🔬 Microcolony Lysogeny Gradient Explorer

📖 Background

Temperate phages and the lysis–lysogeny decision

Temperate phages can, upon infecting a bacterial cell, either immediately initiate lysis — killing the host and releasing progeny virions — or establish lysogeny, integrating their genome into the host chromosome as a dormant prophage. This choice, governed by phage regulatory circuits interacting with host physiological signals, is one of the most studied decisions in virology (Erez et al., 2017; Ptashne, 2004).

For phage λ, the archetypal model system, the decision was shown to be probabilistic and influenced by host nutritional state and by the cellular multiplicity of infection (MOI): more co-infecting phage particles in a single cell increase the probability of lysogeny (Ptashne, 2004). Single-cell imaging studies subsequently quantified this relationship directly, showing P(lys) rising from roughly 20% with a single infecting phage to roughly 80% with five co-infecting phages (Zeng et al., 2010).

The arbitrium communication system

A distinct class of lysis–lysogeny decision mechanism was discovered in phages of the SPbeta group infecting Bacillus subtilis. Erez et al. (2017) showed that these phages encode a small signaling peptide — the arbitrium peptide — which is released during infection and accumulates in the environment. Subsequent infecting phages take up the peptide, and at sufficiently high intracellular concentrations it inhibits the transcriptional regulator AimR, blocking expression of the anti-lysogeny gene aimX and tipping the decision toward lysogeny. The system thus functions as a molecular memory of past infections: early infections in a naïve population are predominantly lytic; as the peptide accumulates, later infections become progressively more lysogenic.

Subsequent work by Aframian et al. (2022) demonstrated that signal-producing lysogens — prophages still carrying an intact aimP gene — continuously secrete arbitrium peptide, biasing infecting phages toward lysogeny even without ongoing lytic cycles. In coculture experiments, the number of newly formed lysogens was 17-fold higher in the presence of signal-producing lysogens than signal-null lysogens. This "prophage signaling" mechanism is central to the spatial dynamics modeled here.

The molecular mechanism of the switch was clarified further by Zamora-Caballero et al. (2024), who showed that in phage phi3T the arbitrium system interfaces with the host toxin–antitoxin system MazE–MazF: when AimP concentrations are low (early infection), the phage expresses AimX and YosL, which inactivate MazF and favor lysis; as AimP accumulates and inactivates AimR, MazF activity is restored, promoting lysogeny. This is an inherently threshold-like switch at the molecular level.

Biofilm context and spatial structure

In liquid culture, the factors that influence lysis–lysogeny decisions — host density, MOI, and arbitrium concentration — are approximately uniform across the population. Biofilms and surface-attached microcolonies are fundamentally different: they are spatially structured, with steep gradients in nutrient availability, metabolic activity, cell density, and potentially phage accumulation as a function of depth from the surface (Shivam et al., 2022).

A phage virion arriving at the surface of a microcolony will first encounter actively growing, nutrient-replete peripheral cells. As infection propagates inward, successive layers experience conditions more characteristic of stationary phase: nutrient limitation, reduced growth rate, and — critically — accumulating phage particles and signaling molecules that cannot freely diffuse out. These gradients create a spatial environment in which the factors favoring lysogeny (high MOI, high arbitrium concentration, stationary- phase host physiology) are concentrated in the interior of the structure.

The adaptive logic of lysis vs. lysogeny within a microcolony

The fitness consequences of lytic versus lysogenic infections differ fundamentally within a spatially structured microcolony. Lytic infections at or near the biofilm periphery generate progeny virions that can disperse into the surrounding environment, enabling infection of new host cells in other microcolonies or elsewhere in the habitat. The primary adaptive value of peripheral lytic infection is thus virion dissemination — conquering new territory.

Lysogenic infection, by contrast, offers a different kind of value: incorporation into the physical fabric of the microcolony or biofilm. A phage that successfully lysogenizes a cell within an established biofilm acquires, in effect, a biofilm lifestyle — shelter, structural support, nutrient access mediated by the biofilm matrix — without its lysogenic form having had to build that biofilm from scratch. We hypothesize that this represents a meaningful fitness benefit distinct from anything available to a phage that only ever lyses its hosts.

The transition from predominantly lytic to predominantly lysogenic infection with increasing depth reflects several converging pressures. Deeper cells are more likely to be nutrient- limited and in stationary phase, conditions that favor lysogeny. The effective local MOI tends to increase at depth as released virions accumulate in a spatially restricted environment where outward diffusion is impeded — high MOI strongly promotes lysogeny. There are also diminishing returns from further lytic infections in the interior: nutrient- depleted cells produce smaller burst sizes and may yield non-productive infections, while the probability of multiple simultaneous phage adsorptions increases, wasting virions on cells that are already infected. An early transition to lysogeny in the interior therefore could be favored both by minimizing microcolony or biofilm impact and by declines in the utility of continued lytic infection in a resource-exhausted space.

This tool explores what the resulting spatial gradient in lysogeny probability might look like under different mechanistic assumptions, and which gradient shape is most consistent with the known biology. See the Which Is Realistic? tab for a detailed evaluation of each model against the literature.

🧮 Methodology

Conceptual framework

The explorer treats a microcolony as a series of discrete cell layers numbered from 0 (surface, exposed to the bulk environment) to D−1 (deepest interior layer), where D is the total number of layers set by the global parameter slider. Each layer index d represents a position in the biofilm. A single temperate phage virion arrives at layer 0 and infection propagates inward; the question is: what is the probability P(lys) that an infection event at layer d results in lysogeny rather than lysis?

All models are constrained to share a common surface baseline P_base (typically low, reflecting the lytic-favoring conditions at the biofilm periphery) and a common interior maximum P_max (typically high, reflecting lysogeny-favoring conditions in the nutrient-depleted interior). Only the shape of the transition between these bounds varies across models.

Model equations

Obligately lytic control

A reference model with no lysogeny at any depth. Not biologically interesting for temperate phages, but anchors visual comparisons and represents obligately lytic phage behavior.

Linear model

The null/default model. Lysogeny probability increases by a fixed amount per cell layer. Serves as the baseline against which nonlinear models are compared. The first simulation results (linear lysogeny gradient) used in the associated modeling work correspond to this shape.

Sigmoidal model

A logistic sigmoid scaled to the global bounds, parameterized by midpoint m and steepness k. The scaling ensures the curve still passes through P_base at the surface and P_max at the interior regardless of parameter values. This is the functional form used by Shivam et al. (2022) for the arbitrium response function in liquid culture models.

Threshold / step model

A linear ramp from 0 to 1 over a transition zone of width w centered on cell layer d_step, clamped outside the zone. When w = 0 (minimum slider position), this produces a true discontinuous step. This model most directly represents the argument that abrupt lysis-to-lysogeny transitions are more adaptive than gradual ones, and is most consistent with the optimization-theoretic prediction that the switch from lysis to lysogeny should always be sharp (Shivam et al., 2022).

Power / convex model

A power-law scaling of depth, parameterized by exponent n. When n = 1, this reduces to the linear model. When n < 1 (concave), lysogeny probability rises quickly in outer layers then levels off. When n > 1 (convex), probability remains low across most of the biofilm before rising sharply near the interior — a shape consistent with the strongly nonlinear MOI-dependence of lysogeny reported for phage λ (Ptashne, 2004; Zeng et al., 2010).

Arbitrium-amplified model

Identical in mathematical form to the sigmoidal model but parameterized to represent the steeper, tipping-point dynamics expected when arbitrium peptide accumulation is the dominant driver. The higher steepness reflects: (i) the threshold-like molecular switch architecture of AimR inhibition (Zamora-Caballero et al., 2024); (ii) the positive feedback of prophage signaling, in which existing lysogens continuously secrete peptide, creating local hotspots that recruit further lysogeny in a nucleation-like process (Aframian et al., 2022); and (iii) the optimization-theoretic result that the switch from lysis to lysogeny is always sharp under selective pressure for near-term phage reproduction (Shivam et al., 2022). Displayed with a dashed line to distinguish it visually from the plain sigmoid.

Parameter choices and limitations

🧬 Which Gradient Shape Is Most Biologically Realistic?

None of these models has yet been directly measured in a microcolony or biofilm — spatial gradients of lysogeny probability as a function of depth remain an open empirical question. But the known biology of lysis–lysogeny decision-making provides converging lines of evidence pointing toward a steep, threshold-like transition as the most realistic shape. The following summarizes the argument for each model in turn.

Linear — the null model, not the right answer

A linear gradient is the most parsimonious assumption and a useful computational starting point, but nothing in the known biology of lysis–lysogeny decision-making produces a uniform, proportional increase in lysogeny probability with depth. The known drivers — host physiology, MOI, and chemical signaling — are all nonlinear in character. Linear is appropriate as a baseline for simulation and comparison, but should be treated as a deliberate simplification rather than a biologically realistic representation.

Sigmoidal — a reasonable middle ground, but undersells the biology

A gentle sigmoid is a reasonable phenomenological description of how lysogeny probability responds to a gradually accumulating signal, and it is the functional form used in most mathematical modeling of arbitrium-based decisions in liquid culture (Shivam et al., 2022). The issue is that a gentle sigmoid describes the relationship between signal concentration and lysogenic outcome at the level of a single cell; it does not directly predict what the spatial gradient across biofilm cell layers looks like, because the gradient of the signal itself is likely to be steep and nonlinear. A gentle sigmoid also fails to capture the positive-feedback dynamics that are central to how arbitrium actually works in a structured community.

Threshold / step — mechanistically motivated, supported by optimization theory

The threshold model is well supported by the optimization literature. Shivam et al. (2022) found that for every optimal probability function they computed under near-term fitness maximization, the switch from lysis to lysogeny was always sharp — not gradual. This theoretical prediction aligns with the central argument of spatially explicit biofilm models: that abrupt lysis-to-lysogeny transitions are more adaptive than gradual ones, and that the "unincorporated lysogen" phenomenon — spatially isolated lysogens cut off from the microcolony interior by surrounding lytic infections — can only arise when the transition is sufficiently abrupt that peripheral lytic zones can form before interior lysogeny is complete.

Power / convex — captures MOI accumulation dynamics

The convex power model (n > 1) captures something real: the probability of lysogeny is known to rise steeply with cellular MOI, with P(lys) going from roughly 20% with a single infecting phage to roughly 80% with five co-infecting phages (Zeng et al., 2010). That relationship is strongly accelerating at low MOI and then saturating — a convex function of phage number. As phages penetrate deeper into a biofilm, local MOI at any given cell is expected to rise due to accumulation effects (multiple infection fronts, restricted drainage of released virions). This suggests outer layers see low P(lys) while interior cells encounter a sharp rise once local MOI crosses a critical level — precisely the convex pattern.

Arbitrium-amplified — the most biologically complete model

The arbitrium-amplified steep sigmoid is the most realistic model for biofilms in which the arbitrium system is active, for several converging reasons.

First, the molecular mechanism is itself a switch, not a rheostat. The AimP peptide accumulates intracellularly and at sufficient concentration inactivates AimR, blocking aimX expression — a mutually exclusive regulatory transition (Zamora-Caballero et al., 2024). This architecture is inherently threshold-like at the molecular level.

Second, arbitrium peptide builds up cumulatively with successive infections. The conditioned medium experiments of Erez et al. (2017) showed that media from prior phi3T infections bias subsequent infections toward lysogeny, and this effect accumulates. In a biofilm where phage penetration is progressive and virion diffusion is restricted, interior layers experience an elevated peptide environment that surface layers never did.

Third, Aframian et al. (2022) showed that signal-producing lysogens continuously secrete arbitrium peptide, pushing nearby infecting phages toward lysogeny — a 17-fold increase in newly formed lysogens compared to signal-null conditions. In a structured biofilm, this creates local hotspots of elevated signal around existing lysogens, a nucleation-like positive feedback that converts a smooth spatial gradient into a sharper front: once a few lysogens form at some depth, they amplify the signal locally, recruiting further lysogeny in neighboring cells, reinforcing the spatial boundary rather than allowing it to drift gradually inward.

🧮 Phage Biology, Phage Therapy, and Teaching Calculators