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MBW:Senescence Can Explain Microbial Persistence

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Article: Senescence can explain microbial persistence Authors: I. Klapper, P. Gilbert, B. P. Ayati, J. Dockery and P. S. Stewart Reviewed by: Ryan Brown

Executive Summary

Many antimicrobial medications like penicillin have been observed to significantly reduce bacterial populations, though they are unable to completely eliminate them. Regardless of the exposure time, some bacterial cells, called persisters, manage to survive. After exposure to the antimicrobial has subsided the persisters are able to reproduce intolerant daughter cells. Persister cells possess many interesting qualities and the mechanism of persister cell formation is of interest. Most theories suggest that the bacteria are able to switch between levels of persister states depending on environmental conditions. In their article Klapper et al. introduce a mechanism where cells senesce with time and this increase in senescence results in increased tolerance to the drug. Using this idea as a foundation they develop a mathematical model to simulate populations of persisters vs time in a culture.

Introduction

After a population of bacteria is exposed to some antimicrobial any bacteria that survive are called persisters. It is important to discern the difference between resistance and persistence. The persister cells do not have an immunology to the antimicrobial medications but rather possess the capability to tolerate exposure by not dying. These persister cells have many interesting characteristics including:

  • persister cells enable repopulation
  • tolerance is not inherited by daughter cells
  • persister cells do not grow or grow slowly in presence of an antimicrobial
  • persister cells are tolerant to multiple antimicrobial agents
  • cultures experience biphasic killing patterns due to the antimicrobial (See Keren et al[1] Figure 2)
  • in continous culture experiments, persister fractions increase with decreasing dilution rates
  • changes in the population fraction of persister cells are growth phase dependent

The basis of the model relies on previous work observations of microbial senescence. Senescence can be thought of as maturity or old age and it is believed that with increased senescence, mother cells display different behaviors than their daughters. Classical views of cell division describe a parent cell dividing into two identical daughters with no age associated with them. Certain yeasts like Saccharomyces cerevisiae have assymetric cell division (physically larger mother cell) and have been the focus of studies of microbial aging [2]. Stewart et al. were able to track aging in the morphologically symmetric dividers E. Coli and concluded that it is a result of a physiological asymmetry in the cells[2]. They tracked roughly 35,000 E. coli to show that the cells had an increase in senescence in the form of slower growth rate. Their model assumed that the cells contained poles of different ages and this age was increased with each division. See Figure A[2]

Figure A. Age is represented by numbers. Red = older, Blue = younger

The purpose Klapper et al.'s article is to create a mathematical model to describe the behaviors of persisters using the concept of senescence. They make several assumptions to develop their model. Below they are listed straight from the original article[3].

  1. bacterial cells age
  2. older cells are more tolerant than younger cells of antimicrobial challenge
  3. growth manifests as a production of new cells
  4. production of new cells decrease with age but remains greater than zero
  5. cell death occurs at a constant rate (k_{{d}}=const)
  6. a given concentration of applied antimicrobial will kill young cells, but not older cells
  7. substrate usage depends on age and concentration, but in a separable way

Assumptions 1-3 are essential to the model, while 4-7 are for completeness of the model and used for simulation purposes. Older cells are regarded as the persisters and grow more slowly than the younger, non persister cells. New cells are produced from older cells as shown in Figure 1[3].

Figure 1. Mother cell age n. Each child is age 0 and the mother ages by one unit
.

Theory

Microbial rates

The Table in the section below describes all important variables and parameters that will be used in the following discussion. Populations are given in units of c.f.u. They start by defining the substrate usage per c.f.u at time t and cell age a, r_{{s}}(a,c(t)), as

r_{{s}}(a,c(t))={\begin{cases}k_{{s}}[(1-{\frac  {a}{\lambda }})+\xi ]c(t)\qquad a\leq \lambda \\k_{{s}}\xi c(t)\qquad \qquad \qquad \quad a>\lambda \end{cases}}

where c(t) is the growth media concentration, k_{{s}} is the substrate usage factor, \xi is the minimum substrate usage factor,\lambda is what they refer to as the senescence time parameter. They then define a birth rate r_{{x}}(a,c(t)) as

r_{{x}}(a,c(t))=Y_{{s}}r_{{s}}(a,c(t))\qquad \qquad (1)

where Y_{{s}} is the cell yeild coefficient. Notice that with increasing age the cell will use less substrate and therefore yield less offspring as asserted by assumption 4. Finally, by assumption 5 the death rate r_{{d}} is

r_{{d}}=k_{{d}}

with k_{{d}} being the death rate constant.

Age Structure

The differential equation used to describe the bacterial population density, b(a,t), is

{\frac  {\partial b}{\partial t}}+{\frac  {\partial b}{\partial a}}=-k_{{d}}b\qquad \qquad (2)

This equation is not valid for b(a=0,t) i.e. for the cells that are born from the existing population at a given time t. To account for this the authors show that

b(0,t)=\int _{0}^{\infty }r_{{x}}bda\qquad \qquad (3)

Similarly, as these new bacteria are created more substrate is used so there is a changing concentration given by the following

{\frac  {dc}{dt}}=-\int _{0}^{\infty }r_{{s}}bda\qquad \qquad (4)

At this point we can introduce the senescence factor, s(a) and the senescence weighted total bacteria population B(t)

s(a)={\begin{cases}1-{\frac  {a}{\lambda }}+\xi \qquad a\leq \lambda \\\xi \qquad \qquad \quad \quad a>\lambda \end{cases}}
B(t)=\int _{0}^{\infty }s(a)b(a,t)da

Now equations (2-4) are rewritten as

{\frac  {\partial b}{\partial t}}+{\frac  {\partial b}{\partial a}}=-k_{{d}}b\qquad \qquad (5)
b(0,t)=Y_{{s}}k_{{s}}cB\qquad \qquad (6)
{\frac  {dc}{dt}}=-k_{{s}}cB\qquad \qquad (7)

The initial conditions chosen by the authors as follows

b(0,a)=b_{{a}}(a)
c(0)=c_{{0}}
b_{{0}}=(10^{{2}}\,c.f.u.)\lambda ^{{-1}}(1-a/2a) for a\leq \lambda else 0.


The above formulation is identical to what the authors produce in their paper. With this groundwork and a proper choice of parameters one can use their favorite differential equation solving technique to simulate the population.

Antimicrobial

To simulate the exposure to an antimicrobial Klapper et al. modify equation (5) with a killing rate, r_{{K}}(a,d), that only applies to young and therefore non persistent cells. If k_{{K}} is a killing rate coefficient and d the antimicrobial concentration then equation (5) becomes

{\frac  {\partial b}{\partial t}}+{\frac  {\partial b}{\partial a}}=-(k_{{d}}+r_{{K}})b\qquad \qquad (8)

with

r_{{K}}(a,d)={\begin{cases}k_{{K}}\qquad a\leq \delta d\\0\qquad a>\delta d\end{cases}}

The tolerance coefficient parameter, \delta , is a value that corresponds to how old a cell has to be before it is considered persistent. For small values of \delta a cell becomes persistent at a relatively young age. This parameter is not experimentally found but is estimated by the authors. They discuss their choice its value in the Methods section of the original article. Notice also that they chose tolerance age to have a linear relationship with antimicrobial concentration. This choice is consistent with the linear relation that senescence has with time and the authors explain that this is for definiteness and that some other relationship can be used as long as senescence and tolerance age are consistent with each other. Equation 8 is numerically solved in the simulation with the initial conditions and parameter values given in the table below.

Table of Variables

Senescence Table.jpg

Note: This is a copy of Table 1 from the original article.

Deciding values of each parameter are discussed in the Methods section of the article. The important constraints are:

  • k_{{K}}>k_{{d}} : antimicrobial killing rate is greater than cell death rate
  • \delta >0 : cells more tolerant with age
  • \xi >0 : persister cells capable of repopulation

Results

Using numerical methods[4] equations (6)-(8) are solved for population vs time. Figure 2 shows the result. The antimicrobial is applied to the batch during different phases of the bacteria growth. After some time of exposure it is removed, the sample is recultured (simulated) and the cells are allowed to repopulate. In Figure (a) the antimicrobial is applied during a stationary growth phase while in (b) it is applied during the exponential growth phase. The plots are log-log plots hence the linear relation in (b) for t<5h. Both graphs tell similar stories. The overall population of cells increases rapidly in the initial stages before the introduction of the antimicrobial. Notice that while the overall population increases dramatically the amount of persister cells does not follow the same sharp increase. When the antimicrobial is applied there is a drastic decline in the population of all cells (solid line), to the point where all cells present are persister cells (dashed line). The persisters then create several daughter cells and this causes the population to shoot back up. The persister cell population does not grow significantly until the stationary phase when the total cell population begins to level out. This is a consequence of the time delay (dependant on the value of \delta ) when the daughter cells age to persisters. It appears at large time all cells will become persisters, but will likely die to limited resources and/or natural causes.

SenescenceFig2.jpg

Extensions

  • The results above were assuming a chronological time used for the aging of the cells. If instead one wanted to use cell divisions to measure time as Stewart et al. do with E. coli some modifications can be made to equations (2)-(4). The authors do so in the Senescence Structure section and the results after simulation are shown in Figure 3. As you can see there is little to no difference.

SenescenceFig3.jpg


  • The article also contains a section labeled Chemostat model where they modify equations (5)-(7) by adding the parameter D, the chemostat dilution rate. The conclusion they come to after this model is simulated is that decreasing the chemostat dilution rate increases the persister fraction in the culture. This is in qualitative agreement with observations made by Sufya et al.[5]

Conclusions

Knowing the mechanism that allows for the presence of persisters after exposure to an antimicrobial has obvious implications to medicine. Understanding the mechanism that causes persistence could lead to the development of more powerful antibiotics. The idea that cells develop persistence as a consequence of senescence might be a step toward a better understanding. With the senescence model, Klapper et al. have been able to mathematically simulate some of the characteristics observed by persisters.


References

  1. I. Keren, N. Kaldalu, A. Spoering, Y. Wang, K. Lewis. Persister cells and tolerance to antimicrobials. FEMS Microbiology Letters 230 13-18. 2004
  2. 2.0 2.1 2.2 2.3 C. Stephens. Senescence: Even Bacteria Get Old. "Current Biology" 15(8) R308. 2005
  3. 3.0 3.1 I. Klapper, P. Gilbert, B. P. Ayati, J. Dockery, and P. S. Stewart. Senescence can explain microbial persistence. Microbiology, 153:3623�3630, 2007.
  4. Described in Numerical Methods section of the paper "as a moviing-grid Galerkin method..."
  5. N. Sufya, D.G. Allison, P. Gilbert. Clonal variation in maximum specific growth rate and susceptibility towards antimicrobials. J Appl Microbial 95, 1261-1267