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Journal of Theoretical Biology 241 (2006) 928–938
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Optimality models of phage life history and parallels in disease evolution
J.J. Bull
The Institute for Cellular and Molecular Biology, Section of Integrative Biology, The University of Texas at Austin, Austin, TX 78712, USA
Received 20 October 2005; received in revised form 6 January 2006; accepted 24 January 2006
Available online 17 April 2006
Abstract
Optimality models constitute one of the simplest approaches to understanding phenotypic evolution. Yet they have shortcomings that
are not easily evaluated in most organisms. Most importantly, the genetic basis of phenotype evolution is almost never understood, and
phenotypic selection experiments are rarely possible. Both limitations can be overcome with bacteriophages. However, phages have such
elementary life histories that few phenotypes seem appropriate for optimality approaches. Here we develop optimality models of two
phage life history traits, lysis time and host range. The lysis time models show that the optimum is less sensitive to differences in host
density than suggested by earlier analytical work. Host range evolution is approached from the perspective of whether the virus should
avoid particular hosts, and the results match optimal foraging theory: there is an optimal ‘‘diet’’ in which host types are either strictly
included or excluded, depending on their infection qualities. Experimental tests of both models are feasible, and phages provide concrete
illustrations of many ways that optimality models can guide understanding and explanation. Phage genetic systems already support the
perspective that lysis time and host range can evolve readily and evolve without greatly affecting other traits, one of the main tenets of
optimality theory. The models can be extended to more general properties of infection, such as the evolution of virulence and tissue
tropism.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Evolution; Optimality; Bacteriophage; Virulence; Adaptation; Model
0. Introduction
‘‘Natural selection is not evolution.’’ Fisher prefaced his
classic book with this caution to remind us that the two
phenomena are not the same and to justify developing a
theory of natural selection (Fisher, 1930). It is now widely
acknowledged that the specific phenotypes favored by
natural selection may not evolve, because of genetic
constraints, for example. Nonetheless, models assuming
natural selection of phenotypes are commonly used to
generate predictions and explanations of evolution in the
absence of genetic details. An extreme form of this
cause–effect inference is the use of optimality models to
generate predictions of life history phenotypes (Charnov,
1982, 1993; Krebs and Davies, 1993, 1997; Maynard Smith,
1982; Parker, 1978; Stephens and Krebs, 1986; Williams,
1966). Without knowing mutation rates, distributions of
fitness effects, pleiotropy, or even population size, the
E-mail address: [email protected].
0022-5193/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2006.01.027
phenotypic optima predicted by those models are used as
the expected trait values of organisms. Use of optimality
models in this fashion is not without its harsh critics
(Gould and Lewontin, 1979; Lewontin, 1978a, b; Pierce
and Ollasen, 1987), but there has been no resolution of
the disagreement nor much attempt at resolution. Perhaps
the main reason for this stalemate is that the potential
problems with the use of optimality models lie in the
genetic architecture of the phenotypes being modeled, and
there are few phenotypes for which we know the
details well enough to evaluate the possible failings of
optimality.
This paper develops optimality models of two phenotypes important to phage fitness, lysis time and host range.
The present work provides important extensions of lysis
time evolution and develops a new perspective on host
range: selection may favor avoidance of certain hosts, even
when infection of those hosts generates viable offspring.
The available, detailed understanding of the genetic bases
of both phenotypes is reviewed briefly, allowing for
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J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938
unusual evaluation of the limitations and value of
optimality models. In a more general context, these models
of phage evolution can be extrapolated to pathogenic
organisms and models of disease evolution.
1. Lysis with one host type
Most characterized phages are lytic (Adams, 1959). Lysis
is the phage equivalent of ‘‘big bang’’ or semelparous
reproduction, a dissolution of the cell that releases all
phage progeny produced in the cell up to that point, ending
the life of the parent phage. Naively, one might suppose
that lysis should be delayed until the phage has produced
the maximum possible number of offspring from that cell.
This optimum in fact applies in an extreme case, but it
ignores the possibility that progeny released early—before
offspring number has peaked—may be able to take
advantage of an abundance of hosts and produce even
more grandchildren than if the phage delayed lysis to gain
a few more offspring (Abedon, 1989).
Although the basic principles of lysis have been known
for at least half a century, the first models of lysis time
evolution were published in the last two decades (Abedon,
1989; Wang et al., 1996). Wang et al. provided a simple,
graphical model to calculate the optimal lysis time. That
model assumed that hosts permanently outnumbered
phages and existed at a constant density, so that the
phage population was expanding exponentially and indefinitely. In the long-term, therefore, phage population
growth rate is
b1=g
per unit time, where b is burst size (fecundity) and g is
generation time. Generation time is faster with shorter lysis
times, and shorter lysis times are better for this reason.
However, burst size increases with time to lysis, so longer
lysis times are better for this reason. These two effects of
lysis time impact fitness in opposite ways, and an
intermediate lysis time maximizes fitness under appropriate
conditions.
Solving the Wang et al. optimum should be easy.
However, a problem arises in calculating the appropriate
generation time, because generation time includes the time
from burst until the progeny phage encounter a new host,
and this component is a random variable. Use of the
average time to infection leads to a bias, perhaps because
time to infection follows an exponential distribution, and
the exponential is highly asymmetric around its mean.
Abedon et al. (2001) simulated an infection process to
empirically determine the optimum and avoid this bias; by
this method, the bias in the Wang et al. optimum (using the
mean time to infection) was serious at low cell densities,
although it was minor at high cell densities. It is thus
desirable to obtain an analytical expression for this more
exact optimum.
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1.1. The optimum lysis time
Here, an approach to the optimum lysis time is offered
that overcomes the problem in calculating generation time.
This approach parallels that of Lenski and May (1994) in a
somewhat different context. As with Wang et al. (1996), we
will calculate the optimum lysis time for a phage cohort in
an environment with a constant density of hosts (C), and
with a phage density such that multiple infections of the
same host are rare.
The following differential equation describes the dynamics of phage density, assuming no change in cell density
(modified from Levin et al., 1977):
P_ ¼ Pðm þ kCÞ þ b edL kCPL ,
(1)
where a superior dot indicates the derivative with respect
to time; P is the density of free phage [P(t)]; PL is the
density of free phage L time units in the past [P(tL)]; C is
the density of uninfected cells; d is intrinsic death rate of
cells, whether infected or not; m is the death rate of free
phage; k is adsorption rate of phage to cells; b is burst size
(number of phage per burst); L is the latent period, the time
from infection to lysis (minutes).
In contrast to the usual SIR models of infectious disease,
this equation describes the free, infectious particles
themselves. Infected hosts do not need to be described,
because they invariably burst and disappear L minutes
after infection, at which point they contribute only to the
free-phage pool.
Eq. (1) is linear. With constant host density, infections
rare, and adsorptions distributed in time, the dynamics will
tend to exponential growth (oscillations will decay), so that
PðtÞ ¼ Pð0Þ elt . Eq. (1) then becomes (Bull et al., 2006)
l elt ¼ ðm þ kCÞ elt þ kbC edL elðtLÞ
(2a)
or
l ¼ m þ kCðb eLðdþlÞ 1Þ.
(2b)
The long-term, asymptotic population growth rate of the
phage is l per unit time. For values of l that deviate from
zero, the phage population would either go extinct (lo0)
or exceed the host density (l40). The latter would either
cause a collapse of host density (Levin et al., 1977) or
multiple infections per host (Bull et al., 2006). In either
case, the model assumptions no longer apply. Thus, the
model is best interpreted as revealing the momentary effect
of selection when hosts are rarely encountered by phage.
One context in which an excess of hosts might be sustained
for long periods of phage growth is a spatial one—the
spreading wave of phage across a spatially uniform
population of hosts. The model would then apply to the
leading edges of the epidemic wave. The conditions
assumed by the model can also be achieved experimentally,
however.
The transcendental equation (2b) is not tractable for
further simplification. However, it may be solved for the
lysis time optimum by setting the partial derivative of l
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with respect to L equal to zero and treating b as a function
of L. This approach yields
b_
^
¼ d þ l,
b^
(3)
where a hat 4 indicates a value at the optimum (Bull et al.,
2006). We denote the quantity d þ l^ as the intrinsic growth
rate of the phage, because it describes how fast the phage
population is capable of growing, i.e. when cell death rate d
is zero. The intrinsic growth rate is specific to cell density
C, of course. The intrinsic phage growth rate likewise
equals the maximum cell death rate at which the phage can
avoid extinction.
Wang et al. (1996) extracted published data that
indicated an approximately linear accumulation of progeny
phage within the cell, starting at the end of the eclipse
period and extending well beyond the normal lysis time
if lysis was abolished. Therefore, letting b ¼ RðL EÞ,
where E is the length of the eclipse period and R is the
slope of progeny accumulation after eclipse (Fig. 1),
Eq. (3) becomes
L^ ¼ E þ
1
d þ l^
.
(4)
The optimum lysis time is the sum of the eclipse period
and the inverse of the intrinsic growth rate. This latter
quantity is the time for the phage to increase its numbers
by one log (or by a factor e), and represents a type of phage
‘‘generation time,’’ which shortens as the growth rate
increases. Although the formula appears simple, l^ is a
function of many parameters including lysis time, so it is
not trivial to solve. The formula does become tractable if
the phage population is at dynamic equilibrium, because
l^ ¼ 0, and the optimum is then independent of all but
eclipse and intrinsic bacterial death rate. Even when the
burst
size
(slope R)
growth rate is not zero, the formula is easily solved
numerically. Curiously, this formula applies even when
lo0 (provided d þ l40), yet the phage population is
declining.
Abedon et al. (2001) compared optima obtained by their
method with optima obtained from the Wang et al.
formula, using the linear accumulation function and the
average time to adsorption. As an example, Wang et al.
(1996) offered a specific set of values whose optimum
was 281.06 min. The corresponding Abedon et al. calculation was 48 min (their Fig. 3). The numerical solution
of our Eq. (4) is 47.75 to the nearest fourth-minute
(k ¼ 1:04 109 ; E ¼ 10; R ¼ 5; C ¼ 5 105 ). Calculations of optima using (4) for other cell densities largely
agree with those of Abedon et al. (their Fig. 3, using the
linear progeny function), although at cell densities less than
105, our method gives slightly shorter optimum latent
periods (not shown). Thus, our method agrees with the
simulation method of Abedon et al. (2001) instead of b1/g,
although as Abedon et al. noted, the discrepancy is small at
high cell densities. Indeed, the discrepancy even at low cell
density is only of quantitative interest. Neither of those
previous studies allowed for an intrinsic cell death rate,
independent of phage killing, and the cell death rate puts
an upper limit on the lysis time optimum in a stable or
growing phage population. A cell death rate might obtain
for various reasons—washout in a chemostat, predation
from non-phage sources, or death from stress or lethal
chemicals (e.g. bacteriocins), or infection by other phages
that blocked reproduction by the phage of interest.
The optimal phenotypes in these and the following
models optimize the growth rate of the phage in a
dynamically mixed population of hosts. Constant cell
densities have been assumed for mathematical convenience
as well as to yield results that can be easily interpreted, but
constant cell density is not a realistic assumption for a
natural phage environment. It was noted above that, if
l40, the phage will outgrow the bacterial population and
either reduce the cell density or cause multiple infections.
Greater realism would be added, therefore, by allowing cell
density to change in response to phage density, as in the
early models of Levin et al. (1977). It might seem that l ¼ 0
is the ultimate dynamic equilibrium in such a model, but
undamped oscillations are predicted to occur under wide
ranges of parameter values in single-host populations, and
it is not clear how oscillations affect lysis time optima.
1.2. Numerical examples
E
Lysis time
Fig. 1. Tradeoff of burst size and lysis time. The lower axis is the time
since infection at which lysis occurs. No progeny are produced prior to E
(eclipse), but progeny output rises linearly thereafter.
Optima of phages with three different progeny accumulation functions are shown for different cell concentrations
(Fig. 2, top). As shown here and was also evident in
Abedon et al. (2001), the optimum changes little across a
wide range of high cell densities but changes more
dramatically at low cell density. At high cell densities, the
differences in optima among different types of phages are
due mostly to differences in eclipse period—eclipse puts an
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Fig. 2. (Top) Lysis time optimum versus cell concentration, by numerical
solution of (4) to find the value of lysis time (L) that maximizes phage
growth rate l, subject to the burst size function b ¼ RðL EÞ, where E is
the eclipse period and R is the rate at which phage progeny are produced
following eclipse. The optimal lysis time functions are shown for three
different burst size functions. (Bottom) Intrinsic growth rate (d+l) versus
cell concentration for the same lysis optima curves in the top. In both
figures, parameters are ðE ¼ 20 min; R ¼ 200 progeny= minÞ for the solid
curve ðE ¼ 20; R ¼ 25Þ for the dashed curve, and ðE ¼ 40; R ¼ 10Þ for
the dotted curve. k ¼ 2 109 ml/min for all curves. The highest value of
R illustrated here (200) leads to unrealistically high burst sizes but is
included to reveal the pattern across a broad range of values.
absolute lower bound on lysis time. Overall, changes in cell
density have the least impact on lysis time optima for those
phages with the highest growth rates (compare top and
bottom figures).
The lower part of Fig. 2 shows intrinsic growth rates
ðd þ lÞ of phages at different cell densities. Lysis time
optima in the top panel apply to the growth rates in the
bottom panel at the same cell density for the corresponding
phage. The optima are not affected by the cell death rate
(d), provided the sum (d þ l) remains the same. If l is
negative, the phage goes extinct, but the optimum still
applies as long as d þ l40.
The strength of selection for the optimum may be
evaluated for different parameter values and host densities
(Fig. 3). The strength of selection for the optimum
increases with the curvature at the optimum (in each curve
of Fig. 3, the optimum is the highest point). Phages with
high growth rate experience stronger selection for the
optimum; selection for the optimum of the same phage
weakens merely as cell density is lowered (compare the top
and middle curves in Fig. 3). Selection is in fact so weak at
the optimum on the lower two curves that it would be
931
Fig. 3. Three curves show the growth rates of phages with different lysis
times. The three curves differ in the progeny function b ¼ kðL EÞ or in
the cell density. From top to bottom, the optima are 26, 34.75, and 66 min.
Per minute, the relative growth advantage of a phage expressing the
optimum lysis time (‘‘opt’’) over a phage expressing an alternative time
(‘‘alt’’) is determined as eðlopt lalt Þ . For these three optima, a rare phage
genotype expressing the optimum lysis time (starting frequency of 106) in
a population where the common type lysis time was 4 min later than
optimum would require 45, 390, and 1903 h (from top to bottom) to reach
50% frequency, so it would not be practical to experimentally evolve the
lower two populations close to the optimum. Parameters are ðE ¼
20 min; R ¼ 25 phage= min; C ¼ 108 cells=mlÞ for the top curve ðE ¼ 20;
R ¼ 25; C ¼ 106 Þ for the middle curve, and ðE ¼ 40; R ¼ 10; C ¼ 106 Þ
for the bottom curve. k ¼ 2 109 for all curves.
tedious experimentally to select a phage population better
than to within a few minutes of the optimum.
1.3. Joint evolution of lysis time and lysis time tradeoff
The model assumes that all parameters are evolutionarily invariant except burst size and lysis time, and these are
forced to covary along a single tradeoff function. In
adaptations of phages to novel conditions, it may also be
that the lysis time function evolves—a shorter eclipse
period, which may in turn result in a different (shallower)
slope. In this case, the lysis time can evolve along
one function or jump to another. Suppose, for example,
that one burst size function is B1 ¼ R1 ðL E 1 Þ and the
other is B2 ¼ R2 ðL E 2 Þ. If E 1 oE 2 (without loss of
generality), but the slopes obey the reciprocal relationship
R2 4R1 , then the two functions will intersect at
Lint ¼ ðR1 E 1 R2 E 2 Þ=ðR1 R2 Þ, as in Fig. 4. The question
is how lysis time will evolve on this discontinuous surface.
The lysis time optimum for a single function is
^ At the point of intersection, both
L^ ¼ E þ 1=ðd þ lÞ.
functions have the same burst size and lysis time, hence
^ Since the two functions
have the same value of ðd þ lÞ.
have different eclipse times, it is impossible for the
intersection to satisfy the optima for both lines. What is
observed in numerical trials is that lysis time optima can
exist anywhere along the uppermost function except in a
zone around the intersection. As conditions are changed to
change the lysis time optimum toward the intersection, a
point is reached at which the optimum leaps across Lint
onto the other function (the zone of instability around Lint
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(slope R2)
burst
size
possible optima
(slope R1)
E1
E2
Lysis time
Fig. 4. Relationship between burst size and lysis time when the phage can
evolve on either of two linear tradeoff functions (see Fig. 1 for a single
tradeoff). The set of feasible optima are given by the shaded gray zones
along the two lines. There is a zone around the intersection of the two lines
which is never optimal, because points on one or the other tradeoff yield
superior fitness (growth rate).
depends on the parameters). Thus, the evolution of lysis
time is discontinuous and, at this discontinuity, exhibits a
more dramatic shift in lysis time than would occur on a
single tradeoff function.
1.4. Genetics of lysis
Lysis is not a passive deterioration of the cell. It is an
active, rapid dissolution of the cell wall, which enlists the
osmotic pressure of the cell to cause a complete liberation
of the cell components into the surrounding media. Not all
phages lyse their host, however. The filamentous phages
(e.g. M13) are assembled at and secreted through the cell
membranes without killing (Model and Russel, 1988).
Temperate phages (the most-studied example being phage
l) can infect and adopt a quiescent state of benign
coexistence as a prophage, but whenever they switch from
their lysogenic state to produce phage progeny, they exit
the cell lytically (Johnson et al., 1981).
Two classes of lysis mechanisms have been described
(Bernhardt et al., 2002; Young, 1992). The simplest one
uses a single phage gene to block cell wall synthesis. In
rapidly dividing cells, lysis occurs at cell division (the
septum cannot form), so lysis time is only partly controlled
by the phage. These mechanisms are found in phages with
the smallest genomes, including the RNA phages (e.g. Qb)
and the isometric phages of the microviridae (e.g. fX174).
The other and most widespread class of lysis mechanism
has two components, a lysin (muralytic enzyme) that
degrades the cell wall and a transmembrane protein called
a holin that permeabilizes the host inner membrane to
allow the lysin access to the cell wall (which lies outside the
inner membrane). Lysins vary in which chemical bonds of
the cell wall they attack, so they have multiple origins.
However, the timing of lysis is attributed to the holin,
because it is not until the holin creates pores in the inner
membrane that lysis can occur; a variety of genetic
manipulations of phages indicate that the holin indeed
controls lysis time. Holins act in a threshold manner, so
that no lysin escapes through the inner membrane until the
threshold density of membrane-bound holin is reached, but
then there is widespread, rapid escape of lysin (Grundling
et al., 2001).
In principle, a two-component system times lysis via the
onset and level of expression of the holin but also on the
holin’s threshold setpoint, the molecular basis of which is
not yet understood (Wang et al., 2000; Young, 2002). That
the holin threshold is tunable is suggested by extensive sitedirected mutagenesis experiments of holin genes in phages
T4 and lambda (Chang et al., 1995; Grundling et al., 2000;
Ramanculov and Young, 2001a, b). Not uncommonly, the
system is not this simple. Anti-holins are often found as
well, non-essential genes (often in-frame with the holin)
whose only observed phenotypic effect is a modification of
lysis time. Extensive work on these genetic systems
indicates that lysis time can evolve with little impact on
other phage fitness components, so the assumptions of the
models regarding phage genetics appear to be met (i.e.
there is little pleiotropy accompanying changes in lysis
time).
A description of the genes involved in lysis provides a
useful understanding of lysis time as pertains to the model
here, but it should be realized that this understanding and
the model address only a small part of lysis time expression
in a natural environment. For an obligately lytic phage, as
assumed in our model, there is the complicated issue of
lysis time plasticity—the spectrum of lysis times in different
hosts and under different physiological conditions. It was
noted early that lysis time scales with bacterial growth rate
(Ellis and Delbruck, 1939; over a six-fold change in
bacterial doubling time effected by temperature), and
much of the observed lysis time plasticity in phages is
apparently optimal from the evolutionary perspective
(Abedon et al., 2001). Our model ignores plasticity by
virtue of treating all cells the same. At an even higher level
of complexity, lysis inhibition (known only in the T-even
phages) is a special type of plasticity in which lysis is
delayed if the cell is co-infected by many phages before the
normal time of lysis (Abedon, 1990; Adams, 1959). Again,
our model ignores lysis inhibition because the rate of
infection per cell is assumed to be low. Temperate phage
face two lysis timing decisions: (i) how quickly to lyse
following an infection adopting the lytic cycle, and (ii) how
quickly to lyse after a lysogen is induced into the lytic
pathway. The control of lysis genes in the temperate
phage lambda, which need to be fully suppressed until
induction, has been extensively studied (e.g. Nickels et al.,
2002; Roberts et al., 1998). These various complexities lie
outside the simple model developed thus far, but they
are clearly relevant to lysis time evolution in a natural
context and are relevant to interpreting lysis functions in
natural phages.
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1.5. Implications for evolution of virulence
2. Evolution in populations of heterogeneous hosts
Abedon et al. (2003) suggested that phage lysis time
was analogous to the virulence of an infection in an
animal host: shortening phage latent period increases
virulence. This parallel may seem puzzling at first, because
a phage infection requires cell/host death for transmission, whereas evolution of virulence theories have
been based on parasites that require a live host for
transmission. Evolution of virulence theories may thus be
most appropriately applied to the non-lethal, filamentous
phages that do not lyse their hosts. However, the virulence
analogy can also be construed to extend in principle to lytic phages, because progeny production requires
a live host, and by halting phage production, the time
of lysis determines how much transmission can occur.
Indeed, our qualitative results are the same as those
of Lenski and May (1994), which addressed the virulence of an infection that transmitted from live hosts
continuously.
Whether or not one accepts lysis time evolution as a
close analogy to virulence evolution, the mathematical
structure of both problems is similar. The simple, tractability of phage systems may thus help in exploring some
of the issues in the evolution of virulence. The chief
architect and advocate of evolution of virulence theories
was Ewald (1994), and one of the main evolutionary
determinants of virulence in his verbal models was the
availability (density) of uninfected hosts: high virulence
had little cost if the parasite was transmitted to many
new hosts before it killed its previous host. Yet as Lipsitch
et al. (1995) pointed out, host density is not a factor in
virulence evolution when the parasite reaches dynamic
equilibrium, because at equilibrium each infection gives
birth to only one new infection during its lifespan, on
average, regardless of host density. Result (4) explicitly
shows that growth per se of the parasite population selects
‘‘higher virulence.’’ Thus, as is well known from life history
theory (Lewontin, 1965; Mertz, 1971), it is only in a
growing population that the geometric advantage of early
reproduction is favored at the expense of total fecundity.
For infections, this translates into a benefit of early
transmission at the cost of high host death rate. This point
was made previously for virulence evolution (Bull, 1994;
Lenski and May, 1994), although the formula here is a
more explicit codification of this principle than has been
developed before.
The numerical illustrations of (4) highlight a subtle
difficulty with qualitative applications of evolution of
virulence theories. Substantial changes in host density (e.g.
100-fold) have a notable impact on the lysis time optimum
for some progeny-lysis time functions but not others, and
the impact also depends on which end of the host-density
spectrum is considered. Without knowing a variety of
quantitative details, effects of changing host density may be
too small to detect. Yet those details are almost never
known in natural settings.
Simple extensions of the preceding model may also be
applied to phenotype evolution in populations of mixed
hosts. Host quality and host strain no doubt vary
in the natural environments phages experience, and in
view of the recent discoveries that bacteria in lab cultures
exhibit a range of physiologies (physiological tolerance:
Wiuff et al., 2005), the phenomenon is also potentially
relevant even in the most controlled, homogeneous
conditions. Furthermore, when multiple host types are
present, lysis time is no longer the only phage phenotype
of interest: host range and other forms of specialization
can evolve.
We first derive the relevant equation for phage growth
in a population with multiple host types (two). Assuming fixed densities of two hosts, C1 and C2, and that
parameter values are specific to each type of infection (subscripted accordingly), the two-host equation
corresponding to (1) is
P_ ¼ Pm þ k1 C 1 ðb1 edL1 PL1 PÞ
þ k2 C 2 ðb2 edL2 PL2 PÞ.
ð5Þ
lt
Again assuming that PðtÞ ¼ Pð0Þ e , the solution takes
the form
l ¼ m þ k1 C 1 ðb1 eL1 ðdþlÞ 1Þ
þ k2 C 2 ðb2 eL2 ðdþlÞ 1Þ.
ð6Þ
This equation will be used to address various questions
of interest.
2.1. Lysis time
With multiple host types, selection on lysis time offers
interesting challenges, and intuition is difficult to trust.
There is also the messy issue of whether the phage
can evolve lysis time on one host independently of the
other type—whether lysis time plasticity can be achieved.
At one extreme, if the phage can evolve lysis time on one
host type fully independently of lysis time on other host
types, the single-host results (3) and (4) describe the
optimal phage response to each host in the mixed
population (eclipse time in the formula is specific to the
host type being infected). This result may seem counterintuitive because different host types may have vastly
different abundances and thus, by this result, seem to select
very different lysis times. But this result is not that lysis
time is optimized according to each host independently of
the abundance of other hosts. The effects of different hosts
on phage growth are united through l, which depends on
all infections, not just the type over which lysis time is
being optimized. Thus, the growth rate of the phage
population affects the optimum equally for rare as well as
for common host types.
At the other extreme, the phage may be forced to adopt
the same lysis time for all hosts. In this case, the result
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analogous to (3) is
P _
1
j bj k j C j
,
¼
P ^
bi ki C i d þ l^
J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938
(7)
i
where i, j ¼ 1, 2. This result is similar to (3) except that the
b_i and bi functions are averaged across the different hosts,
weighted by the force of the infection from that host ðki C i Þ.
If the slope of the progeny accumulation function and/or
eclipse period varies among host types, then this result has
no easy simplification, although a numerical solution
should be attainable when the parameters are specified.
2.2. Host range
Formula (6) also permits analysis of selection on host use
or host range. The evolution of host range in viruses and
other organisms has sometimes been considered from the
perspective that a broad range is better than a narrow one,
except that breadth impairs the ability to infect any one
host well (a tradeoff, Agrawal and Lively, 2002; Agrawal,
2000; Bohannan and Lenski, 2000; Frank, 2000). An
alternative perspective, adopted here, is to suppose that a
phage can potentially infect a wide range of hosts without
penalty, but that some hosts are better for phage fitness
than others. As an extreme example, it behooves a phage
to avoid infecting a cell that will abort its infection or is
otherwise not productive, since no progeny will emerge.
The same principle may apply to more subtle differences
in host quality. The question posed about phage host range
is analogous to the diet of a foraging animal (see
Discussion).
Evolution of host range in this model is strictly evolution
of ki, although other forms of phage specialization on host
type may be addressed as well (as derived below). When the
adsorption rate drops as low as 1011 on a host, that host is
effectively avoided, because most collisions between phage
and bacterium do not result in infection. As in our model
of lysis time, the use of a single parameter for adsorption
rate is a great oversimplification of moderately complicated
biology. Adsorption is minimally dependent on specific
receptors on the bacterial surface, which may be proteins
or LPS constituents. The densities of these receptors as well
as the overall size of the bacterium (as a target for collisions
with phage) may change with bacterial nutrition, altering
the adsorption rate physiologically, without any evolution
(Adams, 1959). In most phages, the primary determinants
of host range are the tail fibers, although chaperones and
other tail-associated proteins may be involved as well (e.g.
Drexler et al., 1989, 1991; Hashemolhosseini et al.,
1994a, b; Letarov et al., 2005). Phage adsorption may
respond not only to the presence of specific bacterial
receptors, but also to various chemical constituents of the
environment (e.g. metal ions, pH: Adams, 1959; Letarov et
al., 2005). Phenotypically, adsorption often proceeds in
two discernable stages, reversible adsorption and a later,
irreversible adsorption (Adams, 1959), and it may be that
this temporal partitioning enhances a phage’s ability to
discriminate among hosts. In phage T1, the reversible step
requires certain cellular receptors, but the irreversible step
requires energy provided by the cell (Hancock and Braun,
1976).
Evolution of host range can be inferred from the sign of
ql=qki , and the main result can be anticipated by
inspection of (6): selection favors
avoidance
of host i if
bi eLi ðdþlÞ o1, equivalent to ln bi =Li od þ l. Since bi
and Li are properties only of the infection and are
independent of bacterial abundance or phage growth rate,
multiple types of infection/host can be aligned along one
axis, and their positions are fixed regardless of dynamics;
host quality in this context is thus measured just by
ln bi =Li . The only variable is on the right side of the
inequality, l, which changes with host density and with
infection rates. The phage
is selected to infect both hosts if
the inequality ln bi =Li 4d þ l applies to both hosts. If
instead d þ l falls in between the ln bi =Li , then the phage is
selected to avoid the poorer host. Avoidance of both hosts
can be selected only if the phage population is going
extinct, and we will not consider that extreme.
Further simplification of conditions favoring the avoidance of one host type have not been obtained, but some
special cases and qualitative results are evident from (6):
(i) If the phage population is at dynamic equilibrium
(l ¼ 0), inclusion of host i in the optimal diet requires
that each infection of that host produce at least one
descendent phage on average, accounting for the
possible death of the host before lysis ðbi eLi d X1Þ.
(ii) If infection rates of the better host are sufficiently
frequent, selection favors avoidance of the poorer
host. This result may be argued on the grounds that,
when the better host (i) is very common, the phage
growth rate will approach ln bi =Li if it infects that host
exclusively (at this high-density extreme, essentially no
time is spent between lysis of the previous host and
infection of a new one). Infection of another host with
an inferior ratio must therefore slow growth rate. This
result stems partly from the assumption that collisions
with one type of host do not delay encounters with the
other host type.
(iii) If bi eLi ðdþlÞ o1 for a host, selection favors avoidance
of that host, and no amount of change in density of
that host can reverse the inequality. Decreasing the
abundance of the preferred host can shift selection to
include the poorer host, however.
2.3. Specializing on one host type when both are infected
Even when a phage infects both hosts and is selected to
maintain both hosts, selection on phage parameters will
typically be stronger on one host type than on the other.
This effect could lead to subtle degrees of specialization on
one host type. For illustration of this general principle,
consider selection for improvements in burst size on the
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J.J. Bull / Journal of Theoretical Biology 241 (2006) 928–938
different hosts. The relative strength of selection on burst
size of host type 1 relative to host type 2 is (found as
ðql=qb1 Þ=ðql=qb2 Þ)
k1 C 1 eL1 ðdþlÞ
.
k2 C 2 eL2 ðdþlÞ
935
different physiological states of the same strain. Adaptation to specialize on one host type may lead to even greater
selection for specialization, as parameters evolve to become
progressively more favorable for the better host.
(8)
If this ratio exceeds unity, selection is stronger for
increasing b1 than b2, and if genetic variation were equal
for both burst sizes, selection would tend to increase b1
more than b2, possibly even at the expense of b2. The force
of infection kiCi is a major determinant of this selection,
but so is lysis time, and lysis time is more important in
growing than stable phage populations. Qualitatively
similar conclusions apply to selection of adsorption rates
and lysis times: selection will be weaker on one infection
type than on the other; selection for avoidance of a host
type is merely part of a continuum of selection on the ki.
2.4. Implications
These results offer a special caution to protocols that
adapt phages to populations of heterogeneous hosts and
select rapid growth. Such a protocol might be used to
improve phage therapy, for example, by passaging a phage
from one infected mammal to another, to obtain a phage
variant that grows faster on bacteria inside the infected
mammal. In such a protocol, the phage might well evolve
to avoid poor quality host types, yet those host types could
in principle be the therapeutically important ones. Inadvertent selection to avoid poor host types can be mitigated
by selecting for phage numbers rather than phage growth
rate, hence by allowing the infected culture to proceed until
phage numbers have reached a plateau. This practice
allows the non-specialist time to grow on the poor hosts
after the specialist has exhausted the good hosts.
An interesting consequence of selection for avoidance of
the poorer host may operate in a spatial context. If an
epidemic wave spreads out through a population of hosts
and cell density is sufficiently uniform and high across a
large region, selection could favor phages at the leading
edge of the epidemic that specialize on the better host
(if the better host is sufficiently dense). Because of their
superior growth rate, these specialists will spread ahead of
‘‘generalist’’ phage that do not specialize. In turn, the
generalist will then encounter a population that consists
largely of the poorer host, and if there is any appreciable
tradeoff in ability to infect one host type versus the other,
the generalist may evolve to specialize on the poor host.
If infection of both bacterial types is maintained,
specialization can evolve to improve performance on one
host type more than on the other (Section 2.2, above). As
one example of how this specialization could be accomplished, if the two host types differed in concentration of a
factor used by the phage (e.g. a sigma factor), the phage
could evolve to grow best in the concentration of that
factor found in the better host. Thus, specialization could
evolve even when the different host types are merely
3. Discussion
The models developed above are optimality models,
commonly used and commonly criticized in evolutionary
biology. For experimental purposes, the genetic and
evolutionary tractability of phages exceeds that of virtually
all other ‘‘organisms’’ that have been pursued with an
optimality perspective, so it seems important to consider
how phages can inform us about optimality approaches.
There are at least three different levels of optimality model
tests to consider: (A) natural history: a comparison of
phenotypes of natural organisms to predicted optima;
(B) genetic systems: whether the genetic bases of the
phenotypes match expectations from the model; and (C)
experimental evolution. We consider each of these in turn.
(A) A major goal in biology is to understand natural
variation, and optimality models offer a simple and useful
step in that direction. In a typical application, an optimum
predicted by the model is compared to observed phenotypes of organisms in the appropriate environmental
contexts (Charnov, 1982; Parker, 1978). The populations
being tested are assumed to have experienced a constant
environment long enough that their phenotypes have
evolved to equilibrium. Regardless of whether this
approach is reasonable for some organisms, at face value,
it cannot be easily applied to phages for two reasons. First,
the most basic properties of phage natural environments
are unknown (host strains, host physiologies, host densities), so that it is not possible to predict an optimum for a
wild phage. Second, the models developed here make such
unrealistic assumptions about the phage environment (e.g.
constant cell density) that they certainly would not apply to
any natural phage.
If we did know the natural environment of a phage, it
might be possible to develop numerical approaches to
predict optima. Yet this accomplishment poses a new
problem: in ignorance of which phenotype set is available
to the organism, it becomes difficult to interpret the
optimality of observed phenotypes, except in the simplest
of environments. Consider lysis time. The simplest
environment would be a single host maintained at constant
density and physiology, for which there is a straightforward optimum. Once multiple host types are infected, the
optimality prediction depends on how well the phage can
evolve lysis time on one host independently of others. If the
relevant infection parameters were known for the different
host types but the degree of lysis time independence was
unknown, one could fit a modified optimality model with a
parameter specifying the degree of lysis time independence
across host types. But as less and less of the underlying
biology of lysis is known, this approach becomes increasingly problematic. Phenotypes that qualitatively or even
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quantitatively match a model may do so for the wrong
reason (as shown for lysis: Heineman et al., 2005), and this
problem presumably increases as more parameters are
required to fit the model.
These impediments stand in the way of simple applications of the comparative method. In specific applications, it
may be feasible to obtain information that enables progress
in understanding evolution in the natural environment, as
when the environment is well defined and/or the susceptible
host density can be ascertained. Furthermore, experimental
systems might be developed to approximate some natural
environments (e.g. see Weld et al. (2004) for a unique blend
of empirical and mathematical applications to phage
therapy in a gut).
(B) Optimality models may also shed light on the
evolution of genetic systems (Bull, 1983; Bull et al., 2004;
Charnov and Bull, 1977). If the typical environment
experienced by a phage is heterogeneous for host type
and/or is temporally variable in host abundance and host
type (as seems plausible), we can infer from the models that
lysis time and host range are both often selected for change
and/or plasticity. In some cases, the only conceivable
response is evolution, but in others it is plasticity. At a
minimum, therefore, phages should have evolvable, modular genetic systems for lysis time and host range that allow
phenotypic change with few mutations and without major
pleiotropic consequences on other traits. The mechanism
for achieving lysis time should also be responsive to cell
physiology (hence allowing plasticity). As described in
Section 1.3 above and elaborated previously, this perspective is supported for lysis time mechanisms in all but the
smallest phage genomes (Bull et al., 2004). The ‘‘genetic
systems’’ perspective is also compatible with the mechanisms known to affect host ranges of tailed phages. Tailed
phages have highly evolvable, modular proteins for
attachment (the tail fibers) that have been shown to be
important in expanding host range (Drexler et al., 1989,
1991; Hashemolhosseini et al., 1994a, b; Tetart et al., 1996,
1998; Werts et al., 1994; Yoichi et al., 2005). Of course,
such ad hoc, correlational evidence can be interpreted in
multiple ways. In our case, the evolvability and modularity
of phage host range mechanisms may have evolved more
from selection for host range expansion than from selection
for host avoidance. Nonetheless, this use of optimality
models should be valuable in generating novel hypotheses
that can be subjected to further testing.
(C) Phages offer the advantage of enabling short-term
selection experiments, and optimality models can be tested
directly by this approach. In the simplest design that is
relevant to the model here, cells would be grown to a
standard density and physiological state before phage are
added. By controlling the initial phage density and time
between transfers, each passage can avoid high multiplicity
while cells are maintained in a relatively constant state,
thus closely matching the assumptions underlying our
calculation of l. These artificial conditions are clearly
simplistic, yet if evolution fails to meet our expectations in
these types of studies, it is feasible to explore why.
Precedents exist for the experimental evolution of lysis
time (Abedon et al., 2003; Heineman et al., 2005; Wang,
2006), but only a couple conditions and phage types have
been explored with this approach. Host range is also
amenable to experimental selection. In the extreme, a
phage can be selected to avoid one host by engineering that
host to abort infections. Once phages that avoid a host
have been isolated, more subtle tests of the models can be
conducted in which mixtures of discriminating and nondiscriminating phages are grown on mixes of host types
differing in density and quality. Testing optimality models
with direct selection experiments is relatively new, but
when used with microbes, it offers some of the most
comprehensive and powerful analyses of evolution to be
found (Abedon et al., 2003; Dekel and Alon, 2005;
Dykhuizen, 1993; Suiter et al., 2003; Wang, 2006).
3.1. Extensions: virulence, foraging, and multicellular hosts
Parallels between optimal lysis time and evolution of
virulence were described above. Although the evolution of
virulence of human pathogens has been an active topic over
the last two decades (Dieckmann et al., 2002; Ewald, 1994;
Frank, 1996), support for evolution of virulence theories is
questionable (Ebert and Bull, 2003). Does the empirical
work on phages, which so far supports the optimality
models, bolster models for evolution of virulence in human
pathogens? I suggest that the support is at best weak simply
because there are major biological differences between
phage lysis time and virulence of the pathogens of
multicellular organisms. Phage lysis time provides an ideal
fit to optimality because it matches the assumptions of
those models well. The tradeoff between lysis time and
burst size in phages corresponds to the virulence-transmission tradeoff in the simplest evolution-of-virulence models,
and for phages, this tradeoff is well established and does
not appear complicated by other dimensions of the life
history. The problem in extrapolating from lysis time to
pathogens of higher organisms is precisely that those tradeoffs are more complex in higher organisms. Thus for a
human pathogen, a higher parasite load may increase its
transmission, all else equal, but the higher load may also
elicit a stronger host immune response and thus decrease
transmission. Likewise, virulence may manifest only after a
significant amount of transmission has occurred (Day,
2003), and virulence may stem from the infection of tissues
that are not important to transmission. None of these
potential complications applies to phages. Thus, phages
offer the best match of biology to the tradeoff between
virulence and transmission assumed in many of the
evolution of virulence models, and any success phages
enjoy in this respect does not extend to more complicated
host–parasite systems.
The host range model considered here offers some direct
parallels to the earliest optimal foraging models (Emlen,
1966; MacArthur and Pianka, 1966). In the original
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foraging models, the function maximized is energy intake,
and the classic result is that the optimal diet is a set of
‘‘best’’ food items, ranked according to energy content and
handling time. An optimal diet consists of all items whose
energy/handling time ratio exceeds a threshold and no
items whose ratio is below the threshold. An unintuitive
result from that model is that changes in the abundance of
items included in the optimal diet can influence the items
included in the optimal diet, but the changes in the
abundance of food items excluded from the optimal diet
does not. In simple terms, an item that is unsuitable
because of its energy/handling time ratio remains unsuitable whether it is common or rare. However, increasing
the abundance of a preferred item can drive lesser items out
of the optimal diet. The phage results here match these
optimal foraging results, even though phage growth rate
(fitness) is the phage property that is maximized in the
model.
The host range results for phages may not apply to most
viruses of terrestrial organisms. A key assumption in the
models is that a virus avoiding host type 1 will not only
ultimately contact host type 2 but will experience no delay
in infecting host type 2 by its collisions with host type 1.
This assumption suits a liquid environment of host
encounters, but it likely does not fit many terrestrial
environments, in which the virus is unlikely to have more
than one opportunity to infect a host. The model may have
parallels to viral infections within complex hosts, however.
Whenever the tissue initially infected by a virus differs from
the tissue disseminating the virus to the next host, the
infection can be represented as an epidemic population of
virus spreading from one tissue to another within a host
and ultimately out of that host. (Analogous models have
been developed for persistent infections, which are
equilibrium rather than epidemic pathogen populations
within the host: Kelly et al., 2003; Orive et al., 2005.) If the
viral population within the host can potentially infect
multiple tissues, the evolution of tissue tropisms should
follow somewhat similar principles as phage epidemics in
heterogeneous host populations—not all tissues will contribute equally to the end product of transmission. Of
course, ‘‘short-sighted’’ adaptation of viral populations
within a complex host may evolve tissue tropisms that do
not benefit transmission (Levin and Bull, 1994).
Acknowledgments
I thank R. Heineman, I. Wang, S. Abedon and two
anonymous reviewers for comments. Steve Abedon kindly
sent the values used to generate Fig. 3 in Abedon et al.
(2001). This work was supported by NIH GM 57756 and
the University of Texas Miescher Regents Professorship.
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