The extracellular search at its most basic consists of a process of virion diffusion. Such diffusion, due to the comparatively small size of phages, occurs at a rate that is substantially greater than the diffusion of free-floating bacteria. As a result, phage extracellular movement towards an idealized bacterium can be described [24] (p. 104) as “simple diffusion to a single sphere”. The likelihood of phage-bacterium collision, even in an environment lacking in spatial structure, thus is a function of phage diffusion much more so than the diffusion of target bacteria. Given the substantially larger size of bacteria relative to phages, the likelihood of phage-bacterium encounter is governed by bacterial target size much more so than phage diameter [35]. Free phages thus can be considered to rapidly diffuse among relatively large and stationary bacterial targets (Figure 1).

Rates of virus diffusion along with the size of target bacteria are not the only factors affecting rates of encounter between these entities, as so too do their environmental densities. For instance, and as considered in some detail by Murray and Jackson [24], the specific rate of virus adsorption is a function of both the size as well as number of adsorbable targets. Bacteria thus represent smaller targets relative to protozoa and as a consequence individual viruses are less likely to encounter individual bacteria in comparison to individual protozoa. Protozoa, however, tend to display lower population densities than do bacteria, resulting in lower rates of virus encounter with protozoa despite the latter’s larger individual sizes. These lower rates of encounter can have the effect of keeping prey densities below ‘winner’ concentrations (Section 1) despite larger target sizes for individual prey organisms.

Ioannou et al. [36] provide both theory and experimental evidence that, in light of predation, the costs to prey of increasing in size—and therefore becoming more absolutely visible to predators—may be offset by prey decrease in abundance. In particular, larval prey that are present at lower densities will be relatively less visible due to, on average, their existing at greater distances from their three‑spined stickleback predator (a type of fish), just as bacteria existing at lower densities too, on average, are present at greater distances from individual phages relative to bacteria found in higher density bacterial populations. Consistent with these considerations, in subsequent sections I suggest that bacterial existence as arrangements or as microcolonies can result in greater bacterial vulnerability to phages due to resulting increases in overall target size. I suggest in addition that such vulnerabilities may be avoided at least in part when bacterial populations are relatively rare, that is, should bacteria enter into what have been described, for free bacteria, as numerical refuges [28,29].

The higher the density at which phage virions are present within environments then the more likely that a given bacterium will encounter a phage, potentially resulting in bacterial conversion from phage uninfected to phage infected. Thus,

where N_t is the density of bacteria that are not phage infected at the end of some interval (t), N₀ the density of uninfected bacteria at the beginning of that interval, P is the density of free phages (that is, phages that are both unadsorbed and no longer associated with their parental infection), and kPt represents an actual multiplicity of phage infection [9,19,25,37], that is, MOI_actual as defined by Kasman et al. [38]. The phage adsorption rate constant, k, is the probability that a single phage within a specified volume will encounter and then adsorb a single bacterium. This value is based in part on the rate of phage diffusion along with the size of bacterial targets (Section 2.1.1). Note in this equation that phage densities are presumed to remain constant over the course of the interval, t, a situation that may be readily approximated particularly when bacterial densities are low [39].

The more bacteria that are present within a given environment then the more likely that a specific phage will encounter some bacterium, such that,

where P₀ is the initial phage density and P_t is that density after time, t. This equation in particular describes the loss of free phages as a function of bacterial adsorption. Substantial declines in phage titers will occur due to bacterial adsorption, however, only if bacterial densities are relatively high or t is relatively large. Consequently, and as is true also with Equation (1), for this study I employ the simplifying assumption that phage densities do not vary over time. Operationally, this means that I am placing greater emphasis on consideration of bacterial vulnerability to phages than I am on the dynamics of phage generation and loss.

The likelihood of a bacterium encountering a phage, as indicated in Equation (1), is kPt, where k is a function in part of the bacterium’s target size [24,35]. If bacteria form into arrangements, then the likelihood that a specific bacterium encounters a phage may be lower due to partial shading of bacteria by other bacteria [42] or, alternatively, because of shading that results from bacterial association with surfaces. To reflect these issues, I will use the term to describe reductions in phage adsorption rates to bacteria that stem from shading, such that < k. Note that the umlaut’s intention is to imply a description of properties associated with bacterial arrangements, with the double dots literally suggestive of a diplococcus.

The rate of phage adsorption to a bacterial arrangement can be described as nP, where n is the number of bacteria making up an individual arrangement. That is, the target size of an arrangement increases by a factor of n relative to free bacteria while at the same time decreases by a factor of /k. The increase due to n, however, likely is greater than the decrease described by /k, at least so long as arrangements are not sequestered within phage-excluding volumes such as (perhaps) defects in the glass walls of chemostats [29]. Larger arrangements, in other words, almost inevitably will tend to serve as larger targets for phage encounter than will either individual bacteria or smaller arrangements.

Generally it is the diameter or ‘breadth’ of bacterial targets that is crucial to determining viral contact rates [24,35]. For example, the target size of paired, spherical bacteria (diplococci) will range between ~1.26 (= 2^1/3 = n), which is the increased diameter of a two-fold larger volume, and approximately two (n) times larger than the target size of individual cocci. These values in other words range from where shading is substantial (1.26 times) to where shading instead is minimal (~2 times; for illustration, see Figure 2). Given diversity in arrangement shape it is clear that using arrangement diameter as a proxy for target size is a simplification, though one which I retain both for the sake of mathematical convenience and because assuming that targets are spherical may be the most reasonable of default assumptions. Clearly though, and as indicated in the above calculation (Figure 2), surface area (as equivalent to the “~2 times” calculation) provides a more intuitive perspective on target size and particularly so given non-spherical as well as relatively immobile targets. The larger and more important point, however, is that in terms of target size, arrangements should be inherently more vulnerable to phage encounter than individual bacteria.

To visualize the impact of forming into arrangements, with shading affecting phage adsorption rates, compare Equation (1) with

Here A stands for arrangement and A_t is the number of arrangements that have not been phage adsorbed over an interval, t, given a constant phage density, P. So long as n > k holds, then N_t/N₀ > A_t/A₀. That is, fewer arrangements will remain fractionally unadsorbed (A_t/A₀) than would individual, free bacteria (N_t/N₀ ), holding bacterial size and adsorption susceptibility otherwise constant. Here nPt is equivalent to MOI_actual for arrangements. Note though that it is my preference to instead use the term multiplicity of adsorption, i.e., MOA, rather than multiplicity of infection because while an arrangement can be wholly adsorbed by a phage, subsequent infection of the whole arrangement is a more complicated process versus the infection of individual phage-adsorbed bacteria.

A complementary perspective on the above assertion—that is, that fewer arrangements will remain unadsorbed by phages relative to free bacteria, N_t/N₀ > A_t/A₀—is that MOA for arrangements can be up to n-fold higher than that for individual cells. A quantity that I will call MOA_input (M) can, after Kasman et al. [38], be set equal to the density of phages divided by the density of phage targets. The density of arrangements (A₀), as phage targets, is expected to be n-fold lower than that of free bacteria, i.e., A₀ = N₀/n, assuming a constancy in both cell size and total species biomass [36,43]. Holding phage numbers constant, then M for arrangements (M_A) is expected to be n-fold higher than M for free bacteria (M_N), since M_A = P/(N₀/n) whereas M_N = P/N₀. The fraction of targets expected to remain unabsorbed, in turn, is readily calculated as e^-M, which is the frequency of the zero category—bacteria (N_t/N₀) or arrangements (A_t/A₀) experiencing no phage adsorption—given a Poisson distribution of phages adsorbing to targets. The larger M then the smaller the fraction of cells or arrangements remaining unadsorbed, and therefore N_t/N₀ > A_t/A₀ if M_A > M_N. More precisely, we can consider instead MOA_actual, which are M_A = nPt versus M_N=kPt. With M defined in this manner, then the fraction of phage targets expected to remain unadsorbed is equal to (= A_t/A₀) and e^-kPt (= N_t/N₀), respectively, which are restatements of Equations (3) and (1), respectively. Note that < e^-kPt if as expected nPt > kPt, implying that A_t/A₀ < N_t/N₀.

These considerations come with the caveat that increases in the likelihood of arrangement adsorption that occur as a function of n, that is, as n contributes to arrangement diameter and therefore to target size, may be slowed to the extent that adsorption rates to the individual cells making up an arrangement, , also may decline as n increases. It may be harder, that is, for environmental phages on average to encounter individual bacteria that are found within larger arrangements (such as a large microcolony) versus individual bacteria that are found in smaller arrangements (such as a diplococcus). Arrangement vulnerability to adsorption, given this tendency, therefore might increase less rapidly as a function of the number of bacteria that they contain.

The expression nPt should adequately describe the likelihood of arrangement encounter with a phage. Further, nPt defines the actual multiplicity, a.k.a., multiplicity of adsorption, of a bacterial arrangement with the value specifying inefficiencies in this phage adsorption process in comparison to free bacteria. In addition, phage adsorption to arrangements is not identical to phage adsorption to free bacteria because only a fraction of the number of bacteria making up an arrangement become initially phage adsorbed (i.e., 1/n) rather than all of the bacteria making up a free bacterium (1/1). Furthermore, with bacterial arrangements an initial (“primary”) phage adsorption could give rise to a variety of subsequent outcomes including infection of only the adsorbed bacterium, subsequent infection of a fraction of the bacteria found within the arrangement, or indeed subsequent infection of all of the bacteria making up an arrangement. The latter result, of course, is the more costly of outcomes to the affected bacteria, just as it is in terms of prey aggregation more generally [36]. The result is that an additional assumption must be made to argue that the vulnerability of bacterial arrangements to phages can be greater than that seen for individual bacteria. This assumption, in particular, is that the efficiency of phage propagation among bacteria found within arrangements must be greater than that which can be sustained among free bacteria.

When phage densities are higher, then the likelihood of phage adsorption of a given bacterium also should be higher, as described by Equation (1). Importantly, then, the density of phages immediately surrounding a lysing bacterium, that is, as made up predominantly of those phages released from that bacterium, will be the highest phage densities that can be readily attained within a given environment. The phages released in this burst will then diffuse outward, declining in density as they do. The result should be a higher rate of phage adsorption to any susceptible bacterium found within the immediate vicinity of a lysing bacterium, but lower rates at increasing distances (assuming that phage intrinsic adsorption ability does not substantially increase over the course of environmental diffusion).

While free bacteria can randomly find themselves in the vicinity of a lysing bacterium, bacteria that are found in arrangements can be spatially constrained to that vicinity. The result is a higher likelihood of phage infection of other bacteria found within the same phage-infected arrangement than to other environmental bacteria (Figure 3). This argument is similar to an observation made by Babic et al. [44] that transfer of conjugative transposons among bacteria found in arrangements (chains) too can be quite efficient and this is for similar reasons, i.e., a constraining of bacterial location in arrangements to within the vicinity of agents infecting the same arrangement (p. 1): “Since many bacterial species grow naturally in chains, this intrachain transfer is likely a common mechanism for accelerating the spread of conjugative elements within microbial communities.”

Analogously, greater bacterial densities found within bacterial arrangement can be viewed as possessing a higher “mass” relative to that associated with planktonic bacteria, where by “mass”—potentially confusingly—I am referring to immediately local densities of bacteria. This “mass” of bacteria within a bacterial arrangement would be more likely to exceed a “critical” level, such that phage propagation can be sustained, than may be achieved by an equivalent number of more locally dilute, free bacteria. This perspective is just as one can consider for nuclear fission, i.e., radioactive decay, and associated chain reactions [45], which occurs more rapidly given higher “fuel” densities. Indeed, the idea of a “critical mass” can be directly equated with phage proliferation thresholds, that is, those bacterial densities at which rates of phage population growth are perfectly balanced by rates of phage loss [19,25]. Each is a description of target densities (atoms or bacteria) that can be sufficiently high that collision with targets (by neutrons or phages), in combination with subsequent proliferation (via fission or infection), balances any losses that can occur due either to interactions with nontargets or movement away from the focus “mass”.

A target bacterium that is found in the immediate vicinity of a phage, in other words, displays a much higher local density from the perspective of that phage than may be case for bacteria that are randomly dispersed throughout an environment. A local concentration of bacterial “mass” thus can result in a high propensity for bacterial adsorption by phages that have been generated within the same “mass”. See Abedon [34], by contrast, for illustration of the relatively low propensity for phages to randomly encounter free-floating bacteria found within fluid environments; see too Hagens and Loessner [46] as well as Goodridge [47]. As default assumptions, therefore, we can view bacterial arrangements as both larger targets for phage adsorption than individual bacteria and as locally higher bacterial densities, densities that may be better able to support local phage propagation and population growth than more diffuse populations of free bacteria. Consistent with the analogy with nuclear fission, which can be controlled by the insertion of neutron-absorbing substances, we can question the efficiency of phage acquisition and then infection of seemingly adjacent bacteria, e.g., perhaps as may be separated by extracellular matrix.

Notwithstanding proposed tendencies for phages to more readily acquire bacteria that are within their immediate vicinity, it is as noted possible for inefficiencies to exist in the sequential phage infection of bacteria co-occupying the same arrangements. To incorporate such inefficiencies into models of arrangement vulnerability to phages, I employ the term, . This represents the number of bacteria within an arrangement that will be lost, on average, as a consequence of phage adsorption of a single bacterium within that arrangement. This number, , and can range up to the total number of bacteria making up an arrangement (n). Another way of viewing , however, is that > 1 implies a phage reproductive number within a bacterial arrangement that is greater than 0 such that some degree of phage propagation within an arrangement occurs along with consequent bacterial death. If insufficient phage release from infected bacteria and/or insufficient subsequent bacterial infection occurs within an arrangement, or subsequent infections don’t happen fast enough, then complete eradication of an arrangement by an infecting phage may not happen, such that < n.

These ideas can be expressed as,

Where (1 - ) is the fraction of arrangements that become phage adsorbed over some interval, t, and is the number of bacteria per arrangement that are lost to this adsorption (assuming, for simplicity, that is independent of the actual multiplicity of phage adsorption to a given arrangement). Were = 1, then though arrangements are more likely to be adsorbed than free cells, nevertheless no more bacteria would be lost per arrangement adsorption. Indeed, to the extent that the initial bacterial infection is less likely, that is, given < k along with = 1, then overall existence as an arrangement could result in less vulnerability to phages rather than more. Such a situation would occur, for example, were phage infections abortive or perhaps could result instead were infections substantially reduced in burst size or extended in latent period such that phage propagation through an arrangement were substantially impaired. Alternatively, the equality = n would imply complete arrangement loss following each phage adsorption of an arrangement. The parameter, , is thus a description of arrangement vulnerability to phages post-adsorption, ranging from minimal ( = 1, or even = 0) to maximal ( = n ). A visual summary of the models represented by Equations (1) and (4) is presented in Figure 4.

Equation (4) is less applicable given higher levels of phage adsorption, as can occur over longer periods. This is because, as noted, Equation (4) fails to take into account the impact, on the overall fraction of bacteria that are lost, of multiple phage adsorptions of individual arrangements. That is, for instance, we might have an expectation of greater bacterial loss with greater levels of arrangement adsorption by environmental phages under conditions where otherwise < n, but Equation (4) does not reflect that if one adsorbing phage fails to clear an arrangement then perhaps more than one adsorbing phage will, with greater likelihood, succeed in doing so. One way to address this concern is simply by setting = n, that is, an assumption of complete arrangement loss per primary phage adsorption.

Alternatively, one can limit one’s considerations to circumstances in which phage multiplicities of adsorption to arrangements are relatively low (<<1). In such cases the equation can be simplified as

which compares, for a free bacterium, with,

Note that N_t (as defined by Equation (5)) is greater than N_t (as defined by Equation (6)) when > k/. That is, when this inequality holds then bacteria found within bacterial arrangements are more vulnerable to phages than are phage-susceptible bacteria that are “free”. In words: Bacteria found within arrangements are more susceptible to phage attack if losses due to existence within a phage‑adsorbed arrangement, , are greater than increases in individual bacterial vulnerability to “primary” adsorptions that come from not existing within an arrangement (that is, k/ where k > ).

If a bacterium is half as likely to be subject to primary phage adsorption when found within an arrangement (/k = 0.5 such that k/ = 2) but each primary adsorption results in the loss of ten bacteria ( = 10), then the resulting 10 > 2 would imply that arrangements are more vulnerable to phages than are free bacteria. Indeed, in this example bacteria found within arrangements would be five times more vulnerable. Note further that if arrangements can reduce to 1, then arrangements will be expected to display lower vulnerability to phages than free bacteria so long as individual bacteria found within arrangements are less vulnerable to primary phage adsorptions than free bacteria (i.e., again, such that k > ). This is an observation that could very well explain the utility of abortive infection systems to bacteria, i.e., phage-resistance mechanisms in which both adsorbed bacteria and adsorbing phages die [30]. Alternatively, for = 0, which is the case given successful bacterial display of, for example, antibacterial restriction-modification systems, then biologically the inequality no longer holds, i.e., 0 > k/. Logically, though, in this case arrangements should be no more or less vulnerable to phages than free bacteria since, in fact, neither would be vulnerable. See Table 1, under “Higher” phage densities for summary. Note, though, that calculations relevant to the “Lower” phage densities portion of the Table, i.e., Equation (7), are not discussed until Sections 2.3.3 and 2.3.4.

Fitness advantages accrued by bacteria from living within groups generally can result from increased short-term growth rates, greater long-term rates of reproduction, increased population resistance to extinction, or greater competitive ability in terms of more effectively sustaining population densities. The latter might be accomplished by gaining better access to nutrients. Adherence to surfaces in conjunction with subsequent growth as a microcolony, for instance, can retain a bacterial population within the vicinity of key resources, such as flowing water [48].

Continued association of cells following division similarly may allow for more effective penetration into resource-supplying substrates, i.e., burrowing, particularly as seen among filamentous microorganisms growing in spatially structured environments such as soils. Aggregated cells also will tend to have larger collective activity domains [49]. This may be beneficial to bacteria by allowing better retention of extracellular regulatory molecules, particularly such as towards quorum sensing [1,50], or for the concentrating of other beneficial extracellular factors such as exoenzymes or exotoxins. Reduced levels of sharing with unrelated organisms of any molecules generated by these extracellular factors may be possible given bacteria association within arrangements and this may be so simply given greater densities of those cells within a given microvolume.

Growing in a single location can allow for more effective interspecific interactions [49] including crossfeeding [51] as well as development of closely spatially integrated microbial consortia [52,53]. Such associations might be enhanced by cells that grow to higher densities within a specific location, including as cell arrangements or instead as cells that simply fail to disperse and thereby form into microcolonies (or, instead, simply don’t disperse very far). More generally, the development of favorable physiochemical gradients might be more readily achieved when cells are living within groups consisting at least in part of related individuals, i.e., as consistent with the above-noted idea of cellular aggregates possessing larger collective activity domains versus bacteria that are not found in association with related bacteria. The result may be a higher potential to contribute to synergistic interspecific interactions. Indeed, “Any transfer between two such organisms will be immensely inefficient unless the concentrations of cells and substrates are high… Such interactions can take place between groups of bacteria where the size of the group means an enlarged activity domain and the retention of reactants in the vicinity of the group.” [49] (pp. 56 and 58).

Bacteria can display movement when found within disfavorable environments, such as can be associated with planktonic or actively dispersing cells, and then cease such movement, such as through adherence to a surface, when found within a favorable environment. This is a behavior that, minimally, is equivalent to the concept of orthokinesis, that is, where the speed of an organism is modified by external cues or, indeed, thigmokinesis, where speed is modified by degree of physical contact with something else. Included among favorable environments and external cues can be other microbial species from which a given bacterium can derive benefits. Association with other microorganisms on a surface, rather than being stable, can as a consequence of thigmokinesis instead be associated with the invasion of later-arriving organisms. The result can be an interspecific competition for space or resources that can put a premium on competitive ability, which in turn may benefit from more effective arrangement or microcolony formation [50,54,55]. Full or partial replacement of one microbial species with another on surfaces more generally is an example of ecological succession, as has been documented particularly well in terms of dental biofilms [49,56].

Cellular clones growing in a single location have the potential to interact with each other cooperatively towards mutual benefit and do so in ways that are less available to free bacteria. This can include elevation of colonies into more nutrient-rich microenvironments [48], certain degrees of cellular differentiation and/or physiological specialization, fruiting body formation, and even seemingly intentional cell cannibalism under starvation conditions [1,57]. Perhaps consistent with this idea of bacteria being able to mutually cooperate when living within single-species groups, cells growing as biofilms are known to display an increased resistance to toxic chemicals found in their environments, which clinically can include both antibiotics and disinfectants [3,4]. Increased resistance may also be seen given association other microbial species [58]. Extracellular polymer production itself is often described as a potentially cooperative activity among related, physically associated bacteria [55].

Growth as arrangements may give rise to a greater resistance to engulfment by protozoa, and biofilms otherwise may be more resistant to protozoa-mediated predation than planktonic bacteria [59]. The resulting adaptations, such as “resisting ingestion, by becoming too large or too long… making themselves inaccessible, by growing in aggregates or biofilms”, [41] (p. 3), however, may conflict with avoidance of phages by these same bacteria. Biofilm formation can also serve as a means of immune system resistance during infections by bacterial pathogens and this is due at least in part to interference with the action of phagocytes, though not necessarily solely via a direct blocking of bacterial engulfment [60,61,62]. More generally, utility that comes with being larger, whether individually per cell or instead as a consequence of cell-to-cell associations, is exploited by most non-bacterial cellular organisms including animals, plants, fungi, and protists.

To what degree are bacteria living in arrangements in fact phage susceptible? This question must be addressed particularly since assertions have been made that biofilm formation serves bacteria as an inherently phage-resistant state, as I review elsewhere [9]. Costerton et al. [63] in particular noted that (p. 440), “the gellike state of the predominantly polysaccharide biofilm matrix limits the access of antibacterial agents, such as… bacteriophage… Therefore, biofilm bacteria are substantially protected from… bacteriophage…” Elsewhere [8,9], however, I review the substantial potential, under experimental conditions, for phages to in fact considerably impact biofilms. Phages may not be well equipped to drive biofilm bacteria completely to extinction in the course of propagating on those bacteria, and phages that specialize in targeting biofilm bacteria may not always be highly prevalent in environments. Nevertheless, there is no evidence that I know of that would appear to indicate that biofilms are not relatively susceptible, or in many cases even highly susceptible to exploitation by phages.

As with the phage potential to clear bacterial biofilms, evidence also exists that phages can propagate through bacterial arrangements. Barron et al. [64], for example, state that “the phage released from a single coccus may infect other cocci in that chain.” The evidence supporting that claim is an observation by Friend and Slade [65]. They found that one-step growth curves on a group A streptococcal strain in fact were two-step in practice, implying an initial arrangement adsorption followed by post-burst adsorption within the same arrangement; see also earlier work by Kjem, 1958 and 1964, as cited by Friend and Slade, as well as work by Fischetti et al. [66]. In the Friend and Slade study, two-step curves were then reduced to one-step curves through the separation of streptococci into individual, that is, “free” bacteria using sonication.

The tendency for phage plaques to be clear, particularly in their centers, also can be viewed as an indication of the tendency for phages to prevent phage-sensitive bacteria from propagating in the immediate vicinity of phage bursts. Indeed, plaque formation explicitly occurs within a context of phage penetration into and subsequent clearance of the bacterial microcolonies that make up bacterial lawns [67]. Consistently, Doolittle et al. [68] observed phage propagation within single-species bacterial biofilms, describing the dynamics as plaque-like. Conversely, plaque cloudiness can signify limitations on the ability of phages to clear bacterial microcolonies [9,13,69]. It should not be controversial therefore that phages can propagate to at least some extent into bacterial arrangements or microcolonies. Indeed, in phage therapy it is often assumed that phages can be quite adept at propagating through bacterial biofilms, a process that I have described elsewhere as an ‘active penetration’ [19]. These claims all come with the caveat, however, that such active penetration is not necessarily going to be the case for every combination of phage, bacterial arrangement, and circumstance.

For the sake of mathematical convenience, I consider especially the phage impact on instantaneous bacterial population growth rates rather than other aspects of bacterial fitness in addressing the extent to which phages might affect bacterial arrangements. Focusing on instantaneous growth rates, in particular, greatly simplifies the mathematics while at the same time avoids consideration of the difficult issue of exactly how bacterial arrangements propagate over longer time periods. The conclusions I reach, however, should be qualitatively applicable to other situations.

These considerations of phage impact on the instantaneous population growth of bacteria existing within arrangements can be easily formulated as a differential equation in which changes in bacterial density (N) are considered in terms of rates of cell division (μ) versus declines due to phage adsorption,

which compares with

for free bacteria. Note that μ_A and μ_N are growth rates of bacteria associated with arrangements and free bacteria, respectively.

The expression nPN/n in Equation (7), or simply kPN in Equation (8), describes those bacteria that have been lost from the unadsorbed bacterial pool, N, as a consequence of phage adsorption either of themselves (Equation (7) and Equation (8)) or of the arrangement in which those bacteria are located (Equation (7)). As noted above (i.e., A₀ = N₀/n in Section 2.2.2), the expression N / n as found in Equation (7) is a description of the density of arrangements consisting of n bacteria that are found in the environment in question. Also as above, the parameter describes the number of bacteria that will be lost to phage infection given phage adsorption of an arrangement. Lastly, nP is a description of the per-arrangement rate of bacterial adsorption by phages given an environmental phage density of P.

Per bacterium, Equation (7) thus describes adsorptions that occur at a rate of P and each of those adsorptions results in a loss of bacteria. This compares with the rate of loss of individual bacteria as described by kP in Equation (8). The expression, μN with either subscript, by contrast, describes in both equations the gains in bacterial density that occur as a consequence of bacterial replication. Thus, If P > µ_A , then the bacterial population will experience a net decline in number whereas P < µ_A indicates net gains and P = µ_A defines a steady state. For Equation (8) the equivalent expressions instead are respectively kP > μ_N , kP < μ_N , and kP = μ_N, where P in the latter can be described as an inundation threshold or even phage minimum inhibitory concentration, that is, of bacteria [25]. In the absence of phages, the bacterial population will simply grow at rates specified by μ_A or μ_N.

In Equation (7) the dynamics of microcolony formation are not considered and nor are various other complications such as multiple adsorption by “environmental” phages of individual bacterial arrangements, where by “environmental” I am distinguishing those phages defining P (=environmental) from phages that instead are explicitly propagating through bacterial arrangements. What Equation (7) nevertheless indicates is that the phage impact on bacterial arrangements varies as a function of phage density (P) in combination with the susceptibility of individual bacteria making up an arrangement to phages (). Specifically, the larger P then the more bacteria found within arrangements that are lost to phage infection. For example, twice as many bacteria will be lost per unit time for bacterial arrangements (P from Equation (7)) versus free bacteria (kP from Equation (8)) if P /k = 2.

Alternatively, a total of P/k-fold more arrangement-associated bacteria will be lost versus free bacteria for any given phage density, P. Forming into an arrangement, and thereby incurring costs of additional vulnerability to phages, therefore should be worthwhile to bacteria only to the extent that μ_A, or some other measure of bacterial fitness, increases as a consequence of group living to a larger extent than bacterial fitness decreases as a result of incurring a greater spatial vulnerability to phages, i.e., as described by P /k.

The extreme situations with regard to Equation (7) are as follows: (i) If no phages are present in an environment (P = 0) then there will be no phage-associated cost to arrangement, microcolony, or biofilm formation (in which case no advantage is required from group living to offset costs of phage adsorption) or (ii) if phages are present at effectively infinite densities (P = ∞), or simply sufficiently high densities, then potentially no amount of phage-independent benefit to group living could offset increases in phage-associated vulnerabilities. The latter situation is just as dire for free bacteria, however, i.e., for which = 1 but nonetheless where it is possible for .tif (Equation (7) versus Equation (8)). At a minimum, however, μ_A = P is necessary for bacterial fitness, as measured here in terms of increases in bacterial growth rates, to offset costs due to phage adsorption, and this compares with μ_N = kP for free bacteria; see Abedon and Thomas-Abedon [19] along with references cited, or Abedon [25], for derivation of the latter.

For circumstances in which P > kP, then the growth rate or other measure of the fitness of bacterial arrangements must be greater than that of free bacteria by that amount, e.g., μ_A-μ_N > P - kP =P( - k), to offset increased phage-associated costs that are borne by bacterial arrangements. This fitness improvement, however, need not be substantial unless phage densities (P) are also substantial. Thus, as or increase so too does the potential for phages to block the evolution of bacterial arrangements, but at the same time such increases do not serve as absolute blocks on this evolution. The alternative perspective is that given sufficiently high phage densities—but not too high, as indicated in the previous paragraph—then evolution could tend to favor reductions in even if bacteria otherwise experience benefits from forming into arrangements, that is, reduced formation of arrangements could serve as a bacterial anti-phage strategy. In simple terms, a coccus might encounter a phage approximately half as often as a diplococcus.

A generalization on these considerations is that bacteria that have formed into arrangements will have to avoid, on a per-bacterium basis, equivalent increases in vulnerability to phages in order to partake of whatever net advantages may be associated with forming into arrangements. This reduced vulnerability, furthermore, can result either from existing in environments in which phage densities are low (where phage-independent bacterial fitness “gains” can dominate competitive dynamics) or, alternatively, from greater resistance by individual bacteria to phage attack when phage densities are high, that is, when phage-mediated losses dominate competitive dynamics [30,31]. This idea of predator-independent aspects to fitness dominating prey evolution when predator densities are low (here bacteria and phages, respectively) but predator-resistance dominating when prey densities are high is a standard conclusion from community ecology. It is also seen in phage-bacterial chemostat studies as phage-sensitive bacteria, if they possess a growth-rate advantage to phage-resistant bacteria, out-compete phage-resistant bacteria when phage densities are lower but those same phage-sensitive bacteria are at a competitive disadvantage if phages instead are more numerous [70,71]. See Table 1 for a general summary of fitness expectations for arrangements versus free bacteria given higher versus lower arrangement vulnerability to phages.

Perhaps the simplest approach to reducing arrangement vulnerability to phages would be for those bacteria to exist at lower population densities, population densities, that is, which are insufficient to support phage population growth to levels at which those phages can substantially impact arrangement fitness. In other words, bacteria that are less able to attain “winner” densities within a given environment—as in “kill the winner” (Section 1)—may as a consequence be better able to exploit biofilm niches. This idea is similar to the conclusions of Ioannou et al. [36], as described in Section 2.1.1, who suggest that one means by which prey species can offset the costs of possessing greater individual sizes is by displaying lower population densities such that, as a prey type, they effectively are a greater distance, on average, from their predator and therefore less likely to be desirable to the predator. A major difference between the scenario presented by Ioannou et al. and that of kill the winner, however, is seen in terms of the degree of specialization of the predator species. For phages, specialization can be extreme such that bacteria phage-susceptibility types can “hide” by failing to support phage replication to inundative densities, that is, by not being “winners”. For the visual predators considered by Ioannou et al., by contrast, it is not that prey avoid predation by impacting predator densities but instead that they avoid being consumed by being less visible to predators that already exist at some more or less fixed density.

Biofilms often can occupy only a small fraction of total environmental volumes, such as in aquatic environments. In such circumstances, biofilm bacteria as a consequence may be inherently unable to achieve winner-level densities. This low potential, furthermore, may be particularly the case to the extent that biofilms consist of mixed populations of bacteria rather than monocultures of specific phage susceptibility types, thereby implying even lower densities of individual bacterial phage susceptibility types than available surfaces might maximally hold. Alternatively, biofilms found within highly structured environments, such as undisturbed soils [18], may over time tend to be mostly sequestered within micro-localities away from phages to which they are susceptible. Physically associated clonal groups of bacteria, such as can make up biofilms, in other words, may not be so much inherently resistant to phage attack as either relatively unexposed to specific phages or inherently less able to support the population growth of those phages to inundative densities [9].

An additional possibility may hold and that is that it is particularly those bacteria which, as a matter of luck, are resistant to all phages present within environments that may attain “winner” densities, either free or in association with arrangements. This latter idea is ecologically similar to the domination of animal-rich ecosystems by plants that are resistant to forage by those herbivores which happen to be present within an environment [72]. That is, bottom-up control on bacterial density (nutrient availability) rather than top-down control (predator prevalence) would be expected to operate within ecosystems where predators are lacking whereas a combination of bottom-up and top-down control may hold when differences exist among prey in terms of their resistance to predation [70]. We might therefore predict a surfeit of bacteria existing as arrangements under three distinguishable circumstances: (i) where the abundance of specific phage susceptibility types of bacteria generally is low (potentially top-down control, ‘kill the winner’, and high bacterial diversity); (ii) where phage‑resistant organisms dominate (combination of bottom-up and top-down control, and potentially lower bacterial diversity); or (iii) where phages simply are absent (bottom-up control with the diversity of bacteria therefore determined by factors other than phage-mediated predation).