Population Dynamics Model and Analysis for Bacteria Transformation and Conjugation
PPopulation Dynamics Model and Analysis for Bacteria Transformation andConjugation
J.J. Dong, J.D. Russo, K. Sampson ∗ Department of Physics and Astronomy, Bucknell University, Lewisburg, PA, USA 17837 (Dated: September 16, 2020)We present a two-species population model in a well-mixed environment where the dynamicsinvolves, in addition to birth and death, changes due to environmental factors and inter-speciesinteractions. The novel dynamical components are motivated by two common mechanisms fordeveloping antibiotic resistance in bacteria: plasmid transformation , where external genetic materialin the form of a plasmid is transferred inside a host cell; and conjugation by which one cell transfersgenetic material to another by direct cell-to-cell contact. Through analytical and numerical methods,we identify the effects of transformation and conjugation individually. With transformation only,the two-species system will evolve towards one species’ extinction, or a stable co-existence in thelong-time limit. With conjugation only, we discover interesting oscillations for the system. Further,we quantify the combined effects of transformation and conjugation, and chart the regimes of stableco-existence, a result with ecological implications. ∗ [email protected] a r X i v : . [ phy s i c s . b i o - ph ] S e p index Reaction Description Typical values and references1 S b S −→ S S growth ∼ . − [9, 10]2 R b R −→ R R growth ∼ (0 . . · b S [11]3 S + P α ( P ) −→ R transformation ∼ − - 10 − hr − [12, 13]4 S + R γ −→ R conjugation ∼ . × − hr − [14]5 R δ −→ P R lysis, releasing a copy of P estimated in modelTABLE I. Interactions involved in the model and the reference reaction rates from experimental studies on various organisms. I. INTRODUCTION
The synthesis of large varieties of antibiotics is one of the most important medical inventions in human history.However, the emergence of antibiotic-resistant bacterial population, infamously known as the “superbugs,” poses analarming challenge to public health authorities [1, 2], whose efforts in combating resistance are outpaced by the rapiddevelopment of new resistance in the bacteria cells [3–5]. Bacteria become resistant to antimicrobial agents as aresult of exchanging genetic materials. Most of the bacterial DNA is contained in the chromosome, which providesthe genetic identification for each cell. In addition to the chromosome, there are plasmids – small circularized DNAmolecules independent of chromosomes – that have essential genes for plasmid functions and accessory genes [6, 7].It is the accessory genes that can confer antibiotic resistance to the host bacteria. There are two main mechanismsof horizontal gene transfer (HGT) through which an individual bacterium cell acquires antibiotic resistance fromplasmids: transformation , where a cell incorporates a plasmid from its surroundings, and conjugation , where a plasmid-carrying cell transfers the plasmid (a single strand of the plasmid DNA) to a plasmid-free one by direct cell contact[8]. A schematic of the two processes is shown in Fig.1. The plasmids are transcribed and translated by the cellularmachinery of the host cell and thus propagate in the population through cell division. transformationconjugation
FIG. 1. Schematics of the two resistance acquisition mechanisms: transformation (top) where the plasmid (circle) in theenvironment is incorporated into the bacterium, and conjugation (bottom) where a plasmid-carrying cell transfers the plasmidto a plasmid-free one through a bridge-like appendage (thick line) on the cell surface.
Due to the short cell doubling cycle of bacteria, a small initial fraction of plasmid-carrying cells will be rapidlyamplified in the population through division. There is an obvious advantage for plasmid-carrying cells in the presenceof antibiotics. In the absence of antibiotics, however, the plasmid-carrying cells can still quickly outnumber theplasmid-free ones as evidenced by both laboratory and clinical findings [15–17]. The domination of either genotypedepends on the respective doubling time as well as the rates of transformation and conjugation, which vary accordingto the plasmid concentration, the medium and the type of bacteria population. It is therefore crucial to quantitativelychart the parameter space and identify how the two populations compete under different conditions, which will bringfurther insights into the emergence of the resistance-dominant population.The dynamics of biological systems has been extensively studied for decades using nonlinear differential equationssuch as the venerable Lotka-Volterra equation [18, 19], agent-based modeling [20, 21], and numerous tools developedin statistical physics [22, 23]. Such interdisciplinary efforts have brought deeper insights into understanding biolog-ical systems of different scales, ranging from neuron networks to global pandemic spread, as well as expanding theunderstanding in generic complex systems. In this study, we take the biological processes of transformation and con-jugation among bacteria cells as motivation, and construct a theoretical model for two growing bacterial populationsof different genotypes inhabiting a well-mixed environment (e.g. a shaken flask): the plasmid-carrying cells that areantibiotic-resistant ( R ), and the plasmid-free ones ( S ) that are susceptible to antibiotics and can be converted to R through either transformation or conjugation. The competition between the two sub-populations depends on thedynamics of the two HGT mechanisms and their respective doubling time, which reflects the plasmid carriage costwith varying benefit according to environmental conditions such as the level of antibiotics. By using a combinationof deterministic calculation and numerical methods of solving coupled nonlinear differential equations, we examinethe effects that these parameters have on the overall population in Sections II and III. One of the main questions weattempt to answer in this study is with the interplay of transformation and conjugation in various growth regimes,how the parameter space is configured such that the overall system displays one species’ extinction (or “fixation” bythe other, which is commonly used when the changes are genotypic) or coexistence with one species dominating. Wesummarize and discuss the biological relevance in Section IV. II. POPULATION DYNAMICS MODEL
We formulate a model at the population level that incorporates growth, death and the mechanisms of HGTbetween the S and R populations in a well-mixed culture. Initially, the system is seeded with S susceptible and R resistant cells, which have growth rates b S and b R . The microscopic growth rates of S and R are accessible throughexperiments by measuring the population doubling time µ S/R of the two genotypes: b S/R = ln 2 · ( µ S/R ) − .We focus on two HGT mechanisms through which cells develop antibiotic resistance: transformation and conju-gation . When there is a large number of plasmids ( P ) in the environment, the transformation rate depends on thecell’s innate ability to take up extracellular DNA. Therefore we choose a constant transformation rate α ( P ) = α to model the unlimited plasmid supply or that the plasmid has a very high affinity to the cell. Accounting for theavailability of free plasmids P in the environment and the kinetics of plasmid uptake, we also study the scenariowhere the transformation rate depends linearly on P when the concentration of plasmids is low and transitions to theconstant rate as P increases, namely α ( P ) = α P/ ( P + K P ). Here K P reflects the plasmid affinity to the cell andis the concentration of plasmids at which transformation occurs at the half-maximum rate. This form is analogousto the Michaelis-Menten kinetics[24] in enzymatic activities. Other P -dependence scenarios can be studied with asimilar approach.During the process of conjugation , the double stranded plasmid DNA in the donor cell becomes two single strands,one of which is transferred from the donor to the recipient through direct cell-to-cell contact mediated by a tubelikestructure called “pilus” that pulls the recipient and perforates it [14, 25]. Once the single strand of plasmid DNA is inthe recipient, the DNA replication process restores the complementary strand and both the donor and the recipientpossess the entire plasmid DNA. We model the conjugation process with constant rate γ in the well-mixed population.Finally, R cells lyse with rate δ . Unlike the phenotypical switching where a bacterium changes from a normalcell to a persistent cell in the presence of antibiotics with the same genetic content [26], a resistant cell dies whenits cell wall breaks down. An R -cell is then removed from the population upon lysis and the plasmid it carries isreleased back to the environment. In the plasmid-limiting scenario, the recycling of plasmids proves to be significantin maintaining a resistant population. When there are antibiotics of various concentrations in the environment, thelysis rates of S and R populations lead to more interesting dynamics which we will not focus on in this article.The above reactions are summarized in Table I. By using a combination of theoretical analysis for insightsand numerical exploration for a wider range of parameters, we aim to provide a comprehensive picture of the rolestransformation and conjugation play. To connect our theoretical model to experiments, we also include typical valuesof the parameters, such as b S , b R , α and γ , in our model. However, the cell lysis process is usually harder to ascertainand we will estimate the parameter δ in our model. More details and discussions will be included in Section III.The dynamics of the two populations, S ( t ) and R ( t ), can be described by the following set of equations afterproperly non-dimensionalize the parameters without loss of generality: dSdt = b S S − α ( P ) S − γSR (1) dRdt = b R R + α ( P ) S + γSR − δR (2) It is possible for S cells to lyse as well. However the death of S effectively reduces b S , and we do not need introduce an additionalparameter here. In addition, the plasmids in the environment vary due to the uptake by S cells, as well as the release by R cellsupon their lysis: dPdt = − α ( P ) S + δR, α ( P ) = α PP + K P (3)To analyze the above set of coupled differential equations, we used a combination of theoretical analysis andnumerical solvers. For the latter, we coded the equations using Python (v3.8) [27]. We used Euler’s method [28] andthe ODE solver odtint from Scipy (v1.5) [29] to perform the numerical simulations, which yielded consistent results.The numerical results presented below are from Euler’s method.Before delving into a systematic analysis of the full model, we analyze the results of two limiting scenarios whichprovide guidance on the choice of parameters in the later simulations. A. Transformation with constant α = α We first consider the transformation-only case, namely γ = 0, and the transformation rate is a constant α . Thiscorresponds to the case where the environment concentration of plasmids is very high, or K P is very small. With theeffective growth rate ˜ b S ≡ b S − α , the dynamics of the S population is an exponential function S ( t ) = S exp (˜ b S t ).Similarly, the growth of the R population involves an exponential term with ˜ b R ≡ b R − δ and the influx of thetransformed S cells: R ( t ) = α ˜ b S − ˜ b R S ( t ) + (cid:20) R − α ˜ b S − ˜ b R S (cid:21) e ˜ b R t = R e ˜ b R t + α ˜ b S − ˜ b R S (cid:104) e ˜ b S t − e ˜ b R t (cid:105) The ratio between the two cell populations f ≡ R/S is given by: f ( t ) = α ε + (cid:20) R S − α ε (cid:21) e − εt , (4)where ε ≡ ˜ b S − ˜ b R , the difference between the two effective growth rates.When ε <
0, namely the effective growth rate of R is greater than that of S , then R population fixates, or f ( t ) = ∞ , in the long-time limit with a characteristic time scale of | /ε | , shown in Fig.2. FIG. 2. The ratio between the two cell populations f ( t ) for different ε <
0. Parameters used: b S = 1 . , b R = 0 . , S = R = 1with ( α , δ, ε ) = (0 . , . , − .
6) for the dotted line, (0 . , . , − .
4) for the dashed line, and (0 . , . , − .
2) for the solid line.Inset: Growth curves of S and R . For the case of ε = 0, f ( t ) = R /S + α t , indicating that the dominance of R increases linearly with t , slowerthan the ε < /f ( t → ∞ ) = 0, leading to R -fixation. Fig.3 shows the linearincrease in f ( t ), with slopes given by α . Note the different scales of f ( t ) in Figs. 2 and 3.When ε >
0, the system displays an S - R coexistence as f ( t → ∞ ) = α /ε . Either S or R can be the majority inthe entire population depending on the competition between transformation and growth differentials: In the case of FIG. 3. The ratio between the two cell populations f ( t ) for ε = 0. Parameters used: b S = 1 . , b R = 0 . , S = R = 1 with( α , δ ) = (0 . , .
7) for the dotted line, (0 . , .
6) for the dashed line, and (0 . , .
5) for the solid line. Inset: Growth curves of S and R . R -dominance, α > ε . Together with the S - R coexistence condition ε >
0, this gives α ∈ (cid:16) ( b S − ˜ b R ) / , ( b S − ˜ b R ) (cid:17) .And similarly, S -dominance is reached when α ∈ (cid:16) , ( b S − ˜ b R ) / (cid:17) . In Fig.4, we show two cases where R is themajority in steady state (dotted and dashed lines), and one where S dominates (solid line). FIG. 4. The ratio between the two cell populations f ( t ) for different ε >
0. Parameters used: b S = 1 . , b R = 0 . , S = R = 1with ( α , δ, ε ) = (0 . , . , .
2) for the dotted line, (0 . , . , .
3) for the dashed line, and (0 . , . , .
5) for the solid line. Inset:Growth curves of S and R . B. Conjugation with constant γ Having seen that transformation-only leads to either R -fixation or stable coexistence, we now turn to the com-plimentary case where α ( P ) = 0 and conjugation is the only HGT mechanism. Eqns.(1) and (2) contain a non-linear term, similar to the Lotka-Volterra type interaction [18, 19]. We find two fixed points: ( S ∗ , R ∗ ) = (0 ,
0) and( S ∗ , R ∗ ) = ( − ˜ b R /γ, b S /γ ).For there to be a coexistence regime where both S and R are non-negative, ˜ b R must be negative. In this case,( S ∗ , R ∗ ) is a saddle point which S evolving away from and R into. Furthermore, from Eqns.(1) and (2) we find thetrace and the determinant of the Jacobian matrix at ( S ∗ , R ∗ ) to be 0 and − b S ˜ b R , respectively. Using the standardstability analysis [30], we find that ( S ∗ , R ∗ ) is a neutrally stable center.In this case, the ratio f ∗ ≡ R ∗ /S ∗ = − b S / ˜ b R does not depend on the conjugation γ. What maybe slightlycounterintuitive is that for f ∗ <
1, i.e. a center with more S -cells than R , b S is constricted to be in (0 , − ˜ b R ),rather than a greater value. We will return to this point in the next section when we consider the combined effect oftransformation and conjugation to the overall population.To determine the individual closed orbits, we first eliminate t by dividing dS/dt by dR/dt and then separate thevariables: (˜ b R S + γ ) dS = ( b S R − γ ) dR (˜ b R ln S + γS ) − ( b S ln R − γR ) = C, where C is the constant of integration. We now have an expression to describe the trajectory for the conjugation-onlysystem: R b S S − ˜ b R e − γ ( R + S ) = e C (5)We show in Fig.5 the phase portraits of the two sub-populations for ˜ b R < . Setting b S = 1 and varying theratio between − ˜ b R and γ using the range of conjugation rates in Table I, we see the competition between growth andconjugation leads to different trajectories, all around the center ( S ∗ , R ∗ ), indicated as a red dot. FIG. 5. Phase portraits for S and R . b S = 1 , γ = 10 − and − ˜ b R /γ = 1 (left) and 10 (right). The red dot indicates the centerof the orbits ( S ∗ , R ∗ ). In Fig.6 we observe the oscillations in S and R on one of the closed orbits. The fact that S - R can coexist when˜ b R < R -cells is through converting more S -cells by conjugation. Acloser comparison between conjugation-only and conjugation with transformation is provided in the following section. FIG. 6. Top: Oscillations of the S and R population (color online) with conjugation only. Bottom: The correspondingoscillations in the sub-population ratio f ( t ). Parameters: S = R = 1 , γ = 0 . , b S = 1 . b R = − . III. COMBINED EFFECTS OF TRANSFORMATION AND CONJUGATION
Informed by the above discussions on the individual effects of transformation and conjugation, we study themodel of which the full dynamics is described in Eqns.(1)-(3) with a range of parameters to chart the different regimesof the sub-populations of S and R cells, with special attention on the S - R coexistence conditions.From the discussions in Section II B, we learned that a physical coexistence of S and R cells requires b R < δ inthe conjugation-only case. With transformation included, first let’s consider the case of constant transformation, or K P (cid:28) P . Thus α ( P ) → α and the new non-trivial fixed points are:( S ∗ , R ∗ ) = (cid:32) ( α b S − · ˜ b R γ , b S − α γ (cid:33) ≡ (cid:32) − ˜ b S ˜ b R γb S , ˜ b S γ (cid:33) , (6)with f ∗ = − b S / ˜ b R . The fixed point at (0 ,
0) remains a saddle point. It is worth noting that the value f ∗ is the sameas in the conjugation-only case. Meanwhile, the overall population of the system approaches ˜ b S γ (cid:32) − ˜ b R b S (cid:33) .Combining transformation and conjugation brings new behaviors in the overall system. Unlike the conjugation-only case where S and R evolve along a closed orbit centered around ( − ˜ b R /γ, b S /γ ) with temporal oscillations, asshown in Figs.5-6, once transformation is introduced, these orbits turn into spirals. We find an S - R coexistence when˜ b R < b S >
0. To determine the stability of ( S ∗ , R ∗ ), we can again examine the Jacobian at this point: the traceand the determinant is ˜ b R αb S < − ˜ b S ˜ b R >
0. Thus ( S ∗ , R ∗ ) will be a stable node or a stable spiral. However,for it to be a stable node, ˜ b R > − b S /α ) ˜ b S . This means when ˜ b R is almost one order of magnitude greater than˜ b S , ( S ∗ , R ∗ ) is a stable node. In other cases where ˜ b R ∼ ˜ b S , the system has a stable spiral towards ( S ∗ , R ∗ ).In Fig.7, the system starts with S = R = 1 and the initial oscillations of the two sub-populations quickly settlesinto ( S ∗ , R ∗ ) = (4 . , S ∗ , R ∗ ). We also show the closed orbit for the same set of parameters except for α = 0 as a reference. Aspredicted, the S - R coexists with a stable ratio of f ∗ = 2 . S and R when transformation is introduced ( α = 0 . S and R , which does not affectthe final coexistence ratio f ∗ . FIG. 7. Timetrace of the S and R population with S = R = 1 , P = 1 , K P = 0 . P , α = 0 . , γ = 0 . , b S = 1 . b R = − . FIG. 8. The trajectory of S - R spirals towards ( S ∗ , R ∗ ) with S = R = 1 , P = 1 , K P = 0 . P , α = 0 . , γ = 0 . , b S = 1 . b R = − .
5, and reaches f ∗ = 2 .
0. The dashed line shows the stable orbit when α = 0 as a reference. As alluded to in Section II B, for S to be the majority in the overall population in steady state, the growth rate b S needs to be less than − ˜ b R . For example, when b S = 0 . − ˜ b R = 0 .
5, as in Fig.9, the final S - R coexistence ratio f ∗ = 0 .
8. When b S < − ˜ b R , even though S is doubling at a lower rate, that also leads to fewer S being conjugated ortransformed into R . The final coexistence state contains more S than R cells. FIG. 9. Left: Timetrace of the S and R population with S = R = 1 , P = 1 , K P = 0 . P , α = 0 . , γ = 0 . , b S = 0 . b R = − .
5. Right: The trajectory of S - R spirals towards ( S ∗ , R ∗ ) with f ∗ = 0 . Next let’s turn to the case where transformation rate is no longer a constant. Here two parameters are at play: theinitial concentration of plasmids in the environment, P , and the transformation kinetics K P . The former obviouslydepends on the condition of the growth medium, while the latter is a biochemical parameter where a higher K P meansa lower plasmid-bacteria affinity. The dynamics of the plasmids in this case is given in Eq.(3).If the lysis rate δ is sufficiently large, or R is the dominating species, then the plasmid concentration willeventually build up and it becomes similar to the α ( P ) = α case as discussed above. However, the transient behaviorin this regime contains some interesting details. For an environment with few low-affinity plasmids, K P (cid:29) P , thetransformation rate is: α ( P ) = α PP + K P ≈ α PK P . The stability condition with conjugation-only case also requires δ > b R while transformation with the low-affinityplasmids is slow. The buildup of plasmids – consequently α ( P ) – therefore sees a “burst”-like behavior, shown inFig.10, where we look at low-affinity plasmids with K P = 10 P . The bursts coincide with the decrease of R due tolysis and increase the fastest when S reaches its minimum so that dP/dt is the greatest. Additionally, the burst cycletracks the S - R oscillations. FIG. 10. Timetrace of S and R , along with plasmid concentration P with its scale to the right. Parameters used: S = R =1 , P = 1 , K P = 10 P , α = 0 . , γ = 0 . , b S = 1 . b R = − . IV. SUMMARY AND DISCUSSION
We studied a two-species population model aimed to explore the interplay between bacterial transformation andconjugation in addition to their natural growth cycles. Using a combination of theoretical and numerical analysis,we analyzed the complete effects of transformation: the overall population will either be fixated by R -cells or instable co-existence. In the latter case, the majority cell type is determined by the transformation rate α and ε , thedifference between the effective growth rates of the two sub-populations. The conjugation-only case gives R -fixation,unless when the effective growth rate of R -cells ˜ b R is less than 0, in which case we can actually observe a stable orbitfor S and R , with each sub-population oscillating. This could potentially be interesting in the biologically-relevantcontext: carrying an additional plasmid confers growth disadvantage for R -cells as ˜ b R <
0, they can neverthelesssurvive steadily only by conjugating more S to R . More interestingly, when transformation and conjugation are bothconsidered, the orbit seen in the conjugation-only case turns into a spiral with a stable center, indicating a steadystate where the final ratio between the two sub-populations f ∗ reaches a constant.In our numerical analysis of the model, the reaction rates tested are not directly comparable to the experimentallyobtained ones listed in Table I because the equations of the system were first non-dimensionalized. However, ourresults point to the important regimes where interesting behaviors, such as stable co-existence and oscillation, emergewhen the relative reaction rates are taken into consideration. In particular, the difference between the two effectivegrowth rates ε in the case of transformation, and the ratio between the two effective growth rates characterized by f ∗ = − b S / ˜ b R in the case of conjugation are direct indication on the overall population behavior.From a modeling perspective, our findings are novel in that we quantitatively analyze the effects of two mainhorizontal gene transfer (HGT) mechanisms at a population level. The results are significant especially with the useof a minimal model with few parameters, mostly experimentally accessible. The lysis rate δ is the only estimatedparameter here, due to its elusiveness because cell lysis often involves various other environmental and physiologicalfactors. And yet it is crucial in determining whether the system has a steady state, or which sub-population dominates.We thus continue to seek experimental breakthroughs on further clarification on the lysis process.The results from this study are potentially helpful to a wider range of explorations. In our study, there is nolimit on the population growth aside from the dynamical parameters. It is conceivable to impose an environmentalconstraint such as a carrying capacity and analyze the difference. For this study, the effects of antibiotic resistanceis binary, in the form of carrying the plasmid, and deterministic. It is possible for cells to lose a plasmid withoutlysing (e.g. [31]), in which case there needs to be an additional reaction to capture R −→ S + P process. And interms of stochasticity, our preliminary study using kinetic Monte Carlo shows consistent results with what we presenthere, and is more powerful to generalize to more reactions with other complications. Further extensions to includethe population spatial structure beyond the existing well-mixed population are also being considered. The spatialaspect is especially compelling to investigate because both transformation and conjugation processes depend on localavailability of free plasmids as well as direct contacts between S and R cells. Thus the diffusion of plasmids as wellas the cells are expected to be relevant, an aspect not necessary for the well-mixed case.0 ACKNOWLEDGMENTS
The authors acknowledge the financial support from National Science Foundation grants DMR-1248387 andDMR-1702321. JJD is grateful for the hospitality of Dr. Stefan Klumpp at the University of G¨ottingen where theinitial stage of this research was carried out. The manuscript benefitted from fruitful discussions with M.D. Eichenlaub. [1] J. Davies and D. Davies. Origins and evolution of antibiotic resistance.
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