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Optimizing bacteriophage plaque fecundity
ARTICLE IN PRESS
Journal of Theoretical Biology 249 (2007) 582–592 www.elsevier.com/locate/yjtbi
Optimizing bacteriophage plaque fecundity Stephen T. Abedon, Rachel R. Culler Department of Microbiology, The Ohio State University, Mansfield, OH 44906, USA Received 21 April 2007; received in revised form 7 August 2007; accepted 7 August 2007 Available online 24 August 2007
Abstract Bacteriophages (phages), the viruses of bacteria, form visible lesions within bacterial lawns (called plaques), which are employed ubiquitously in phage isolation and characterization. Plaques also can serve as models for phage population growth within environments that display significant spatial structure, e.g. soils, sediments, animal mucosal tissue, etc. Furthermore, phages growing within plaques, in experimental evolution studies, may become adapted to novel conditions, may be selected for faster expansion, or may evolve toward producing more virions per plaque. Here, we examine the evolution of the latter, greater plaque fecundity, considering especially tradeoffs between phage latent period and phage burst size. This evolution is interesting because genetically lengthening latent periods, as seen with phage lysis-timing mutants, should increase phage burst sizes, as more time is available for phage-progeny maturation during infection. Genetically shortening latent periods, however, is a means toward producing larger phage plaques since phage virions then can spend more time diffusing rather than infecting. With these larger plaques more bacteria become phage infected, resulting in more phage bursts. Given this conflict between latent period’s impact on per-plaque burst number versus per-infection burst size, and based on analysis of existing models of plaque expansion, we provide two assertions. First, latent periods that optimize plaque fecundity are longer (e.g. at least two-fold longer) than latent periods that optimize plaque size (or that optimize phage population growth within broth). Second, if increases in burst size can contribute to plaque size (i.e. larger plaques with larger bursts), then latent-period optima that maximize plaque fecundity should be longer still. As a part of our analysis, we provide a means for predicting latent-period optima—for maximizing either plaque size or plaque fecundity—which is based on knowledge of only phage eclipse period and the relative contribution of phage burst size versus latent period toward plaque size. r 2007 Elsevier Ltd. All rights reserved. Keywords: Bacteriophage; Plaques; Tradeoffs; Optimization; Latent period
1. Introduction Bacteriophages (phages) are the viruses of bacteria. When phages replicate within well-mixed broth, it is those phages that can obtain bacteria faster that dominate within-culture population growth. Rapid phage population growth, however, does not necessarily correspond to efficient conversion of an environment’s resources into phage progeny (Abedon et al., 2003; Abedon, 2008). Conceptually similar tradeoffs can arise during phage population growth within semi-solid media, such as occurs during phage plaque formation within agar (Abedon and Culler, 2007; Abedon and Yin, 2008). There, more-rapid environmental exploitation is associated not only with Corresponding author. Tel.: +1 419 755 4343; fax: +1 419 755 4327.
E-mail address: [email protected] (S.T. Abedon). 0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2007.08.006
rapid phage population growth and host acquisition, but also with more-rapid spatial spread of the phage population, that is, with the formation of larger plaques. By contrast, phage growth could be biased toward a more efficient conversion of environmental resources into phage progeny, potentially resulting in plaques that contain more phage progeny, i.e. which display a greater fecundity. Abedon and Culler (2007) present an exploration of how changes in phage properties could contribute, under varying conditions, to greater plaque size. Here, we extend that analysis to consider the evolution of plaque fecundity from the perspective of tradeoffs between phage latent period and burst size, the duration of phage infection and number of bacteria produced per infection, respectively. For lytic phages, progeny mature intracellularly only so long as phage infections remain intact, but those same progeny are unable to obtain new bacteria to infect unless
ARTICLE IN PRESS S.T. Abedon, R.R. Culler / Journal of Theoretical Biology 249 (2007) 582–592
infections are terminated with lysis. A number of studies have explored the evolution of the timing of this lysis within well-mixed broth cultures (Abedon, 1989, 1994, 2006, 2008; Abedon et al., 2001, 2003; Bull et al., 2004; Bull, 2006; Kerr et al., 2008; Levin and Lenski, 1983; Wang et al., 1996; Wang, 2006). The general arguments are that lysis timing may be optimized such that rates of phage population growth are maximized. Alternatively, a standard protocol in studies of phage experimental adaptation is to grow phages as non-overlapping plaques and then to homogenize plaques and lawns, prior to subsequent replating, by removing plaquecontaining top agar en masse to diluent (Burch and Chao, 1999, 2000; Chao et al., 1997; Duffy et al., in press; Turner and Chao, 1998). Such an approach, which competes plaques in terms of titer, should select for greater plaque fecundity (Bull, 2008), which can result from more phage infections per plaque (a result of greater plaque size; Kaplan et al., 1981; Poon and Chao, 2005) or from greater per-infection fecundity (greater burst size). These two routes toward greater plaque fecundity are to some extent in conflict, however: Genetically shorter phage latent periods should result in larger plaques (as reviewed by Abedon and Culler, 2007) but simultaneously in smaller burst sizes (e.g. Abedon et al., 2003). Toward developing a greater understanding of selection acting on phages during plaque enlargement, such as during experimental evolution studies, here we develop a method for predicting latent periods that optimize plaque fecundity.
pathogen epidemics or species invasions in general (Conner and Miller, 2005; Murray, 2003), may be viewed—at least at coarser resolutions—as traveling waves. Given tradeoffs between phage latent period and burst size (Section 3.2), we provide estimations of the phage ‘‘optimal’’ latent period that maximizes rates of the phage wavelike propagation during plaque development (Lcopt ; c ¼ plaque wavefront velocity; Sections 3.3 and 3.6) or, alternatively, that latent period which maximizes overall phage productivity within plaques (LFopt ; F stands for plaque fecundity; Sections 3.4 and 3.5). 3.1. Plaque wavefront velocity as functions of latent period and burst size In practice, plaque wavefront velocity, c, is constant over most of the course of visible plaque development, so long as bacterial physiology also remains more or less constant (Kaplan et al., 1981; Koch, 1964; Lee and Yin, 1996; Mayr-Harting, 1958; Yin, 1991). Abedon and Culler (2007) review seven equations, which estimate c as functions of various phage growth parameters. We present these equations below in comparable forms, emphasizing here the contributions of B (burst size) and L (latent period), which, except for Eq. (6), are shown prior to the larger dot, ‘‘‘‘: c ¼ B0=4 L2=4 10 D1=2 ,
We consider the plaque-growth models of Koch (1964), Yin and McCaskill (1992), and Ortega-Cejas et al. (2004) to obtain estimations of what latent period—given standard assumptions regarding burst size-latent-period tradeoffs (e.g. Abedon et al., 2001)—will maximize plaque wavefront velocity (c), that is, the rate of plaque radial expansion. We then combine these models with calculations of plaque volume to derive an equation that allows facile estimation of the phage optimal latent period that maximizes plaque fecundity or, alternatively, the phage optimal latent period that maximizes plaque size. These equations are expressed in terms of the relative contribution of burst size versus latent period to plaque wavefront velocity, or to overall plaque fecundity. 3. Results Anderson (1950) described the spread of phage infection, during plaque growth, as epidemic: ‘‘In the bacterial viruses where the cells dissolve, or lyse, as a result of the disease, such epidemics produce clear areas, or plaques, in an otherwise uniform bacterial smear on agar.’’ The outward spread of these phage epidemics within plaques (e.g. Wei and Krone, 2005; Yin, 1991; Yin and McCaskill, 1992; You and Yin, 1999; Section 3.1), like those of
(1) "
2=4
2. Methods
583
c¼B
2=4
L
1=2
2:0 D
N o k1 ðk1 ð1 þ N o k1 =k1 Þ2 Þ
#1=2 , (2)
c ¼ B1=4 L1=4 1:6 D1=2 ðN o k1 Þ1=4 ,
(3)
c ¼ B2=4 L0=4 2:0 D1=2 ðN o k1 Þ1=2 ,
(4)
c ¼ B4=4 L4=4 2:6 D1=2 ðN o k1 Þ1=2 ,
(5)
2=4
c¼B
0=4
L
1=2
2:0 D
N o k1 ð1 þ L B N o k1 Þ
c ¼ B0=4 L2=4 1:4 D1=2 ,
1=2 ,
(6) (7)
where D is virion diffusivity, No is the lawn bacterial density, k1 is a phage-to-bacterium binding constant, and k1 is a desorption constant of phage from bacteria. The equations were derived by Abedon and Culler (2007) from those of (i) Koch (1964) (Eq. (1)), (ii) Yin and McCaskill (1992) (Eqs. (2)–(5)), and (iii) Ortega-Cejas et al. (2004) (Eqs. (6) and (7)). The Yin and McCaskill equations correspond to conditions of equilibrated virion adsorption (Eq. (2)), large burst size (Eq. (3)), slow phage adsorption (Eq. (4)), and fast phage adsorption (Eq. (5)). The OrtegaCejas et al. (2004) models differ in their applicability as dependent upon various phage growth parameters but also as dependent on host density. In brief, Eq. (7) is applicable
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at higher, arguably more-realistic host densities (Abedon and Culler, 2007). Given the similarity between Eqs. (1) and (7), along with the relative importance of Eq. (7) in the Ortega-Cejas et al. model, Abedon and Culler (2007) argue that the presented models from Koch and Ortega-Cejas et al. are algebraically equivalent under typical phage plaquing conditions. For the sake of achieving comparability, we have modified a number of these equations from those originally published. For a number of the Yin and McCaskill (1992) equations for instance, B–1 has been replaced with B (where the subtracted 1 indicates virion loss to the initiation of phage infections) and the rate constant for phage-induced lysis of infected bacteria, k2, is presented here instead by an approximation, 1/L (which Yin and McCaskill employ when solving their model numerically). We also disregard any consideration of the impact of the occupation of an ‘‘inert volume’’ by hosts on phage diffusivity (an ‘‘effective diffusion’’ as employed by both Yin and McCaskill and Ortega et al.) since these effects do not differ between equations (Abedon and Culler, 2007) nor impact subsequent calculations provided here. 3.2. Defining burst size in terms of latent period Phage burst size (B) can vary as a direct function of phage latent period, (e.g. Abedon et al., 2001): B ¼ ðL EÞ R,
(8)
where E is eclipse period, that is, the period of phagemediated biochemical preparation for phage-virion production (e.g. Abedon, 2006). R is the rate of intracellular phage-progeny maturation once that production begins and Wang et al. (1996) provide evidence for the nearconstancy of R over much of the post-eclipse latent period within phage infections that are lysis defective. Though the magnitudes of B, R, L, and E can all vary as functions of bacterial physiology (Hadas et al., 1997; You et al., 2002), for most phages L also may be both precisely controlled and readily modified, via phage lysis-timing mutations, without affecting either E or R (e.g. Bull, 2006; Wang, 2006; Young and Wang, 2006). 3.3. Determining wavefront-velocity-maximizing optimal latent period In Fig. 1, we solve for plaque wavefront velocity (c) as a function of latent period (L) using Eqs. (1)–(7), replacing burst size (B), when employed, with Eq. (8). Only Eq. (6) shows a peak, indicating a latent period that maximizes wavefront velocity (Lcopt ) without being essentially as short as possible (which would be where latent period is only slightly greater than eclipse period, E). Even with Eq. (6), for No ¼ 108 bacteria/ml this peak is quite close to E (i.e. E ¼ 10 min versus Lcopt ¼ 11.5 min), but is greater if No is reduced. These figures also indicate a seemingly anomalous result for Eqs. (2)–(5), those of Yin and McCaskill (1992), where optimal latent
period neither approaches the eclipse period nor peaks. One explanation for this latter behavior is that the contribution of burst size to plaque size is large in these equations, perhaps too large, as we consider subsequently (Section 3.6). Because Eq. (6) displays a peak (Fig. 1), we were able to calculate Lcopt . To do this we replaced B in Eq. (6) with Eq. (8), differentiated the resulting equation with respect to L, and then solved for L when dc/dL ¼ 0 The resulting equation is complicated but does provide non-trivial estimations of Lcopt (Fig. 2) that are consistent with estimations determined graphically (Fig. 1). Eq. (6), however, is valid only when Noo(L B k1)1 (Abedon and Culler, 2007; Ortega-Cejas et al., 2004). Consequently, Eq. (6) actually is not valid over all combinations of bacterial density, optimal latent period (such that L ¼ Lcopt ), and burst size (¼ ðLcopt EÞ R) shown in Fig. 2, and only becomes valid, at a coarseness of half log increments, at Nop104 bacteria/ml (calculations not presented). These calculations based on Eq. (6), nevertheless, are illustrative of the potential impact of burst size on Lcopt , as we will consider again in Section 3.6 and Appendix A. The various approaches used for estimating Lcopt produce seemingly disparate results (Fig. 1). The models of Koch (1964) and Ortega-Cejas et al. (2004), though, do provide somewhat consistent and biologically plausible determinations. In particular, if B does not contribute to plaque wavefront velocity, then that L which maximizes c should be as short as possible. However, we have an expectation that Eqs. (1)–(7) will not hold either at very short latent periods or for very small burst sizes (Koch, 1964; You and Yin, 1999). Therefore, for Eqs. (1) and (7) reasonable interpretation should be that Lcopt approaches but falls short of reaching E. Under the conditions modeled, a latent period of 15 min provides a burst size of 25. We therefore feel justified in asserting that these equations in fact provide a prediction of an optimal latent period of less than 15 min, but probably equal to or greater than 12 min (where a burst size of less than 10 phage/bacterium may be sufficiently small that plaque size is greatly negatively impacted; Carlson and Miller, 1994). Eq. (6) provides essentially the same prediction for Lcopt , which for reasonable values for No during plaque formation, i.e. greater than 107 bacteria/ml, is one of less than 15 min (Fig. 2). By contrast, estimations by Abedon et al. (2001) of Lbopt , the optimal latent period for phage population growth within broth, are of 22 min for No ¼ 107 bacteria/ml, ranging down to about 15 min at higher bacterial densities (Fig. 2). We suspect that at least part of the explanation for why Lbopt is greater than Lcopt (as calculated here) has to do with burst size making a greater contribution to phage population growth rates in well-mixed broth than it does to plaque wavefront velocity as determined by Eqs. (1), (6), or (7). 3.4. Plaque fecundity In the models of plaque growth presented as Eqs. (1)–(7), bacterial density is either not considered to contribute to
ARTICLE IN PRESS S.T. Abedon, R.R. Culler / Journal of Theoretical Biology 249 (2007) 582–592
10-2
5
Front Velocity (c in cm min-1)
Front Velocity (c in cm min-1)
10-1
4 10-2
2 3 1
peak, Eq. (6)
10-3
No = 10
10-4 0
15
8.0
6 7
bacteria/ml
2 1 10-3
6 7 10-4
30
45
60
75
peak, Eq. (6)
No = 10 0
90 105 120
15
bacteria/ml
30
45
60
75
90 105 120
10-2
Front Velocity (c in cm min-1)
10-2
Front Velocity (c in cm min-1)
7.5
Latent Period (min)
Latent Period (min)
2 1
peak, Eq. (6)
10-3
6 7 10-4
585
No = 10 0
15
7.0
bacteria/ml
1 2 10-3
6 7 10-4
30
45
60
75
90 105 120
peak, Eq. (6)
No = 10 0
15
6.5
bacteria/ml
30
45
60
75
90 105 120
Latent Period (min)
Latent Period (min)
Fig. 1. Predicted plaque wavefront velocity for different bacterial densities assuming latent period impact on burst size as defined by Eq. (8). Numbers refer to Eqs. (1)–(7). Since Eqs. (2)–(5) behave similarly to each other (these are the dotted-line curves in panel A), only Eq. (2) is presented in panels B–D. Peaks are seen only with Eq. (6) (as indicated). Parameter values, where possible, are the same as used by Abedon et al. (2001). These are a 10-min eclipse period (E) and a rate of accumulation of intracellular phage progeny of 5 min (R). The phage binding constant, k1, of 1.5 109 ml/min, is that employed by You and Yin (1999). Additional values are D ¼ 2.4 106 cm2/min (Yin and McCaskill, 1992) and k1 ¼ 0.05/min (You and Yin, 1999).
rates of plaque development or is assumed to not change as a function of time (that is, bacteria are assumed to not replicate). If phage diffusion within such a lawn occurs equally in all three dimensions, then the resulting plaques are either spherical or consist of spheres truncated at the air or hard-agar interfaces (Kaplan et al., 1981; Krone and Abedon, 2008). In the untruncated case, we can define an upper limit on plaque fecundity as a function of plaque radius, F(r), as
and then by plaque volume. Plaque volume is a function of plaque radius, which in turn is the product of the duration of plaque development (t) and plaque wavefront velocity (c). Burst size may be defined using Eq. (8) and c in Eqs. (1)–(7) also may be described as a function of latent period (L), that is, c ¼ c(L). Making these substitutions, we can estimate plaque fecundity as a function of the duration (t) of plaque development, F(t), rather than as a function of a time-dependent plaque radius:
F ðrÞ ¼ B N o 4 p r3 =3,
F ðtÞ ¼ ðL EÞ R N o 4 p cðLÞ3 t3 =3.
3
(9)
where 4 p r /3 is the volume of a sphere of radius r. Note that with increasing size, and therefore truncation, the geometry of a plaque may be instead approximated as a cylinder, the consequences of which we address in Appendix B. We assume, however, that purely spherical phage population expansion represents a good first approximation of diffusion-mediated phage spread within spatially structured natural environments. The number of phage produced, according to Eq. (9), is equal to burst size (B) multiplied by bacterial density (No)
(10)
Note in Eq. (10), as well as Eq. (9), that we ignore fecundity losses due to phage adsorption to uninfected bacteria, which should be constant per burst but proportionally small except when burst sizes are very small (this is the B–1 term discussed in Section 3.1). As is typical for models of phage-plaque formation, we also ignore phage adsorption to infected bacteria. However, unlike adsorption to uninfected bacteria, adsorption to infected bacteria, to a first approximation, should affect F(r) proportionally regardless of burst size, e.g. such that a loss of 25% of free
ARTICLE IN PRESS S.T. Abedon, R.R. Culler / Journal of Theoretical Biology 249 (2007) 582–592
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phages to this latter adsorption would result in a decline in F(r) of essentially 25%, regardless of the magnitude of B. 60 55
45 40
broth
35 30 25
es qu pla
Optimal Latent Period (min)
50
3.5. Plaque-fecundity-maximizing optimal latent period
eclipse period
Given the observation of plaque fecundity peaks in Fig. 3, we can infer that LFopt may be calculated. To ease this effort, we first express Eq. (10) as a function of latent period, L,
20 15 10 5
In Fig. 3, we solve Eq. (10) numerically, using where possible the same parameter values as those employed by Abedon et al. (2001), as well as by Wang et al. (1996) and Bull (2006). We see that even though there is a decrease in plaque size as latent period increases (Fig. 1), when defining c using Eqs. (1), (2), (6), or (7) the same decrease does not occur for overall plaque fecundity (Fig. 3). Instead, with three of the four equations, Eqs. (1), (6), and (7), there is a peak plaque fecundity at 30 min, which is at least two-fold greater than the peak plaque wavefront velocity estimated in Section 3.3. With Eq. (2), by contrast, plaque fecundity, like plaque wavefront velocity (Section 3.3), continues to rise as latent period increases.
F ðLÞ ¼ ðL EÞ cðLÞ3 Z,
0 105
106
107
108
109
Bacterial Density (No in ml-1) Fig. 2. Latent period optima that maximize wavefront velocity (Lcopt ) versus broth population growth (Lbopt ). The curve labeled ‘‘plaques’’ (open squares), calculates Lcopt for Eq. (6), with eclipse period indicated via a dotted line (parameter values are those employed in Fig. 1; see text for calculation details). The optimal latent period in broth (Lbopt ), as determined by Abedon et al. (2001), is indicated by the label, ‘‘broth’’ (open circles). To explore the impact of variance in rates of phage adsorption on Lcopt we bracket the ‘‘plaques’’ curve (where k1, of 1.5 109 ml/min) over a full-log range (where log(1.5 109) ¼ 8.8). Thus, we provide curves where k1 ¼ 5 109 ml/min (dash–dot line, lower, log(5 109) ¼ 8.3) and k1 ¼ 5 1010 ml/min (dash–dot–dot line, higher, log(5 1010) ¼ 9.3).
where Z ¼ R No 4 p t /3. That is, Z is the product of various parameters found in Eq. (10) that are not defined with respect to L. In principle, we could replace c(L) with any of Eqs. (1)–(7). However, we will limit ourselves to Eqs. (1), (2), and (7). In so doing we note that F(L)(1,7) solves for F according to Eqs. (1) or (7) F ðLÞð1;7Þ ¼ ðL EÞ L3=2 Z ð1;7Þ 1/2
3
(12)
3/2
, Z(1) ¼ 10 D Z (for F(L)(1)), and for c(L)pL Z(7) ¼ 1.43 D3/2 Z (for F(L)(7)). Contrasting Eq. (12), F(L)(2) solves for F(L) according to Eq. (2) as F ðLÞð2Þ ¼ ðL EÞ5=2 L3=2 Z ð2Þ
1012
(13)
10 2
1011 1
1010
109
6 7
108
Plaque Fecundity x 10-8 (F )
Plaque Fecundity (F, # phage)
(11)
3
9 8 7
6
6 5 4 3
7
2 1 0
0
15
30 45 60 75 90 105 Latent Period (min)
0
15
30 45 60 75 90 105 Latent Period (min)
Fig. 3. Plaque fecundity as a function of phage latent period assuming latent period impact on burst size as defined by Eq. (8). Parameter values are those employed in Fig. 1. In addition, No was set to 5.0 107 bacteria/ml and t was set to an arbitrary 600 min. Compared, in panel A, are solutions to Eq. (10) with c defined by Eqs. (1), (2), (6), and (7), as indicated. The vertical dashed line is the approximate predicted latent-period optimum for maximizing plaque productivity (which is 30 min for Eqs. (1) and (7), 31 min for Eq. (6), and undefined using Eq. (2)). In panel B, the same data for Eqs. (6) and (7) are graphed on a linear scale to better illustrate the broadness of plaque fecundity peaks.
ARTICLE IN PRESS S.T. Abedon, R.R. Culler / Journal of Theoretical Biology 249 (2007) 582–592
for c(L)pB1/2 ¼ (LE)1/2 R1/2 and F(L)(2)pc(L)3. Therefore, F(L)(2)pB B3/2 and Z(2) ¼ R3/2 2.03 D3/2 (No k1)3/2 (k1 (1+No k1/k1)2)3/2 Z. We generalize Eqs. (11), (12), and (13) to F ðLÞðeÞ ¼ ðL EÞX Y LY ZðeÞ ,
(14)
where ‘‘e’’ stands for some c-defining ‘‘equation’’ (e.g. Eqs. (1)–(7)). Note that X Y is an exponent with value of 1 if burst size does not contribute to plaque wavefront velocity, c, or a value of greater than one if burst size does contribute to plaque wavefront velocity (Appendix A). That L which maximizes F(L) in Eq. (14) represents the latent-period optimum that maximizes plaque fecundity. This maximum, LFopt , is determined by setting dF/dL ¼ 0: LFopt ¼
E . 1X
(15)
Thus, LFopt is a function solely of the eclipse period and the ratio, in Eq. (14), of the exponent found over (L–E) (indicating the L-dependent burst size contribution to plaque fecundity) and the exponent found over L (indicating the direct latent-period contribution to plaque size and therefore to plaque fecundity). Note that the larger the contribution of burst size to wavefront velocity then the larger the exponent over (L–E) relative to that over L and therefore the greater LFopt . More formally, and as derived in Appendix A: LFopt ¼
E , 1 ðð1 þ ð3 W ÞÞ=ð3 UÞÞ
(16)
where W and U are the exponents over B and L, respectively, in Eqs. (1)–(5) and (7).
Equivalent to Eq. (14), but defining plaque wavefront velocity, is (17)
where U is equal to the exponent over L in Eqs. (1)–(5) and (7) while the exponent over B in these same equations, W, is equal to X U. The latent period-independent portions of these same equations is presented as Z(e) (Section 3.1). That L which maximizes c(L) in Eq. (17) represents the wavefront velocity latent-period optimum, Lcopt , which is determined by setting dc=dL ¼ 0: E . (18) 1X Though seemingly identical to Eq. (15), note that the X terms in Eqs. (14) and (15) (henceforth, XF) are different from the X terms in Eqs. (17) and (18) (henceforth, Xc). That is, as derived in Appendix A, Lcopt ¼
Lcopt ¼
E . 1 ðW =UÞ
For Eqs. (1) and (7), Xc U ¼ 0 and U ¼ 0.5. Therefore Xc ¼ 0 and we would find, according to Eq. (18), that Lcopt ¼ E=ð1 0Þ ¼ E, which is to say that Lcopt is as short as possible when burst size minimally contributes to plaque size (this is the same conclusion reached in Section 3.3). Similarly for Eq. (2), Xc U ¼ 0.5 and U ¼ 0.5. Therefore Xc ¼ 1, and Lcopt ¼ E=ð1 1Þ ¼ 1, indicating that burst size contributes sufficiently to c relative to latent period that wavefront velocity continues to increase as latent period increases. To the extent that this latter conclusion is not correct, then we could argue that burst size is given too much weight in Eq. (2) (though with the caveat that Eq. (2) is not identical to the equivalent equation presented by Yin and McCaskill, 1992; see Section 3.1 and Abedon and Culler, 2007, for details). Thus, we can envisage two extremes on a spectrum, where burst size does not contribute to plaque wavefront velocity at one end, and therefore Lcopt approximates E, and where burst size contributes too much to plaque wavefront velocity (XcX1), and therefore latent period is predicted to evolve toward infinity (see Appendix B for additional circumstances where X terms can equal or exceed 1). In the latter case, we note that limitations on burst size could affect these conclusions. Evolution, in particular, should favor the predicted latent-period optimum (either Lcopt or LFopt ) or, assuming that R remains constant, that latent period which supports the maximum possible burst size, whichever is shorter. See Wang et al. (1996) for consideration of burst size limits during phage infection. Finally, as we consider more fully in Fig. 4 and Appendix A, we note that contributions of burst size to plaque wavefront velocity (W) should impact LFopt (as defined by Eq. (16)) more so than Lcopt (as defined by Eq. (19)). 4. Discussion
3.6. Estimating Lcopt as a function of E and X
cðLÞðeÞ ¼ ðL EÞX U LU Z ðeÞ ,
587
(19)
Spatial structure imposes limitations on both phage and bacterial movement, and here we consider the limiting case where bacterial movement is fully constrained and phage movement is limited to diffusion. In the laboratory, such conditions are typically obtained via phage growth within semi-solid agar gels. We speculate that environments where phage movement is mostly limited to phage diffusion, and in which bacterial movement has been constrained to a greater extent than that of phage, may be observed within soils, sediments, animal mucosal tissue, bacterial communities in contact with plant tissues, and, indeed, within biofilms, especially as found within relatively quiescent environments (Abedon and Culler, 2007; Abedon and Yin, 2008). 4.1. Optimizing phage latent period within plaques Plaque wavefront velocity, depending on circumstances, may be enhanced by modifying phage growth parameters, such as by increasing diffusivity or by decreasing latent period (Abedon and Culler, 2007). Not all phage growth
ARTICLE IN PRESS S.T. Abedon, R.R. Culler / Journal of Theoretical Biology 249 (2007) 582–592
588
1.1 1.0 LFopt
c
L opt
102
101
eclipse period
Relative Plaque Fecundity
Optimal Latent Period (min)
103
W=0.00
0.9
W=0.10 W=0 .0
5
0.8 0.7 0.6
W=0.15
0.5 0.4 0.3 0.2 0.1
100
0.0 0.0
0.1 0.2 0.3 0.4 0.5 Burst Size Contribution to c (=W )
0 10 20 30 40 50 60 70 80 90 100 Latent Period (min)
Fig. 4. Calculating latent period optima, LFopt versus Lcopt , while accounting for contributions of burst size to wavefront velocity (c). In panel A the dashed line indicates a calculation of LFopt employing Eq. (16) while the solid line indicates a calculation of Lcopt employing Eq. (19). Eclipse period is set equal to 10 min and U, the exponent over latent period (L) as found in Eqs. (1)–(5) and (7), is set equal to 0.5. W is equivalent to the exponent over burst size (B) found in those same equations and the vertical line indicates the minimum W estimation, 1/4, provided by You and Yin (1999). In panel B, Eq. (15) is employed to calculate plaque fecundity as a function of latent period and relative to that seen at LFopt . The vertical line indicates 20 min, which is twice Lcopt as calculated using Eq. (19) based on Eqs. (1) and (7) such that U ¼ 0.5 and W ¼ 0.0.
parameters, however, may always be varied independently, and in fact numerous tradeoffs likely exist that could impinge upon the co-optimization of a number of phage properties (e.g. Breitbart et al., 2005). This theme of tradeoffs, during plaque development and between phage latent period and burst size, is what we have considered here. Based on our analysis, we conclude that the optimal phage latent period (LFopt ) that serves to maximize plaque fecundity (number of phage produced per plaque; Fig. 3, Section 3.4, and Eq. (15)) can be at least twice as long as that phage latent period (Lcopt ) that serves to maximize plaque wavefront velocity (Figs. 1 and 2, Section 3.3, and Eq. (18)) as well as longer than that latent period which maximizes phage broth growth (Lbopt ; Fig. 2). Application of this assertion to actual phage evolution within plaques likely is complicated, however, by considerations of phage within-plaque competition. Withinplaque competition occurs when one harvests only a single plaque (or part of a plaque) and then plates for phage number. We expect those phages that can acquire uninfected bacteria fastest to contribute the most to rates of phage population growth within plaques, as well as within broth. Thus, the product of within-plaque competition should be an increase in the representation of those phages that attain the greatest frequency (relative prevalence) within the harvested plaque, even if this greater frequency comes at the expense of overall plaque fecundity. By contrast, competition between plaques may be observed by harvesting multiple (typically well-isolated) plaques from a lawn and then plating for phage number. We expect those phages that can produce the most progeny overall to dominate phage competition as it can occur between individual plaques (Bull, 2008), or as it can occur between individual broth cultures (Abedon et al., 2003).
Phages that acquire more bacteria do so at the expense of the ability of competing phages to acquire those same bacteria (Abedon, 1994; Turner and Duffy, 2008). This competition within plaques potentially results in a decline in overall plaque fecundity if plaque size is increased through reductions in phage latent period and if the resulting greater plaque sizes cannot compensate for associated declines in phage burst size. That is, given within-plaque competition we have an expectation of Lcopt pL pLFopt where L* is the actual latent period obtained following between-plaque selection. Variables which should bias L* toward Lcopt rather than LFopt include higher phage mutation rates (thereby increasing withinplaque genetic heterogeneity), longer periods of plaque growth (especially as this contributes to longer periods during which population growth of phage mutants may occur), and specific details of the plaque transfer protocols experimentally employed. In addition, see Appendix B for consideration of the impact on latent period evolution of assuming that plaques are shaped like cylinders rather than as spheres. 4.2. Modification of selection as a function of plaquetransfer protocol Whether or not one succeeds in biasing selection for greater plaque fecundity (toward LFopt ) over greater plaque size (toward Lcopt ) depends on the timing of plaque transfer (early versus late relative to the timing of lawn maturation), one’s approach to plaque sampling (e.g. entire plaques versus specific locations within plaques), whether plaques are pooled prior to replating, and to what extent phage genetic heterogeneity during plaque growth may be controlled. In general, earlier transfers, which are then
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fully mixed with other plaques prior to replating, should yield the greatest selection for LFopt versus Lcopt . Early transfer minimizes the accumulation of phage mutations and therefore reduces plaque genetic heterogeneity plus reduces the duration of within-plaque competition between the resulting non-identical phages. Sampling just from a plaque’s interior could serve as an alternative means of biasing against the transfer of more Lcopt -like phage variants that have arisen and been selected during later stages of plaque growth, though with the caveat that at least some phage mixing within plaques, due to virion diffusion, is expected to occur. By contrast, allowing plaques to grow for longer periods, sampling from a plaque’s periphery, and experimentally evolving phage lineages without pooling plaques should all result in selection for phages that are more Lcopt like, even absent any biases in actual plaque sampling (such as sampling noticeably larger plaques). The latter are equivalent to those mechanisms that could contribute to the evolution of greater phage virulence during broth propagation (Abedon, 2008). The experiments of Yin (1993) and simulations of Wei and Krone (2005) both suggest a biasing of selection toward faster wavefront velocity when sampling is made from the periphery of large plaques (see also, Bull, 2008). Given growing lawns, a nearly equivalent result may occur were one to evolve phage populations through singleplaque bottlenecks (Chao, 1990) via transfer of individual (subsequently homogenized) plaques, since it is later during plaque development that the majority of phage infections within plaques will occur (Kaplan et al., 1981; see also Wei and Krone, 2005). A similar effect should occur given replica plating (Krone and Abedon, 2008), which has the effect of artificially enhancing plaque size and therefore increasing both plaque heterogeneity and the duration of within-plaque competition between these genetically heterogeneous phages. Alternatively, lawn homogenization, which indiscriminately transfers plaque-containing top agar to diluent (Burch and Chao, 1999, 2000; Chao et al., 1997; Duffy et al., in press Turner and Chao, 1998), should bias selection more toward LFopt than the other techniques discussed above, though potentially less so to the degree that phages are allowed to evolve, within plaques, toward faster wavefront velocities. In such cases—and unless care is taken to otherwise avoid such an eventuality—it may be a suboptimal latent period that evolves, one which overly emphasizes neither the optimization of plaque size nor the optimization of plaque fecundity. This result may be especially likely given the broadness of plaque-fecundity peaks as seen in Fig. 3B such that much of the fecundity advantage associated with longer latent periods is obtained with latent periods that are well short of LFopt . Whatever latent period ultimately evolves in the midst of these conflicting forces, LFopt should be greater, perhaps substantially so, the more increases in burst size alone can contribute to greater plaque wavefront velocities (Fig. 4).
589
Acknowledgments Thank you to Bruce Rothschild for helpful discussion of the Kaplan et al. model, to Cameron Thomas who read and commented on this manuscript, and to two anonymous reviewers for their helpful comments. Appendix A. Burst-size contribution to wavefront velocity lengthens latent-period optima What is the impact of burst size contributing positively to plaque wavefront velocity (i.e. larger burst size resulting in greater wavefront velocity) but contributing more modestly than B does to wavefront velocity according to Eq. (2)? By way of example, consider an exponent burst size contribution to plaque wavefront velocity (W) of 0.1. That is, cpB0.1. In determining Lcopt , this would be equivalent to setting Xc U to 0.1 in Eq. (17). If U ¼ 1/2 (as with Eqs. (1), (2), and (7)), then Xc ¼ 0.2 ¼ 0.1 2 ¼ W/U and Lcopt ¼ 12:5 min (by Eq. (18)) given an eclipse of 10 min. Plaque fecundity (F), by contrast, is a function of plaque volume where volume pc3 such that FpB1.3 when cpB0.1. That is, XF Y ¼ 1+(3 0.1) in Eq. (14), where the 1 represents the wavefront velocity-independent contribution of burst size to plaque fecundity. If Y ¼ 3/2 (Eqs. (12) and (13)), then XF ¼ 1.3 2/3 ¼ 0.87 and therefore LFopt ¼ 75 min (by Eq. 15) given E ¼ 10 min. Thus, a burst size contribution to plaque size will have a substantial impact on that latent period which maximizes plaque fecundity (LFopt ), but a more modest impact on that latent period which maximizes plaque wavefront velocity (Lcopt ). We can generalize the above calculations as XF Y ¼ 1+(3 W) where W is the exponent over burst size’s contribution to c, i.e. such that FpB1+(3 W) with cpBW and with the 1 coming from the direct contribution of burst size to F. Therefore, in defining LFopt via Eq. (15), X F ¼ ð1 þ 3 W Þ=ð3 UÞ,
(20)
where U, as above, is the opposite of the exponent over latent period (L) in defining c (see Appendix B for further consideration of Eq. (20)). Alternatively, in defining Lcopt , X c ¼ ð0 þ W Þ=U ¼ W =U.
(21)
These values produced by Eqs. (20) and (21) can be plugged into Eqs. (15) or (18), respectively, to determine LFopt or Lcopt , which we do in generating Eqs. (16) and (19) as well as Fig. 4. Our conclusion is that a positive contribution of burst size to plaque wavefront velocity should have a substantially larger positive impact on LFopt than on Lcopt (Fig. 4). Appendix B. Impact of changes in plaque geometry on latent period optima The volume, V, of a spherical plaque, of radius r, is equal to 4 p r3/3. This calculation comes with a caveat,
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590
however, that plaques initiated near the air or hard-agar interface will be to some degree truncated. One way of addressing this truncation is by dividing by some amount, e.g. an otherwise spherical plaque displaying maximal truncation would have a volume of 2 p r3/3 (that is, half a sphere) and the average volume might be assumed to reside somewhere between 2 p r3/3 and 4 p r3/3 (i.e. see Kaplan et al., 1981), with the position on this spectrum depending on the magnitude of plaque radius (r) relative to soft agar thickness (h). A problem with this approach is that the constant employed (ranging from 2/3 to 4/3) will in fact change as a function of r. Nevertheless, the spherical geometry of the plaque is retained, which is necessary for Eq. (20) to remain valid. The above approximation works reasonably well so long as rph/2, where h is the thickness of a soft–agar overlay.
Prior to this point a plaque is not sufficiently large for it to be doubly truncated, i.e. truncated at both the air and the hard-agar interface. For r4h, on the other hand, all plaques will be expected to be doubly truncated. For h/2orph, therefore, we can expect the number of plaques which are doubly truncated to increase as r increases. When double truncation occurs, then plaque geometry is less spherical. For example, the volume of an otherwise spherical plaque, initiated halfway between the air and hard-agar interfaces, is described by Z
þh=2
p ðr2 y2 Þ dy ¼ h p r2 h3 =12.
V¼
Since h is a constant, it becomes increasingly small relative to r as r becomes larger. As a consequence,
20
50.0 Plaque Volume (h3 units)
untruncated
Plaque Volume (h3 units)
(22)
h=2
15 cylinder
10
5 average sphere
0
untruncated
10.0 5.0
1.0 0.5
average sphere cylinder
0.1 0.0
1.0 2.0 Plaque Radius (r in h units)
0.1
3.0
3
1.0 Plaque Radius (r in h units)
50.0 Plaque Volume (h3 units)
2
average sphere
Plaque Volume (h3 units)
untruncated∗3/4
1
0
untruncated∗3/4
10.0 5.0
1.0 0.5
average sphere
0.1 0.0
0.2 0.4 0.6 0.8 Plaque Radius (r in h units)
1.0
0.3 0.4 0.5 0.7 1.0 2.0 Plaque Radius (r in h units)
3.0
Fig. 5. Plaque volumes calculated by various methods (soft agar depth equals h). Four lines are shown: (i) volume calculated as an untruncated sphere (dash–dot line, labeled ‘‘untruncated’’), (ii) volume calculated as a cylinder (solid line, labeled ‘‘cylinder’’; this curve is shown in panels A and B but not in panels C and D), (iii) volume calculated as a truncated sphere (dash–dot–dot line, unlabeled; this curve in panel A completely overlaps that labeled ‘‘cylinder’’, is unobscured in panel B only in the lower portion of the graph, and is not shown in panels C and D), and (iv) volume calculated as the average volume of spheres (potentially truncated) initiated at a series of points (101 in total, evenly spaced) starting at the soft agar–hard agar interface and ending at the soft agar–air interface (dotted line, labeled ‘‘average sphere’’). The latter is the best approximation of average plaque volume of the four approaches. Especially in panel B, note that the ‘‘average sphere’’ curve is at first approximated by the ‘‘untruncated’’ sphere curve and then by the ‘‘cylinder’’ curve, with the transition occurring when the plaque’s radius is approximately three-quarters the top agar thickness (vertical line, panel B; that is, when h/2orph). For 4 ml of soft agar poured into a 100 mm Petri dish, that works out to a plaque diameter of only approximately 0.8 mm, which, though small, is a good approximation of the plaque sizes studied by Kaplan et al. (1981). Panels C and D compare the ‘‘average sphere’’ with a 3/4 ‘‘untruncated’’ sphere (V ¼ p r3) with only these two curves shown.
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V ¼ h p r2h3/12-h p r2 as r-N, where the expression h p r2 is explicitly that of the volume of a cylinder. For plaques, r is a function of c. With a cylindrical plaque, the exponent over wavefront velocity in Eqs. (10) and (11) thus becomes a 2 (the exponent 2 in h p r2) rather than a 3 (the exponent 3 in 4 p r3/3). This change automatically increases the relative importance of burst size to plaque fecundity. That is, as derived from Eq. (20), ¼ ð1 þ 2 W Þ=ð2 UÞ, X cylinder F
(23)
X sphere ¼ ð1 þ 3 W Þ=ð3 UÞ, F
(24)
and 4X sphere X cylinder F F
(25)
for all non-negative values of W and positive values of U. Per Eq. (15), as XF becomes larger then so too does LFopt . Thus, the latent period optimum for plaque fecundity would be expected to increase for a change in plaque geometry from sphere to cylinder. Caution is advised, however. For U ¼ 1/2 (that is, for cpL1/2), modeling a plaque as a cylinder rather than as a truncated sphere would result in X cylinder ¼ ð1 þ 2W ÞX1 F (for WX0) and therefore LFopt ¼ 1 (Eq. (15)). Thus, on the one hand, for small plaques which are not doubly truncated and therefore which have a geometry that is predominantly spherical (such that 2 p r3/3oVo4 p r3/3), then Eq. (20) predicts that LFopt will be substantially larger than Lcopt (Appendix A and Fig. 4). On the other hand, for larger plaques, ones, which are substantially truncated at both interfaces (such that V-h p r2), then LFopt is even larger, though perhaps unrealistically large (i.e. ¼ N). Splitting the difference, for intermediate levels of truncation, where a mixture of singly and doubly truncated plaques exist, then we would predict a value for LFopt which is intermediate between these two extremes. See Fig. 5 for graphical consideration of these assertions.
References Abedon, S.T., 1989. Selection for bacteriophage latent period length by bacterial density: a theoretical examination. Microb. Ecol. 18, 79–88. Abedon, S.T., 1994. Lysis and the interaction between free phages and infected cells. In: Karam, J.D. (Ed.), The Molecular Biology of Bacteriophage T4. ASM Press, Washington, DC, pp. 397–405. Abedon, S.T., Herschler, T.D., Stopar, D., 2001. Bacteriophage latentperiod evolution as a response to resource availability. Appl. Environ. Microbiol. 67, 4233–4241. Abedon, S.T., Hyman, P., Thomas, C., 2003. Experimental examination of bacteriophage latent-period evolution as a response to bacterial availability. Appl. Environ. Microbiol. 69, 7499–7506. Abedon, S.T., 2006. Phage ecology. In: Calendar, R., Abedon, S.T. (Eds.), The Bacteriophages. Oxford University Press, Oxford, pp. 37–46. Abedon, S.T., 2008. Phage population growth: constraints, games, adaptations. In: Abedon, S.T. (Ed.), Bacteriophage Ecology. Cambridge University Press, Cambridge, UK. Abedon, S.T., Culler, R.R., 2007. Bacteriophage evolution given spatial constraint. J. Theor. Biol. 248, 111–119.
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Abedon, S.T., Yin, J., 2008. Impact of spatial structure on phage population growth. In: Abedon, S.T. (Ed.), Bacteriophage Ecology. Cambridge University Press, Cambridge, UK. Anderson, T.F., 1950. Bacteriophages and their action on host cells. Res. Med. Sci., 1–16. Breitbart, M., Rohwer, F., Abedon, S.T., 2005. Phage ecology and bacterial pathogenesis. In: Waldor, M.K., Friedman, D.I., Adhya, S.L. (Eds.), Phages: Their Role in Bacterial Pathogenesis and Biotechnology. ASM Press, Washington, DC, pp. 66–91. Bull, J.J., Pfening, D.W., Wang, I.-W., 2004. Genetic details, optimization, and phage life histories. Trends Ecol. Evol. 19, 76–82. Bull, J.J., 2006. Optimality models of phage life history and parallels in disease evolution. J. Theor. Biol. 241, 928–938. Bull, J.J., 2008. Patterns in experimental adaptation of phages. In: Abedon, S.T. (Ed.), Bacteriophage Ecology. Cambridge University Press, Cambridge, UK. Burch, C.L., Chao, L., 1999. Evolution by small steps and rugged landscapes in the RNA virus f6. Genetics 151, 921–927. Burch, C.L., Chao, L., 2000. Evolvability of an RNA virus is determined by its mutational neighbourhood. Nature (London) 406, 625–628. Carlson, K., Miller, E.S., 1994. Enumerating phage: the plaque assay. In: Karam, J.D. (Ed.), Molecular Biology of Bacteriophage T4. ASM Press, Washington, DC, pp. 427–429. Chao, L., 1990. Fitness of RNA virus decreased by Muller’s ratchet. Nature (London) 348, 454–455. Chao, L., Tran, T.T., Tran, T.T., 1997. The advantage of sex in the RNA virus f6. Genetics 147, 953–959. Conner, M.M., Miller, M.W., 2005. Movement patterns and spatial epidemiology of a prion disease in mule deer population units. Ecol. Appl. 14, 1870–1881. Duffy S., Burch, B., Turner, P.E. Evolution of host specificity drives reproductive isolation among RNA viruses. in press doi:10.1111/ j.1558-2007.00226.x Hadas, H., Einav, M., Fishov, I., Zaritsky, A., 1997. Bacteriophage T4 development depends on the physiology of its host Escherichia coli. Microbiology 143, 179–185. Kaplan, D.A., Naumovski, L., Rothschild, B., Collier, R.J., 1981. Appendix: a model of plaque formation. Gene 13, 221–225. Kerr, B., West, J., Bohannan, B.J.M., 2008. Bacteriophage: models for exploring basic principles of ecology. In: Abedon, S.T. (Ed.), Bacteriophage Ecology. Cambridge University Press, Cambridge, UK. Koch, A.L., 1964. The growth of viral plaques during the enlargement phase. J. Theor. Biol. 6, 413–431. Krone, S.M., Abedon, S.T., 2008. Modeling phage plaque growth. In: Abedon, S.T. (Ed.), Bacteriophage Ecology. Cambridge University Press, Cambridge, UK. Lee, Y., Yin, J., 1996. Detection of evolving viruses. Nat. Biotechnol. 14, 491–493. Levin, B.R., Lenski, R.E., 1983. Coevolution in bacteria and their viruses and plasmids. In: Futuyama, D.J., Slatkin, M. (Eds.), Coevolution. Sinauer Associates, Inc., Sunderland, MA, pp. 99–127. Mayr-Harting, A., 1958. Die Entwicklung von Phagenloechern und der mechanismus der Phagenwirkung in festen Naehrboeden. Zbl. f. Bakt. Paras. Infek. u. Hyg. 171, 380–392. Murray, J.D., 2003. Mathematical Biology II: Spatial Models and Biomedical Application. Springer, Berlin. Ortega-Cejas, V., Fort, J., Me´ndez, V., Campos, D., 2004. Approximate solution to the speed of spreading viruses. Phys. Rev. E 69, 0319091–031909-4. Poon, A., Chao, L., 2005. The rate of compensatory mutation in the DNA bacteriophage fX174. Genetics 170, 989–999. Turner, P.E., Chao, L., 1998. Sex and the evolution of intrahost competition in RNA virus f6. Genetics 150, 523–532. Turner, P.E., Duffy, S., 2008. Evolutionary ecology of multi-phage infections. In: Abedon, S.T. (Ed.), Bacteriophage Ecology. Cambridge University Press, Cambridge, UK. Wang, I.-N., Dykhuizen, D.E., Slobodkin, L.B., 1996. The evolution of phage lysis timing. Evol. Ecol. 10, 545–558.
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Journal of Theoretical Biology 249 (2007) 582–592 www.elsevier.com/locate/yjtbi
Optimizing bacteriophage plaque fecundity Stephen T. Abedon, Rachel R. Culler Department of Microbiology, The Ohio State University, Mansfield, OH 44906, USA Received 21 April 2007; received in revised form 7 August 2007; accepted 7 August 2007 Available online 24 August 2007
Abstract Bacteriophages (phages), the viruses of bacteria, form visible lesions within bacterial lawns (called plaques), which are employed ubiquitously in phage isolation and characterization. Plaques also can serve as models for phage population growth within environments that display significant spatial structure, e.g. soils, sediments, animal mucosal tissue, etc. Furthermore, phages growing within plaques, in experimental evolution studies, may become adapted to novel conditions, may be selected for faster expansion, or may evolve toward producing more virions per plaque. Here, we examine the evolution of the latter, greater plaque fecundity, considering especially tradeoffs between phage latent period and phage burst size. This evolution is interesting because genetically lengthening latent periods, as seen with phage lysis-timing mutants, should increase phage burst sizes, as more time is available for phage-progeny maturation during infection. Genetically shortening latent periods, however, is a means toward producing larger phage plaques since phage virions then can spend more time diffusing rather than infecting. With these larger plaques more bacteria become phage infected, resulting in more phage bursts. Given this conflict between latent period’s impact on per-plaque burst number versus per-infection burst size, and based on analysis of existing models of plaque expansion, we provide two assertions. First, latent periods that optimize plaque fecundity are longer (e.g. at least two-fold longer) than latent periods that optimize plaque size (or that optimize phage population growth within broth). Second, if increases in burst size can contribute to plaque size (i.e. larger plaques with larger bursts), then latent-period optima that maximize plaque fecundity should be longer still. As a part of our analysis, we provide a means for predicting latent-period optima—for maximizing either plaque size or plaque fecundity—which is based on knowledge of only phage eclipse period and the relative contribution of phage burst size versus latent period toward plaque size. r 2007 Elsevier Ltd. All rights reserved. Keywords: Bacteriophage; Plaques; Tradeoffs; Optimization; Latent period
1. Introduction Bacteriophages (phages) are the viruses of bacteria. When phages replicate within well-mixed broth, it is those phages that can obtain bacteria faster that dominate within-culture population growth. Rapid phage population growth, however, does not necessarily correspond to efficient conversion of an environment’s resources into phage progeny (Abedon et al., 2003; Abedon, 2008). Conceptually similar tradeoffs can arise during phage population growth within semi-solid media, such as occurs during phage plaque formation within agar (Abedon and Culler, 2007; Abedon and Yin, 2008). There, more-rapid environmental exploitation is associated not only with Corresponding author. Tel.: +1 419 755 4343; fax: +1 419 755 4327.
E-mail address: [email protected] (S.T. Abedon). 0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2007.08.006
rapid phage population growth and host acquisition, but also with more-rapid spatial spread of the phage population, that is, with the formation of larger plaques. By contrast, phage growth could be biased toward a more efficient conversion of environmental resources into phage progeny, potentially resulting in plaques that contain more phage progeny, i.e. which display a greater fecundity. Abedon and Culler (2007) present an exploration of how changes in phage properties could contribute, under varying conditions, to greater plaque size. Here, we extend that analysis to consider the evolution of plaque fecundity from the perspective of tradeoffs between phage latent period and burst size, the duration of phage infection and number of bacteria produced per infection, respectively. For lytic phages, progeny mature intracellularly only so long as phage infections remain intact, but those same progeny are unable to obtain new bacteria to infect unless
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infections are terminated with lysis. A number of studies have explored the evolution of the timing of this lysis within well-mixed broth cultures (Abedon, 1989, 1994, 2006, 2008; Abedon et al., 2001, 2003; Bull et al., 2004; Bull, 2006; Kerr et al., 2008; Levin and Lenski, 1983; Wang et al., 1996; Wang, 2006). The general arguments are that lysis timing may be optimized such that rates of phage population growth are maximized. Alternatively, a standard protocol in studies of phage experimental adaptation is to grow phages as non-overlapping plaques and then to homogenize plaques and lawns, prior to subsequent replating, by removing plaquecontaining top agar en masse to diluent (Burch and Chao, 1999, 2000; Chao et al., 1997; Duffy et al., in press; Turner and Chao, 1998). Such an approach, which competes plaques in terms of titer, should select for greater plaque fecundity (Bull, 2008), which can result from more phage infections per plaque (a result of greater plaque size; Kaplan et al., 1981; Poon and Chao, 2005) or from greater per-infection fecundity (greater burst size). These two routes toward greater plaque fecundity are to some extent in conflict, however: Genetically shorter phage latent periods should result in larger plaques (as reviewed by Abedon and Culler, 2007) but simultaneously in smaller burst sizes (e.g. Abedon et al., 2003). Toward developing a greater understanding of selection acting on phages during plaque enlargement, such as during experimental evolution studies, here we develop a method for predicting latent periods that optimize plaque fecundity.
pathogen epidemics or species invasions in general (Conner and Miller, 2005; Murray, 2003), may be viewed—at least at coarser resolutions—as traveling waves. Given tradeoffs between phage latent period and burst size (Section 3.2), we provide estimations of the phage ‘‘optimal’’ latent period that maximizes rates of the phage wavelike propagation during plaque development (Lcopt ; c ¼ plaque wavefront velocity; Sections 3.3 and 3.6) or, alternatively, that latent period which maximizes overall phage productivity within plaques (LFopt ; F stands for plaque fecundity; Sections 3.4 and 3.5). 3.1. Plaque wavefront velocity as functions of latent period and burst size In practice, plaque wavefront velocity, c, is constant over most of the course of visible plaque development, so long as bacterial physiology also remains more or less constant (Kaplan et al., 1981; Koch, 1964; Lee and Yin, 1996; Mayr-Harting, 1958; Yin, 1991). Abedon and Culler (2007) review seven equations, which estimate c as functions of various phage growth parameters. We present these equations below in comparable forms, emphasizing here the contributions of B (burst size) and L (latent period), which, except for Eq. (6), are shown prior to the larger dot, ‘‘‘‘: c ¼ B0=4 L2=4 10 D1=2 ,
We consider the plaque-growth models of Koch (1964), Yin and McCaskill (1992), and Ortega-Cejas et al. (2004) to obtain estimations of what latent period—given standard assumptions regarding burst size-latent-period tradeoffs (e.g. Abedon et al., 2001)—will maximize plaque wavefront velocity (c), that is, the rate of plaque radial expansion. We then combine these models with calculations of plaque volume to derive an equation that allows facile estimation of the phage optimal latent period that maximizes plaque fecundity or, alternatively, the phage optimal latent period that maximizes plaque size. These equations are expressed in terms of the relative contribution of burst size versus latent period to plaque wavefront velocity, or to overall plaque fecundity. 3. Results Anderson (1950) described the spread of phage infection, during plaque growth, as epidemic: ‘‘In the bacterial viruses where the cells dissolve, or lyse, as a result of the disease, such epidemics produce clear areas, or plaques, in an otherwise uniform bacterial smear on agar.’’ The outward spread of these phage epidemics within plaques (e.g. Wei and Krone, 2005; Yin, 1991; Yin and McCaskill, 1992; You and Yin, 1999; Section 3.1), like those of
(1) "
2=4
2. Methods
583
c¼B
2=4
L
1=2
2:0 D
N o k1 ðk1 ð1 þ N o k1 =k1 Þ2 Þ
#1=2 , (2)
c ¼ B1=4 L1=4 1:6 D1=2 ðN o k1 Þ1=4 ,
(3)
c ¼ B2=4 L0=4 2:0 D1=2 ðN o k1 Þ1=2 ,
(4)
c ¼ B4=4 L4=4 2:6 D1=2 ðN o k1 Þ1=2 ,
(5)
2=4
c¼B
0=4
L
1=2
2:0 D
N o k1 ð1 þ L B N o k1 Þ
c ¼ B0=4 L2=4 1:4 D1=2 ,
1=2 ,
(6) (7)
where D is virion diffusivity, No is the lawn bacterial density, k1 is a phage-to-bacterium binding constant, and k1 is a desorption constant of phage from bacteria. The equations were derived by Abedon and Culler (2007) from those of (i) Koch (1964) (Eq. (1)), (ii) Yin and McCaskill (1992) (Eqs. (2)–(5)), and (iii) Ortega-Cejas et al. (2004) (Eqs. (6) and (7)). The Yin and McCaskill equations correspond to conditions of equilibrated virion adsorption (Eq. (2)), large burst size (Eq. (3)), slow phage adsorption (Eq. (4)), and fast phage adsorption (Eq. (5)). The OrtegaCejas et al. (2004) models differ in their applicability as dependent upon various phage growth parameters but also as dependent on host density. In brief, Eq. (7) is applicable
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at higher, arguably more-realistic host densities (Abedon and Culler, 2007). Given the similarity between Eqs. (1) and (7), along with the relative importance of Eq. (7) in the Ortega-Cejas et al. model, Abedon and Culler (2007) argue that the presented models from Koch and Ortega-Cejas et al. are algebraically equivalent under typical phage plaquing conditions. For the sake of achieving comparability, we have modified a number of these equations from those originally published. For a number of the Yin and McCaskill (1992) equations for instance, B–1 has been replaced with B (where the subtracted 1 indicates virion loss to the initiation of phage infections) and the rate constant for phage-induced lysis of infected bacteria, k2, is presented here instead by an approximation, 1/L (which Yin and McCaskill employ when solving their model numerically). We also disregard any consideration of the impact of the occupation of an ‘‘inert volume’’ by hosts on phage diffusivity (an ‘‘effective diffusion’’ as employed by both Yin and McCaskill and Ortega et al.) since these effects do not differ between equations (Abedon and Culler, 2007) nor impact subsequent calculations provided here. 3.2. Defining burst size in terms of latent period Phage burst size (B) can vary as a direct function of phage latent period, (e.g. Abedon et al., 2001): B ¼ ðL EÞ R,
(8)
where E is eclipse period, that is, the period of phagemediated biochemical preparation for phage-virion production (e.g. Abedon, 2006). R is the rate of intracellular phage-progeny maturation once that production begins and Wang et al. (1996) provide evidence for the nearconstancy of R over much of the post-eclipse latent period within phage infections that are lysis defective. Though the magnitudes of B, R, L, and E can all vary as functions of bacterial physiology (Hadas et al., 1997; You et al., 2002), for most phages L also may be both precisely controlled and readily modified, via phage lysis-timing mutations, without affecting either E or R (e.g. Bull, 2006; Wang, 2006; Young and Wang, 2006). 3.3. Determining wavefront-velocity-maximizing optimal latent period In Fig. 1, we solve for plaque wavefront velocity (c) as a function of latent period (L) using Eqs. (1)–(7), replacing burst size (B), when employed, with Eq. (8). Only Eq. (6) shows a peak, indicating a latent period that maximizes wavefront velocity (Lcopt ) without being essentially as short as possible (which would be where latent period is only slightly greater than eclipse period, E). Even with Eq. (6), for No ¼ 108 bacteria/ml this peak is quite close to E (i.e. E ¼ 10 min versus Lcopt ¼ 11.5 min), but is greater if No is reduced. These figures also indicate a seemingly anomalous result for Eqs. (2)–(5), those of Yin and McCaskill (1992), where optimal latent
period neither approaches the eclipse period nor peaks. One explanation for this latter behavior is that the contribution of burst size to plaque size is large in these equations, perhaps too large, as we consider subsequently (Section 3.6). Because Eq. (6) displays a peak (Fig. 1), we were able to calculate Lcopt . To do this we replaced B in Eq. (6) with Eq. (8), differentiated the resulting equation with respect to L, and then solved for L when dc/dL ¼ 0 The resulting equation is complicated but does provide non-trivial estimations of Lcopt (Fig. 2) that are consistent with estimations determined graphically (Fig. 1). Eq. (6), however, is valid only when Noo(L B k1)1 (Abedon and Culler, 2007; Ortega-Cejas et al., 2004). Consequently, Eq. (6) actually is not valid over all combinations of bacterial density, optimal latent period (such that L ¼ Lcopt ), and burst size (¼ ðLcopt EÞ R) shown in Fig. 2, and only becomes valid, at a coarseness of half log increments, at Nop104 bacteria/ml (calculations not presented). These calculations based on Eq. (6), nevertheless, are illustrative of the potential impact of burst size on Lcopt , as we will consider again in Section 3.6 and Appendix A. The various approaches used for estimating Lcopt produce seemingly disparate results (Fig. 1). The models of Koch (1964) and Ortega-Cejas et al. (2004), though, do provide somewhat consistent and biologically plausible determinations. In particular, if B does not contribute to plaque wavefront velocity, then that L which maximizes c should be as short as possible. However, we have an expectation that Eqs. (1)–(7) will not hold either at very short latent periods or for very small burst sizes (Koch, 1964; You and Yin, 1999). Therefore, for Eqs. (1) and (7) reasonable interpretation should be that Lcopt approaches but falls short of reaching E. Under the conditions modeled, a latent period of 15 min provides a burst size of 25. We therefore feel justified in asserting that these equations in fact provide a prediction of an optimal latent period of less than 15 min, but probably equal to or greater than 12 min (where a burst size of less than 10 phage/bacterium may be sufficiently small that plaque size is greatly negatively impacted; Carlson and Miller, 1994). Eq. (6) provides essentially the same prediction for Lcopt , which for reasonable values for No during plaque formation, i.e. greater than 107 bacteria/ml, is one of less than 15 min (Fig. 2). By contrast, estimations by Abedon et al. (2001) of Lbopt , the optimal latent period for phage population growth within broth, are of 22 min for No ¼ 107 bacteria/ml, ranging down to about 15 min at higher bacterial densities (Fig. 2). We suspect that at least part of the explanation for why Lbopt is greater than Lcopt (as calculated here) has to do with burst size making a greater contribution to phage population growth rates in well-mixed broth than it does to plaque wavefront velocity as determined by Eqs. (1), (6), or (7). 3.4. Plaque fecundity In the models of plaque growth presented as Eqs. (1)–(7), bacterial density is either not considered to contribute to
ARTICLE IN PRESS S.T. Abedon, R.R. Culler / Journal of Theoretical Biology 249 (2007) 582–592
10-2
5
Front Velocity (c in cm min-1)
Front Velocity (c in cm min-1)
10-1
4 10-2
2 3 1
peak, Eq. (6)
10-3
No = 10
10-4 0
15
8.0
6 7
bacteria/ml
2 1 10-3
6 7 10-4
30
45
60
75
peak, Eq. (6)
No = 10 0
90 105 120
15
bacteria/ml
30
45
60
75
90 105 120
10-2
Front Velocity (c in cm min-1)
10-2
Front Velocity (c in cm min-1)
7.5
Latent Period (min)
Latent Period (min)
2 1
peak, Eq. (6)
10-3
6 7 10-4
585
No = 10 0
15
7.0
bacteria/ml
1 2 10-3
6 7 10-4
30
45
60
75
90 105 120
peak, Eq. (6)
No = 10 0
15
6.5
bacteria/ml
30
45
60
75
90 105 120
Latent Period (min)
Latent Period (min)
Fig. 1. Predicted plaque wavefront velocity for different bacterial densities assuming latent period impact on burst size as defined by Eq. (8). Numbers refer to Eqs. (1)–(7). Since Eqs. (2)–(5) behave similarly to each other (these are the dotted-line curves in panel A), only Eq. (2) is presented in panels B–D. Peaks are seen only with Eq. (6) (as indicated). Parameter values, where possible, are the same as used by Abedon et al. (2001). These are a 10-min eclipse period (E) and a rate of accumulation of intracellular phage progeny of 5 min (R). The phage binding constant, k1, of 1.5 109 ml/min, is that employed by You and Yin (1999). Additional values are D ¼ 2.4 106 cm2/min (Yin and McCaskill, 1992) and k1 ¼ 0.05/min (You and Yin, 1999).
rates of plaque development or is assumed to not change as a function of time (that is, bacteria are assumed to not replicate). If phage diffusion within such a lawn occurs equally in all three dimensions, then the resulting plaques are either spherical or consist of spheres truncated at the air or hard-agar interfaces (Kaplan et al., 1981; Krone and Abedon, 2008). In the untruncated case, we can define an upper limit on plaque fecundity as a function of plaque radius, F(r), as
and then by plaque volume. Plaque volume is a function of plaque radius, which in turn is the product of the duration of plaque development (t) and plaque wavefront velocity (c). Burst size may be defined using Eq. (8) and c in Eqs. (1)–(7) also may be described as a function of latent period (L), that is, c ¼ c(L). Making these substitutions, we can estimate plaque fecundity as a function of the duration (t) of plaque development, F(t), rather than as a function of a time-dependent plaque radius:
F ðrÞ ¼ B N o 4 p r3 =3,
F ðtÞ ¼ ðL EÞ R N o 4 p cðLÞ3 t3 =3.
3
(9)
where 4 p r /3 is the volume of a sphere of radius r. Note that with increasing size, and therefore truncation, the geometry of a plaque may be instead approximated as a cylinder, the consequences of which we address in Appendix B. We assume, however, that purely spherical phage population expansion represents a good first approximation of diffusion-mediated phage spread within spatially structured natural environments. The number of phage produced, according to Eq. (9), is equal to burst size (B) multiplied by bacterial density (No)
(10)
Note in Eq. (10), as well as Eq. (9), that we ignore fecundity losses due to phage adsorption to uninfected bacteria, which should be constant per burst but proportionally small except when burst sizes are very small (this is the B–1 term discussed in Section 3.1). As is typical for models of phage-plaque formation, we also ignore phage adsorption to infected bacteria. However, unlike adsorption to uninfected bacteria, adsorption to infected bacteria, to a first approximation, should affect F(r) proportionally regardless of burst size, e.g. such that a loss of 25% of free
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phages to this latter adsorption would result in a decline in F(r) of essentially 25%, regardless of the magnitude of B. 60 55
45 40
broth
35 30 25
es qu pla
Optimal Latent Period (min)
50
3.5. Plaque-fecundity-maximizing optimal latent period
eclipse period
Given the observation of plaque fecundity peaks in Fig. 3, we can infer that LFopt may be calculated. To ease this effort, we first express Eq. (10) as a function of latent period, L,
20 15 10 5
In Fig. 3, we solve Eq. (10) numerically, using where possible the same parameter values as those employed by Abedon et al. (2001), as well as by Wang et al. (1996) and Bull (2006). We see that even though there is a decrease in plaque size as latent period increases (Fig. 1), when defining c using Eqs. (1), (2), (6), or (7) the same decrease does not occur for overall plaque fecundity (Fig. 3). Instead, with three of the four equations, Eqs. (1), (6), and (7), there is a peak plaque fecundity at 30 min, which is at least two-fold greater than the peak plaque wavefront velocity estimated in Section 3.3. With Eq. (2), by contrast, plaque fecundity, like plaque wavefront velocity (Section 3.3), continues to rise as latent period increases.
F ðLÞ ¼ ðL EÞ cðLÞ3 Z,
0 105
106
107
108
109
Bacterial Density (No in ml-1) Fig. 2. Latent period optima that maximize wavefront velocity (Lcopt ) versus broth population growth (Lbopt ). The curve labeled ‘‘plaques’’ (open squares), calculates Lcopt for Eq. (6), with eclipse period indicated via a dotted line (parameter values are those employed in Fig. 1; see text for calculation details). The optimal latent period in broth (Lbopt ), as determined by Abedon et al. (2001), is indicated by the label, ‘‘broth’’ (open circles). To explore the impact of variance in rates of phage adsorption on Lcopt we bracket the ‘‘plaques’’ curve (where k1, of 1.5 109 ml/min) over a full-log range (where log(1.5 109) ¼ 8.8). Thus, we provide curves where k1 ¼ 5 109 ml/min (dash–dot line, lower, log(5 109) ¼ 8.3) and k1 ¼ 5 1010 ml/min (dash–dot–dot line, higher, log(5 1010) ¼ 9.3).
where Z ¼ R No 4 p t /3. That is, Z is the product of various parameters found in Eq. (10) that are not defined with respect to L. In principle, we could replace c(L) with any of Eqs. (1)–(7). However, we will limit ourselves to Eqs. (1), (2), and (7). In so doing we note that F(L)(1,7) solves for F according to Eqs. (1) or (7) F ðLÞð1;7Þ ¼ ðL EÞ L3=2 Z ð1;7Þ 1/2
3
(12)
3/2
, Z(1) ¼ 10 D Z (for F(L)(1)), and for c(L)pL Z(7) ¼ 1.43 D3/2 Z (for F(L)(7)). Contrasting Eq. (12), F(L)(2) solves for F(L) according to Eq. (2) as F ðLÞð2Þ ¼ ðL EÞ5=2 L3=2 Z ð2Þ
1012
(13)
10 2
1011 1
1010
109
6 7
108
Plaque Fecundity x 10-8 (F )
Plaque Fecundity (F, # phage)
(11)
3
9 8 7
6
6 5 4 3
7
2 1 0
0
15
30 45 60 75 90 105 Latent Period (min)
0
15
30 45 60 75 90 105 Latent Period (min)
Fig. 3. Plaque fecundity as a function of phage latent period assuming latent period impact on burst size as defined by Eq. (8). Parameter values are those employed in Fig. 1. In addition, No was set to 5.0 107 bacteria/ml and t was set to an arbitrary 600 min. Compared, in panel A, are solutions to Eq. (10) with c defined by Eqs. (1), (2), (6), and (7), as indicated. The vertical dashed line is the approximate predicted latent-period optimum for maximizing plaque productivity (which is 30 min for Eqs. (1) and (7), 31 min for Eq. (6), and undefined using Eq. (2)). In panel B, the same data for Eqs. (6) and (7) are graphed on a linear scale to better illustrate the broadness of plaque fecundity peaks.
ARTICLE IN PRESS S.T. Abedon, R.R. Culler / Journal of Theoretical Biology 249 (2007) 582–592
for c(L)pB1/2 ¼ (LE)1/2 R1/2 and F(L)(2)pc(L)3. Therefore, F(L)(2)pB B3/2 and Z(2) ¼ R3/2 2.03 D3/2 (No k1)3/2 (k1 (1+No k1/k1)2)3/2 Z. We generalize Eqs. (11), (12), and (13) to F ðLÞðeÞ ¼ ðL EÞX Y LY ZðeÞ ,
(14)
where ‘‘e’’ stands for some c-defining ‘‘equation’’ (e.g. Eqs. (1)–(7)). Note that X Y is an exponent with value of 1 if burst size does not contribute to plaque wavefront velocity, c, or a value of greater than one if burst size does contribute to plaque wavefront velocity (Appendix A). That L which maximizes F(L) in Eq. (14) represents the latent-period optimum that maximizes plaque fecundity. This maximum, LFopt , is determined by setting dF/dL ¼ 0: LFopt ¼
E . 1X
(15)
Thus, LFopt is a function solely of the eclipse period and the ratio, in Eq. (14), of the exponent found over (L–E) (indicating the L-dependent burst size contribution to plaque fecundity) and the exponent found over L (indicating the direct latent-period contribution to plaque size and therefore to plaque fecundity). Note that the larger the contribution of burst size to wavefront velocity then the larger the exponent over (L–E) relative to that over L and therefore the greater LFopt . More formally, and as derived in Appendix A: LFopt ¼
E , 1 ðð1 þ ð3 W ÞÞ=ð3 UÞÞ
(16)
where W and U are the exponents over B and L, respectively, in Eqs. (1)–(5) and (7).
Equivalent to Eq. (14), but defining plaque wavefront velocity, is (17)
where U is equal to the exponent over L in Eqs. (1)–(5) and (7) while the exponent over B in these same equations, W, is equal to X U. The latent period-independent portions of these same equations is presented as Z(e) (Section 3.1). That L which maximizes c(L) in Eq. (17) represents the wavefront velocity latent-period optimum, Lcopt , which is determined by setting dc=dL ¼ 0: E . (18) 1X Though seemingly identical to Eq. (15), note that the X terms in Eqs. (14) and (15) (henceforth, XF) are different from the X terms in Eqs. (17) and (18) (henceforth, Xc). That is, as derived in Appendix A, Lcopt ¼
Lcopt ¼
E . 1 ðW =UÞ
For Eqs. (1) and (7), Xc U ¼ 0 and U ¼ 0.5. Therefore Xc ¼ 0 and we would find, according to Eq. (18), that Lcopt ¼ E=ð1 0Þ ¼ E, which is to say that Lcopt is as short as possible when burst size minimally contributes to plaque size (this is the same conclusion reached in Section 3.3). Similarly for Eq. (2), Xc U ¼ 0.5 and U ¼ 0.5. Therefore Xc ¼ 1, and Lcopt ¼ E=ð1 1Þ ¼ 1, indicating that burst size contributes sufficiently to c relative to latent period that wavefront velocity continues to increase as latent period increases. To the extent that this latter conclusion is not correct, then we could argue that burst size is given too much weight in Eq. (2) (though with the caveat that Eq. (2) is not identical to the equivalent equation presented by Yin and McCaskill, 1992; see Section 3.1 and Abedon and Culler, 2007, for details). Thus, we can envisage two extremes on a spectrum, where burst size does not contribute to plaque wavefront velocity at one end, and therefore Lcopt approximates E, and where burst size contributes too much to plaque wavefront velocity (XcX1), and therefore latent period is predicted to evolve toward infinity (see Appendix B for additional circumstances where X terms can equal or exceed 1). In the latter case, we note that limitations on burst size could affect these conclusions. Evolution, in particular, should favor the predicted latent-period optimum (either Lcopt or LFopt ) or, assuming that R remains constant, that latent period which supports the maximum possible burst size, whichever is shorter. See Wang et al. (1996) for consideration of burst size limits during phage infection. Finally, as we consider more fully in Fig. 4 and Appendix A, we note that contributions of burst size to plaque wavefront velocity (W) should impact LFopt (as defined by Eq. (16)) more so than Lcopt (as defined by Eq. (19)). 4. Discussion
3.6. Estimating Lcopt as a function of E and X
cðLÞðeÞ ¼ ðL EÞX U LU Z ðeÞ ,
587
(19)
Spatial structure imposes limitations on both phage and bacterial movement, and here we consider the limiting case where bacterial movement is fully constrained and phage movement is limited to diffusion. In the laboratory, such conditions are typically obtained via phage growth within semi-solid agar gels. We speculate that environments where phage movement is mostly limited to phage diffusion, and in which bacterial movement has been constrained to a greater extent than that of phage, may be observed within soils, sediments, animal mucosal tissue, bacterial communities in contact with plant tissues, and, indeed, within biofilms, especially as found within relatively quiescent environments (Abedon and Culler, 2007; Abedon and Yin, 2008). 4.1. Optimizing phage latent period within plaques Plaque wavefront velocity, depending on circumstances, may be enhanced by modifying phage growth parameters, such as by increasing diffusivity or by decreasing latent period (Abedon and Culler, 2007). Not all phage growth
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1.1 1.0 LFopt
c
L opt
102
101
eclipse period
Relative Plaque Fecundity
Optimal Latent Period (min)
103
W=0.00
0.9
W=0.10 W=0 .0
5
0.8 0.7 0.6
W=0.15
0.5 0.4 0.3 0.2 0.1
100
0.0 0.0
0.1 0.2 0.3 0.4 0.5 Burst Size Contribution to c (=W )
0 10 20 30 40 50 60 70 80 90 100 Latent Period (min)
Fig. 4. Calculating latent period optima, LFopt versus Lcopt , while accounting for contributions of burst size to wavefront velocity (c). In panel A the dashed line indicates a calculation of LFopt employing Eq. (16) while the solid line indicates a calculation of Lcopt employing Eq. (19). Eclipse period is set equal to 10 min and U, the exponent over latent period (L) as found in Eqs. (1)–(5) and (7), is set equal to 0.5. W is equivalent to the exponent over burst size (B) found in those same equations and the vertical line indicates the minimum W estimation, 1/4, provided by You and Yin (1999). In panel B, Eq. (15) is employed to calculate plaque fecundity as a function of latent period and relative to that seen at LFopt . The vertical line indicates 20 min, which is twice Lcopt as calculated using Eq. (19) based on Eqs. (1) and (7) such that U ¼ 0.5 and W ¼ 0.0.
parameters, however, may always be varied independently, and in fact numerous tradeoffs likely exist that could impinge upon the co-optimization of a number of phage properties (e.g. Breitbart et al., 2005). This theme of tradeoffs, during plaque development and between phage latent period and burst size, is what we have considered here. Based on our analysis, we conclude that the optimal phage latent period (LFopt ) that serves to maximize plaque fecundity (number of phage produced per plaque; Fig. 3, Section 3.4, and Eq. (15)) can be at least twice as long as that phage latent period (Lcopt ) that serves to maximize plaque wavefront velocity (Figs. 1 and 2, Section 3.3, and Eq. (18)) as well as longer than that latent period which maximizes phage broth growth (Lbopt ; Fig. 2). Application of this assertion to actual phage evolution within plaques likely is complicated, however, by considerations of phage within-plaque competition. Withinplaque competition occurs when one harvests only a single plaque (or part of a plaque) and then plates for phage number. We expect those phages that can acquire uninfected bacteria fastest to contribute the most to rates of phage population growth within plaques, as well as within broth. Thus, the product of within-plaque competition should be an increase in the representation of those phages that attain the greatest frequency (relative prevalence) within the harvested plaque, even if this greater frequency comes at the expense of overall plaque fecundity. By contrast, competition between plaques may be observed by harvesting multiple (typically well-isolated) plaques from a lawn and then plating for phage number. We expect those phages that can produce the most progeny overall to dominate phage competition as it can occur between individual plaques (Bull, 2008), or as it can occur between individual broth cultures (Abedon et al., 2003).
Phages that acquire more bacteria do so at the expense of the ability of competing phages to acquire those same bacteria (Abedon, 1994; Turner and Duffy, 2008). This competition within plaques potentially results in a decline in overall plaque fecundity if plaque size is increased through reductions in phage latent period and if the resulting greater plaque sizes cannot compensate for associated declines in phage burst size. That is, given within-plaque competition we have an expectation of Lcopt pL pLFopt where L* is the actual latent period obtained following between-plaque selection. Variables which should bias L* toward Lcopt rather than LFopt include higher phage mutation rates (thereby increasing withinplaque genetic heterogeneity), longer periods of plaque growth (especially as this contributes to longer periods during which population growth of phage mutants may occur), and specific details of the plaque transfer protocols experimentally employed. In addition, see Appendix B for consideration of the impact on latent period evolution of assuming that plaques are shaped like cylinders rather than as spheres. 4.2. Modification of selection as a function of plaquetransfer protocol Whether or not one succeeds in biasing selection for greater plaque fecundity (toward LFopt ) over greater plaque size (toward Lcopt ) depends on the timing of plaque transfer (early versus late relative to the timing of lawn maturation), one’s approach to plaque sampling (e.g. entire plaques versus specific locations within plaques), whether plaques are pooled prior to replating, and to what extent phage genetic heterogeneity during plaque growth may be controlled. In general, earlier transfers, which are then
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fully mixed with other plaques prior to replating, should yield the greatest selection for LFopt versus Lcopt . Early transfer minimizes the accumulation of phage mutations and therefore reduces plaque genetic heterogeneity plus reduces the duration of within-plaque competition between the resulting non-identical phages. Sampling just from a plaque’s interior could serve as an alternative means of biasing against the transfer of more Lcopt -like phage variants that have arisen and been selected during later stages of plaque growth, though with the caveat that at least some phage mixing within plaques, due to virion diffusion, is expected to occur. By contrast, allowing plaques to grow for longer periods, sampling from a plaque’s periphery, and experimentally evolving phage lineages without pooling plaques should all result in selection for phages that are more Lcopt like, even absent any biases in actual plaque sampling (such as sampling noticeably larger plaques). The latter are equivalent to those mechanisms that could contribute to the evolution of greater phage virulence during broth propagation (Abedon, 2008). The experiments of Yin (1993) and simulations of Wei and Krone (2005) both suggest a biasing of selection toward faster wavefront velocity when sampling is made from the periphery of large plaques (see also, Bull, 2008). Given growing lawns, a nearly equivalent result may occur were one to evolve phage populations through singleplaque bottlenecks (Chao, 1990) via transfer of individual (subsequently homogenized) plaques, since it is later during plaque development that the majority of phage infections within plaques will occur (Kaplan et al., 1981; see also Wei and Krone, 2005). A similar effect should occur given replica plating (Krone and Abedon, 2008), which has the effect of artificially enhancing plaque size and therefore increasing both plaque heterogeneity and the duration of within-plaque competition between these genetically heterogeneous phages. Alternatively, lawn homogenization, which indiscriminately transfers plaque-containing top agar to diluent (Burch and Chao, 1999, 2000; Chao et al., 1997; Duffy et al., in press Turner and Chao, 1998), should bias selection more toward LFopt than the other techniques discussed above, though potentially less so to the degree that phages are allowed to evolve, within plaques, toward faster wavefront velocities. In such cases—and unless care is taken to otherwise avoid such an eventuality—it may be a suboptimal latent period that evolves, one which overly emphasizes neither the optimization of plaque size nor the optimization of plaque fecundity. This result may be especially likely given the broadness of plaque-fecundity peaks as seen in Fig. 3B such that much of the fecundity advantage associated with longer latent periods is obtained with latent periods that are well short of LFopt . Whatever latent period ultimately evolves in the midst of these conflicting forces, LFopt should be greater, perhaps substantially so, the more increases in burst size alone can contribute to greater plaque wavefront velocities (Fig. 4).
589
Acknowledgments Thank you to Bruce Rothschild for helpful discussion of the Kaplan et al. model, to Cameron Thomas who read and commented on this manuscript, and to two anonymous reviewers for their helpful comments. Appendix A. Burst-size contribution to wavefront velocity lengthens latent-period optima What is the impact of burst size contributing positively to plaque wavefront velocity (i.e. larger burst size resulting in greater wavefront velocity) but contributing more modestly than B does to wavefront velocity according to Eq. (2)? By way of example, consider an exponent burst size contribution to plaque wavefront velocity (W) of 0.1. That is, cpB0.1. In determining Lcopt , this would be equivalent to setting Xc U to 0.1 in Eq. (17). If U ¼ 1/2 (as with Eqs. (1), (2), and (7)), then Xc ¼ 0.2 ¼ 0.1 2 ¼ W/U and Lcopt ¼ 12:5 min (by Eq. (18)) given an eclipse of 10 min. Plaque fecundity (F), by contrast, is a function of plaque volume where volume pc3 such that FpB1.3 when cpB0.1. That is, XF Y ¼ 1+(3 0.1) in Eq. (14), where the 1 represents the wavefront velocity-independent contribution of burst size to plaque fecundity. If Y ¼ 3/2 (Eqs. (12) and (13)), then XF ¼ 1.3 2/3 ¼ 0.87 and therefore LFopt ¼ 75 min (by Eq. 15) given E ¼ 10 min. Thus, a burst size contribution to plaque size will have a substantial impact on that latent period which maximizes plaque fecundity (LFopt ), but a more modest impact on that latent period which maximizes plaque wavefront velocity (Lcopt ). We can generalize the above calculations as XF Y ¼ 1+(3 W) where W is the exponent over burst size’s contribution to c, i.e. such that FpB1+(3 W) with cpBW and with the 1 coming from the direct contribution of burst size to F. Therefore, in defining LFopt via Eq. (15), X F ¼ ð1 þ 3 W Þ=ð3 UÞ,
(20)
where U, as above, is the opposite of the exponent over latent period (L) in defining c (see Appendix B for further consideration of Eq. (20)). Alternatively, in defining Lcopt , X c ¼ ð0 þ W Þ=U ¼ W =U.
(21)
These values produced by Eqs. (20) and (21) can be plugged into Eqs. (15) or (18), respectively, to determine LFopt or Lcopt , which we do in generating Eqs. (16) and (19) as well as Fig. 4. Our conclusion is that a positive contribution of burst size to plaque wavefront velocity should have a substantially larger positive impact on LFopt than on Lcopt (Fig. 4). Appendix B. Impact of changes in plaque geometry on latent period optima The volume, V, of a spherical plaque, of radius r, is equal to 4 p r3/3. This calculation comes with a caveat,
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however, that plaques initiated near the air or hard-agar interface will be to some degree truncated. One way of addressing this truncation is by dividing by some amount, e.g. an otherwise spherical plaque displaying maximal truncation would have a volume of 2 p r3/3 (that is, half a sphere) and the average volume might be assumed to reside somewhere between 2 p r3/3 and 4 p r3/3 (i.e. see Kaplan et al., 1981), with the position on this spectrum depending on the magnitude of plaque radius (r) relative to soft agar thickness (h). A problem with this approach is that the constant employed (ranging from 2/3 to 4/3) will in fact change as a function of r. Nevertheless, the spherical geometry of the plaque is retained, which is necessary for Eq. (20) to remain valid. The above approximation works reasonably well so long as rph/2, where h is the thickness of a soft–agar overlay.
Prior to this point a plaque is not sufficiently large for it to be doubly truncated, i.e. truncated at both the air and the hard-agar interface. For r4h, on the other hand, all plaques will be expected to be doubly truncated. For h/2orph, therefore, we can expect the number of plaques which are doubly truncated to increase as r increases. When double truncation occurs, then plaque geometry is less spherical. For example, the volume of an otherwise spherical plaque, initiated halfway between the air and hard-agar interfaces, is described by Z
þh=2
p ðr2 y2 Þ dy ¼ h p r2 h3 =12.
V¼
Since h is a constant, it becomes increasingly small relative to r as r becomes larger. As a consequence,
20
50.0 Plaque Volume (h3 units)
untruncated
Plaque Volume (h3 units)
(22)
h=2
15 cylinder
10
5 average sphere
0
untruncated
10.0 5.0
1.0 0.5
average sphere cylinder
0.1 0.0
1.0 2.0 Plaque Radius (r in h units)
0.1
3.0
3
1.0 Plaque Radius (r in h units)
50.0 Plaque Volume (h3 units)
2
average sphere
Plaque Volume (h3 units)
untruncated∗3/4
1
0
untruncated∗3/4
10.0 5.0
1.0 0.5
average sphere
0.1 0.0
0.2 0.4 0.6 0.8 Plaque Radius (r in h units)
1.0
0.3 0.4 0.5 0.7 1.0 2.0 Plaque Radius (r in h units)
3.0
Fig. 5. Plaque volumes calculated by various methods (soft agar depth equals h). Four lines are shown: (i) volume calculated as an untruncated sphere (dash–dot line, labeled ‘‘untruncated’’), (ii) volume calculated as a cylinder (solid line, labeled ‘‘cylinder’’; this curve is shown in panels A and B but not in panels C and D), (iii) volume calculated as a truncated sphere (dash–dot–dot line, unlabeled; this curve in panel A completely overlaps that labeled ‘‘cylinder’’, is unobscured in panel B only in the lower portion of the graph, and is not shown in panels C and D), and (iv) volume calculated as the average volume of spheres (potentially truncated) initiated at a series of points (101 in total, evenly spaced) starting at the soft agar–hard agar interface and ending at the soft agar–air interface (dotted line, labeled ‘‘average sphere’’). The latter is the best approximation of average plaque volume of the four approaches. Especially in panel B, note that the ‘‘average sphere’’ curve is at first approximated by the ‘‘untruncated’’ sphere curve and then by the ‘‘cylinder’’ curve, with the transition occurring when the plaque’s radius is approximately three-quarters the top agar thickness (vertical line, panel B; that is, when h/2orph). For 4 ml of soft agar poured into a 100 mm Petri dish, that works out to a plaque diameter of only approximately 0.8 mm, which, though small, is a good approximation of the plaque sizes studied by Kaplan et al. (1981). Panels C and D compare the ‘‘average sphere’’ with a 3/4 ‘‘untruncated’’ sphere (V ¼ p r3) with only these two curves shown.
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V ¼ h p r2h3/12-h p r2 as r-N, where the expression h p r2 is explicitly that of the volume of a cylinder. For plaques, r is a function of c. With a cylindrical plaque, the exponent over wavefront velocity in Eqs. (10) and (11) thus becomes a 2 (the exponent 2 in h p r2) rather than a 3 (the exponent 3 in 4 p r3/3). This change automatically increases the relative importance of burst size to plaque fecundity. That is, as derived from Eq. (20), ¼ ð1 þ 2 W Þ=ð2 UÞ, X cylinder F
(23)
X sphere ¼ ð1 þ 3 W Þ=ð3 UÞ, F
(24)
and 4X sphere X cylinder F F
(25)
for all non-negative values of W and positive values of U. Per Eq. (15), as XF becomes larger then so too does LFopt . Thus, the latent period optimum for plaque fecundity would be expected to increase for a change in plaque geometry from sphere to cylinder. Caution is advised, however. For U ¼ 1/2 (that is, for cpL1/2), modeling a plaque as a cylinder rather than as a truncated sphere would result in X cylinder ¼ ð1 þ 2W ÞX1 F (for WX0) and therefore LFopt ¼ 1 (Eq. (15)). Thus, on the one hand, for small plaques which are not doubly truncated and therefore which have a geometry that is predominantly spherical (such that 2 p r3/3oVo4 p r3/3), then Eq. (20) predicts that LFopt will be substantially larger than Lcopt (Appendix A and Fig. 4). On the other hand, for larger plaques, ones, which are substantially truncated at both interfaces (such that V-h p r2), then LFopt is even larger, though perhaps unrealistically large (i.e. ¼ N). Splitting the difference, for intermediate levels of truncation, where a mixture of singly and doubly truncated plaques exist, then we would predict a value for LFopt which is intermediate between these two extremes. See Fig. 5 for graphical consideration of these assertions.
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