA replication
generations per infected flask will
be
|
(A1)
|
| |
ACKNOWLEDGMENTS |
This work was supported by grant PM97-0060-C02-02 from the
Spanish Dirección General de Enseñanza Superior. R.M. was
supported by a fellowship from the Ministerio de Educación y Cultura.
We thank Paul E. Turner for insightful comments and critical reading of
the manuscript and Olga Cuesta for excellent technical assistance.
| |
FOOTNOTES |
*
Corresponding author. Mailing address: Institut
Cavanilles de Biodiversitat i Biología Evolutiva,
Edifici d'Instituts de Paterna, Universitat de València,
Apartado 2085, 46071 València, Spain. Phone: (34) 963 983 666. Fax: (34) 963 983 670. E-mail: santiago.elena{at}uv.es.
| |
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Journal of Virology, April 2000, p. 3566-3571, Vol. 74, No. 8
0022-538X/00/$04.00+0
Copyright © 2000, American Society for Microbiology. All rights reserved.
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Abstract
Introduction
