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MBW:Effects of Persister Formation on Bacterial Response to Dosing

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A review of the article Effects of persister formation on bacterial response to dosing by N.C. Cogan


Summary

Biofilms have become a major focus of study within the past few decades when examining bacterial growth as they are responsible for several chronic diseases that are difficult to treat. The fact that biofilm bacteria show a much greater resistance to antibiotics than their free-living counterparts has lead N.G. Cogan to investigate the mechanistic basis of this phenomenon. His study is based on the hypothesis that specialized ‘persister’ cells, which are extremely tolerant of antimicrobials, are the source of resistance for bacteria living within biofilms. Cogan uses results from a series of experiments describing the dynamics of persister cells for his mathematical model of ordinary different equations to demonstrate the problematic bacterial resistance and the positive effects of a periodic dosing regiment to kill all the susceptible and persister cells. Analysis indicates that the relative dose/withdrawal times are important in determining the effectiveness of such a treatment.

Background

In most natural settings, bacteria are found predominantly in biofilms. The fact that biofilms can be found in a myriad of places, from water pipes to indwelling devices in hospital patients, has led to an increased interest in investigating the molecular mechanisms underlying the formation and maintenance of these communities. Bacteria that attach to surfaces aggregate in a hydrated polymeric matrix of their own synthesis to form biofilms. These adherent cells are embedded within a self-produced matrix of exoplysaccharides (EPS), proteins, and sometimes nucleic acids. Biofilms then protect bacteria from an array of environmental threats such as antibiotics, predators, and the human immune system.

JBiofilm.jpg

There are currently three different hypotheses concerning resistance mechanisms that can be placed into three broad categories: transport limitation, physiological tolerance and phenotypic resistance. Based on the results of prior investigations that have eliminated transport limitation and physiological tolerance as possible means of bacterial resistance to antimicrobials, Cogan focuses on physiological tolerance via persister cells.


Bacterial populations produce cells that neither grow nor die in the presence of microbial antibiotics. Persisters are largely responsible for high levels of biofilm tolerance to antimicrobials, leading Cogan to target these cells. The presence of antimicrobials causes some susceptible cells to become persister cells. Then in the absence of the medication, the persisters revert back to the original cells that are susceptible to the antimicrobial.

Mathematics Used

Cogan utilizes two models:

  • Nonlinear and non-autonomous system of ODEs
  • Linear and non-autonomous system of ODEs

The first system allows Cogan to create a number of simulations of various combinations of dosage and withdrawal times that are intended to provide an accuracy guide for the second system. This second system is a reduced version of the nonlinear equations that can be solved analytically and the dynamics of the populations of susceptible and persistent bacteria reduced to a two-dimensional map. The success of the dosing strategy is determined by the eigenvalues of the map.

Mathematical Model

The following mathematical model describes the dynamics of the susceptible and persister bacterial cells with one growth limiting substrate. In the absence of antibiotic, the susceptible bacteria consume substrate and reproduce. When the antibiotic is added, a fraction of the susceptible cells are killed while another fraction converts to persister cells. Persister cells are not killed by the antibiotic, nor do they grow. Instead, if there is no antibiotic, persister cells revert to susceptible cells at a fixed rate.


The following variables are used: Bs - susceptible density Bp - persister density S - growth-limiting substrate A - one antibiotic

Below is a table that lists the parameters and values. Cogan uses typical values for maximum growth rate, yield, and Monod coefficient. The four parameters which depend on the antimicrobial agent, kd, α, kl and kg were estimated.

Cogan1.jpg

The equation governing the dynamics of the susceptible populations is given qualitatively as:

Cogan2.jpg

Growth is described by Monod kinetics with maximum specific growth rate, Monod coefficient and yield denoted μmax, Ks, and Y respectively. Thus the growth term is

Cogan3.jpg

Disinfection depends on the type of antibiotic used. Fleoroquinolones are known to be partially effective in killing non-growing bacteria and beta-lactams kills growing bacteria. Therefore, if the antibiotic is a beta lactam then the disinfection rate is assumed to be proportional to the growth rate. If the antibiotic is a fluoroquinolone we allow for disinfection in the absence of growth, although at a reduced ate. Therefore the disinfection term is

Cogan4.jpg

where α is zero for a beta-lactam and non-zero for a fluoroquinolone. The function kd(A,t) depends on the antibiotic concentration. In particular, kd = 0 if A = 0 and is nonzero otherwise.


Cogan assumes that the loss of susceptible cells to the persister population occurs at a rate that depends on both the growth rate and the antibiotic concentration. Mathematical we have,

Cogan5.jpg

When the population is in the stationary phase there is essentially no persister formation. During the exponential growth there is a relatively high rate of persister formation.


We assume that persister cells only revert to susceptible cells if there is no applied antibiotic. Mathematically we have r(Bp, S, A) = kg(A, t)Bp.


Putting these together gives the equations governing the dynamics of the susceptible population as

Cogan6.jpg

Persister cells are not killed by the antibiotic, instead the population changes as cells convert to and from susceptible cells,

Cogan7.jpg

We assume that substrate is being consumed only by the susceptible population so the equation governing the substrate concentration is,

Cogan8.jpg


Simplified Model

The following simplified model reduces the original model from a system of three coupled nonlinear, non-autonomous equations to a system of two linear, non-autonomous equations by assuming that the nutrient concentration is constant in time. This gives us

Cogan9.jpg

in which

Cogan10.jpg

The solutions to this simplified model need to demonstrate that the antibiotic level switches instantaneously from application to withdrawal. While the antibiotic is being applied, kd and kl are zero, and non-zero during withdrawal. The rate of reversion from persister to susceptible, kg, is non-zero only during withdrawal of the antibiotic. Solutions to the above differential equations with initial conditions Bs(0) and Bp(0) are

Cogan11.jpg

Now we need two sets of equations: the first for the dynamics of the bacteria in the presence of the antibiotic, and the second in the absence of the antibiotic. When the antibiotic is present, we use the above equations for Bs and Bp in which A > 0 evaluated at Td, which is the length of times the antibiotic is applied. This gives the population of each phenotype as

Cogan12.jpg

Because we want each cycle of antibiotic presence and withdrawal to build upon the previous, we must use the above values (Bs(Td) and Bp(Td)) as initial conditions for the dynamics with A = 0 for a length of time Tw (the length of time for antibiotic withdrawal). This gives the new populations

Cogan13.jpg

A matrix M is now needed to map (BS(0), Bp(0)) to (Bs(Tw), Bp(Tw)), since we define one cycle ending by the completion of the withdrawal period. Cogan defines the entries of this matrix as

Cogan14.jpg

Simulation Results & Analysis

The experimental results for this paper are purely a result from simulations of the above mathematical models whose parameters are based on biological experiments. In order to provide a general picture of the dynamics the models should resemble, Cogan provides the following figure of growth-stage dependence of persister formation. The solid curve shows the time course of an unchallenged population of bacteria. At designated times samples were taken and enumerated after 3 h exposure to ampicillin (triangles) or oflaxacin (diamonds). These two antibiotics were chosen because of the cells they target. Ampicillin is a fluoroquinolone, which is known to be effective against non-growing cells. Oflaxacin is a ß-lactam which kills the growing bacterium.

Cogan15.jpg

This figure affirms previously established conclusions about the nature of persister cells in the presence of an antibiotic. The antibiotic is able to prevent the bacteria from growing for the first 3 hours. But when the bacterial growth enters the exponential phase, the persister cells are able to accumulate and create a dominating presence on the population of bacteria. Eventually the level of bacteria levels out as the availability of nutrients diminishes. Simulations of the first nonlinear and non-autonomous model create the same picture. The below figure depicts the growth-stage dependence of persister formation for bacteria challenged with a ß-lactam (growth-rate dependent) antibiotic. The squares represent the cell count before the challenge and the triangles are the cell count after the challenge. The solid curve corresponds to the untreated population, so the disinfection parameters kd and kl are zero. The dashed curve is generated by measuring bacteria at the indicated times and exposing them to a constant level of antibiotic (kd and kl are non-zero) for 3 h and interpolating the data.

Cogan16.jpg

We see an abrupt increase in the number of surviving bacteria at approximately 5 h indicating an increase in the number of persister type bacteria.


The below image depicts growth-stage dependence of persister formation for bacteria challenged with a fluoroquinolone (non-growth-rate dependent).

Cogan17.jpg


Cogan also uses the nonlinear non-autonomous model to create simulations of the dosing experiment. The simulation begins with a population of susceptible cells which are exposed to nutrient and an antibiotic for a fixed length of time, denoted Td. The nutrient is initially twise Ks. The susceptible cells are quickly eliminated and a small population of persister cells in produced. The antibiotic is removed allowing the cells to grow for a fixed length of time, Tw. Persister cells revert to susceptible cells, which then consume nutrient and reproduce. This completes on dose/withdrawal period. After one dose/withdrawal period, fresh nutrient is added and the cycle is repeated.


Below are results from numerical experiments with a dose period of 10 h and varying withdrawal periods: 1 h, 10 h, and 8 h respectively.

Cogan18.jpg

Cogan19.jpg

Cogan20.jpg


For 1 h withdrawal periods, the persister population in being killed very slowly and actually quickly re-grows if the treatment is discontinued. This treatment is unsuccessful. For 10 h withdrawal periods neither the susceptible not the persister cells are eliminated. This treatment is unsuccessful as well as the persisters become a source of susceptible cells once the antibiotic is withdrawn for the 10 h period, which is apparently too long for success. Cogan is able to demonstrate successfully killing all the bacteria with a withdrawal time of 8 h. Now that we know dose/withdrawal treatment can successfully kill all the susceptible and persister bacterium, Cogan solves for the eigenvalues of the map M in the simplified model. Because the elements of M depend on Td and Tw, each combination of Td and Tw has a corresponding pair of real eigenvalues. The success or failure of the dosing strategy can actually be determined by the eigenvalues of M. If the magnitude of each eigenvalue is less than one, the repeated dose/withdrawal will eventually eradicate both populations. Otherwise there will be growth of one or both of the phenotypes, indicating an unsuccessful outcome. The below curve in the (Td, Tw)-axis is for when the eigenvalue is equal one. Above this curve the eigenvalue is less than one, while below this curve the eigenvalue is greater than one.

Cogan21.jpg

The value of the eigenvalue for fixed dose time and varying withdrawal time is also shown. This curve (below) agrees qualitatively with the dose/withdrawal simulations. In particular, for short withdrawal times, we have successful treatment. For long withdrawal times the eigenvalue is greater than one, indicating unsuccessful treatment.

Further Analysis

According to the eigenvalue analysis, a dose/withdrawal experiment with dose time equal to 2 h should eradicate the susceptible and persister bacterium fastest when a withdrawal time of 7.5 h is used. Furthermore, the eradication time should be faster than an 8 h withdrawal period as shown in the successful dose/withdrawal simulation. In order to re-create a dose/withdrawal simulation, SimBiology was used with the nonlinear reaction equations. As expected, all the bacteria is killed at an optimal time period of 9.5 days, which is quite an improvement from 14.25 days with 8 h withdrawal periods.

JPicture2.jpg

  • Bp – Dose: 2hrs, Withdrawal: 7.5hrs. Ended 9.5days

Cogan23.jpg

  • Bs – Dose: 2hrs, Withdrawal: 7.5hrs. Ended 9.5days

Cogan24.jpg

Discussion

The simulation results confirm the dose/withdrawal treatment hypothesis that is used to eradicate both the susceptible and persister bacterium from the microbial biofilm. However, this model has made numerous grand simplifications regarding the formation and behavior of persisters within a biofilm; not to mention the complications associated with the transport of nutrients and antibiotics. It remains to be determined whether the dose/withdrawal strategy will be effective in a biofilm setting.

References

N.C. Cogan. Persister, tolerance and dosing. J. Theroretical Biology, 238(6): 694 703, 2006.