Nutrient depletion is critical for formation of the lysis pattern

We designed a series of quantitative experiments based on our previously described phage drop assay [37], which allow a spatially propagating bacterial population to encounter phages. Salmonella enterica serovar Typhimurium 14028s and χ phage were inoculated 1 cm apart (Fig 1A) on 0.3% agar plate containing bacterial growth medium [30, 38]. As the bacterial population grew, nutrients were consumed. Due to the low agar concentration, bacteria swam through the matrix and followed the self-generated nutrient gradient via chemotaxis, causing spreading of the bacterial population and the appearance of a swim ring. As the bacterial swim ring expanded, it reached the phage inoculation point. The phages then infected the bacteria and generated a lysis area with low bacterial density in the swim ring (Fig 1B). This experiment gave rise to an intriguing sector-shaped lysis pattern (Fig 1B). Most strikingly, as the bacterial swim ring expanded, the radial boundaries of the lysis area stayed unchanged behind the expanding front, resulting in a frozen or immobilized lysis pattern (Fig 1B, S1 Movie). Once the spreading of the swim ring stopped at the plate wall, the lysis pattern persisted for at least 48 hours.

To understand the formation of this lysis pattern, we constructed a mean-field partial differential equation (PDE) model for the phage-bacteria system (Eqs (1) ~ (4)). Like the previous phage plaque models [31–36], our model depicts the basic processes underlying the proliferation and propagation of phages and bacteria. Namely, the bacteria consume nutrients, divide, and move up the nutrient gradient via chemotaxis-directed swimming motility. Once infected by phages, the bacterium is lysed after a latent period, and releases new phage progeny. Note that the run-and-tumble mechanism of bacterial chemotaxis results in a biased random walk of the bacterial cells up the nutrient gradient. The random walk is expressed in the model as the cell diffusion terms and the bias as the cell drift terms (Eqs (1) and (2)). The diffusion of bacteria characterizes the overall effect of active motility and random tumbling of bacteria. Therefore, diffusion and motility of bacteria will be used interchangeably in the rest of this work. In addition, we incorporated the following new assumptions about phage-bacteria interactions in the model, which are critical elements for generating the sector-shaped lysis pattern (S1 Fig).

1. Nutrient deficiency inhibits phage replication (Fig 1C). Because phage replication in the host bacteria requires energy, it is likely reduced at low nutrient levels.
2. High bacterial density inhibits phage production (Fig 1C). Inhibition of phage attack by quorum sensing signals has been documented in various bacteria, including E. coli [39], Vibrio [40, 41], and Pseudomonas [42], and could be a widespread phenomenon. This inhibition stems from reduction of phage receptors (in E. coli and Vibrio) or other mechanisms. High bacterial density, which induces production of the quorum sensing signal, could hence decrease phage production.

Our model reproduces the lysis pattern observed in the phage drop assay (Fig 1B, S1 Movie) and quantitatively matches the bacterial and phage density profiles throughout different areas of the agar plate (Fig 1D). Both experimental and modeling results displayed the highest bacterial density at the inoculation point (area 4), followed by areas outside the lysis sector (areas 1 and 3), along the radial boundaries of the lysis pattern (area 5), in the middle of the lysis pattern (area 2), and the lowest at the outer edge of the lysis sector (area 6). The predicted phage densities also matched the experimental results, i.e., highest in the middle of the lysis sector and lowest at the edge of the swim ring near the wall of the plate (Fig 1D). It should be noted that the lysis area was not entirely void of bacteria. In both experimental and modeling results, a low density of bacteria remained within the lysis area. In the model, nearly all bacteria in this area are infected bacteria (Fig 1E and 1F). In reality, this subpopulation could also include phage-resistant bacteria, which has not been encompassed in our current model.

The model further reveals local nutrient depletion as the key reason for the lysis pattern to immobilize behind the expanding front of the bacterial swim ring. According to the simulation results, as the swim ring expands, nutrients are depleted within the ring (Fig 1H). Nutrient depletion inhibits both phage production (Fig 1I) and bacterial motility (Fig 1J–1M). Note that phages rely on the infection of motile bacteria to propagate in space, because the passive diffusion of phage particles (D_P~1 μm²h⁻¹) is negligible compared to the “active” diffusion of bacteria resulting from run-and-tumble (D_B~10⁵ μm²h⁻¹). Therefore, inhibition of bacterial motility, especially motility of the infected bacteria (Fig 1L and 1M), also hinders spatial propagation of phages. Together, the inhibition of phage production and propagation due to local nutrient depletion and reduction of bacterial motility results in immobilization of the lysis pattern at the interior of the bacterial swim ring. The lysis pattern only actively grows at the expanding front of the swim ring, where nutrient supply from the unoccupied periphery can support active phage production and propagation (Fig 1I–1M).

The lysis pattern reflects radial projection of phage initiation zone

Interestingly, the angle of the lysis sector decreased in the experiment when phages were inoculated further away from the bacterial inoculation point and vice versa (Fig 2A). This observation was successfully reproduced and explained by our model (Fig 2A). The lysis patterns in these cases approximately reflect the radial projection from the bacterial inoculation point over an approximately 0.7 cm circle centered at the phage inoculation point (Fig 2A, cartoon). This circle roughly corresponds to the model-predicted area that is occupied by phages when nutrients initially get depleted at the phage inoculation point (Fig 2A, 3rd and 4th rows, nutrient depleted to 5% of initial level). We hereby term this area the “phage initiation zone”, which marks the initialization of the steady expansion of the lysis pattern. Specifically, after the phage initiation zone is established, the phage and bacterial densities at the expanding front remain at a steady level throughout the rest of the pattern formation (S2 Fig).

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Fig 2. The lysis pattern reflects the radial projection of the phage initiation zone and is insensitive to extrinsic factors.

Model and experimental results with (a) phages inoculated at different distances from the bacterial inoculation point, (b) phages inoculated with rods of different lengths, (c) different initial nutrient levels, and (d) different initial phage numbers. In (a-d), the phage initiation zones represent the model-predicted areas occupied by phages when the nutrient level at the center of the phage inoculation area drops below a certain threshold (set at 5% of the initial nutrient level in model simulations). Cartoons summarize scenarios leading to different lysis patterns. Black dots: bacterial inoculation point. Solid dark grey and light grey areas: illustration of phage initiation zones. Solid lines: radial boundaries of lysis patterns. Dashed lines: extension from the bacterial inoculation points to the boundaries of the lysis areas.

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Our model further predicts how the phage initiation zone and the projected lysis pattern rely on additional factors, which have all been confirmed by experiments (Fig 2). Firstly, the phage initiation zone is predicted to encompass the original phage inoculation area (Fig 2A and 2B). In the corresponding experiments, when phages were inoculated with rods that were significantly longer than the size of the phage initiation zone during point inoculation, the phage initiation zone became dominated by the rod size, and the lysis pattern roughly reflected the radial projection of the phage inoculation area (Fig 2B). Secondly, the size of the phage initiation zone is predicted to remain roughly the same and result in similar angles of the lysis sector, despite changes in total nutrient concentration (Fig 2C). This prediction was also validated by experiment (Fig 2C). Thirdly, the phage initiation zone is predicted to enlarge and result in larger angles in the lysis sector, as the initial phage particle number increases; again, this was validated by experiment (Fig 2D). This result can be understood in that lowering the initial phage number reduces the number of phages being produced by the time nutrients get depleted at the phage inoculation point and shrinks the phage initiation zone (Fig 2D). When the initial phage number is too low, phages fail to establish the initiation zone and subsequently the lysis sector (Fig 2D, P₀ = 1.5×10²). The model also predicts that the initial bacteria number does not affect the lysis pattern (S3 Fig). Overall, our modeling and experimental results show that the lysis pattern maintains straight radial boundaries (Fig 2) despite changes in the extrinsic factors tested above, i.e., distance between inoculation point, size of inoculation area, overall nutrient level, and initial phage/bacteria number.

Competition between phages and bacteria determines shape of lysis pattern

Although the straight radial boundaries of the lysis pattern are maintained under various external conditions like nutrient level and initial inoculation, our model predicts a significant change in the lysis pattern when intrinsic biological parameters are altered. In the simulation results, promoting the proliferative efficiency of phages (by increasing phage adsorption rate, phage burst size or lysis rate of infected bacteria) causes the lysis pattern to flare out, and decreasing phage proliferation causes the lysis pattern to close up (Fig 3, horizontal axes). Meanwhile, promoting bacterial proliferation causes the lysis pattern to curve inward and close up, and vice versa (Fig 3, vertical axes). To understand this result, note that both bacterial and phage proliferation depend on nutrient (expressed as increasing functions of local nutrient concentration in the model, see Materials and Methods). As previously shown, proliferation of bacteria causes nutrient to be depleted inside the bacterial swim ring (Fig 1H). Therefore, the bacterial proliferation rate at the expanding front determines how fast nutrients get depleted locally. This time further determines angular spreading of phages along the expanding front of the swim ring, because phage proliferation only thrives before local nutrient depletion. Therefore, either stronger phage proliferation or a weaker bacterial proliferation (causing slower nutrient consumption) allows phages to spread in an accelerated fashion as the swim ring expands, resulting in a flared-out lysis pattern. Vice versa, a weaker phage proliferation relative to bacterial proliferation results in a lysis pattern with edges closing inwards. The sensitivity of the predicted lysis pattern to phage and bacterial proliferation suggests that the experimentally observed lysis pattern requires a balance between the proliferation of the bacterial and phage strains tested.

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Fig 3. Competition between bacterial and phage proliferation determines the shape of lysis patterns.

Simulated lysis patterns with various bacterial growth rate constants versus (a) phage adsorption rate constants, (b) phage burst sizes, and (c) phage-induced lysis rate constants.

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Bacterial motility and chemotaxis affect the lysis pattern

We next used the model to investigate how bacterial motility and chemotaxis influence the shape of the lysis pattern. Bacterial motility is reflected by the bacterial diffusion coefficient in the model. A higher cell speed corresponds to a larger diffusion coefficient [43]. Expectedly, a larger diffusion coefficient causes faster expansion of the bacterial swim ring in the model (Fig 4A). The chemotactic efficiency, on the other hand, characterizes the bias of bacterial diffusion. Chemotaxis promotes the directed motility of bacteria in the radial direction due to the nutrient gradient formed by bacterial nutrient consumption (Fig 1H). Consistently, the model predicts that higher chemotactic efficiency expedites expansion of the bacterial swim ring (Fig 4A), because the moving cells at the expanding front can follow the nutrient gradient more efficiently.

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Fig 4. Effects of bacterial motility and chemotaxis on the shape of lysis patterns.

(a) Simulated lysis patterns with various bacterial diffusion coefficients and chemotactic efficiencies. The diffusion coefficient in the model reflects the efficiency of bacterial motility. (b) Illustration of how bacterial diffusion and chemotaxis affect the evolution of lysis patterns. The cartoon illustrates a hypothetical history of expansion and mixing of two bacterial patches that would occur along the expanding front of the swim ring. The dark and light grey patches start with infected and susceptible bacteria, respectively. Because nutrient is depleted behind the expanding front, bacterial expansion along the radial axis only occurs in the outward direction. Strong bacterial diffusion promotes expansion equally in all directions, which enhances mixing between the infected and susceptible bacteria and leads to more effective phage propagation and flare-out of the lysis pattern (top row). In contrast, high chemotactic efficiency promotes expansion only against the nutrient gradient in the radial direction, which effectively parallelizes bacterial motion, reduces their mixing in the angular direction, and causes the lysis pattern to close up (bottom row).

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The effects of bacterial motility and chemotactic efficiency on the lysis pattern, however, are predicted to be exactly opposite to each other. According to the simulation results, increasing the bacterial diffusion coefficient or decreasing the chemotactic efficiency causes the pattern to flare out (Fig 4A). Vice versa, decreasing the bacterial diffusion coefficient or increasing chemotactic efficiency causes the pattern to close up (Fig 4A). The model generated this result because increasing bacterial diffusion coefficient, i.e., increasing bacterial motility, promotes mixing of infected and susceptible bacteria (Fig 4B, top row). Such mixing is critical for spatial propagation of phages and angular expansion of the lysis pattern, because phages cannot move on their own and rely on infected bacteria to spread in space. In contrast, enhancing chemotaxis inhibits such mixing, because it effectively promotes parallel motion of the bacteria along the radial direction towards the high-nutrient area outside the swim ring (Fig 4B, bottom row). Taken together, bacterial motility and chemotaxis need to be in balance to generate a lysis pattern with straight radial boundaries. Collectively, these findings and those from the model in the previous section indicate that bacterial motility and chemotaxis are required to be in balance with bacterial and phage proliferation rate to generate straight radial boundaries in the lysis pattern (S4 Fig).

To test these model predictions, we performed the phage drop assay with strains of S. Typhimurium 14028s containing deletions in two chemoreceptor encoding genes, tar and tsr. Strains with deletions in tar or tsr did not significantly change the lysis pattern, whereas the strain containing a deletion of both genes produced a moderate flare-out of the lysis pattern (S5 Fig). This result is qualitatively consistent with the model prediction that weaker chemotactic efficiency causes a pattern with a wider angle (Fig 4A).

We further tuned bacterial motility through varying the agar density. We only experimented with lower agar densities, because higher agar densities are known to switch the mode of bacterial motility to surface swarming, and the results would not be comparable to those obtained from soft agar. In our experiments, softer agar indeed increased bacterial motility, as the bacterial swim ring took a shorter time to reach the edge of the plate (7 h in 0.2%, 9 h 0.25% agar and 14 h in 0.3% agar). However, we found similar sector-shaped lysis patterns regardless of the agar density (S6A Fig). To understand why higher bacterial motility in softer agar did not change the lysis pattern, it is important to note that lowering the agar density also increases the passive diffusion of small molecules such as nutrients [44, 45]. In our model, when bacterial and nutrient diffusion coefficients are proportionally varied, the sector-shaped lysis pattern is indeed maintained (S6B Fig, diagonal from bottom left to top right). The subtle increase of the angle in the pattern in softer agar was also reproduced by the model (S6B Fig, diagonal from bottom left to top right). It is worth noting that agar density per se is an extrinsic factor. Although agar density affects bacterial motility, an intrinsic property, the effect of the latter on the lysis pattern is canceled by the accompanying changes in nutrient diffusion. These experimental and modeling results further confirm our previous conclusion that the lysis pattern is insensitive to extrinsic factors.

Straight radial boundary of lysis pattern is a telltale sign for extended co-propagation

A closer look at the model results reveals that the straight radial boundaries in the lysis pattern imply co-propagation of bacteria and phages over extended periods (Fig 5). Unlike the sector-shaped pattern with straight radial boundaries, a flared-out or closed-up lysis pattern indicates that one species would outcompete the other during the co-propagation (Fig 5A). For example, the result at the upper right corner of Fig 5A shows a case where phages encircle bacteria and block their further propagation in space. Vice versa, the result at the lower left corner of Fig 5A shows the opposite case where phage propagation is blocked by bacteria. For the less extreme flared-out or closed-up lysis patterns (e.g., Fig 5A, the last column on the 2nd row), one species would eventually encircle and block the other, if the simulation had been run on a larger spatial domain that allow further spatial expansion.

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Fig 5. Co-propagation between phages and bacteria is reflected by the straight radial boundaries of the lysis pattern.

(a) Lysis patterns and corresponding spatial patterns of phages with various bacterial diffusion coefficients and phage adsorption rate constants. Red/grey gradient in the background illustrates the relative rate at which phages encircle bacteria (red) or bacteria encircle phages (grey). Red shade in simulated patterns: density of phages. Grey shade in simulated patterns: density of bacteria. Grey shadowed staircase: potential trajectory of evolutionary arms race on which phages and bacteria maintain balance with each other. Refer to S7 Fig for clearer, separate view of the lysis patterns and phage patterns. (b) Experimental results from pairs of different bacterial species and their cognate phages.

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Remarkably, similar sector-shaped lysis patterns with straight radial boundaries were observed in different bacteria-phage pairs (Fig 5B). First, we performed the phage drop assay using E. coli and phage χ and found a similar sector-shaped lysis pattern (Fig 5B, first row). Since χ is a flagellotropic phage that targets the flagella of the host bacteria [46–48], we further examined whether infection of motile E. coli by a non-flagellotropic phage, λ, would generate a similar lysis pattern. Phage λ was chosen because of an overlapping bacterial host range with χ. Interestingly, similar sector-shaped lysis patterns were observed independent of the utilized phage type (Fig 5B). Combined with the model findings above and earlier results on sensitivity of the pattern to intrinsic parameters, these highly similar sector-shaped lysis patterns indicated that both Salmonella and E. coli are in intrinsic biological balance with both phages, λ and χ.

Bacterial strains and phages

The strains of bacteria and phages used are listed in Table 1.

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Table 1. Biological materials used in this study.

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Media and growth conditions

Salmonella enterica serovar Typhimurium 14028s was grown in MSB at 37°C. MSB is a modified LB medium (1% tryptone, 0.5% yeast extract, and 0.5% NaCl) supplemented with 2 mM MgSO₄ and 2 mM CaCl₂. Escherichia coli strains were grown at 30°C in T-broth containing 1% tryptone and 0.5% NaCl at 30°C.

Construction of mutant strains

The protocol for lambda-Red genetic engineering [58, 59] was followed to make S. Typhimurium mutant lacking tar.

Phage drop assay

Swim plates containing MSB medium (for S. Typhimurium) or T-broth (for E. coli) and 0.3% bacto agar were inoculated with 2.5 μl of a stationary phase bacterial culture in the center of the plate along with 2.5 μl of phage suspension (MOI = 25.4) at a 1 cm distance from the inoculation point. A 2.5 μl spot of 0.85% saline was placed at the same distance from the bacterial inoculation point, opposite from the phage suspension, as a control. Plates were incubated at 37°C (S. Typhimurium) or 30°C (E. coli) for 14 hours. All plates were imaged using the Epson Perfection V370 scanner. Phage drop assays with slight modifications were conducted to test different variables. For the rod-shaped inoculations, sealed glass capillaries of different lengths were immersed in phage suspension and pressed against the soft agar at a distance of 0.5 cm from the bacterial inoculation point. To test the effect of varying inoculation distances, phage suspensions were inoculated at 0.5, 1.0, 1.5, 2.0, and 2.5 cm from the bacterial inoculation point. For altered nutrient concentration experiments, the initial concentrations of tryptone and yeast extract were adjusted to be 0.05, 0.1, 0.5, 0.75, or 2.0 times of the regular nutrient concentration, which is referred to as a concentration of 1. To evaluate the effect of phage number, the initial phage stock was serially diluted ten-fold and then spotted on the plate. In experiments conducted with λ phage, the swim plates were supplemented with 10 mM MgSO₄ and 0.2% maltose.

Phage titer

Serial dilutions of the phage stock were made and 100 μl of each dilution was added to host bacterial cells with an OD₆₀₀ of 1.0. Bacteria-phage mixtures were incubated for 10 min at room temperature. Each mix received 4 ml of pre-heated 0.5% soft agar and was then overlaid on LB plates. Plates were incubated at 37°C for 4–6 h. The titer of the phage stock was determined by counting the plaques on the plate that yielded between 20 to 200 plaque forming units and multiplying the number by the dilution factor.

χ phage preparation

Dilutions of phage suspensions mixed with bacteria were plated to achieve confluent lysis as described in the phage titer protocol using 0.35% agar for the overlay. Following formation of plaques, 5 ml of TM buffer (20 mM Tris/HCl [pH = 7.5], 10 mM MgSO₄) was added to each plate and incubated on a shaking platform at 4°C for a minimum of 6 h. The soft agar/buffer mixture was collected, pooled, and bacteria were lysed by adding chloroform to at final concentration of 0.02%. Samples were mixed vigorously for 1 min, transferred to glass tubes, and centrifuged at 10,000 x g for 15 min at room temperature. The supernatant was passed through a 0.45 μm filter and NaCl was added to a final concentration of 4%. The protocol of phage preparation was followed as described in [60]. The final phage stock was stored in TM buffer at 4°C.

Bacteria and phage quantifications

Phage drop assays were conducted as described above. At the 14-hour end point, different areas of the plate were sampled by taking agar plugs using a 10 ml syringe barrel with plunger. Each agar plug was placed in 1 ml of 0.85% saline and incubated at room temperature for 10 min with shaking to allow even mixture of the agar. Serial dilutions of each sample were plated on LB agar plates and incubated at 37°C overnight. For phage quantifications, 100 μl of chloroform was added to each sample. The number of phage particles present in each sample was quantified as described in the phage titer protocol. Densities reported correspond to plaque forming units (for phage) or colony forming units (for bacteria). To compare with model results, the volume densities were converted to area densities, based on 0.5 cm thickness in the agar, i.e., area density (cm^-2) = volume density (CFU/cm³ or PFU/cm³) × 0.5 cm.

Model setup

We constructed a mean-field diffusion-drift-reaction model for the bacteria-phage system. Our model includes four variables: density of susceptible bacteria B(x,t), density of infected bacteria L(x,t), density of phages P(x,t), and nutrient concentration n(x,t). The motility and chemotaxis of bacteria are represented by the Keller-Segel type diffusion and advective terms widely used in the literature [61, 62]. The equations governing the spatiotemporal dynamics of bacteria, phages and nutrient read as Eqs (1) ~ (4).

Susceptible bacteria: (1)

Infected bacteria: (2)

Phages: (3)

Nutrient: (4)

Eqs (1) ~ (4) incorporate the following model assumptions.

1. The division rate of susceptible bacteria follows the Monod rate law [63].
2. Division of the infected bacteria is neglected because they are likely lysed before dividing. But they consume nutrients at the same rate as susceptible bacteria (changing this rate does not affect the qualitative behavior of the model).
3. The phage adsorption rate decreases with increasing bacterial density. This is how New Assumption (ii) in Results is implemented.
4. Multi-adsorption is considered, i.e., phages can be adsorbed onto bacteria that are already infected.
5. Because phage assembly requires energy, we assume that the lysis period elongates as nutrient level decreases. This is how New Assumption (i) in Results is implemented.
6. Because bacterial motility requires energy, it depends on nutrient level. This dependence is reflected by the fraction containing K_v in both the diffusion and chemotaxis terms. This applies to both susceptible and infected bacteria.

The variables and parameters of the model are summarized in Table 2.

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Table 2. Parameters of mathematical model.

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