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PLoS One. 2011; 6(11): e27796.
Published online Nov 22, 2011. doi:  10.1371/journal.pone.0027796
PMCID: PMC3222649

The Phenotype-Fitness Map in Experimental Evolution of Phages

Barry L. Williams, Editor

Abstract

Evolutionary biologists commonly interpret adaptations of organisms by reference to a phenotype-fitness map, a model of how different states of a phenotype affect fitness. Notwithstanding the popularity of this approach, it remains difficult to directly test these mappings, both because the map often describes only a small subset of phenotypes contributing to total fitness and because direct measures of fitness are difficult to obtain and compare to the map. Both limitations can be overcome for bacterial viruses (phages) grown in the experimental condition of unlimited hosts. A complete accounting of fitness requires 3 easily measured phenotypes, and total fitness is also directly measurable for arbitrary genotypes. Yet despite the presumed transparency of this system, directly estimated fitnesses often differ from fitnesses calculated from the phenotype-fitness map. This study attempts to resolve these discrepancies, both by developing a more exact analytical phenotype-fitness map and by exploring the empirical foundations of direct fitness estimates. We derive an equation (the phenotype-fitness map) for exponential phage growth that allows an arbitrary distribution of lysis times and burst sizes. We also show that direct estimates of fitness are, in many cases, plausibly in error because the population has not attained stable age distribution and thus violates the model underlying the phenotype-fitness map. In conjunction with data provided here, the new understanding appears to resolve a discrepancy between the reported fitness of phage T7 and the substantially lower value calculated from its phenotype-fitness map.

Introduction

Fitness is the ultimate metric of natural selection. Our understanding and appreciation of how fitness is determined relies on partitioning an organism into components known as phenotypes, such as fecundity, survival, behavior, physiology, and many others. A phenotype-fitness map is the specific relationship between fitness and the phenotype states of individuals (e.g., a particular beak size or number of offspring). From these relationships, we can anticipate how the phenotypes will evolve under natural selection.

It is often pragmatic, if not necessary, to limit phenotype-fitness maps to just a few of the phenotypes determining total fitness. Inferences of evolution based on these disembodied phenotypes should be robust if the variation affecting those phenotypes does not have pleiotropic effects outside those phenotypes or affect trade-offs involving other phenotypes. Yet some model systems enable the measurement of a set of phenotypes that completely determines fitness, and these systems should be of special importance in understanding the strengths and weaknesses of approaches that use phenotypes isolated from the whole. One such model is the growth of a bacterial virus (bacteriophage or ‘phage’) on a continual excess of hosts. In this system, phage fitness is thought to be completely determined by a mere three phenotypes, all easily measurable [1] and translatable into total fitness [2][6]. In addition, the system allows independent determination of phage fitness as an empirical check on the complete phenotype-fitness map. This system thus seems well suited to explore the strengths and weaknesses of phenotype-fitness maps as a general tool in evolutionary biology.

Yet despite its simplicity, the currently understood phage phenotype-fitness map does not match reality. In the few cases that a comparison is possible between direct measures of fitness and fitness calculated from the full set of phenotypes, there are serious discrepancies [3], [7], [8] (and see below). Given the importance of phenotype-fitness maps in evolutionary biology and the accessibility of this unique system to empirical and analytical exploration, it seems justified to identify reasons for the incongruities, which we attempt here.

Growth rate as phage fitness: biology and mechanics

Bacteriophages are viruses that reproduce by infecting and killing bacteria. Under suitable growth conditions, they typically multiply much faster than bacteria and can quickly exhaust the host population. In the lab, however, they can be propagated by dilution and serial transfer to new cultures of hosts so that the phage population is undergoing continual expansion without ever overwhelming the host population. In this environment, the relevant measure of phage fitness is growth rate of its population. In turn, growth rate is determined by host density and three phage parameters: burst size (fecundity), lysis time (time between infection and cell bursting to release progeny), and adsorption rate. When using large numbers of cells and phage, the process is appropriately modeled by mass action dynamics [1], [2], [5] unless one is concerned with the fate of rare mutants [6]. Furthermore, all 3 phage parameters can be measured in a single assay [1].

Under serial transfer, growth of the culture eventually becomes emancipated from initial conditions, and an approximate formula for growth rate of the phage population is trivially An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e001.jpg, where An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e002.jpg is burst size and An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e003.jpg is average generation time [2], [4]. Average generation time is determined from lysis time, adsorption rate, and cell density. Unfortunately, use of the average generation time is known to underestimate actual phage growth rate, seriously so when cell density is low [9]. A more explicit formula has overcome this bias [5], [6], reviewed below, although even that model makes the unrealistic assumption that the phage parameters do not vary.

Incongruity between the phenotype-fitness map and direct measures

In a study of phage T7, observed fitness of a genotype adapted to lab conditions was nearly 48 doublings/hr [8], whereas the calculated fitness is 39 (using the formula of [5]). We have experienced other discrepancies of 3–5 doublings/hr (unpublished and [10]). Wang [7] and Shao and Wang [3] have also observed discrepancies, although their method of propagation does not lend itself easily to the analytic formula. These are major discrepancies, especially when considering that doublings/hr is an exponential measure of fitness, whereby even small differences have large effects on population behavior. These incongruities motivate this paper.

It may seem that any discrepancy between a direct measure of fitness and a measure calculated from a phenotype-fitness map must result from a flawed map instead of a flawed direct measure. Despite this intuition, both can be at fault. The phenotype-fitness map can predict the wrong fitness either because the phenotypes are measured incorrectly or because they are converted into fitness incorrectly. Alternatively, empirical measures of fitness can be wrong because the model used to fit the data is in error. This paper is organized along this dichotomy. The first section in Results reviews the existing analytical model developed to calculate fitness (the theoretical phenotype-fitness map). Then (in ‘An Amended Model’), the existing model is extended to include greater realism; the new results are both analytical and numerical. The third Results section identifies a problem in the direct measurement of fitness. The fourth section revisits the T7 example in light of this new understanding.

Results

The Basic Model

The basic model describes the growth rate of a large population of phage, all of the same genotype. The process is thus one of dynamics, not evolution. With constant cell density (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e004.jpg), adsorption rate (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e005.jpg), burst size (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e006.jpg), and a strictly invariant lysis time (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e007.jpg), we may write the rate of change of phage density (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e008.jpg) as

equation image
(1)

The subscript An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e010.jpg in An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e011.jpg indicates that phage density is a function of time An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e012.jpg. An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e013.jpg is the phage density An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e014.jpg time units in the past. This equation assumes a closed environment and ignores phage loss from death, washout, and all other causes except infection – assumptions appropriate for lab propagation. The equation also assumes phage density is sufficiently low that cell density is not reduced by infection.

An easy solution applies under steady-state conditions, when growth of the phage population becomes exponential. Under exponential growth, we can write An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e015.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e016.jpg, with An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e017.jpg the solution to

equation image
(2)

[5], [6]. The quantity An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e019.jpg is commonly referred to as the intrinsic growth rate. Although (2) is transcendental, it is easily solved numerically for a given set of parameters.

An Amended Model

A general approach to variation in lysis time: the An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e020.jpg function

Although the assumption of fixed lysis time is a sufficiently accurate approximation for many phages, it is violated for many others and is violated even for mutants and novel growth environments of some phages that otherwise show little variance in lysis time. Burst size is highly variable, presumably across all phages [11]. Variance in lysis time and burst size are easily accommodated in (1) above:

equation image
equation image
(3)

An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e023.jpg denotes infections that occurred An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e024.jpg time units before time An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e025.jpg. An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e026.jpg is the burst size function, the rate of progeny production, per cell, of cells infected An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e027.jpg time units in the past, accounting for the possibility that only a subset of infected cells will lyse after An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e028.jpg units. An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e029.jpg is independent of the state of the culture and hence of time An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e030.jpg. The actual number of progeny released from the average infected cell from An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e031.jpg to An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e032.jpg minutes post infection is An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e033.jpg and the average burst size of an infected cell is An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e034.jpg. An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e035.jpg will be zero for times less than some threshold, because infected cells require a minimum time after infection to assemble the first viable progeny (known as the eclipse period, An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e036.jpg). When there is no variance in lysis time, An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e037.jpg is simply a spike (delta function).

An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e038.jpg is found as the derivative of the function of phage titer over time obtained from a standard, one-step growth curve. A one-step curve provides the time course since infection of progeny produced per infected cell, rising from An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e039.jpg and reaching an asymptote at the burst size (Fig. 1). (In a strict sense, a one-step curve used for deriving An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e040.jpg must be corrected for any asynchronous infection.)

Figure 1
The burst function An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e041.jpg and its derivation from a 1-step growth curve.

An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e046.jpg can be thought of as a composite function derived from separate lysis time and burst size functions. Yet although it is possible to derive An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e047.jpg from separate functions of burst size (conditioned on lysis time) and lysis probability over time, it is not possible to work backward from An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e048.jpg to obtain separate lysis and burst functions unless burst size is independent of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e049.jpg (in which case, the shape of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e050.jpg matches the shape of the lysis function). Thus, if An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e051.jpg is used to calculate the mean, variance, or other moments of lysis time, those moments will be weighted by any systematic change in burst size with An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e052.jpg. Since An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e053.jpg gives the average number of progeny produced per unit time, it does not provide information about variance in burst size among cells lysing at the same time; that variance is in fact irrelevant to the growth rate equations because of the mass action assumption, but is relevant to the fate of genotypes when rare [6].

An analytical solution for the amended model

Assuming the population has attained exponential growth, equation (3) can be solved analytically by using the aforementioned fact that An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e054.jpg. Then An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e055.jpg and

equation image
(4)

If we rewrite An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e057.jpg as An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e058.jpg, where An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e059.jpg is the average, total burst size, then An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e060.jpg can be treated as a probability density function (pdf) of burst. If the constant An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e061.jpg is moved outside, the resulting integral is merely the moment generating function of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e062.jpg, with the dummy variable set to An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e063.jpg. Writing again An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e064.jpg, we can simplify (4) to

equation image
(5)

where An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e066.jpg is the moment generating function of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e067.jpg.

This decomposition of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e068.jpg requires that burst size be independent of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e069.jpg. If average burst size changes with An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e070.jpg, a scaling function must be included inside the integral.

Special cases

Given that no progeny can be produced prior to some minimum time after infection (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e071.jpg), greater flexibility in fitting the model may be offered by modifying the time scale of progeny production. In particular, we may let An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e072.jpg and confine the burst pdf to the interval An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e073.jpg. Assuming that An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e074.jpg is a gamma pdf, the solution for An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e075.jpg obeys

equation image
(6)

where An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e077.jpg is the mean lysis time and An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e078.jpg the variance in lysis time (measured on An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e079.jpg). If instead An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e080.jpg is uniform on An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e081.jpg, as in Fig. 1D, the solution obeys

equation image
(7)

Extensions of the amended model

Two additional modifications of (3) may be needed for realism in serial transfer. One is to include changes in cell density, which can increase from cell division and decrease from infection:

equation image
equation image
equation image
(8)

where cell density An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e086.jpg is now a function of time An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e087.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e088.jpg is the growth rate of cells. The fluctuations in cell density will preclude the strict attainment of exponential phage growth, so the previous analytical results will be only approximate [6]. The importance of accounting for changes in cell density can be evaluated by simulation.

A second dimension of realism is to allow variance in the adsorption rate, An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e089.jpg. Here, we do not model this effect, because we have no evidence that it is relevant in T7, at least over timescales that should dominate serial transfer dynamics. However, we acknowledge that it might be necessary to include this effect for some systems. There are different ways that nature might vary adsorption rate, and only some are amenable to easy inclusion. One type of variation is to suppose that each burst produces an array of progeny with different adsorption rates, as would result if some virions had 6 tail fibers, others 5, and so on. With this type of variation, each phage particle has a specific adsorption rate that it maintains for life, and the model requires a separate phage equation for each adsorption rate category, all bursts contributing to each category. The cell equation likewise needs a separate loss term for each phage type. Another form of adsorption rate variation is due to phage aging. If adsorption rates decline with time, separate equations for each adsorption rate category are also needed, but one phage type decays into another phage type, rather than all types arising within single bursts. Finally, phage particles may undergo maturation after release, so that their adsorption rate increases with time [12].

Simulations: effect of lysis time variance and cell growth

A simulation of equations (8) revealed the effects of various elaborations of (1) (Table 1). Results were contrasted between the basic model (fixed An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e090.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e091.jpg) and the amended model with variation in lysis time (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e092.jpg) and cell growth (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e093.jpg, a doubling every 23 minutes, which approximates our system). The An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e094.jpg function was either a spike spanning An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e095.jpg minutes (the minimum interval possible under the numerical integration scheme) or, for the An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e096.jpg column, was a rectangular function like that in Fig. 1 D. For the An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e097.jpg calculation in each row of this table, the base and height of the rectangular An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e098.jpg was chosen to achieve the indicated lysis time variance while maintaining the same burst size and average lysis time (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e099.jpg) used in the other two columns of that row. The simulation results for the basic model column were found to match analytical results for equations (2) [variance in An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e100.jpg near zero]; likewise, simulation results for the An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e101.jpg column (large variance) matched (7).

Table 1
Simulated phage growth rate per parameter modifications.

Cell growth (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e124.jpg)

Allowing cells to grow during single cultures increased the fitness estimate, provided that phage densities were maintained orders of magnitude below cell densities. This qualitative result is obvious for the same reason that phage fitness increases with higher initial cell density, all else equal: higher cell density reduces generation time by reducing the time a phage spends in the free state. The magnitude of the effect seen in Table 1 is not so easily understood, however. It must first be appreciated that the simulation mimicked actual serial transfer in using a dilution and transfer schedule similar to those used in practice: the dilution factor was between An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e125.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e126.jpg, and transfer interval between An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e127.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e128.jpg minutes, so that phage densities remained low relative to cell densities. Thus rapid transfer must be used when fitness is high, or the phage will quickly outnumber cells (e.g., An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e129.jpg minute transfers are required when fitness is An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e130.jpg doublings/hr). Since each new culture starts with the same initial cell density, there is less time for cell growth when transfers are rapid than when they are slow. So a larger effect of cell growth is expected for phages with low growth rates because those cultures provide more time for cells to grow before transfer.

In addition, however, the effect of cell growth is larger for phages with low adsorption rates. This counter-intuitive effect of adsorption rate may stem from the fact that the duration of a phage's life that is spent in the free state varies according to the inverse of the product of adsorption rate and cell density (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e131.jpg). When the product is high, phages spend little time in the free state, so further increases in cell density do not appreciably shorten the free state; when the product is low, further increases in cell density do appreciably shorten the free state, and thus shorten generation time. Thus the benefit of higher cell density stemming from culture growth is realized more at low adsorption rates than at high ones.

Variable lysis time (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e132.jpg)

Allowing variation in An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e133.jpg invariably increased fitness, even though the variation was symmetric around the mean lysis time. The reason for this effect is that early bursts contribute more to fitness than late bursts detract. The benefit of variance in lysis time was greater with higher baseline fitness, but this effect may be a trivial consequence of differences in mean lysis time. All trials in this column used the same burst function relative to the mean lysis time, so the early bursts shortened generation time proportionally more when lysis time was short than when long. We also studied the effect of an asymmetric An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e134.jpg function, but fitness effects could not be attributed solely to skew because other moments varied when skew was changed.

Estimating fitness directly from sequential titers

Sampling phage titers temporally seems to provide the most relevant and convenient method of measuring fitness directly. Typically, one adds phage to a bacterial culture, transfers a small volume to a new culture at the appropriate time and repeats the process as long as needed. Titers used in the fitness calculation are taken from the culture at the time of transfer to a new culture, the sample being treated with chloroform to kill the cells and stop phage growth. The first sample for the fitness calculation is typically taken after 30 minutes or more of phage growth (which is not usually the first culture in the series), and the second sample is taken an hour or more after the first sample. The growth rate is determined by the difference in titers between the two samplings, adjusted for the time and any transfer dilutions between them. The only issue with this approach is to ensure that the population is growing exponentially. Attainment of exponential growth rests on the population having attained a stable age-of-infection distribution (henceforth “stable age distribution”). A stable age distribution implies that the fraction of infected cells that has been infected for just 1 minute, just 2 minutes, and so on does not change into the future.

When conducting an assay, the age distribution is clearly not stable at first, because phage are added all at once to a culture of cells. Most infections from those phage will occur rapidly, and then there will be a long interval before the infected cells burst. During this interval, few new infections are occurring. If lysis time is largely invariant, bursts will then also be highly synchronous, and the asynchrony of infection times will persist for many generations. Subtle deviations from a stable age distribution can persist for generations (Fig. 2).

Figure 2
Deviations from stable age distribution in simulations.

On an intuitive level, we may think of a stable age distribution as represented by a large variance of infection times in the culture relative to the duration of the phage life cycle. Starting from a newly inoculated culture, the variance of infection times (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e138.jpg) comes both from the variance in lysis time and the variance in time to adsorption. Under exponential decay of free phage from infection, the variance of infection times is An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e139.jpg; the variance in lysis time depends on An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e140.jpg. With multiple rounds of infection through time, the variance in infection times will increase linearly (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e141.jpg) to a first approximation. Thus, one can obtain a qualitative sense of how quickly a stable age distribution will be attained by knowing both the lysis time variance and the loss to adsorption relative to average lysis time.

To illustrate the error introduced by fitting phage titers from two time points, the simulation model of the previous section was run for various conditions and is illustrated for three cases: one with a high adsorption rate and short lysis time, one with a low adsorption rate and short lysis time, and one with a low adsorption rate and long lysis time (L1, L3, and L4, basic model, Table 1). Strong effects of deviation from stable age distribution are seen in all runs, with moderate to large effects on fitness estimation across the first few hours (Fig. 3). The pattern of deviation differs radically between the three conditions, precluding any obvious empirical generalization.

Figure 3
Deviations from stable age distribution in simulated serial transfer protocols and their effects on direct fitness estimation.

Revisiting T7

The previous section offered possible reasons for a discrepancy between fitness calculated from the basic model and fitness estimated directly from sequential titers of a serial transfer experiment. We now return to the discrepancy between the two methods of fitness determination specifically for the strain referred to as T7An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e145.jpg in [8]. Cell density at the time of phage addition was intended to be An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e146.jpg/mL. The estimated lysis time was An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e147.jpg min, burst size was An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e148.jpg phage, and adsorption rate was An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e149.jpg mL/min. With a delta function for An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e150.jpg, the calculated fitness is An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e151.jpg doublings/hr, whereas the directly estimated value is An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e152.jpg.

For the present study, we considered the possibility that some of the prior parameter estimates were in error. Fitness calculations are highly sensitive to lysis time and adsorption rates because of the effect on generation time (less so for burst size), so lysis time and adsorption rates were re-evaluated. There was no suggestion that lysis time was different from that reported, but adsorption rate was re-estimated as An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e153.jpg, a 2.7-fold increase. We also checked for the possibility of adsorption rate variation when measured over different intervals; there was no evidence that the adsorption rate constant measured over two minutes differed from that over 4 minutes, and no evidence of an effect of aging the lysate for 24 hr before the assay; thus we use a model of fixed An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e154.jpg.

The fitness calculated from the basic model for these parameters is An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e155.jpg, the increase over An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e156.jpg due to the higher adsorption rate. The calculation rises modestly to An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e157.jpg by including cell growth (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e158.jpg, a doubling time of 23 min) and variance in lysis time (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e159.jpg, as per [8], using the An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e160.jpg function of Table 1). To illustrate the sensitivity of calculated fitness to lysis time, the estimate of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e161.jpg rises to An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e162.jpg merely by decreasing mean lysis time An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e163.jpg minute to An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e164.jpg, all else equal. It is not practical to estimate mean lysis time to the nearest An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e165.jpg minute, thus the calculated fitness cannot be precise.

The direct fitness estimate could be affected by the failure of the culture to obtain a stable age distribution. An inflated direct estimate would help bring the two numbers into agreement. A simulation shows that the oscillations in the estimate of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e166.jpg are potentially large enough to account for the high reported value (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e167.jpg) if the variance of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e168.jpg is as small as reported, but not if it is slightly larger (Fig. 4).

Figure 4
Fitness estimates in a simulation of T7 infection during serial transfer every 15 minutes.

To assess the possibility of deviations from stable age distribution, we performed new serial transfers of T7An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e173.jpg, this time recording phage titer at each transfer out to 2 hr; fitness estimates in the previous study were based on titers taken at An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e174.jpg min and at An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e175.jpg min. Methods were as before [8], except that transfers this time were made every An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e176.jpg min instead of every An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e177.jpg min. Sequential titers reveal a behavior characteristic of the failure to have attained stable age distribution (Fig. 5), and calculations of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e178.jpg based on these data fail to stabilize before 2 hr. Importantly, these new data suggest a direct fitness estimate of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e179.jpg that is compatible with the calculated one and considerably lower than the published one.

Figure 5
Violation of stable age distribution in laboratory transfers of T7 An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e180.jpg.

At face value, there is reconciliation between the former directly estimated fitness of T7An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e184.jpg and the current one (the previous one was plausibly inflated by a violation of stable age distribution); there is also agreement between the calculated and directly estimated fitnesses provided here. Some qualifications are warranted, however. First, it is not strictly possible to revisit the former fitness estimate; frozen cell stocks were not the same for the two sets of assays, and the ingredients used for LB broth now and then possibly differ (LB is not a defined medium). Thus, conditions used in the assays are subject to unknown variation. Second, the reconciliation is likely no better than approximate, because the simulations show that fitnesses are sensitive to deviations in parameter values that lie well within measurement error. Third, manual serial transfer assays are cumbersome and do not provide the level of resolution needed for a detailed resolution of within-culture dynamics. A more complete resolution of the empirical dimensions of this problem thus awaits automation of serial transfer and a reporter system to monitor infection rates in real time.

The temporal pattern of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e185.jpg values calculated from simulations parameterized with T7 data in Fig. 4 does not match the empirical pattern, at least in the nature of the oscillations. The basis for this discrepancy is not clear, but there are some obvious possibilities to consider. First, as shown in Fig. 3, the temporal pattern is qualitatively sensitive to at least modest variations in individual parameters. It is thus plausible that combinations of T7 parameters statistically compatible with the observed values could be found that lead to a better fit of the An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e186.jpg calculations. More fundamentally, however, the biological titers differ from the simulated titers. Simulated titers are of free phage. The titers estimated in an experiment are taken from a sample of the culture treated with chloroform and thus include free phage and phage released by cells partway through infection when exposed to chloroform. It is challenging to model this reality: the number of phage released from an infected cell depends on the number of progeny produced since infection and on how many of those progeny are released from the cell by chloroform treatment. The effect of chloroform is not easily calculated because chloroform merely creates lesions in cells but does not lyse them unless the infection has proceeded long enough that lysin has accumulated inside the cell.

Discussion

This study provides the most accurate model to date of the phenotype-fitness map of bacteriophage growth. This map applies specifically to the idealized environment of serial propagation on a continual excess of hosts. The model developed here extends a previous analytical solution by allowing distributions of lysis time (rather than a fixed time) and in using simulation to measure sources of error in fitness measurement. Although the specifics apply to bacteriophage growth, some of the principles are likely to be general. In particular, a deviation from stable age distribution and the use of full parameter distributions rather than parameter means is likely of broad relevance to fitness measures in other experimental microbial systems.

The model developed here is dynamic rather than evolutionary, but its chief utility is evolutionary prediction. By specifying phage fitness as a function of three parameters (or parameter distributions), the model is easily used to understand how fitness changes with changes in the parameters, hence how selection operates on each of the three phenotypes. The immediate value of such a model is to improve our ability to apply experimental studies of phages in testing evolutionary models and in interpreting laboratory evolution of phages and other viruses. Such experimental systems often offer the only real hope of testing models intended for viruses relevant to human health, so the correct interpretation of phage studies is important. More broadly, work such as this enhances an appreciation of the strengths and limitations of phenotype-fitness maps of other organisms.

For the parameter combinations considered here, the cumulative effect of errors can easily lead to a 5% difference between measured and calculated fitness, and even a 10% error is feasible. As growth rate is a geometric measure of fitness, any error compounds rapidly over generations. However, with an awareness of the possible problems, it is also feasible to anticipate and perhaps reduce errors by including appropriate extensions of the basic model, with simulations if necessary. Application of this new perspective to T7 data seemingly explained the large discrepancy between the two methods of fitness estimation evident in prior work. It should also be appreciated that, even with this new understanding, the methods of parameter estimation are too coarse to provide accuracy beyond 1–2 doublings/hr when phage fitness is as high as in T7.

As noted, a large potential error arises in the direct calculation of an exponential growth rate if the culture has not neared a stable age-of-infection distribution. The deviation from stable age distribution decays with time, but the rate of decay depends highly on the variance in lysis time and adsorption rate in the culture. Thus the extent to which deviation from a stable age distribution is a problem will vary from phage to phage and also vary with culture conditions. Various empirical methods to hasten the approach to stable age distribution may be attempted. For example, reducing the concentration of cells in the initial culture has the effect of slowing infections and thus of dispersing them over time, compared to a culture with high cell density. Thus it might be supposed that a more rapid approach to stable age distribution could be achieved with a protocol in which the first culture used low cell density. Yet this protocol does not appear to work well (simulations not shown), as it leads to a high density of free phage in the first culture, and those free phage are quickly adsorbed when transfered to a culture with high cell density. An alternative protocol might be to inoculate phage incrementally into the first culture, spreading their infections over time (suggested by a reviewer). We have not evaluated this approach, but as with any attempt to bypass natural decay, it runs the risk of delaying decay instead of hastening it; such a protocol might need to be tailored specifically to each system.

The existence of large estimation errors has implications for the interpretation of some phage studies. One is that the measurement of small fitness effects must be interpreted with caution, at least when those fitnesses constitute a form of growth rate [13][16]. Small fitness effects may be real for the exact measurement assay, but the present study suggests that small effects measured in one assay may not translate across even minor variations in protocol. Use of competition strains to estimate a relative fitness may alleviate some of the problem, but if the two strains in the same culture experience different approaches to stable age distribution, then even competitor strains will face the problem identified here.

In many studies, errors in fitness estimation may have little impact. For example, the optimal lysis time evolves as An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e187.jpg [5]. To first order, a small percent error in An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e188.jpg leads to the same percent error in An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e189.jpg. Thus, if An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e190.jpg is An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e191.jpg minutes, a 10% error is An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e192.jpg minutes, barely detectable in lysis time assays, and potentially much smaller than the deviation between the predicted optimum and the observed endpoint of evolution (as in [8]). Much of phage experimental evolution is about large magnitudes of evolution, and the types of errors identified here will not change conclusions based on those effects.

Our analysis underscores that mapping specific phenotypes into fitness is complicated, and that predicted fitness values are highly sensitive to details of individual phenotypes (e.g., the exact lysis time). These findings are reminiscent of results by Wahl and coauthors [6], [17], [18], who found that fixation probabilities can differ widely for rare mutations affecting different phenotypes, even if these mutations have the same effect on long-term growth rate (i.e., fitness). They also found that fixation probabilities were dramatically altered by the exact timing of dilutions [6]. In summary, theirs and our results show that mapping phenotypes to fitness and evolutionary success is a difficult and non-trivial undertaking, even in a system as simple as bacteriophage grown in a chemostat.

An obvious, if difficult next step is to model the intracellular determinants of the three phage parameters of burst size, lysis time, and adsorption rate. This deeper model would provide the basis for understanding the evolution of intracellular mechanics through their effects on the higher-level phenotypes. This level of understanding appears within reach: Endy et al. [19] generated a kinetic simulation model of the life cycle of phage T7, with over 100 genetic elements and based entirely on empirically determined parameter values. As methods improve for measuring intracellular concentrations and localization of phage proteins, we may be able to move to this next level and create a molecular-based phenotype-fitness map.

Methods

Assays of sequential titers of phage T7 used the serial transfer methods of Heineman and Bull [8]. Briefly,E. coli K-12 cells (strain IJ1133) were grown in 10 mL of LB broth in 125 mL flasks in an orbital waterbath with agitation for 1 hr at An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e193.jpg C to a density of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e194.jpg cells/mL. Phage were added, grown for 15 min, and a sample of the culture was transfered to another culture of cells that had been incubating for 1 hr. A sample of the completed flask was extracted over chloroform and used for titer determination. Phage titers were maintained two orders of magnitude below cell densities to ensure that host density was only negligibly depressed by infection.

Adsorption assays were conducted by adding phage to a culture of cells that had been incubated for 1 hr to a density of An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e195.jpg cells/mL (following the same protocol used for sequential titer measures). Cell density (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e196.jpg) was measured just prior to phage addition by plating to determine colony forming units. At time An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e197.jpg (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e198.jpg) minutes after phage addition, a sample was removed. Total phage titer (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e199.jpg) was determined by plating an aliquot immediately, and free phage titer (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e200.jpg) was determined by plating an aliquot of the supernatant of a centrifuged culture (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e201.jpg rpm for An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e202.jpg sec). The adsorption rate constant (An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e203.jpg) was estimated from

equation image
(9)

in units mL/min. This ‘end-point’ determination runs the risk of underestimating An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e205.jpg when An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e206.jpg is large (S. Abedon, pers. comm.), but should be largely free of this problem for the time intervals used here.

Numerical analyses of phage population dynamics (equations (8)) were used for Figs. 3 and and44 and Table 1. Computer programs to conduct these analyses were written in C, with variables updated every An external file that holds a picture, illustration, etc.
Object name is pone.0027796.e207.jpg minute.

Acknowledgments

We thank two reviewers for helping us clarify the ms and S. Abedon for clarification of aspects of parameter estimation.

Footnotes

Competing Interests: The authors have declared that no competing interests exist.

Funding: This work was supported by National Institutes of Health (NIH) GM 57756 and the Miescher Regents Professorship (to JJB) and National Science Foundation (NSF) Grant EF-0742373, NSF Cooperative Agreement No.\ DBI-0939454, and NIH R01 GM088344 (to COW). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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