Systematic design of pulse dosing to eradicate persister bacteria

A small fraction of infectious bacteria use persistence as a strategy to survive exposure to antibiotics. Periodic pulse dosing of antibiotics has long been considered a potentially effective strategy towards eradication of persisters. Recent studies have demonstrated through in vitro experiments that it is indeed feasible to achieve such effectiveness. However, systematic design of periodic pulse dosing regimens to treat persisters is currently lacking. Here we rigorously develop a methodology for the systematic design of optimal periodic pulse dosing strategies for rapid eradication of persisters. A key outcome of the theoretical analysis, on which the proposed methodology is based, is that bactericidal effectiveness of periodic pulse dosing depends mainly on the ratio of durations of the corresponding on and off parts of the pulse. Simple formulas for critical and optimal values of this ratio are derived. The proposed methodology is supported by computer simulations and in vitro experiments.


Author summary
Administering antibiotics in periodic pulses that alternate between high and low concentration has long been known as a possible dosing strategy to treat stubborn infections caused by bacteria known as persisters. Such bacteria use clever mechanisms to survive otherwise lethal temporary exposure to antibiotics and to resume normal activity upon antibiotic removal. Persisters pose a serious health problem. Recent studies have elucidated mechanisms of persistence and have confirmed that pulse dosing, if designed appropriately, can indeed be effective. However, effective pulse dosing design has been mainly handled by trial and error, requiring relatively extensive experimentation. Here we develop a method for rapid systematic design of effective pulse dosing. The method relies on a simple mathematical model and a minimal amount of standard experimental data. We derive corresponding design formulas that explicitly characterize the shape of generally effective or optimal periodic pulses. We tested our method through computer simulations and in vitro experiments, as well as on prior literature data. In all cases, the outcomes on persister bacteria eradication predicted by our method were confirmed. These results pave the way for ultimately developing effective pulse dosing regimens in realistic situations in vivo.

Introduction
Persister cells are a small fraction of a bacterial population in a physiological state that enables them to survive otherwise lethal doses of antibiotics. While these cells remain in the state of persistence, they cannot be killed by conventional antibiotics, unless the cells phenotypically switch to the normal cell state and become susceptible to antibiotics again [1]. Persisters are enriched in biofilms [2] and implicated in many chronic infections such as tuberculosis and relapse of infections such as recurrent urinary tract infection or cystic fibrosis [3][4][5][6][7][8]. Unlike antibiotic-resistant mutant cells, persisters are phenotypic variants that survive treatments without acquiring heritable genetic changes [9]. However, prolonged persistence creates favorable conditions for the emergence of the mutant cells [10,11]. Although the term persister was coined in the 1940s [12], our fundamental knowledge of persisters has accelerated only in the last two decades with the advent of new technologies enabling us to study cell heterogeneity [1,13]. Persisters survive via a plethora of putative molecular mechanisms [14] and recent studies have started shedding light on how diverse and multifaceted phenomena can link effects of initial states and environmental factors to phenotypic changes during persister formation, survival, and return to normal state [14][15][16][17][18][19][20]. In view of that complexity, developing anti-persister therapeutics remains a challenge. Such development can be broadly classified into two categories: (a) developing new anti-persister drugs, and the most common approach (b) manipulating the dosing regimen of approved antibiotics, used either individually or in combination. The former relies on detailed knowledge about persister mechanisms (formation, survival, and resuscitation) and is certainly significantly more time-consuming and resourceintensive than the latter. Within the latter category, strategies can be based on bacterial population dynamics without complete knowledge of molecular mechanisms, capitalizing instead on the back and forth switching of persisters from normal state to persistence and vice-versa. The idea of periodic pulse dosing to kill persisters is as old as the term persister itself [12]. Yet, there are only sporadic studies on pulse dosing, examining in vitro efficacy [21,22] and model fitting of experimental data or characterization of effective pulse dosing strategies [23][24][25][26][27]. A remaining challenge is a simple systematic design of an effective periodic pulse dosing regimen, comprising periods of antibiotic administration at high and low concentrations successively. Indeed, the alternating periods of antibiotic application (on) and removal (off) are critical to the success of pulse dosing strategies [22,28], as use of an inappropriate strategy fails to achieve eradication [29,30]. Attempts to connect experiments and modelling [28] have underscored the importance of characterizing optimal dosing regimens in a simple quantitative fashion.
The present study aims at addressing this issue. The specific contributions of this study are (a) rigorous theoretical justification that the efficacy of pulse dosing with alternating on/off periods of antibiotic administration depends on the ratio of the corresponding on/off periods of a pulse rather than on their individual values; (b) explicit formulas for robustly optimal values of this ratio in terms of easily estimated parameters; and (c) experimental confirmation in vitro of both positive and negative model predictions (bacterial eradication or not, respectively). In the rest of the paper, we describe our experimental and modeling studies, present our main results, and close with suggestions for future studies. Proofs and details, to the extent that they provide insight, are included in S1 Text.

Media and chemicals.
Luria-Bertani (LB) broth was used for all liquid cultures. LB broth was prepared by dissolving its components (10g Tryptone, 10 g Sodium Chloride, 5 g Yeast Extract) in 1 L distilled water and sterilized with an autoclave. Ampicillin (Sigma Aldrich) was used to treat cells at a constant dose of 100 μg/mL . Phosphate Buffered Saline (PBS) was used to wash the cells to remove Ampicillin. LB agar medium was prepared by dissolving 40 g LB agar premix in 1 L DI water and sterilized with an autoclave. LB agar medium was used to enumerate colony forming units (CFUs) of E. coli [13,31,32].
2.1.3 Constant (control) and pulse dosing experiments. Bacteria were exposed to Ampicillin in two ways: at constant Ampicillin concentration (control) and at pulsed antibiotic concentration of the same amplitude (pulse on/off dosing, Fig 1).
Each pulse experiment was started by inoculating (1,100-fold) an overnight (24 h) culture of E. coli into 25 ml of LB. For selection and retention of plasmids in bacterial cells, 50 μg/mL kanamycin was added in culture media [32]. To induce fluorescent protein expression, 1 mM IPTG was used [32]. The overnight culture was prepared from frozen glycerol stock (-80˚C). All cells were cultured in a shaker at 37˚C and 250 rpm. The pulse dosing schedule was: (a) expose bacteria to Ampicillin (100 μg/mL) for t on h, and (b) wash treated cells and grow in fresh media for t off h. Treated cells were washed with PBS buffer solution to remove the antibiotics. Cells were serially diluted in PBS using 96-well plates, spotted on LB agar, and incubated at 37˚C for 16 h to enumerate CFUs. Bacteria population size was assessed by colony counting on LB-Agar plates.

Modelling and simulation
A two-state model [1], comprising two cell-balance differential equations for normal and persister cells, was used in all analysis: " # a = switch rate from normal to persister state b = switch rate from persister to normal state K n ≝ μ n −k n −a = net decline / growth rate of normal cells K p ≝ μ p −k p −b = net decline / growth rate of persister cells μ n , μ p = growth rate of normal or persister cells, respectively k n , k p = kill rate of normal or persister cells, respectively The parameters a, b, K n , K p in Eqs (1) and (2) are generally distinct when the antibiotic is administered or not (on/off), resulting in corresponding matrices A on , A off in Eq (3). In the constant dosing (control) experiment the antibiotic remained always on, whereas for pulse dosing experiments the antibiotic alternated between on (administered) and off (not administered) with corresponding durations t on , t off . Therefore, to fit the full data set (at constant and pulse dosing) by Eqs (1) and (2), parameter estimation generally entailed eight values for estimates of {a, b, K n , K p } off and {a, b, K n , K p } on during on and off periods, respectively. In addition, because data fit relied on measurements of the total number of cells, a ninth parameter, namely the initial fraction of persister cells, f 0 , was estimated. Mathematica and MATLAB were used for all modelling, parameter estimation, and analysis computations.

Results
We present first a theoretical analysis for characterization of optimal pulse dosing regimens based on the model of Eqs (1) and (2), and subsequently present a series of experiments for validation of analysis results in vitro. (1) and (2) (Appendix A in S1 Text) suggests that successive local peaks of the bacterial population size, c(t), at times t 2ℓ ≝ ℓ(t on +t off ), as generally depicted in Fig 1, are characterized as

Characterizing {t on , t off } for decline of a bacterial population. Αnalysis of Eqs
where λ 1 , λ 2 are the eigenvalues of the matrix and p 1 , p 2 are coefficients depending on the model parameters and initial conditions. It can be shown (Appendix A in S1 Text) that λ 1 , λ 2 in Eq (5) are inside the unit disk. As a result, the pattern of c(t 2n ) in Eq (5) is downward if and only if the respective on/off periods of pulse dosing, t on , t off , satisfy the inequality Note that the critical value (t off /t on ) c is positive, because K n,on <0, K n,off >0.
As an example, for the estimates K n,on , K n,off shown in Table 1, Eq (7) yields for peak-to-peak decline, as shown in Fig 2.

Characterizing {t on , t off } for rapid peak-to-peak decline of the bacterial population.
For pulse dosing with fixed on/off periods t on , t off that satisfy the inequality in Eq (7), it can be shown (Appendix B in S1 Text) that Eq (3) implies that successive peaks of c(t) at times t 2n (Fig 1) decline exponentially over time at a rate, k characterized as k≝ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ðK n;off x þ K n;on or, equivalently, with a time constant, τ = 1/k. It follows immediately (Appendix A in S1 Text) that the maximum peak-to-peak decline rate for k (as shown in Eq (9)) is attained at where R p ≝ K p;on K p;off ; R n ≝ K n;on K n;off ð12Þ and the approximation in Eq (11) is based on Note that (t off /t on ) opt in Eq (11) depends on the ratios R p ≝ K p,on /K p,off , R n ≝ K n,on /K n,off , rather than on the individual values of K p,on , K p,off , K n,on , K n,off . Note also that combination of Eqs (11) and (7) immediately connects the optimal and critical values of the ratio t off /t on as Application of Eq (11) yields a profile or the optimal ratio (t off /t on ) opt in terms of the ratios R p , R n in Eq (12) as shown in  Table 1 are used to mark the indicated point.
To visualize Eqs (7), (9), and (11), estimates of {K n , K p } off and {K n , K p } on in Table 1 were used to produce a characterization of the peak-to-peak decline rate as a function of t on , t off , as shown in Fig 4. A remarkable implication of Eq (11) is that an optimal ratio t off /t on (in the sense defined in this section) can be approximately assessed by mere knowledge of the ratio K n,on /K n,off , which,  Table 1. Values of t on , t off that keep both λ 1 , λ 2 below 1 (equivalently ln(λ 1 ), ln(λ 2 ) below 0) are shown. The white line corresponding to (t off /t on ) c �2.8, Eq (8), on the gray plane at 0 indicates the threshold (critical) value of t off /t on , above which λ 1 >1.
https://doi.org/10.1371/journal.pcbi.1010243.g002 in turn, can be easily obtained from two simple short-term experiments, namely standard time-growth and time-kill. The corresponding initial slopes in such experiments immediately yield K n,off (time-growth) and K n,on (time-kill), as discussed next. This is a significant simplification of the parameter estimation task for the purpose of pulse dosing regimen design, because it reduces the number of parameters to be estimated from nine (cf. section 2.2) to two. In addition, these two parameters are much easier to estimate than the remaining seven, whose accurate estimates are hard to obtain, as discussed in section 3.3.

Constant and pulse dosing experiments
Detailed statistics are provided in Appendix C in S1 Text. Note that the pattern shown in Fig 5B indicates exponential decline (corresponding to a straight line of the bacterial population logarithm) as the almost horizontal part of the biphasic trend, which is typical of persisters [2], has not yet been reached within 3 hours. That trend is reached through continuation of the experiment after 3 hours, as indicated by the red data points and corresponding bend of the line fitted in  Using the above two estimates for K n,off , K n,on in Eq (12) to calculate R n and substituting the resulting value of R n into Eq (11) yields

PLOS COMPUTATIONAL BIOLOGY
Systematic design of pulse dosing to eradicate persister bacteria Therefore, to be close to the value of 0.6, pulse dosing with was implemented in validation experiments in vitro.
The outcome of pulse dosing using the above t on , t off , along with additional data for continuation of the constant dosing (control) experiment to 13h, are shown in Fig 6. In the case of constant dosing, colonies were seen in all replicates at the end of the 13h treatment (5−100 CFU/mL) whereas in the case of pulse dosing no colony was observed in any replicate in the latter part of the last t on cycle. Note that both Figs 5 and 6 indicate a 5 log decrease of viable cells over the first 3 hours.

Model parameter estimation with full data set
To confirm that the model used in the analysis for development of the formulas applied in the approach proposed to pulse dosing design, the parameters in Eqs (1) and (2) were fit to the full set of experimental data of Fig 6, with corresponding curves shown in Fig 6 and parameter estimates shown in Table 1.

Pulse dosing design validation
Eqs (7) and (8) indicate that bacteria eradication would be achieved if the pulse dosing period ratio t off /t on remained below 2.8, based on the short-term data fit (Fig 5), or below 2.4, based on full data fit (Fig 6). This assessment is confirmed via both simulation and experiment.

Experiment.
In addition to the experimental data in Fig 6, which confirmed rapid bacterial eradication for pulse dosing at ratio t off /t on = 2/3, example cases of pulse dosing at ratios t off / t on anticipated by the developed theory and by computer simulations not to result in bacterial eradication were also tested through in vitro experiments, as shown in Fig 8. This figure shows the outcome (dots) of two pulse dosing experiments corresponding to the same t off /t on = 6, above the critical value (t off /t on ) c = 2.8 (Eq (8)), for t on = 0.5h and t on = 3h. Model predictions (Eqs (1) and (2) with parameter estimates from Table 1 based on data of Fig 6) are also shown.
Note that for {t on , t off } = {3h, 18h} the linear model of Eqs (1) and (2) correctly predicts the observed peak-to-peak upward trend, but the model crosses the bacterial population saturation limits (Eq (SI-23)) which are at about 9 log(CFU/mL) (Appendix C in S1 Text) hence the quantitative discrepancy.
Additional experiments were also conducted to test the robustness of pulse dosing in case of non-uniform perturbations of t on around 3h and t off around 2h. The results, shown in

Summary of results
In summary, the presented method to pulse dosing regimen design relies on Eq (11), which expresses the optimal ratio (t off /t on ) opt as approximately a function of R n ≝ K n,on /K n,off , where the parameters K n,off , K n,on are easy to obtain from the early parts of time-growth and time-kill experiments, respectively, as demonstrated in Fig 5. Starting with this, the efficacy of a near optimal proposed design is experimentally validated in Fig 6, the sub-optimality of designs with t off /t on other than (t off /t on ) opt is illustrated in Fig 7, predictions for the complete failure of pulse dosing regimens with t off /t on above the critical value (Eq (7)

Characterizing {t on , t off } for decline of a bacterial population.
The analysis presented relies on the linear model of Eqs (1) and (2). That model is clearly not valid when the bacterial population reaches its saturation limits after long growth periods (e.g. Fig 5A and Eq (SI-23) in Appendix C in S1 Text). Therefore, Eq (7), which places an upper bound on the ratio t on /t off of pulse dosing, is understood to hold for values of t off that do not drive the bacterial population beyond logarithmic growth to saturation.

Characterizing {t on , t off } for rapid peak-to-peak decline of the bacterial population.
Selection of the geometric average of two decline rate constants, in Eq (9), captures well an overall decline rate (Fig 10) when early parts of the decline are important.
An alternative, focusing more on the long-term decline rate, would be to choose the dominant (smaller) of the two decline rates k 1 , k 2 (Fig 10). The results would be somewhat different quantitatively, but of similar nature. This is exemplified in Fig 11, which is the counterpart of The quantitative analysis presented in section 3.1 focuses on the rate of peak-to-peak decline, with the intent to design pulse dosing regimens for rapid bacterial eradication. A somewhat more accurate characterization of time to eradication would be provided by focusing on the rate of dip-to-dip decline (c.f. Figs 6 and 7). It can be shown (Eqs (SI-1) and (SI-2) in Appendix A in S1 Text), however, that both peaks and dips of the bacterial population follow the same rate, governed by the eigenvalues λ 1 , λ 2 of the matrix M (Eq (6)). Because tracking peaks is simpler to analyze than tracking dips, the choice was made to focus on the former, leaving the latter for a future study.
In addition to the quantitative analysis that resulted in the selected values t on = 3h, t off = 2h (Eq (17)) for the pulse dosing experiment, heuristic analysis was also used to corroborate that choice, as follows: In the early part of the constant dosing (control) experiment, it is evident (Fig 6) that around 3h persisters start becoming an increasingly significant fraction of the bacterial population (as manifest by the bend in the declining population logarithmic size) and subsequently dominate the population. Therefore, keeping the antibiotic on beyond 3h would result in reduced killing rate. Furthermore, while turning the antibiotic off starts driving persisters back to normal cells, keeping it off beyond 2h would reach the limits of that drive. While the above analysis suggests that the selected of t on , t off values are sensible, it cannot provide a general characterization of optimal values or establish the importance of the ratio t on /t off .
Related experiments [33] have found that persisters treated for 3h with Ampicillin were able to resuscitate after growth in fresh media for as little as 1h. In our experiments also, a manifold increase in kill rates when treated again with Ampicillin suggested that persisters resuscitated to normal cells indeed within an hour of growth in fresh media. Exponential peak-to-peak decline rate modes corresponding to each individual rate k 1 , k 2 (Eq (18)) and the geometric average rate k ≝ (k 1 k 2 ) 1/2 (Eq (9)). https://doi.org/10.1371/journal.pcbi.1010243.g010

Modeling and parameter estimation
Estimation of the four model parameters in Eqs (1) and (2) faces the challenge that measurements of bacterial population size, Eq (4), are linear combinations of the two solution modes, q 1 exp(ρ 1 t)+q 2 exp(ρ 2 t) (Appendix D in S1 Text) with ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 4ab þ ðK n À K p Þ which makes it difficult to estimate all four {K n , K p , a, b} with good accuracy from data. Nevertheless, assumptions based on fundamental knowledge can facilitate estimation. These assumptions are in accordance with relevant literature [9] and similar observations in our own experiments. When the antibiotic is on, it follows that b�0, because persisters remain dormant, and a≳0, as some normal cells switch to persisters due to antibiotic stress. Estimation of a nonzero a yielded a very small value, in agreement with the assumption a�0, i.e. that persisters can be sourced to the initial pool of cells.
When there is no antibiotic and cells are grown in fresh media, it follows that a�0, as the tendency of normal cells to become dormant is negligible during the early exponential phase (e.g. in less than 3h of growth), whereas b>0, as persisters will be resuscitated shortly (e.g. within an hour) after inoculation in fresh media [13,33,34]. In addition, K p,on � −b, because μ p,on � 0 and k p,on � 0.  Table 1. The crease line (white), where the two surfaces intersect, corresponds to Eq (SI-16) and characterizes the highest peak-to-peak actual-time decline rate, for t off /t on = 1.5. The different bactericidal outcomes produced by constant and pulse dosing are evident in Fig 6A, where the (one-standard-error) confidence bands towards the end of the experimental period of 13h are clearly separated, with pulse dosing yielding no detectable bacterial load and constant dosing leaving a load of about 20 log(CFU/mL) at 13h. In addition, Fig 6B shows the fitted model projection until the bacterial population outcome resulting from constant dosing reaches 0 at about 30h, long after pulse dosing has reached the same outcome in less than 12h.

Application of the proposed theory to literature data
To further test the ability of the proposed theory to predict the bactericidal efficacy of pulse dosing regimens, we analyzed experimental data from two studies in literature [28,29]. These studies presented pulse dosing regimens on bacterial populations with persister bacteria.
To perform the test, estimates of K n,on (decline slope of the logarithmic population of normal cells under antibiotic exposure) were obtained from data presented in corresponding figures for both studies (Tables 2 and 3); and estimates of K n,off (growth slope of the logarithmic population of normal cells in growth) for [29] were either obtained from a corresponding figure in [28] (Table 2) or estimated based on past experience ( Table 3). The critical values of the ratio t off /t on were subsequently calculated for both references from Eqs (7) and (11), respectively (Tables 2 and 3). These values were then compared to the actual values used in experiments presented in each reference cited. Corresponding predictions from this comparison ere consistent with data in all cases. Specifically, Table 2 indicates that bacterial populations declined when t off /t on <(t off /t on ) c (Eq (7)) and grew when t off /t on >(t off /t on ) c . In fact, the peakto-peak slopes shown in Fig 4 of [28] are in agreement with the discrepancy between the values of t off /t on used and the critical or optimal values. Similarly, Table 3 indicates that t off /t on in [29] is higher than the critical value (t off /t on ) c for all four strains focused on, in agreement with observed increasing persister percentages in the bacterial populations studied.
For completeness, the optimal ratios (t off /t on ) opt are also shown in Tables 2 and 3.

Conclusions and future work
We have developed a methodology for systematic design of pulse dosing regimens that can eradicate persistent bacteria. The methodology relies on explicit formulas that make use of easily obtainable data from time-growth and time-kill experiments with a bacterial population exposed to antibiotics. Several extensions of this work can be pursued, including: • Test the outlined strategy on various pairs of pathogenic bacterial strains and antibiotics, including combinations of antibiotics for stubborn infections.
• Extend and test the developed methodology, both theoretically and experimentally, to � Clinically relevant pharmacokinetic profiles of antibiotic administration, e.g. periodic injection followed by exponential [35].
� Ultimately in vivo studies • The experiments presented used Ampicillin (a β-lactam antibiotic) which is a time-dependent antibiotic, namely it exhibits best efficacy if administered in periodic injections of as high concentration as possible. It is worth exploring the performance of the proposed pulse dosing design methodology to concentration-dependent antibiotics such as aminoglycosides and quinolones [36].
• Models developed using data from flow cytometry experiments [33] can better monitor the heterogeneity of a bacterial population, with potential improvements in modeling and, as a result, pulse dosing design.
• Extension to viable but not culturable cells: In addition to persisters, a well-known bacterial phenotype that can survive exposure to antibiotics is viable but not culturable (VBNC) cells [19,37,38]. The pulse dosing methodology presented here can, in principle, be applied to such cells, as they can be restored to normal growth and susceptibility to antibiotics upon provision of appropriate stimuli [39,40]. However, experimentally studying this case poses different challenges, as standard cell cultures cannot be routinely used, and this is left for future exploration.
Supporting information S1 Text. Appendix A. Derivation of Eq (7); Appendix B. Optimal rate of decline for bacterial population peaks characterized by Eq (9); Appendix C. Estimation of K n,off , K n,on from data in Fig 5; Appendix D. Analytical solution of Eqs (1) and (2). (DOCX) Table 3. Literature data analysis to test proposed theory for pulse dosing regimen design.