Optimality of the spontaneous prophage induction rate

Abstract

Lysogens are bacterial cells that have survived after genomically incorporating the DNA of temperate bacteriophages infecting them. If an infection results in lysogeny, the lysogen continues to grow and divide normally, seemingly unaffected by the integrated viral genome known as a prophage. However, the prophage can still have an impact on the host’s phenotype and overall fitness in certain environments. Additionally, the prophage within the lysogen can activate the lytic pathway via spontaneous prophage induction (SPI), killing the lysogen and releasing new progeny phages. These new phages can then lyse or lysogenize other susceptible nonlysogens, thereby impacting the competition between lysogens and nonlysogens. In a scenario with differing growth rates, it is not clear whether SPI would be beneficial or detrimental to the lysogens since it kills the host cell but also attacks nonlysogenic competitors, either lysing or lysogenizing them. Here we study the evolutionary dynamics of a mixture of lysogens and nonlysogens and derive general conditions on SPI rates for lysogens to displace nonlysogens. We show that there exists an optimal SPI rate for bacteriophage λ and explain why it is so low. We also investigate the impact of stochasticity and conclude that even at low cell numbers SPI can still provide an advantage to the lysogens. These results corroborate recent experimental studies showing that lower SPI rates are advantageous for phage-phage competition, and establish theoretical bounds on the SPI rate in terms of ecological and environmental variables associated with lysogens having a competitive advantage over their nonlysogenic counterparts.

Graphical Abstract

1. Introduction

Bacteriophages (or phages for short) are viruses that infect bacterial cells. By infecting bacteria, the phages can replicate themselves using the host’s cellular machinery. The replicated phages can then destroy the host cell and are released into the environment, which is known as the lytic or lysis pathway. Lysis is the typical outcome of infection, but some phages can follow the alternative pathway of lysogeny after infection. Phage λ is a well-known example for such “temperate” phages [1, 3, 14, 16, 18, 26, 31, 33, 40]. In the lysogenic pathway, the phages integrate their viral DNA into the host’s genome. The host cell, now called a lysogen, is then seemingly unaffected and continues to grow and divide normally. Typically this lysogenic state is very stable and the cell remains a lysogen after many cell divisions. Sometimes the lysogen can revert back to the lytic pathway in a process called prophage induction [29]. Prophage induction can be triggered by various environmental factors, such as DNA damage by UV radiation that induces the host’s SOS response, or it can also occur spontaneously in a process termed spontaneous prophage induction or SPI [29]. SPI is likely due to spontaneous accumulation of DNA damage initiating the host’s SOS response during cell replication [11, 27, 29, 32]. This SPI process plays a role in the population biology of temperate phages in mixtures of susceptible cells and lysogens. While modeling has indicated that the temperate phage strategy has benefits over virulence [40], the conditions and limits on when specific SPI rates can be beneficial have been less well explored.

When a bacterial cell converts to a lysogen, its genome contains viral genes, becoming genetically different from the nonlysogens. Consequently, the lysogen and nonlysogen genomes can be subject to natural selection. Expression of integrated viral genes within the lysogen can have phenotypic and fitness effects. For example, lysogens can have increased antibiotic resistance, biofilm formation ability, and virulence [7, 24, 25, 42, 43]. The viral genes could also slow host cell growth because viral protein expression could affect host cell resource availability and thereby host cell division [34, 37] or host cell transcription and translation [17, 23, 41]. Taken together, these facts imply that lysogens could outcompete nonlysogens or vice versa because of their differing growth rate (fitness) in certain environments. From the perspective of phage fitness, the temperate strategy appears to be superior relative to a strictly lytic or less temperate strategy [4, 6, 26, 38, 40]

In addition to any growth rate differences, the process of SPI should additionally affect competition in a mixture of lysogens and nonlysogens [29]. SPI directly causes lysogenic cell death, and thereby the release of new phages that can kill the nonlysogens or convert them to lysogens. A higher SPI rate allows the lysogens to kill the nonlysogens faster, but it is also costly because it directly causes lysogenic cell death.

Evidence for competitive advantage of lysogens over nonlysogens comes from experiments showing that mixtures of lysogens and lysogen-cured bacteria tend towards a state in which the lysogen starts to displace the lysogen-cured strain [8, 13]. This advantage could be caused by the SPI process, differences in growth rates, or both.

SPI can also play a role in the competition between two lysogenic strains, each having different rates of SPI. Berngruber et al experimentally competed two phage strains, the wild-type and the virulent cI857 mutant [5]. The virulent mutant is like the wild-type, but it has a higher post-infection lysis probability as well as a higher SPI rate. The experiments showed that the virulent mutant had an early but transient advantage, but at longer times the temperate phage steadily displaced the virulent one. Thus, for phages with different SPI rates (and post-infection lysogeny rates) the more virulent phage is at a disadvantage in the long run. Although focused on the competition among phages, this study suggests that lysogens with lower SPI rates could outcompete lysogens with higher SPI rates.

Overall, these experimental findings imply that lysogens could use SPI to displace nonlysogens, and that lysogens with lower SPI rates should possess a competitive advantage against other lysogens with higher SPI rates. However, these ideas are not well understood theoretically and under general conditions. At what point, if at all, does a high or low SPI rate become a disadvantage to the lysogens? Is there an optimal SPI rate? How do these relationships arise quantitatively and how are they affected by differences in cell growth rates and other ecological parameters?

In this work, we use replicator dynamics and simulation [30] to gain a theoretical understanding of how SPI influences the competition between lysogens and nonlysogens under general parameter sets, and to identify an optimal SPI rate enabling the lysogens to be naturally selected over the nonlysogens, which is comparable with experimental values. The model is intended to be general for temperate phages, such that it can theoretically describe the long-term competition between any lysogen and its nonlysogen counterpart so long as the lysogen can undergo SPI. We quantitatively compare our model to data for bacteriophage λ infecting E. coli because many parameters for the phage λ - E. coli system have been measured previously [31]. We find that the SPI rate should be as low as possible but still greater than a lower bound which we derive analytically. For phage λ we find that the experimentally measured SPI rate matches the model’s prediction. We also derive expressions to describe how stochasticity affects the competition between nonlysogens and lysogens undergoing SPI.

2. The LUV model

To understand how SPI plays into the natural selection of lysogenic cells over their uninfected counterparts, we developed a model summarized by Equation 1 - Equation 5. This model simulates evolutionary dynamics of lysogens (L), uninfected cells (U) and viruses (V) using replicator equations [30] such that the total cell population size remains constant at K = L+U ≈ 10⁶/mL. The lysogens and uninfected cells grow exponentially in the experimentally reasonable volume of 1 mL, in the same environment, at rates r and g, respectively. We use cell generation times of ≈ 30 minutes per cell [2] as time units. Since the lysogenic genome contains viral genes, there are many ways in which lysogenized cells could be different from nonlysogens. These differences could influence host cell physiology by directly affecting the cell growth rate, thus we assume r and g to be slightly different. The lysogens undergo SPI at a rate σ, and the resulting free viruses, V, infect the nonlysogens U with a success rate α per cell per phage per generation per mL. Infections enter the lysogenic pathway at rate p and enter the lytic pathway at rate 1 − p. The lytic pathway releases b free viruses, which is the phage burst size. The viruses degrade at rate γ on the order of 0.1 per day per phage [12], which converts to γ ≈ 0.001 per generation per phage. We also model superinfections, consisting of free phages infecting lysogenic cells at the same rate α, and disappearing without any additional effects. Overall, our model is somewhat similar to the early and well-known model proposed by Stewart and Levin [40], except that we do not consider resistant viruses separately from lysogens and we enforce constant population size by always removing the cells in excess of K proportionally with their current fraction in the population through the function ϕ. Keeping the total cell number L + U = K fixed limits the cell population size, approximating the periodic resuspensions of long-term evolution or competition experiments [9, 15], while allowing lysogens and uninfected cells to compete.

$$\dot{L}=rL-\sigma L+p\alpha UV-\varphi L$$ (1)

$$\dot{U}=gU-\alpha UV-\varphi U$$ (2)

$$\dot{V}=(1-p)b\alpha UV-\gamma V+b\sigma L-\alpha VL$$ (3)

$$\varphi =\frac{(r-\sigma )L+gU-\alpha UV(1-p)}{K}$$ (4)

$$K=L+U$$ (5)

The variables and parameter values for the phage λ - E. coli system are summarized in detail in Table 1. The probability of lysogeny is known to be host cell volume- and MOI-dependent, ranging from 20% to 80% [5, 19, 20, 39, 44], so we take p ≈ 0.3. We require p > 0 since we are modeling temperate phages. Cell doubling times can range from 20 mins to 40 mins [2, 45], so we take growth rates of g = 1 cell division per cell per generation time (=30 minutes). Similarly, we take r to be ≈ 1 cell per generation time, but we investigate different r values in the range 0.9 ≤ r ≤ 1 to determine how the growth-cost due to lysogeny affects the dynamics. In earlier models, lysogens and nonlysogen growth rates were typically assumed to be identical [5], although the difference could be less than 0.001 [13], implying that g − r ≤ 0.001. Also, experiments competing λ lysogens against their nonlysogenic counterparts showed that their growth rates are not statistically different, suggesting that r ≈ g [13].

Table 1:

LUV model parameters

Parameter	Symbol	Value
Lysogen replication rate	r	0.9 ≤ r ≤ 1 / gen	[2, 45]
Nonlysogen replication rate	g	1 / gen	[2, 13, 45]
Constant population size	K	≥ 106 / ml
Phage-cell binding/infection rate	α	≈ 10−7 mL / phage / cell / gen	[10, 28, 35]
Phage burst size	b	≈ 150 to 200	[36]
Lysogenization rate	p	≈ 0.30	[5, 20, 39, 44]
Phage degradation rate	γ	≈ 0.001 / gen	[12]
SPI rate	σ	≈ 10−3 to 10−5 / gen	[5, 21, 22, 45]

Still, the lysogen’s genome contains additional genes from the integrated prophage. Maintenance of the lysogenic state generally requires continued expression of a key viral repressor protein which for phage λ is the CI repressor. It is possible that the cell incurs some growthrate cost by maintaining expression of this viral repressor. Thus, it is still possible that r could be different from g by a very small amount, such as g−r ≤ 0.001 that could be difficult to detect. Overall, we assume that the lysogenic growth rate r is at most g or otherwise slightly smaller. Note that since g ≥ r then g > σ as well. Also note that if r > g then the lysogens would displace the nonlysogens even without SPI in the long run, and the inclusion of SPI would only accelerate the process. Since this observation is somewhat trivial, we do not specifically investigate it here.

The parameter α measures the rate at which phages infect nonlysogens successfully. Its units are 1 mL/cell/phage/generation-time = 5.55 × 10⁻⁴ mL/s. This includes the phage infection/adsorption rate and the rate at which infected cells develop lytically or lysogenically per generation. We estimated α from phage adsorption experiments [10, 28] by fitting a small mass-action model of the phage-cell infection dynamics to the raw data. This α estimate allowed lysogen fixation to occur on the timescale of a few days, similar to the experimental results for competing phage λ lysogens against their nonlysogenic counterparts [13]. Additionally, this value of α is in line with the model proposed in [35], which is similar to ours. The results are relatively robust to this parameter value as it varies from 10⁻⁷ to 10⁻⁸. The intrinsic SPI rate of phage λ in a recA⁻ background is < 10⁻⁸, but in a recA⁺ background the SPI rate is estimated to be 10⁻⁴ - 10⁻⁵, or less per generation [5, 21, 22, 45]. Since recA⁻ mutants probably do not exist in nature, we use the recA⁺ SPI rate for σ to model naturally relevant scenarios.

We can take advantage of the differing time scales of the reactions to impose some conditions (inequalities) on the parameters. From the parameter estimates, we find that the quantity αbK/γ ≈ 1.5 · 10⁴ is quite large. The numerator ν₁ = αbK is essentially the rate of phage release, equal to the total number of phages released per generation from successful infections of K nonlysogens. The denominator ν₂ = γ is the rate at which a single phage degrades per generation. Since phage degradation is on the order of days and lysis-lysogeny decision-events occur on the order of hours, it is no surprise that the ratio ν₁/ν₂ is quite large for literature-estimated parameters. Thus, we assume that αbK/γ ≫ 1, which should safely hold even with some uncertainty in the parameter estimates. We also assume that r > σ because if σ > r then the net lysogen growth rate would be negative, indicating an unsustainable population.

With the model and its parameters specified, we numerically simulated the LUV model using the Runge Kutta integration scheme (see Figure 1) for the parameter values in Table 1 to understand how SPI contributes to the competition between lysogens and nonlysogens. We chose a step-size Δt of approximately 0.001, which gave smooth time trajectories. The simulations were robust to the exact value of Δt, such that even at Δt ≈ 0.1 we observed qualitatively accurate trajectories. The initial conditions we chose were L(t = 0) = 0.01K and U = 0.99K, i.e., the lysogens initially comprise 1% of the population. Since K ≥ 10⁶, the initial values of L and U were at least 10⁴ which means that deterministic models can well-approximate the behavior of this system.

Figure 1:

LUV model simulated using parameters from Table 1. Specifically, we set r = 0.99 and σ = 10⁻⁴. The results are robust to varying the model’s parameters.

The simulations show that this system tends towards a state in which the lysogens completely displace the nonlysogens in the long run (Figure 1) even though the lysogenic growth rate is lower (g > r). Consistent with experimental observations [8, 13, 29], this suggests that SPI allows the lysogens to subvert the nonlysogens even if the lysogenic state is costly to cell growth. Essentially, a low but steady rate of lysogens undergoing SPI allows the released phages to either kill off the nonlysogen strain or convert them to the lysogenic state.

3. Lower bound on SPI rate for lysogenic advantage

To determine the exact bounds on the SPI rate necessary for the lysogens to displace the nonlysogens, we analytically solved the model (Equation 1 - Equation 5) at steady state to obtain its equilibrium points and then tested their stability by calculating the eigenvalues of the system’s associated Jacobian matrix. Despite the nonlinearity of the model, it admits analytical solutions for its equilibrium points. We used Mathematica to check all calculations and derivations.

First, we set $\dot{V}=0$ in Equation 3 and solve for V. This gives us the following relationship.

$$V=\frac{b\sigma L}{\gamma +\alpha [K-(1+b(1-p))U]}$$ (6)

Next, we set $\dot{U}=0$ using Equation 2.

$$0=U[g-\alpha V-\varphi ]$$ (7)

The only way for Equation 7 to be zero is if either U = 0 or if [g − αV – ϕ] = 0. We will analyze both cases.

If U = 0 it immediately requires that L = K by constraint Equation 5. This then immediately implies that V = bσK/(γ + αK) by Equation 6. Thus, the point $P_1^{LUV}=(L,U,V)=(K,0,b\sigma K/(\gamma +\alpha K))$ is one equilibrium point of this system.

Otherwise, if g−αV −ϕ = 0 then the expression in Equation 4 for ϕ immediately implies that Equation 8 must hold.

$$\frac{(g-r+\sigma )L}{\alpha [K-U(1-p)]}=\frac{b\sigma L}{\gamma +\alpha [K-(1+b(1-p))U]}$$ (8)

We can immediately note that L = 0 is a possible solution of Equation 8, implying that U = K by Equation 5. Then, using Equation 6 we immediately see that V = 0 if U = K. Thus, the second equilibrium point of the system is given by $P_2^{LUV}=(L,U,V)=(0,K,0)$.

If U ≠ 6 K, then the resulting system becomes difficult to solve by hand. As a result, we used Mathematica to solve for the system’s remaining equilibrium points to discover that there are 2 additional solutions for the case U ≠ 6 K. The third solution to the system, $P_3^{LUV}$ is quite complicated, but it gives L = gp(γ +αK)/α[(σ − r)(1 + b) + bpr]. It turns out that this expression for L is only positive if σ > r [1 − p(b/(b + 1))] ≈ r[1 − p], but this is not consistent with the fact that σ is on the order of 10⁻⁴ or less. Therefore, $P_3^{LUV}$ is generally an unphysical solution of the system, causing L to be negative.

The final solution, $P_4^{LUV}$, is extremely long and complicated so we do not show it here. This fourth solution results in negative population counts when σ is above 10⁻⁴ to 10⁻⁵. Below 10⁻⁵ this solution results in a steady-state mixture of both lysogens and nonlysogens. This suggests that when σ is exceedingly small, the effect of SPI is not sufficient to cause the lysogens to completely displace the nonlysogens. However, when it is higher than some threshold then the lysogens will displace the nonlysogens.

3.1. Stability of equilibrium points for LUV model

To determine the conditions required for the lysogens to displace the nonlysogens, we investigated the stability of the LUV model’s equilibrium points (steady states) by computing the eigenvalues of the associated Jacobian matrix J^LUV. For an equilibrium point to be stable, all associated eigenvalues must be negative.

$$J^{LUV}=\left(\begin{array}{lll}\partial \dot{L}/\partial L\hfill & \partial \dot{L}/\partial U\hfill & \partial \dot{L}/\partial V\hfill \\ \partial \dot{U}/\partial L\hfill & \partial \dot{U}/\partial U\hfill & \partial \dot{U}/\partial V\hfill \\ \partial \dot{V}/\partial L\hfill & \partial \dot{V}/\partial U\hfill & \partial \dot{V}/\partial V\hfill \end{array}\right)$$ (9)

We calculate the partial derivatives in J^LUV using the LUV model (Equation 1 - Equation 5) and we define $J_i^{LUV}$ to be J^LUV evaluated at equilibrium point $P_i^{LUV}$.

First, we analyze the stability of equilibrium point $P_1^{LUV}$, namely (L, U, V) = (K, 0, bσK/(γ+ αK)), a state of the system in which the lysogens have displaced the nonlysogens. Using these values for L, U, and V we calculate $J_1^{LUV}$.

$$J_1^{LUV}=\left(\begin{array}{ccc}-r+\sigma & -g+\alpha bK\sigma /(\gamma +\alpha K)& 0\\ 0& g-r+\sigma -\alpha bK\sigma /(\gamma +\alpha K)& 0\\ b\gamma \sigma /(\gamma +\alpha K)& \alpha b^{2}K\sigma (1-p)/(\gamma +\alpha K)& -(\gamma +\alpha K)\end{array}\right)$$ (10)

The eigenvalues of $J_1^{LUV}$ result from the characteristic equation det($J_1^{LUV}-\lambda I$) where I is the identity matrix, which implies for the eigenvalues λ:

$$0=(-r+\sigma -\lambda )(g-r+\sigma -\alpha bK\sigma /(\gamma +\alpha K)-\lambda )(\gamma +\alpha K+\lambda )$$ (11)

The eigenvalues are the values of λ which solve Equation 11, namely λ₁ = −r + σ, λ₂ = g−r−σ [αbK/(γ + αK) − 1], and λ₃ = −(γ +αK). The equilibrium point (K, 0, bKσ/γ) is stable if all eigenvalues are negative. Since (γ +αK) > 0 we know that λ₃ is negative. Since σ < r we have that λ₁ is negative since λ₁ = −(r − σ) < 0. The eigenvalue λ₂ is negative if and only if σ > σ_LB with σ_LB given by Equation 12.

$${\sigma }_{LB}=\frac{g-r}{\alpha bK/(\gamma +\alpha K)-1}$$ (12)

Using literature estimates for the parameter values from Table 1, and specifically with g − r = 0.01, we estimate σ_LB ≈5 · 10⁻⁵. If we use g − r ≤ 0.001 (as we noted earlier) then σ_LB ≈ 5 · 10⁻⁶ or lower. Thus, must be larger than this lower bound (defined as σ_LB) in order to establish (K, 0, bKσ/γ) as a stable equilibrium point, otherwise it is unstable. Note that this is consistent with the experimentally measured SPI value between 10⁻⁴ and 10⁻⁵ (or less) per generation for phage λ. Thus, the experimentally measured SPI rate may be sitting at or near this estimated lower bound σ_LB, above which lysogens completely displace nonlysogens.

Recall that we assume g ≥ r so there are two possibilities, either g = r or g > r. If g = r then the lysogenic state does not hurt cell growth rate and, as a consequence, any σ > σ_LB = 0 would cause the point (L, U, V) = (K, 0, bKσ/γ) to be a steady equilibrium point.

Next, we examine equilibrium point $P_2^{LUV}=(0,K,0)$. This point describes a state in which the nonlysogens have displaced the lysogens. The Jacobian matrix $J_2^{LUV}$ for (L, U, V) = (0, K, 0) is given in Equation 13.

$$J_2^{LUV}=\left(\begin{array}{ccc}-g+r-\sigma & 0& \alpha Kp\\ -r+\sigma & -g& -\alpha Kp\\ b\sigma & 0& \alpha bK(1-p)-\gamma \end{array}\right)$$ (13)

The eigenvalues of $J_2^{LUV}$ are obtained by calculating its determinant, which results in a characteristic polynomial (Equation 14).

$$0=(-g-\lambda )[{\lambda }^2-\lambda (-g+r-\sigma +\alpha bK(1-p)-\gamma )+(-g+r-\sigma )(\alpha bK(1-p)-\gamma )-\alpha Kpb\sigma ]$$ (14)

The eigenvalues are then obtained in a straightforward manner. First, λ₁ = −g < 0. For λ₂ and λ₃, we see that these are the roots of the quadratic inside the square brackets in Equation 14. The roots of this quadratic take the form $\lambda =q\pm \sqrt{q^2+\epsilon }$ with q = (−g + r − σ + αbK(1 − p) − γ)/2 and ε = (αKpbσ − (− g + r − σ)(αbK(1 − p) − γ)). If ε > 0 then one of the solutions λ > 0, causing this equilibrium point to be unstable. So, for $P_2^{LUV}$ to be stable, we must have ε < 0. However, if ε < 0 then it implies σ < −(g − r)[(αbK(1 − p)/γ − 1)/(αbK/γ − 1)]/γ, and the term in the square-brackets is positive which implies σ must be negative. Since σ is never negative we conclude that $P_2^{LUV}$ is not a stable equilibrium point.

Moving on to the last two equilibrium points, we note that we do not need to assess the stability of PLUV₃ because it generally results in negative population counts, so regardless of its stability it is unphysical. For $P_4^{LUV}$ we found that when σ < σ_LB this point became the only steady state of the system. If σ > σ_LB then not only is this point unstable but it also returns negative population counts, making it also unphysical. This result makes sense because, when σ < σ_LB then only stable equilibrium point is $P_1^{LUV}$. When σ < σ_LB then $P_1^{LUV}$ becomes unstable while $P_2^{LUV}$ remains unstable and $P_3^{LUV}$ continues to return negative population counts. Thus, $P_4^{LUV}$ must become stable since it is the only state left, as we indeed found numerically.

Overall, we have shown that lysogens will displace nonlysogens even if the lysogenic state is costly to cell growth provided that σ is at least (g − r)/[Kαb/γ − 1]. If σ drops below this lower bound then the lysogens and nonlysogens will coexist. We have also shown that experimental wild-type phage λ SPI rate measurements are consistent with this lower bound, suggesting that it has evolved towards this value. These results only establish a lower bound on σ necessary for lysogens to displace the nonlysogens, and any σ > σ_LB would produce the same results. Thus, it is natural to ask what would drive the SPI rate down towards this lower bound.

4. The LUV2 model

In the previous section we showed that if r > σ > (g − r)/[Kαb/(γ + αK) − 1] = σ_LB then a mixture of lysogenic and nonlysogenic cells will tend towards a steady state in which the lysogens displace nonlysogens. Note that this inequality places no strong restriction on the exact value of σ for lysogens to displace nonlysogens. The only requirement is that it must be strictly greater than σ_LB and must be less than the intrinsic cell replication rate r. To understand what value of σ may be optimal, we adjusted the LUV model so that it included two lysogen strains, L₁ and L₂, which differ only in their SPI rates σ₁ and σ₂ respectively. These lysogens will compete with each other and with the nonlysogenic cells U. These lysogen strains also produce their own phages, denoted V₁ and V₂, respectively. All other parameters and interactions between the two lysogen strains and the nonlysogens remain the same to ensure we are only comparing different SPI rates. Our goal is to identify an SPI rate which would allow a lysogen strain to outcompete another lysogen strain while also being able to displace the nonlysogens. The LUV2 model is summarized by Equation 15 - Equation 21.

$${\dot{L}}_1=r{L}_1-{\sigma }_1{L}_1+p\alpha U{V}_1-\varphi L_1$$ (15)

$${\dot{L}}_2=r{L}_2-{\sigma }_2{L}_2+p\alpha U{V}_2-\varphi L_2$$ (16)

$$\dot{U}=gU-\alpha U\left(V_1+V_2\right)-\varphi U$$ (17)

$${\dot{V}}_1=(1-p)b\alpha U{V}_1-\gamma V_1+b{\sigma }_1{L}_1-\alpha V\left(L_1+L_2\right)$$ (18)

$${\dot{V}}_2=(1-p)b\alpha U{V}_2-\gamma V_2+b{\sigma }_2{L}_2-\alpha V\left(L_1+L_2\right)$$ (19)

$$\varphi =\frac{1}{K}\left[\left(r-{\sigma }_1\right)L_1+\left(r-{\sigma }_2\right)L_2+gU-\alpha U\left(V_1+V_2\right)(1-p)\right]$$ (20)

$$K=L_{\text{1}}+L_{\text{2}}+U$$ (21)

In this model, L₁ denotes the number of lysogens of strain 1 with SPI rate σ₁ and L₂ denotes the number of lysogens of strain 2 with SPI rate σ₂. The variables V₁ and V₂ are the phages emitted from lysogen strains 1 and 2, respectively, via the SPI process. All parameters are the same as in the previous model. Without loss of generality, we assume that σ₁ > σ₂ so that lysogen strain 1 has a higher SPI rate than lysogen strain 2. Note that here we are also assuming that both L₁ and L₂ can displace the nonlysogens in separate competitions, implying that σ_i > σ_LB for i = 1, 2. We also assume that g − r ≥ 0, and r > σ₁ (as in the LUV model). Since σ₁ > σ₂ we also have r > σ₂.

To understand the behavior of this system, we simulated it using the parameter values in Table 1. We set r = 0.99 and σ₁ to be 10 times greater than σ₂ = 10⁻⁴ so that both σ₁ and σ₂ were larger than σ_LB. In Fig 2 we show a sample trajectory of the system using initial conditions L₁(t = 0) = 0.005K, L₂(t = 0) = 0.005K, and U(t = 0) = 0.99K so that the lysogens initially only comprise 1% of the population, and this 1% is split evenly among the two lysogen strains. We see that early on, L₁ has a clear advantage in the sense that it greatly outnumbers both L₂ and the nonlysogens U. However, in the long run we see that L₂ actually starts to overtake L₁, and that this switch occurs after U decreases towards 0.

Figure 2:

LUV2 model simulation using default parameters from Table 1. We used r = 0.99, σ₂ = 10⁻⁶, and σ₁ = 10⁻⁵.

Numerical simulations of various parameter sets showed similar behavior, with L₂ always displacing both L₁ and U in the long run. This suggests that lower SPI rates are generally favorable in the long run. Intuitively, we can reason that once the nonlysogens go extinct there is no more benefit to having a higher SPI rate. Thus, we can conclude that if both lysogen strains L₁ and L₂ survive after U reaches 0, then the lysogen with the lower SPI rate will be selected for in the long run because there is no more benefit to the SPI process.

5. Slower SPI rates are advantageous

To prove that the lysogenic strain with the lower SPI rate is the one which is favored by natural selection, we calculate the equilibrium points of the model and assess their stability. Each lysogen strains can separately outcompete nonlysogens, implying that both σ₁ > σ_LB and σ₂ > σ_LB. As in the LUV model, we first begin with setting the free phage rate equations to zero (Equation 18 and Equation 19) i.e. ${\dot{V}}_1={\dot{V}}_2=0$. This results in the following relationships for V₁ and V₂.

$$V_1=\frac{b{\sigma }_1{L}_1}{\gamma +\alpha K-\alpha [b(1-p)+1]U}$$ (22)

$$V_2=\frac{b{\sigma }_2{L}_2}{\gamma +\alpha K-\alpha [b(1-p)+1]U}$$ (23)

Next, we set Equation 17 to 0 so that $\dot{U}=0$. This implies that:

$$0=U\left[g-\alpha \left(V_1+V_2\right)-\varphi \right]$$ (24)

Similar to the LUV system, Equation 24 can be solved if either U = 0 or if g − α(V₁ + V₂) – ϕ = 0. We will examine both cases. If U = 0 then we immediately see that V₁ and V₂ are determined by their respective lysogen strain levels. That is, if U = 0 we have that V₁ = bσ₁L₁/(γ + αK) and V₂ = bσ₂L₂/(γ + αK). Thus, we calculate L₁ and L₂ next. If U = 0 then K = L₁ +L₂ by the constraint Equation 21. This means that L₁ = K −L₂. We can plug this into the rate equation for ${\dot{L}}_1$ (Equation 15) and set ${\dot{L}}_1=0$. After simplifying, this results in an equation with only 1 unknown variable, namely L₂.

$$0=L_2\left[1-\frac{L_2}{K}\right]$$ (25)

The two solutions to this equation are L₂ = 0 or L₂ = K. Along with the constraint (Equation 21), we know that (L₁, L₂) = (K, 0) and (L₁, L₂) = (0, K) are both solutions of Equation 25. Using Equation 6 and Equation 23, the two equilibrium points of the LUV2 system (L₁, L₂, U, V₁, V₂), are $P_1^{LUV2}=\left(K,0,0,b{\sigma }_{1}K/(\gamma +\alpha K),0\right)$ and $P_2^{LUV2}=\left(0,K,0,0,b{\sigma }_{2}K/(\gamma +\alpha K)\right)$.

If instead g − α(V₁ + V₂) – ϕ = 0 then we can combine this equation with Equation 22, Equation 23, and Equation 20 to show that the following equation must hold.

$$\frac{L_1\left(g-r+{\sigma }_1\right)+L_2\left(g-r+{\sigma }_2\right)}{\alpha [K-(1-p)U]}=\frac{b{\sigma }_1{L}_1+b{\sigma }_2{L}_2}{\gamma +\alpha K-\alpha [b(1-p)+1]U}$$ (26)

We can immediately see that if L₁ = L₂ = 0 then Equation 26 holds. If L₁ = L₂ = 0 then U = K by the constraint Equation 21. Furthermore, if L₁ and L₂ are both zero then we also have that V₁ = V₂ = 0 by Equation 6 and Equation 23. Thus, the third equilibrium point of the system is given by $P_3^{LUV2}=(0,0,K,0,0)$.

If both L₁ and L₂ are not zero, then we need to solve the Equation 26 for both L₁, L₂, and U along with Equation 15 - Equation 21. However, this is very difficult to do by hand, and even with the use of symbolic algebra we were not able to arrive at simple expressions for the remaining solutions. Nevertheless, the remaining solutions were probed via numerical simulations. We found that, as long as the model’s parameters obeyed the aforementioned inequalities, L₁ always displaced L₂ in the long run in a manner similar to that shown in Figure 2. Otherwise, if either σ₁ or σ₂ (or both) were less than σ_LB, then more complex dynamics resulted with a steady-state mixture of all three species in the long run. However, these solutions were not of particular interest to us because we are competing two lysogens which can individually displace the nonlysogens if competed separately, which requires σ₁ and σ₂ to be greater than σ_LB.

We summarize all equilibrium points of the LUV2 model which exist under this parameter regime in Table 2. Note that these equilibrium points are similar to the ones we found for the LUV model. Similar to the LUV model, we analyzed the stability of the LUV2 equilibrium points (see the APPENDIX). Overall these results collectively indicate that L₂ will displace both L₁ and U if σ_LB < σ₂ < σ₁. The second half of the inequality σ₂ < σ₁ gives L₂ a competitive advantage against L₁, and the first half of the inequality σ_LB < σ₂ gives L₂ the competitive advantage against the nonlysogens. Thus, a lower SPI rate is advantageous over a higher SPI rate in the long run, indicating that phages of lower SPI rates can invade and fix until the lower bound on the SPI rate is reached.

Table 2:

LUV2 model equilibrium points

equilibrium point	L1	L2	U	V1	V2
equilibrium point 1	K	0	0	bσ1K/(γ + αK)	0
equilibrium point 2	0	K	0	0	bσ2K/(γ + αK)
equilibrium point 3	0	0	K	0	0

6. Stochastic factors influencing selection of SPI rate

To understand how stochastic factors could influence the competition between lysogens and nonlysogens, we simulated the LUV model with some modifications. Since the SPI rate is fairly small, at low lysogen counts (e.g. L ≈ 100 or less) it is possible for several generations to pass before a single SPI event takes place. This is difficult to capture in the LUV model because it is simulated using ODEs which are deterministic. In this deterministic setting, some SPI events ”partially” occur from the start of the dynamics since the net rate of SPI, L · σ, is > 0 at t = 0. Thus, we modified the LUV model so that the SPI reactions would occur stochastically during the dynamics. Over a small time interval Δt over which we integrate, the probability that a single lysogen undergoes SPI is σ · Δt. For L lysogens, the probability that ℓ ≤ L of them undergo SPI in Δt is given by a binomial distribution as in Equation 27 with the average number of SPI events in a time interval Δt given by $\overline{\mathcal{l}}=L\sigma \Delta t$.

$$\mathit{\text{Pr}}(\mathcal{l})=\left(\begin{array}{l}L\hfill \\ \mathcal{l}\hfill \end{array}\right){[\sigma \Delta t]}^{\mathcal{l}}{[1-\sigma \Delta t]}^{L-\mathcal{l}}$$ (27)

Figure 3 shows 2 sample simulations of this modified model at an initial lysogen count of L₀ = L(0) = 0.001 · K. In the right panel we observe the lysogens being slowly displaced by the nonlysogens (since r < g) but when an SPI reaction occurs this “saves” the lysogens and they end up taking over the population. However, it can also happen that the SPI reaction does not occur at all. If an SPI reaction fails to fire early enough (as in the left panel), the probability L·σ that an SPI reaction will fire later approaches 0 as L is gradually outcompeted. Thus, if stochasticity matters (e.g. at low population counts) and if r < g, then the total SPI rate, L · σ, must not be too low if the lysogens are to displace the nonlysogens.

Figure 3:

Stochastic LUV model simulations using the parameters from Table 1, specifically using r = 0.9 and σ = 10⁻⁵. In the left panel, the nonlysogens displaced the lysogens because no SPI reaction occurred. In the right panel an SPI reaction occurred early enough for the lysogens to displace the nonlysogens.

At high initial lysogen counts, the lysogens almost always displaced the nonlysogens because an SPI event almost always occurs during the simulation. On the other hand, when the initial lysogen count is low (e.g. 10 or 100) then they often were displaced by the nonlysogens. A similar trend was observed for σ, namely that high σ allowed the lysogens to displace the nonlysogens more frequently relative to the case of a very low σ value. Additionally, if the lysogenic growth-cost was small (e.g. g ≈ r), then the lysogens were able to displace the nonlysogens with greater probability over a much wider range of L(0) and σ values. To systematically understand the effect of L(0), σ, and r on the probability that the lysogens displace the nonlysogens, we estimated the probability of lysogen-sweeping events from repeated stochastic simulation runs. In Figure 4 we show the probability of outcomes with L = K, U = 0 over a range of L(0) and σ values for r = 0.9 and r = 0.9999. Generally we see that at low L(0) and σ values the lysogens usually cannot displace the nonlysogens. On the other hand, at higher L(0) and σ the lysogens were always able to more probably displace the nonlysogens. For intermediate values of L(0) and σ, the probability that the lysogens displace the nonlysogens ranges from 0 to 1. But as r approaches g (right panel with r = 0.9999), we see that the lysogens can displace the nonlysogens over a much wider range of L(0) and σ values, even if L(0) and σ are low. When g −r = 0, the lysogens always displace the nonlysogens even in stochastic simulations. This is because the two populations remain at their initial values until the first SPI reaction occurs, no matter how long it takes. Thus, if the lysogenic growth-cost is small or zero, we expect that the lysogens will generally displace the nonlysogens with high probability.

Figure 4:

Fraction of simulations resulting in lysogens displacing the nonlysogens for different values of L(0) and σ using the stochastic LUV model. The left panel was simulated at r = 0.9 and the right panel was simulated at r = 0.9999. The rest of the parameters were estimated from Table 1

In order for the lysogens to displace the nonlysogens, the SPI reaction must occur before the time T at which the lysogen population drops below 1. To understand how this time T depends on g − r and L₀, we analytically calculated the latest time an SPI reaction must occur by solving the LUV model (Equation 15 - Equation 5) with σ = 0 for L(t). The result is given in Equation 28.

$$L(t)=\frac{K}{1+\left(\frac{K-L_0}{L_0}\right)e^{(g-r)t}}$$ (28)

The latest time T that an SPI reaction can occur and still allow the lysogens to displace the nonlysogens can be calculated by setting L(T) = 1 and solving for T.

$$T=\frac{1}{(r-g)}\text{ln}\frac{K/L_0-1}{K-1}\approx \frac{1}{g-r}\text{ln}{L}_0$$ (29)

The approximation in Equation 29 is quite accurate since K is many orders of magnitude greater than 1, as is K/L₀. If an SPI reaction occurs by time T, then the lysogens will displace the nonlysogens. This establishes some additional constraints on the value of σ necessary for the lysogens to displace the nonlysogens, namely that if there is a lysogenic growth cost then the SPI rate cannot be arbitrarily low in a stochastic setting. In Figure 5 we show the T value at different values of r − g and L₀. We see that T gradually increases with L₀, implying that higher initial lysogen count gives the lysogens more time to undergo an SPI event. However, we see that as r approaches g the value of T increases dramatically. This means that the lysogens have a much longer time to initiate an SPI event when there is a small lysogenic growth rate cost. If the lysogenic growth-cost is arbitrarily small, then $r\approx g\Rightarrow T\to \infty$, implying that eventually an SPI reaction will occur and the lysogens will displace the nonlysogens with probability 1.

Figure 5:

The T value calculated at different values of r and L₀ using K = 10⁶.

To understand how the probability, Ψ, that the lysogens will displace the nonlysogens depends on general parameter values (e.g. not just when r ≈ g), we calculated the probability that at least 1 SPI reaction occurs within time T by analyzing the stochastic model. First, we split up the time interval 0 ≤ t ≤ T into small subintervals denoted by Δt_k of fixed size Δt of which there are N = T/Δt. Within each of these intervals, the number of lysogens is denoted by L_k. The probability of at least 1 SPI reaction over 0 ≤ t ≤ T is equal to 1 minus the probability that 0 SPI reactions take place over 0 ≤ t ≤ T, which is 1 minus the probability that 0 SPI reactions occur within all N = T/Δt subintervals Δt_k. In each subinterval, the probability of 0 SPI reactions is ${[1-\sigma \Delta t]}^{L_k}$. The total number of subintervals is N = T/Δt, which implies that the probability of 0 SPI reactions over the entire interval 0 ≤ t ≤ T is ${\prod }_{k=1}^N{[1-\sigma \Delta t]}^{L_k}$. This can be rewritten as ${[1-\sigma \Delta t]}^{\Sigma L_k}$ where the sum Σ ranges over the L_k from k = 1 to k = N = T/Δt. Using $\Sigma L_k=N\overline{L}$ with $\overline{L}=\text{ time}$ average of L_k, we find that $\Psi ={(1-\sigma \Delta t)}^{\overline{L}T/\Delta t}$. As Δt → 0 one can easily show that Ψ is given by Equation 30.

$$\Psi =1-e^{-\sigma T\overline{L}}$$ (30)

Since we know L(t) over time T from Equation 28, we can calculate $\overline{L}=\frac{1}{T}{\int }_0^{T}L(t)dt$. Applying straightforward integration techniques, we find that $\overline{L}$ is given by Equation 31.

$$\overline{L}=\frac{-K}{T(g-r)}\text{ln}\left[1-\frac{\left(1-e^{-(g-r)T}\right)}{K/L_0}\right]$$ (31)

We can then determine a final expression for Ψ by combining Equation 30, Equation 29, and Equation 31 to derive Equation 32.

$$\Psi =1-{\left[\frac{K-1}{K-L_0}\right]}^{-\sigma K/(g-r)}$$ (32)

The expression for Ψ matches the results of the stochastic simulation remarkably well as shown in Figure 6 using the same parameter values from Table 1. From this expression it is clear that the probability that the lysogens displace the nonlysogens increases with L₀, σ, and K. There is also a strong dependence on g − r. If the lysogenic-growth cost is very small then g − r → 0 and, as a result, Ψ → 1 − 0 = 1. Overall, the lysogens can maximize their probability of displacing nonlysogens by maximizing the quantity σ/(g − r). The value of σ can be arbitrarily small so long as the lysogenic growth-cost (e.g. g − r) is correspondingly small (which it is expected to be). These analytical insights agree with the stochastic simulations. It is worth noting that when r ≈ g the simulations are extremely time consuming, especially at low L₀, because both the lysogen and nonlysogen population remains approximately constant in time until finally an SPI event occurs since σ > 0. However, from the analytical results we can immediately see the effect of various parameters on P without simulation. We take advantage of this by using Ψ to calculate the probability the lysogens displace the nonlysogens over a range of L₀ and σ values for g − r = 10⁻⁵.

Figure 6:

Probability that the lysogens displace the nonlysogens using the Ψ function. Left panel matches with stochastic simulation results in Figure 4, and right panel shows results when g −r is very small (e.g. 10⁻⁴) the lysogens are more likely to displace the nonlysogens for any L₀ and σ.

To understand how the stochastic evolutionary dynamics occurs in the case that there is a single initial lysogen, we created a separate model because the prior models do not adequately capture the case when L₀ = 1. In this case, when t = 0 we have that V = 0, U = K − 1, and L = 1 so the dynamics of L can be described by $\dot{L}=(r-\sigma -g)[1-1/K]\approx (r-\sigma -g)$ using Equation 1, where we used the fact that 1/K is much smaller than 1. This expression for L˙ shows that the initial rate of change for L is made of two parts. The g − r term corresponds to the competitive growth of the lysogens vs the nonlysogens. The second term σ corresponds to the SPI reaction. If g > r then there are only two possibilities for the next reaction, namely i) either the nonlysogens will displace the nonlysogens by outgrowing them or ii) an SPI event will occur and the lysogens will displace the nonlysogens. The time to the next reaction in typical Gillespie simulation is given by τ = −ln[1 − R]/(g −r +σ) with R as a random number from a uniform distribution ranging from 0 to 1. The probability that the next reaction at time τ is the SPI reaction is simply given by the proportion σ/(g−r+σ) and can be written as in Equation 33.

$${\Psi }_{\left\{L_0=1\right\}}=\frac{\sigma /(g-r)}{1+\sigma /(g-r)}$$ (33)

From Equation 33 we clearly see that in the case L₀ = 1 the probability the lysogens displace the nonlysogens approaches 1 as r → g. Thus, at little to no lysogenic growth-cost, the lysogens can displace the nonlysogens with high probability even when there is only 1 initial lysogen. As in Equation 32, the higher the ratio σ/(g − r) is, the more likely the lysogens displace the nonlysogens. It is easy to show that ${\Psi }_{\left\{L_0=1\right\}}\ge 0.5$ if σ ≥ g − r. Thus, with σ = 10⁻⁵ the lysogens have a 50% chance of winning if the lysogenic growth-cost is on the order of 10⁻⁵. If g = r then in the model both L and U remain at their initial population sizes until an SPI reaction fires, and so the lysogens win with probability Ψ = 1 in this case. This is easily shown by letting g → r and taking a limit of Equation 33.

To understand how stochastic factors influence the competition between two lysogen strains, we modified the LUV2 model to be stochastic in the same way as in the LUV model. These insights were similar to the LUV model with one extra detail. As shown in Figure 7, we see that so long as either lysogen undergoes SPI then L₂ will always displace L₁ in the long run. This is because when the SPI event initially occurs, the nonlysogens start to be killed off, causing empty space to appear for both L₁ and L₂ to both grow into. Once both L₁ and L₂ fill this empty space and the nonlysogens disappear, it is simply a competition between only L₁ and L₂. From this point onward it is clear that L₂ has the advantage because its effective growth rate r − σ₂ is slightly greater than the effective growth rate r−σ₁ of L₁ (since σ₂ < σ₁). The V₁ curve jumps up and down a bit towards the end because the stochasticity of SPI reactions becomes more apparent as L₁ drops to lower levels. Notice that L₁ underwent an SPI reaction very early on (marked by the increase in V₁) while L₂’s first SPI event occurred much later on at ≈ 10 generations. Even though L₂ was late to SPI, it still wins in the long run since its effective growth rate is higher.

Figure 7:

A sample simulation from the stochastic LUV2 model. Early on L₁ manages to have an SPI reaction, causing V₁ to rise and U to be killed off. During this, L₂ begins to grow since there is now free space to do so. It does not have its own SPI event until about 10 generations. Eventually, the relatively higher SPI rate of L₁ causes it to slowly but surely drop to 0 as L₂ rises towards the constant population size K = 10⁶. This simulation was run with r = 0.99, σ₁ = 10⁻³, and σ₂ = 10⁻⁴. We observed the same qualitative behavior for other parameter values.

7. Discussion

In this work we explore how SPI influences the natural selection among lysogens and nonlysogens, and also between lysogen variants with different SPI rates. We find a lower bound σ_LB ≈ 10⁻⁵/generation above which lysogens should displace the nonlysogens for general parameter values. Furthermore, we find that lower SPI rates are naturally selected, because although higher SPI rates confer an early advantage, they are costly to growth in the long run. These results collectively show that natural selection should push the SPI rate down towards σ_LB, an optimal value in terms of natural selection, provided that there is a lysogenic growth-cost (i.e. g − r > 0). Such a growth cost is plausible, but it is likely very small such that g − r ≈ 10⁻ⁿ with n > 2. Thus, there should be a lower bound σ_LB strictly greater than 0, but still likely to be very small. Evolution towards this optimal SPI rate could occur by mutations affecting the stability and/or RecA-cleavage rate of the CI protein in response to spontaneous host DNA damage, or mutations altering CI’s expression level or its viral DNA-binding affinity, all of which could influence lysogen stability [21].

We estimate on the lower bound of the SPI rate to be on the order of 10⁻⁴ or 10⁻⁵ lysogens per generation, or smaller according to (g − r), which matches experimental estimates of recA⁺ bacterial SPI rates [21, 22, 45]. After including stochasticity in the dynamics we reach similar conclusions in a probabilistic sense, with lysogenic fixation becoming increasingly probable at higher SPI rates, implying that stochastic effects on natural selection would tend to favor higher SPI rates. In the stochastic setting, however, the SPI rate scales with the lysogenic growth-cost, implying that for very low lysogenic growth-cost the SPI rate can remain arbitrarily low while still maintaining a high probability of lysogenic fixation. In the extreme case of a single initial lysogen with the naturally-occurring SPI rate σ* ≈ 10⁻⁵ in the population, this lysongen will still fix with a probability ≈ 1 if the lysogenic growth-cost is on the order of 10⁻⁵. Overall, we show that deterministic evolutionary forces push the SPI rate down towards σ_LB, while stochastic factors tend to push it higher, but only if g − r is not very small. Otherwise, if g −r is very small then σ can also be very small and still allow lysogens to outcompete nonlysogens with high probability. Thus, the theory we develop suggests that phage λ may be sitting at or near this optimal SPI rate.

Our methods have some limitations which could be addressed in future work. For example, we focused on modeling temperate phages and thus avoided analyzing the case p = 0. A phage with p = 0 would only be able to lyse via SPI if σ > 0. The results of our model seem to be robust to p approaching 0, but we defer a complete analysis of this scenario to future work. Also, we chose to use replicator dynamics with constant cell population size over more traditional phage population dynamics models because the former is apt to simulate long-term evolution and competition scenarios. Since we aimed to understand the outcome of a competition over long, evolutionary timescales, in the replicator dynamics models we approximated the relatively fast phage infection and adsorption as one-step processes. For this reason, our model might have difficulties capturing short-term development or population dynamics. All of these details could be included into future models. Nonetheless, the model’s predictions should be testable by long-term evolution experiments that aim to keep the total cell count constant, as in turbidostats. This would allow experimentally investigating whether there exists a lower bound on the SPI rate for lysogens to outcompete nonlysogens. In principle, multiple mutant host strains with a range of SPI rates can be generated and competed against each other, to test whether lower SPI rates are advantageous. Some of these strains can also be competed with a nonlysogen strain, to test that for SPI rates below some threshold nonlysogens are not displaced. Finally, long-term evolution experiments with a high SPI-rate strain could be initiated to test the mutational paths and speed of SPI rate decrease by naturally arising mutations.

In summary, we demonstrate how mathematical modeling is useful to analyze evolutionary scenarios and explore the consequences of ecological constraints and interactions among competing lysogens and nonlysogens. It will be interesting to further validate this model by applying it to other temperate phages. We show how by a small growth cost a species can gain a competitive advantage over another. This might relate to bacterial persistence, sporulation, and other survival strategies in which a small subpopulation switches to a phenotypically different, growth-reduced state.

• The spontaneous prophage induction (SPI) rate for bacteriophage λ must be higher than a threshold value of ^~10⁻⁴/h for lysogens to outcompete nonlysogens.

• When two lysogens compete in the presence of nonlysogens, the lysogen with lower SPI rate wins, implying that natural selection should lower the SPI rate towards the threshold.

• Stochastic models indicate that SPI is still advantageous even at low cell numbers.

7.1. APPENDIX: Stability analysis of LUV2 equilibrium points

To understand what conditions enable one lysogen strain to outcompete the other, we analyzed the stability of our model’s equilibrium points by calculating the eigenvalues of the Jacobian matrix J^LUV2 for each equilibrium point in Table 2.

$$J^{LUV2}=\left(\begin{array}{lllll}\partial {\dot{L}}_1/\partial L_1\hfill & \partial {\dot{L}}_1/\partial L_2\hfill & \partial {\dot{L}}_1/\partial U\hfill & \partial {\dot{L}}_1/\partial V_1\hfill & \partial {\dot{L}}_1/\partial V_2\hfill \\ \partial {\dot{L}}_2/\partial L_1\hfill & \partial {\dot{L}}_2/\partial L_2\hfill & \partial {\dot{L}}_2/\partial U\hfill & \partial {\dot{L}}_2/\partial V_1\hfill & \partial {\dot{L}}_2/\partial V_2\hfill \\ \partial \dot{U}/\partial L_1\hfill & \partial \dot{U}/\partial L_2\hfill & \partial \dot{U}/\partial U\hfill & \partial \dot{U}/\partial V_1\hfill & \partial \dot{U}/\partial V_2\hfill \\ \partial {\dot{V}}_1/\partial L_1\hfill & \partial {\dot{V}}_1/\partial L_2\hfill & \partial {\dot{V}}_1/\partial U\hfill & \partial {\dot{V}}_1/\partial V_1\hfill & \partial {\dot{V}}_1/\partial V_2\hfill \\ \partial {\dot{V}}_2/\partial L_1\hfill & \partial {\dot{V}}_2/\partial L_2\hfill & \partial {\dot{V}}_2/\partial U\hfill & \partial {\dot{V}}_2/\partial V_1\hfill & \partial {\dot{V}}_2/\partial V_2\hfill \end{array}\right)$$ (34)

For $P_1^{LUV2}=\left(L_1,L_2,U,V_1,V_2\right)=\left(K,0,0,Kb{\sigma }_1/(\gamma +\alpha K),0\right)$, the Jacobian matrix is given by Equation 35. This equilibrium point describes a state in which lysogen strain L₁ (with the higher SPI rate) displaces both the nonlysogens U and lysogen strain L₂.

$$J_1^{LUV2}=\left(\begin{array}{ccccc}-r+{\sigma }_1& -r+{\sigma }_2& -g+\frac{\alpha bK{\sigma }_1}{\gamma +\alpha K}& 0& 0\\ 0& {\sigma }_1-{\sigma }_2& 0& 0& 0\\ 0& 0& g-r+{\sigma }_1-\frac{\alpha bK{\sigma }_1}{\gamma +\alpha K}& 0& 0\\ \frac{b\gamma {\sigma }_1}{\gamma +\alpha K}& -\frac{\alpha bK{\sigma }_1}{\gamma +\alpha K}& \frac{\alpha b^{2}K{\sigma }_1(1-p)}{\gamma +\alpha K}& -(\gamma +\alpha K)& 0\\ 0& b{\sigma }_2& 0& 0& -(\gamma +\alpha K)\end{array}\right)$$ (35)

The eigenvalues of $J_1^{LUV2}$ are determined by solving the equation det $\left(J_1^{LUV2}-\lambda I\right)=0$ which results in the following characteristic polynomial.

$$0=\left(-r+{\sigma }_1-\lambda \right)\left({\sigma }_1-{\sigma }_2-\lambda \right)\left(g-r+{\sigma }_1-\frac{\alpha bK{\sigma }_1}{\gamma +\alpha K}\right){(\gamma +\alpha K+\lambda )}^2$$ (36)

The eigenvalues are λ₁ = σ1 − r, λ₂ = σ₁ – σ₂, λ₃ = −(γ + αK), λ₄ = −(γ+ αK), and λ₅ = g−r+σ₁ [1 − αbK/(γ+ αK)]. Without inspecting each eigenvalue, we can immediately conclude $P_2^{LUV2}$ is an unstable equilibrium point because eigenvalue λ₂ > 0 since σ₁ > σ₂. Thus, the system will not tend to a state in which lysogen strain L₁ displaces lysogen strain L₂ and the nonlysogens U.

Next, we analyze equilibrium point $P_2^{LUV2}=\left(0,K,0,0,Kb{\sigma }_2/(\gamma +\alpha K)\right)$. This point describes a state in which lysogen strain L₂ (with the lower SPI rate) displaces both the nonlysogens U and lysogen strain L₁. The corresponding Jacobian is given by $J_2^{LUV2}$ in Equation 37.

$$J_2^{LUV2}=\left(\begin{array}{ccccc}-{\sigma }_1+{\sigma }_2& 0& 0& 0& 0\\ -r+{\sigma }_1& -r+{\sigma }_2& -g+\frac{\alpha bK{\sigma }_2}{\gamma +\alpha K}& 0& 0\\ 0& 0& g-r+{\sigma }_2-\frac{\alpha bK{\sigma }_2}{\gamma +\alpha K}& 0& 0\\ b{\sigma }_1& 0& 0& -(\gamma +\alpha K)& 0\\ -\frac{\alpha bK{\sigma }_2}{\gamma +\alpha K}& \frac{b\gamma {\sigma }_2}{\gamma +\alpha K}& \frac{\alpha b^{2}K{\sigma }_2(1-p)}{\gamma +\alpha K}& 0& -(\gamma +\alpha K)\end{array}\right)$$ (37)

The corresponding characteristic equation is given by:

$$0=\left(-{\sigma }_1+{\sigma }_2-\lambda \right)\left(-r+{\sigma }_2-\lambda \right)\left(g-r+{\sigma }_2\left[1-\frac{\alpha bK}{\gamma +\alpha K}\right]-\lambda \right){(\gamma +\alpha K+\lambda )}^2$$ (38)

The eigenvalues are given by λ₁, λ₂ = −(γ + αK), λ₃ = −r + σ₂, λ₄ = −σ₁ + σ₂, and λ₅ = (g −r)−σ₂ [1 − αbK/(γ + αK)]. Eigenvalues 1 – 4 are obviously negative since γ > 0, σ₁ > σ₂, and σ₁, σ₂ < r. Eigenvalue 5 can be rewritten as λ₅ = (g − r)[1 − σ₂/σ_LB] using the definition of σ_LB. The factor (g − r) > 0, but since σ₂ > σ_LB the quantity in the square brackets is negative. As a result, we see that λ₅ < 0. Therefore, all eigenvalues are negative, implying that this is a stable equilibrium point of the system. This means that this system will tend towards a state in which lysogen strain L₂ displaces lysogen strain L₁ and the nonlysogens U.

Finally, we examine equilibrium point $P_3^{LUV2}=(0,0,K,0,0)$ which describes a state in which both lysogen strain L₁ and L₂ are displaced by the nonlysogens U. The Jacobian for this equilibrium point is given by $J_3^{LUV2}$.

$$J_3^{LUV2}=\left(\begin{array}{ccccc}-g+r-{\sigma }_1& 0& 0& \alpha Kp& 0\\ 0& -g+r-{\sigma }_2& 0& 0& \alpha Kp\\ -r+{\sigma }_1& -r+{\sigma }_2& -g& -\alpha Kp& -\alpha Kp\\ b{\sigma }_1& 0& 0& \alpha bK(1-p)-\gamma & 0\\ 0& b{\sigma }_2& 0& 0& \alpha bK(1-p)-\gamma \end{array}\right)$$ (39)

The characteristic equation is given by:

$$\begin{gathered} 0=(g+\lambda ) \\ \times \left[\left(-g+r-{\sigma }_1-\lambda \right)(\alpha bK(1-p)-\gamma -\lambda )-\alpha bKp{\sigma }_1\right] \\ \times \left[\left(-g+r-{\sigma }_2-\lambda \right)(\alpha bK(1-p)-\gamma -\lambda )-\alpha bKp{\sigma }_2\right] \end{gathered}$$ (40)

The first eigenvalue is given by $(g+\lambda )=0\Rightarrow {\lambda }_1=-g<0$. The remaining eigenvalues are given by solving the quadratic equations within both square brackets [·]. Each expression can be rewritten in the form 0 = λ²−λθ+ε with solutions $\lambda =\theta /2\pm \sqrt{{(\theta /2)}^2-\epsilon }$. The variable θ = [−g + r − σ_i + αbK(1 − p) − γ] and ε = [(−g + r − σ_i)(αbK(1 − p) − γ) − αbKpσ_i] for i = 1, 2 corresponding to each square bracket in Equation 40. It is straightforward to show that if ε < 0 then one of the eigenvalues λ must be positive. If we suppose instead that ε > 0 we find that the following inequality must hold −(g − r)(αbK(1 − p)/γ − 1) > σ_i (αbK/γ − 1). However, we know that the factors (g−r), (αbK(1 − p)/γ − 1), and (αbK/γ − 1) are all positive. Thus, the only way this inequality could hold is if σ_i < 0 which is unphysical. As a result, we conclude that ε must be negative, which implies that some λ values will be positive, causing this equilibrium point to be unstable. From all of this we conclude that system will not tend towards a state in which the nonlysogens dipsplace both lysogen strains L₁ and L₂.