# Spatial Vulnerability: Bacterial Arrangements, Microcolonies, and Biofilms as Responses to Low Rather than High Phage Densities

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Phage Adsorption to Free Bacteria

#### 2.1.1. Phage Movement towards Bacterial Targets

**Figure 1.**Illustration of phage and bacterial contributions to phage adsorption rates. Generally phages are relatively small and bacteria somewhat larger. Since diffusion rates are inversely proportional to particle size, whereas target size is proportional to particle size, the result is that phage diffusion (larger arrows pointing right) is a more important contributor to phage adsorption than is bacterial diffusion (smaller arrows point left) while bacterial target size is more important than phage target size to the likelihood of phage‑bacterial encounter. An approximate doubling of total bacterial size (lower right) consequently affects target size but has little relevant impact on combined diffusion rates. Note that arrow lengths reflect an assumption that phages are one-tenth the diameter of the coccus and one-twentieth the diameter of the diplococcus.

#### 2.1.2. Basic Adsorption Calculations

_{t}is the density of bacteria that are not phage infected at the end of some interval (t), N

_{0}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].

_{0}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.

#### 2.2. Phage Interaction with Bacterial Arrangements

#### 2.2.1. Increased Target Size

^{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.

**Figure 2.**Shading of bacteria by bacteria. Shown is a progression starting with two “free” coccus-shaped bacteria (left) which is followed by a diplococcus displaying some degree of attachment (middle) that in turn is followed by a diplococcus displaying maximal attachment as well as minimized surface-to-volume ratio (right), i.e., existing as a combined-volume sphere of 2

^{1/3}-fold increased radius over an individual cell (see calculation, below). The left-hand lack of arrangement shows no shading whereas the right-hand arrangement shows an approximation of maximal shading for a combined spherical shape. The middle arrangement displays some intermediate degree of shading and therefore some intermediate overall target size between maximal and minimal (holding cell volumes constant). Note that the volume of a sphere, V

_{1}, is equal to . Twice its volume (V

_{2}) therefore is , which as a sphere is equal to . For , then r

_{2 }= 2

^{1/3}r

_{1}. With such shading, then, diameter increases by only 2

^{1/3}= 1.26 fold.

#### 2.2.2. Increased Multiplicity of Adsorption

_{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}/N

_{0 }> A

_{t}/A

_{0}. That is, fewer arrangements will remain fractionally unadsorbed (A

_{t}/A

_{0}) than would individual, free bacteria (N

_{t}/N

_{0}), holding bacterial size and adsorption susceptibility otherwise constant. Here n Pt 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.

_{t}/N

_{0 }> A

_{t}/A

_{0}—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 (A

_{0}), as phage targets, is expected to be n-fold lower than that of free bacteria, i.e., A

_{0}= N

_{0}/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/(N

_{0}/n) whereas M

_{N}= P/N

_{0}. 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}/N

_{0}) or arrangements (A

_{t}/A

_{0}) 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}/N

_{0 }> A

_{t}/A

_{0}if M

_{A}> M

_{N}. More precisely, we can consider instead MOA

_{actual}, which are M

_{A}= n Pt 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}/A

_{0}) and e

^{-kPt}(= N

_{t}/N

_{0}), respectively, which are restatements of Equations (3) and (1), respectively. Note that < e

^{-kPt}if as expected n Pt > kPt, implying that A

_{t}/A

_{0}< N

_{t}/N

_{0}.

#### 2.2.3. Phage Propagation within Arrangements

**Figure 3.**Illustration of the tendency of phages to display biases towards acquisition of locally available bacteria. Here shown to the right is phage acquisition of a bacterium (blue) that is found as part of the same arrangement as a lysing bacterium (red with dashed border). The green arrows represent outwardly diffusing phage progeny released upon bacterial lysis while the shorter, gray arrows illustrate the tendency of those phages that are released immediately adjacent to an uninfected bacterium to encounter that bacterium. Contrasting this second bacterium looming large in the vicinity of an adjacent phage burst, even at a high plankton bacterial density of 10

^{8}per mL, each free bacterium (left) occupies a total environmental volume of 10

^{4}µm

^{3}(1 cm = 10

^{4}µm, meaning that 1 mL = 1 cm

^{3}= 10

^{12}µm

^{3}, where 10

^{12}µm

^{3}/10

^{8}bacteria = 10

^{4}µm

^{3}/bacterium). This density in turn implies an average distance between bacteria of about 10

^{4/3}(i.e., the cube root of 10

^{4}µm

^{3}), or more than 10 µm, which one may compare with a typical bacterium diameter of about 1 µm. Thus, bacteria in arrangements can be not-unreasonably described as having local densities that should encourage phage adsorption with higher likelihood than that seen among planktonic, individual bacteria.

#### 2.2.4. Inefficiencies in Phage Propagation

#### 2.2.5. An Important Special Case

_{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 > ).

**Figure 4.**The model. Parameters include P (density of phages in environment), k (phage adsorption constant), (phage adsorption constant considering reductions due to shading of bacteria by bacteria found within bacterial arrangements), n (number of bacteria found per arrangement), N (bacterial density of overall environment), L (phage latent period, which is the duration of a phage infection), and (number of bacteria per arrangement lost subsequent to phage infection of one cell in the arrangement). Likelihood of phage adsorption of bacterial arrangements is n and density of arrangements within environments is equal to N/n = N

_{0}/n (or indeed n

_{0}and N

_{0}/n

_{0}, respectively, to reflect that n changes as a function of time in the figure). The inequality t ≥ 2L indicates how phage acquisition of bacteria within a bacterial arrangement, according to this model, involves at least two sequential rounds of phage infection. The absence of cells in the lower right is intentional as too is the reduction in cell number to n

_{t}in the lower left. Both of these reductions in cell number, going from middle to bottom, indicate phage-induced bacterial lysis.

**Table 1.**Summary of predictions as a function of phage densities in environments and phage potential to acquire bacteria sequentially within bacterial arrangements (Recall in interpreting the table that the inequality, N

_{t }/ N

_{0 }> A

_{t }/ A

_{0}, implies greater success over time in the face of phage-mediated predation for free bacteria versus bacteria found within arrangements while N

_{t }/ N

_{0 }< A

_{t }/ A

_{0}implies the opposite). Generally, k/ ≥ 1. Calculations relevant to lower phage densities, as found in the bottom portion of the table, are not discussed until Section 2.3.3 and especially section 2.3.4.

Environmental Phage Density (P) | Phage Propagation Ability Through Arrangements ( ) | |||
---|---|---|---|---|

Higher | Lower | |||

Higher(bacterial losses dominate dynamics) | For [lesser or no impediments to phage propagation within arrangements] | For [impediments less than absolute] | For [e.g., abortive infections] | For [e.g., phage restriction] |

Lower(bacterial gains dominate dynamics) | For [which, as P → 0, is more likely] | [assuming phage-independent advantages to arrangement formation, i.e., μ_{A} - μ_{N} > 0, and that μ_{A} - μ_{N} > P( -k)holds, which is likely given both P → 0 and → 0] |

#### 2.3. Utility of Group Living in Light of Phages

#### 2.3.1. Selective Benefits of Living in Arrangements

#### 2.3.2. Susceptibility of Bacterial Arrangements and Microcolonies to Phage Exploitation

#### 2.3.3. Phage-Mediated Costs of Existing as Arrangements

_{A}and μ

_{N}are growth rates of bacteria associated with arrangements and free bacteria, respectively.

_{0}= N

_{0}/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, n P is a description of the per-arrangement rate of bacterial adsorption by phages given an environmental phage density of P.

_{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}.

_{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.

#### 2.3.4. Importance of Reduced Vulnerability to Phages

_{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.

_{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.

#### 2.3.5. Reduced Bacterial Densities as Phage-Resistance Strategy

## 3. Experimental Section

## 4. Conclusions

## Acknowledgments

## Conflict of Interest

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Abedon, S.T.
Spatial Vulnerability: Bacterial Arrangements, Microcolonies, and Biofilms as Responses to Low Rather than High Phage Densities. *Viruses* **2012**, *4*, 663-687.
https://doi.org/10.3390/v4050663

**AMA Style**

Abedon ST.
Spatial Vulnerability: Bacterial Arrangements, Microcolonies, and Biofilms as Responses to Low Rather than High Phage Densities. *Viruses*. 2012; 4(5):663-687.
https://doi.org/10.3390/v4050663

**Chicago/Turabian Style**

Abedon, Stephen T.
2012. "Spatial Vulnerability: Bacterial Arrangements, Microcolonies, and Biofilms as Responses to Low Rather than High Phage Densities" *Viruses* 4, no. 5: 663-687.
https://doi.org/10.3390/v4050663