Phage-Bacteria Coexistence

Phage-Bacteria Coexistence

AVINOAM RABINOVITCH1, IRA AVIRAM2 AND ARIEH ZARITSKY3

Departments of Physics1 and Life Sciences3,Ben-GurionUniversity of the Negev,

Beer-Sheva, 84105, Israel, and

35 Sderot Ye’elim, Be’er-Sheva, 84730, Israel2

Abstract: -Experiments show that, although virulent phages greatly multiply inside, then break down the host bacteria and ‘savagely’ attack additional bacteria, a lasting equilibrium is finally reached in which a (small) number of susceptible bacteria and a (large) number of phages coexist.

We propose [1]a model for bacterial survival by shielding enacted by bacterial debris which the phage considers to be live bacteria,and die in the attempt to invade the bacterial debris.The model is formulated by a set of four difference equations for susceptible bacteria, contaminated bacteria, bacterial debris and phages.

Simulation results indeed manifest the possibility of coexistence which would have been unattainable without such ‘debris shielding’ and exhibit the domains of existence of stable and unstable solutions of the problem.Under some conditions temporal oscillations appear which can lead either to extinction or to a final equilibrium.

Key-Words: -Debris, Survival, Bacteriophage, Environment, Basin of Attraction, Fixed Points

1. Introduction

Coexistence of virulent bacteriophages (phage) with bacteria is a situation observed in nature and experimentally in batch and chemostat cultures. Theoretically, however, this situation seems improbable if not completely impossible, since both phage multiplication is very fast and its virulence is extremely high. Several models have appeared in recent years that tried to solve this apparent discrepancy (see e.g. [2] and references therein): existence of resistant bacteria, numerical refuge, transitory immuneness and spatial refuge. Analyzing several “refuge” possibilities, Schrag & Mittler concluded that spatial refuge was the preferred survival mechanism [2],a conclusion which was refuted by Bohannan and Lenski [3].

The resistant bacteria hypothesis (e.g. [4]) is based on the spontaneous appearance of resistane mutants[5] which survive while susceptible cells disappear. According to this model, an “arms race” then develops as phage “device” new means of attacking resistant bacteria, the latter mutate into even more sophisticated resistance et cetera. This hypothesisdoes not agree with experimental results since the process is too slow to account for the equilibrium situation which is reached, where susceptible bacteria, albeit of small concentrations coexist with a (large) concentration of phage (e.g. [6]). A different occurrence is the appearance of oscillations in the concentrations of both phage and bacteria. For example, Bohannan and Lenski [7] observed that just by increasing nutrients input to the culture, prolonged oscillations ensued without population extinction.

We have suggested A completely different solution to the problem has recently been proposed[1], according to which the bacteria bursting after lysis by the phage remain in the cultureas debris [8]. As is shown, this constitutes a shielding mechanism by which a phage can mistake the debris to be a live bacterium and inject its DNA into it thus in effect committing suicide. Since this three-prongedbacteria-phage-debris interaction is quite complicated, different outcomes can result depending on the parameters used in the mathematical description and on the initial conditions.

1. Discrete Model

We assume that resources are abundant and practically inexhaustible hence they do not appear in the equations. Phage multiplication starts by penetration of its DNA into a sensitive bacterium, where it spends a latent period, after which the infected bacterium bursts, releasing a number of new phages into the system. We consider here a “generation” model whereby all bursts and new infections occur at discrete events separated by , and the time evolution of the concentrations of all species in the system is monitored only at these time points. Resistant bacteria as well as statistical changes and spreads, usually accounted for in other models, are ignored here. The system consists of four species with the following concentrations at time step g: sensitive () and infected () bacteria, debris (), and phages ().

The controlling, difference equations of these species are easily obtained to be:

(1)

(2)

(3)

(4)

where all rates are calculated per time step; is the bacterial multiplication rate; is the debris dissolution rate; is the natural death rate of the viruses and is the burst size—the number of phages that come out of a lysed bacterium.

Assuming that the adsorption rate is the same on all infectable species N, M, D, one can rescale all concentrations by multiplying Eqs. (1)-(4) by , redefining , etc., and replacing with by Eq. (2).The following set of difference equations is thus obtained:

(5)

(6)

(7)

(8)

where .

The interesting fixed point of this system

(9)

is obtained by taking etc. in Eqs. (5) – (8). The second fixed point is of course the null solution, .

Before analyzing the stability of Eq. (9), let us first consider the case where the debris is neglected, i.e. we discard Eq. (7) and from Eq. (8). The non-zero fixed point becomes

(10)

The characteristic equation for the eigenvalues of the linearized system is now of degree three, for the roots of which there exist explicit, albeit not particularly interesting, algebraic formulae. Instead, we conducted a thorough mapping and solved the eigenvalue problem of the system numerically on a sufficiently dense grid of points. It is well known that the roots of such a problem can be either all three real, or one real and a pair of complex conjugates. The mapping results were quite revealing: over the whole examined grid of points we invariably found at least one eigenvalue for which , making the corresponding fixed point unstable. It can therefore safely be concluded that the system becomes unstable in the absence of debris.This means that no stable steady state solution can exist (under our assumptions) with no debris present. And in fact, numerical tests with no debris present led invariably to final extinction of the bacteria followed by a slow exponential decay of the phages.

On the other hand, a stability analysis of the case in which debris is present (Eq. 9) does lead to a stable steady state solution of Eqs. (5) – (8). The analysis is rather complicated as the characteristic equation of the linearized system is of the 4th order.While it is hard to visualize the entire 4-dimensional space of all parameters, we have chosen to represent some of the stability properties in the subspace for several discrete values of q and . For the T4-Escherichia coli system [9, 10], varies roughly between 2-200 while and yield . Due to their high viability, the dissolution time constant of the phages is small, and a value of per generationis used throughout. The values of q were chosen arbitrarily in these calculations.In Fig. 1 the curves of the stable/unstable transition limit (solid lines) are shown for a set of three values of q, 8 x 10-4, 1.5 x 10-3 and 3 x 10-3. The dashed lines represent the transition in Eq. (9),, to an non-real situation where is negative.

Fig. 1. The parameters’ subspace showing the loci curves (solid lines) across which a transition occurs between regions (s/u) of stable and unstable fixed points of the system, for different values of . Also shown (dashed lines) are the loci of singularity of . (From [1].)

1. Numerical Simulations

In order to understand the qualitative features of this model, we subsequently present the time evolution of the component species, by direct simulation of the time series of the system for a few typical conditions. Only the species N, D and Vare displayed since Mremains very small at all times and its evolution is inconsequential. All simulations were carried out at the point P in the plane, i.e. , Fig. 1.

3.1.An unstable system

We begin with a typical evolution of an unstable system (Fig. 2). The coordinatesNp and Dpof the corresponding unstable fixed point, Eq. (9), are negative.

Fig. 2. A diverging oscillatory time series, corresponding to the point P in Fig. 1(). The fixed point is unstable. Initial conditions are:, , . Species N and V eventually die out abruptly, while D decays exponentially. (From [1].)

Initiated at, and, this simulation illustrates the progress of oscillations during three stages, typified by the amplitude of N: the first starts with a relatively large amplitude oscillations which decrease until, in the second stage, they become again progressively larger. Stage three is marked by a complete and very rapid collapse of both bacteria and phages, and slow exponential decay of D. Under varying initial conditions the species populations may oscillate more or less widely, and the duration of the series may be more or less extended. However, since the fixed point is unstable, all species eventually become extinct.

3.2.A stable system

Next, we consider a stable system for, which situates P above the stable/unstable locus. The stable fixed point is a focus at: , , . The “coexistence ratio” defined here as , in good agreement with experimental results (e.g. [11]). Note that the approach of the dynamical solution towards a stable focus always presents a damped (convergent) oscillatory behavior. Initiated at , and, the time evolution in this case is represented in Fig. 3. The first minimum of any convergent oscillatory N-series, indicated by anarrow, is always the deepest of all.

Fig. 3. A stable focus time series, corresponding to the point P in Fig. 1 and . The coordinates of the fixed point are:,, . Initial conditions, , and, are situated inside the basin of attraction. The series converges towards the fixed point. (From [1].)

The precise boundaries of the basin of attraction (BA) of a given fixed point F are very hard to determine, even partially in the two-dimensional subspace (N,V). It may be infinite or have a finite extent with a very complicated shape. However, since we are dealing with a model of a real system, inherently unable to capture all its details, it is reasonable to impose additional constraints on the BA size, inferred from an understanding of the real system. For example, we may assume that if, during its evolution in time, the population of bacteria N falls below a certain low level, it should be regarded as practically extinct. As a matter of illustration we choose a lower limit of , which, due to the scaling by , corresponds to about 3 individualsml-1 in the sample. Thus, when the the value of N drops below this limit during its first down swing, the species becomes “extinct”, thereby drastically reducing the extent of the corresponding BA.

Fig.4 represents the map of such a “biological” BA of F (diamond) in the (N,V) plane. It was construced by simulating a large number of time series initiated at various points (N0,V0) in the plane. All series initiated inside the contour stayed above the lower limit of 10-10, while those starting outside dropped below this value. A sample of such initial points is shown around the contour, the adjacent negative numbers –s indicate the values 10-s below which the first minima of the corresponding N-seriesdrop. It is seen that the boundaries of BA extend over more than three orders of magnitude of N, while the V range is extremely narrow. The system becomes unstable for initial N populations above 10-4.

Fig. 4. The BA (—–) of the stable focus F2. (See text for definition)., the initial point corresponding to the time series of Fig. 3. , initial points outside theBA, while the adjacent negative numbers –s indicatethatthe first down swing of the correspondingly initiated time series drops bellow 10-s. (From [1].)

1. Conclusion

The relatively simple mathematical model developed in this work has shown that the presence of bacterial debris in the bacteria-bacteriophage system alters its asymptotic solution from an unstable focus to a possibly stable one, thus becoming a haven of survival and coexistence of both species.

References

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