Quantifying Subpopulation Synergy for Antibiotic Combinations via Mechanism-Based Modeling and a Sequential Dosing Design

Abstract

Quantitative modeling of combination therapy can describe the effects of each antibiotic against multiple bacterial populations. Our aim was to develop an efficient experimental and modeling strategy that evaluates different synergy mechanisms using a rapidly killing peptide antibiotic (nisin) combined with amikacin or linezolid as probe drugs. Serial viable counts over 48 h were obtained in time-kill experiments with all three antibiotics in monotherapy against a methicillin-resistant Staphylococcus aureus USA300 strain (inoculum, 10⁸ CFU/ml). A sequential design (initial dosing of 8 or 32 mg/liter nisin, switched to amikacin or linezolid at 1.5 h) assessed the rate of killing by amikacin and linezolid against nisin-intermediate and nisin-resistant populations. Simultaneous combinations were additionally studied and all viable count profiles comodeled in S-ADAPT and NONMEM. A mechanism-based model with six populations (three for nisin times two for amikacin) yielded unbiased and precise (r = 0.99, slope = 1.00; S-ADAPT) individual fits. The second-order killing rate constants for nisin against the three populations were 5.67, 0.0664, and 0.00691 liter/(mg · h). For amikacin, the maximum killing rate constants were 10.1 h⁻¹ against its susceptible and 0.771 h⁻¹ against its less-susceptible populations, with 14.7 mg/liter amikacin causing half-maximal killing. After incorporating the effects of nisin and amikacin against each population, no additional synergy function was needed. Linezolid inhibited successful bacterial replication but did not efficiently kill populations less susceptible to nisin. Nisin plus amikacin achieved subpopulation synergy. The proposed sequential and simultaneous dosing design offers an efficient approach to quantitatively characterize antibiotic synergy over time and prospectively evaluate antibiotic combination dosing strategies.

INTRODUCTION

The emerging global health care crisis caused by multidrug-resistant bacteria and a lack of effective antibiotics presents a major challenge and is one of the three greatest threats to human health (1–5). While resistance is increasing rapidly, the number of new approved antibiotics has declined dramatically since the 1980s (6). Synergistic combinations of available antibiotics offer a promising and timely option to combat multidrug-resistant bacteria. However, quantitative approaches for antibiotic combinations that characterize the time course of synergistic killing or of resistance prevention are scarce (7–9). Existing methods to describe synergy are limited to outcomes at a single time point (often 24 h) and do not implement the mechanism(s) of synergy (10, 11). In vitro experiments and mathematical modeling offer advantages, since they can assess the time course and proposed mechanisms of synergy and can therefore more rationally optimize antibiotic combination therapy.

Bacterial populations that are less susceptible or even resistant to antibiotics may cause failure of antibiotic therapy, in particular in severe infections with a high bacterial burden or infections in patients with an impaired immune system (9, 12, 13). High-inoculum infections likely harbor preexisting populations that are less susceptible to one or both antibiotics. To optimally treat such infections, antibiotic combination strategies may utilize subpopulation synergy with one antibiotic killing the resistant population(s) of the second antibiotic and vice versa (Fig. 1) (7–9). In the absence of a population that is resistant to both antibiotics in monotherapy, subpopulation synergy can yield eradication of a bacterial inoculum without resistance. Such synergy seems particularly promising for combinations of antibiotics that do not share an important resistance mechanism which affects both antibiotics.

Fig 1.

Subpopulation synergy concept. Bacterial cells inside the solid line are susceptible to drug A; bacterial cells inside the long dashed lines are susceptible to drug B. Bacteria within the dotted lines are resistant to either drug A or drug B. The combination of drug A and drug B will eradiate these bacterial populations due to the lack of a population resistant to both drugs.

For the simplest case of subpopulation synergy, bacterial killing by antibiotic A does not affect killing by antibiotic B and vice versa (i.e., independent effects). Such combinations may be designed to maximize killing during the first hours of therapy to prevent bacteria from becoming tolerant or resistant to both antibiotics. Importantly, emergence of resistance is a time-dependent process involving both phenotypic tolerance and genotypic resistance mechanisms (14). To ensure survival of the predominant population during the first hours of therapy, bacteria commonly utilize phenotypic tolerance mechanisms that do not require a mutation.

To minimize both upregulation of tolerance and de novo formation of resistant mutants, we used a rapidly killing first antibiotic (nisin) and tested the efficacy of the second antibiotic against the population(s) surviving the first antibiotic in real time (i.e., without isolation of resistant colonies after growth for ≥24 h on antibiotic-containing agar plates, for example). Some studies assessed the susceptibility (MICs) of colonies resistant to the first antibiotic that were generated via long-term (≥24 h) passaging toward a panel of second antibiotics (15). We are not aware of systematic methods to evaluate subpopulation synergy that accounted for the time-dependent upregulation and minimized the impact of phenotypic tolerance mechanisms during the first hours of therapy. Therefore, methods that can rapidly isolate less-susceptible populations may be a valuable alternative. This may be important to better understand the reasons for success of simultaneously dosed combinations, as bacteria have limited time to become tolerant or resistant after simultaneous dosing of two antibiotics.

Nisin, a lantibiotic, is a peptide antibiotic that induces pore formation in bacterial membranes and inhibits peptidoglycan synthesis. Due to its rapid killing of methicillin-resistant Staphylococcus aureus (MRSA) (16, 17), nisin served as a probe drug.

Our primary objective was to assess an experimental strategy that identifies and characterizes potential types of synergy and to develop quantitative, mechanism-based models for antibiotic combinations. In addition to monotherapy and simultaneously dosed antibiotic combinations, we utilized a new sequential design that exposed bacteria to nisin for 1.5 h to rapidly kill nisin-susceptible bacteria. Subsequently, nisin was removed and the efficacy of amikacin or linezolid was assessed against the bacteria surviving 1.5 h exposure to nisin.

(Part of this work has been presented as a short oral presentation and a poster at the National Institute of General Medical Sciences Quantitative and Systems Pharmacology Workshop II, Bethesda, MD, 9 to 10 September 2010 and as a poster at the American Conference on Pharmacometrics, San Diego, CA, 3 to 6 April 2011.)

MATERIALS AND METHODS

Bacterial strains, media, and susceptibility testing.

An MRSA USA300 strain obtained from the Network on Antimicrobial Resistance was utilized for all experiments (18). Static time-kill studies and MIC tests were performed in Luria-Bertani broth (Difco Laboratories, Detroit, MI) supplemented with 12.5 mg/liter Mg²⁺ and 25 mg/liter Ca²⁺. The MICs were additionally determined in duplicate in cation-adjusted Mueller-Hinton broth. Nisin A and amikacin were obtained from Sigma-Aldrich, St. Louis, MO, and linezolid was provided as a kind gift by Pfizer. Nisin A was dissolved at pH 2 in diluted hydrochloric acid, and the denatured milk solids were removed by centrifugation. Viable counts were determined on Luria-Bertani agar (Difco Laboratories, Detroit, MI).

Time-kill experiments.

Time-kill experiments were performed as described previously (19–21). In brief, fresh bacterial colonies were grown on Luria-Bertani agar for approximately 20 h prior to each experiment. A bacterial suspension of ∼10⁹ CFU/ml in saline was prepared spectrophotometrically and diluted into 20 ml of fresh, prewarmed, sterile broth to obtain an initial inoculum of ∼10^7.7 CFU/ml. Bacteria were then grown for 60 min to ∼10^8.0 CFU/ml prior to addition of the respective antibiotic(s). Viable counts representing the initial inoculum were obtained within less than 10 min prior to dosing.

Monotherapy.

Static time-kill studies assessed nisin (4, 8, 32, and 128 mg/liter), amikacin (1, 4, 16, and 64 mg/liter), and linezolid (2, 8, and 32 mg/liter) in monotherapy.

Combinations.

Sequential combinations with nisin switched to amikacin or linezolid quantified the rate of growth and bacterial killing by amikacin or linezolid against the nisin-intermediate and nisin-resistant populations. Both the nisin-intermediate and nisin-resistant populations were selected by exposing the initial bacterial inoculum to 8 mg/liter nisin over 1.5 h. The nisin-resistant population only was selected separately by exposure to 32 mg/liter nisin over 1.5 h. These nisin concentrations and the 1.5 h duration of exposure were optimized in pilot time-kill studies (results not shown). Nisin was then removed by centrifugation of the conical tubes and careful removal of the supernatant. The bacterial pellets were immediately resuspended in prewarmed broth containing no antibiotic (control), 8 or 16 mg/liter amikacin, or 8 or 32 mg/liter linezolid at approximately 1.75 h. In addition to the sequential combinations with pretreatment by nisin, simultaneous combinations also were studied, where the bacteria were simultaneously exposed to nisin plus amikacin or nisin plus linezolid throughout the whole experiment. The simultaneously dosed combinations were studied for selected informative combinations of nisin (8, 16, and 32 mg/liter) with amikacin (4, 8, and 16 mg/liter) or linezolid (2, 8, and 32 mg/liter).

Viable counting.

For all arms, serial viable counts were quantified over 48 h as described previously (19–21). For sequential combinations, additional viable counts were assessed before centrifugation and ∼5 min after resuspension of the bacterial pellets.

Pharmacodynamic modeling and synergy concept. (i) Subpopulation synergy.

We define two types of synergy: subpopulation synergy and mechanistic synergy. Subpopulation synergy (Fig. 1) occurs if antibiotic A kills the bacteria that are less susceptible to antibiotic B and vice versa. If there are no bacteria resistant to both antibiotics, the combination will eradicate all bacteria. In the simplest case of independent effects, antibiotic A does not affect the rate of killing by antibiotic B and vice versa.

(ii) Mechanistic synergy.

The second type of synergy defined here is mechanistic synergy (Fig. 2), which, in the present analysis, refers to an enhanced rate of killing of a bacterial population due to the simultaneous presence of two antibiotics. Mechanistic synergy was identified via modeling if a bacterial population was killed more rapidly by the combination of antibiotics A and B than by the independent effects of both antibiotics in monotherapy. For example, antibiotic B may be a β-lactamase inhibitor or an efflux pump inhibitor that achieves no (or limited) killing in monotherapy. In combination with antibiotic A, antibiotic B may increase the target site concentration and therefore the killing by antibiotic A. In our modeling analysis, mechanistic synergy was considered for either specific population(s) or all populations.

Fig 2.

Mechanistic synergy with drug B enhancing the rate of killing by drug A for the population susceptible to drug A, the population resistant to drug A, or both populations.

Life cycle growth model.

For each population, a life cycle growth model (22, 23) was applied that accounts for the underlying biology of bacterial growth (24) and contains bacteria that are preparing for replication (state 1) and bacteria immediately before the replication step (state 2). The transition from state 1 to state 2 occurred via a first-order growth rate constant, k₁₂ (Fig. 3), and replication (k₂₁, or doubling) was assumed to be fast. This two-state life cycle growth model offers the advantage to describe a lower growth rate due to inhibition of protein synthesis by linezolid, for example. Alternatively, a less-susceptible population may have a decreased biofitness and a longer mean generation time (1/k₁₂) compared to susceptible bacteria (Fig. 3). The potentially decreased biofitness of less-susceptible populations was estimated as a growth rate factor (fk₁₂) that was multiplied with the growth rate constant (k₁₂) of the susceptible population. For example, the growth rate constant (k_{12_RR}) of a population resistant to both antibiotics was expressed as k_{12_RR} = fk_{12_RR} · k₁₂.

Fig 3.

Life cycle growth model for the population susceptible to both nisin and amikacin. This two-state life cycle growth model was applied to each of the six populations in Fig. 4. k₁₂, first-order growth rate constant; k₂₁, first-order replication rate constant (representing doubling); Inh_Rep, inhibition of the probability of successful replication; Inh_k12, inhibition of the first-order growth rate; other symbols are explained in Table 1 and Materials and Methods.

Population model.

Models with one, two, or three preexisting populations with different susceptibilities to the respective antibiotic were considered similarly to previously described models (8, 14, 19, 22, 25–30). The susceptibility of each population was estimated via a specific rate of bacterial killing by the respective antibiotic (Fig. 4).

Fig 4.

Subpopulation synergy model for nisin and amikacin comprising six populations with different susceptibility to each antibiotic. Nis^s, nisin susceptible; Nisⁱ, nisin intermediate; Nis^r, nisin resistant; Ami^s, amikacin susceptible; Ami^r, amikacin resistant. Thick arrows represent fast growth or fast killing. Solid downward arrows represent killing by nisin, and dashed arrows represent killing by amikacin. See Table 1 for parameter explanations.

Nisin-plus-amikacin combination.

The mathematical model for nisin plus amikacin contained six populations, including all combinations of a nisin-susceptible (Nis^s), -intermediate (Nisⁱ), and -resistant (Nis^r) population and of an amikacin-susceptible (Ami^s) and -resistant (Ami^r) population. With the two states (i = 1 and i = 2 in equation 1) for each population from the life cycle growth model, the total population (CFU_ALL) contained 12 compartments (i.e., two compartments for each of the six populations).

$${\text{CFU}}_{\text{ALL}}={\displaystyle \sum _{i=1}^2{\text{CFU}}_{\text{SS},i}+{\text{CFU}}_{\text{IS},i}+{\text{CFU}}_{\text{RS},i}+{\text{CFU}}_{\text{SR},i}+{\text{CFU}}_{\text{IR},i}+{\text{CFU}}_{\text{RR},i}}$$ (1)

The differential equation for the concentration of bacteria from the Nis^s/Ami^s population in state 1 (CFU_SS,1) comprised the second-order killing by nisin and a saturable killing by amikacin (initial condition described below):

$$\frac{d{\text{CFU}}_{\text{SS},1}}{dt}=\text{REP}·k_{12}·{\text{CFU}}_{\text{SS},2}-\left(k_{12}+k_{2\text{S}}·{\text{C}}_{\text{Nis}}+\frac{{\text{Kmax}}_{\text{S}}·{\text{C}}_{\text{Ami}}^{\text{Hill}}}{{\text{C}}_{\text{Ami}}^{\text{Hill}}+{\text{KC}}_{50}^{\text{Hill}}}\right)·{\text{CFU}}_{\text{SS},1}$$ (2)

where C_Nis is the nisin and C_Ami the amikacin concentration in broth medium. CFU_SS,2 is the concentration of bacteria in state 2 of the Nis^s/Ami^s population. The rate constants k₂₁, k₁₂, and k_2S and the parameters for the Hill-type killing function for amikacin (Kmax_S, KC₅₀, and Hill) are defined in Table 1. The replication factor REP ensures that the total concentration of bacteria in all populations (CFU_ALL) does not exceed the maximum population size (CFU_max):

$$\text{REP}=2·\left(1-\frac{{\text{CFU}}_{\text{ALL}}}{{\text{CFU}}_{\text{max}}+{\text{CFU}}_{\text{ALL}}}\right)$$ (3)

Table 1.

Parameter estimates for nisin combinations with amikacin or linezolid

Parameter	Symbol (unit)	Estimate (relative SE [%])
		Nisin + amikacin		Nisin + linezolid
		S-ADAPT	NONMEM	S-ADAPT	NONMEM
Growth parameters
Log10 (initial inoculum)		7.89 (2.0)c	7.84 (2.2)	7.88 (1.4)c	7.83
Mean generation time
Niss/Amis and Nisi/Amis	k12−1 (min)	57.3 (7.0)b	67.5 (9.5)
Niss/Lins and Nisi/Lins	k12−1 (min)			83.9 (7.5)b	56.6
Growth rate factors
Nisr/Amis and Niss/Amir	fk12_RS_SR	Very lowa	Very lowa
Nisi/Amir	fk12_IR	0.532d	0.519 (14)
Nisr/Amir	fk12_RR	0.413d	0.386 (12)
Nisr/Lini	fk12_R			0.144 (15)	0.102
Replication rate constant	k21 (h−1)	50 (fixed)	50 (fixed)	50 (fixed)	50 (fixed)
Log10 (maximum population size)	CFUmax	9.23 (2.5)c	9.23 (2.0)	9.38 (1.7)c	9.25
Log10 (mutation frequencies)
Nisi/Amis		−4.25 (15)c	−3.63 (8.2)
Nisr/Amis		−3.25 (7.1)	−3.89 (8.0)
Niss/Amir		−2.57 (23)	−2.58 (18)
Nisi/Amir		−4.37 (9.3)	−4.48 (8.2)
Nisr/Amir		−7.42 (7.7)	−7.47 (4.5)
Nisi/Lins				−2.88 (6.2)c	−3.02
Nisr/Lini				−4.15 (7.3)	−3.59
Second-order killing rate constants for nisin
Susceptible population	k2S [liters/(mg · h)]	5.67 (14)b	3.58 (17)	4.49 (12)b	3.08
Intermediate population	k2I [liters/(mg · h)]	0.0664 (6.9)	0.0605 (19)	0.0209 (12)	0.0605
Resistant population	k2R [liters/(mg · h)]	0.00691 (13)	0.00484 (21)	0.00318 (13)	0.00383
Killing by amikacin
Maximum killing rate constants
Susceptible population	KmaxS (h−1)	10.1 (19)	9.10 (21)
Resistant population	KmaxR (h−1)	0.771 (18)	0.635 (10)
Amikacin concn causing 50% of Kmax	KC50 (mg/liter)	14.7 (7.7)	16.5 (7.2)
Hill coefficient	Hill	2.45 (12)b	2.83 (14)
Inhibition of replication and growth rate by linezolid
Inhibition of successful replication by linezolid (InhRep) for Niss/Lins and Nisi/Lins
Maximum inhibition of replication	ImaxRep			1.0 (fixed)	1.0 (fixed)
Linezolid concn causing half-maximum inhibition of protein synthesis	IC50,Prot (mg/liter)			3.92 (19)	7.35
Inhibition of rate of growth (all populations, Inhk12)
Maximum inhibition of rate of growth	Imax,k12			0.858 (12)	0.918
Linezolid concn causing 50% of effect	IC50,k12 (mg/liter)			4.25 (31)	19.4
Hill coefficient	Hillk12			10 (fixed)	10 (fixed)
Turnover rate constant for protein pool	kProt (h−1)			0.72 (7.1)	0.458
Standard deviation of additive residual error on log10 scale	SDCFU	0.395 (9.5)	0.775 (15)	0.307 (7.6)	0.549

^aReplication for these two populations was estimated to be very slow in the present static time-kill experiment. Eventually, fk_{12_RS_SR} was fixed to 0.

^bFor the population PD modeling analysis in S-ADAPT, the biological variability between the viable count profiles (between-curve variability) was set to a coefficient of variation of 15% for all parameters except those estimated on a log₁₀ scale. The between-curve variability was set to 10% for the Hill coefficient.

^cThe between-curve variability for parameters estimated on a log₁₀ scale in S-ADAPT was set to a variance of 0.25. For the maximum population size and the experimentally standardized initial inocula, the variances were estimated to be small and therefore eventually fixed to 0.01.

^dAs these fractions were estimated via a logistic transformation in S-ADAPT, their mean estimate was close to zero on a transformed scale (equivalent to 50% on a linear scale). This resulted in relative standard errors of 222% for fk_{12_IR} and of 65.4% for fk_{12_RR}, since the mean on transformed scale was almost zero. These artificially inflated relative standard errors did not impact the performance of the model or the curve fits, as also suggested by the small standard error in the NONMEM analysis which did not use this transformation.

At low viable counts, REP approaches 2, representing a 100% probability of successful replication (i.e., doubling). As CFU_ALL approaches CFU_max, REP approaches 1, representing a 50% probability of successful replication where bacteria continue to transition between states 1 and 2 but the total viable count is constant (22). The differential equation for CFU_SS,2 also included killing by nisin and amikacin:

$$\frac{d{\text{CFU}}_{\text{SS},2}}{dt}=k_{12}·{\text{CFU}}_{\text{SS},1}-\left(k_{21}+k_{2\text{S}}·{\text{C}}_{\text{Nis}}+\frac{{\text{Kmax}}_{\text{S}}·{\text{C}}_{\text{Ami}}^{\text{Hill}}}{{\text{C}}_{\text{Ami}}^{\text{Hill}}+{\text{KC}}_{50}^{\text{Hill}}}\right)·{\text{CFU}}_{\text{SS},2}$$ (4)

The differential equations for the five populations that were less susceptible to nisin, amikacin, or both are shown in the supplemental material. These differential equations accounted for the decreased susceptibility to nisin (using k_2I or k_2R instead of k_2S; Table 1) or to amikacin (using Kmax_R instead of Kmax_S). Some less-susceptible populations were estimated to have a decreased biofitness described by a longer mean generation time (denoted by the rate constants k_{12_RS_SR}, k_{12_IR}, and k_{12_RR} [Table 1]).

Survival fraction.

The survival fraction after pretreatment with nisin for 1.5 h was calculated for each population based on the second-order killing rate constants (i.e., k₂ was either k_2S, k_2I, or k_2R) as Fr_Survive = exp(−k₂ · C_Nis · 1.5 h).

Initial conditions.

The total inoculum and the mutation frequency of each less-susceptible population were estimated model parameters. The product of the total inoculum and the mutation frequency defined the size of each of the less-susceptible populations at 0 h (i.e., initial condition). The initial condition of the population susceptible to both antibiotics was calculated by subtracting the size of all less-susceptible populations from that of the total initial inoculum. All bacteria of a population were initialized into state 1 of the respective population (i.e., for i = 1 in equation 1). Initial conditions for the compartments representing state 2 (i = 2 in equation 1) were zero, since k₂₁ was rapid.

Nisin-plus-linezolid combination.

The model for nisin plus linezolid contained three populations, including nisin-susceptible and linezolid-susceptible (Nis^s/Lin^s), nisin-intermediate and linezolid-susceptible (Nisⁱ/Lin^s), and nisin-resistant and linezolid-intermediate (Nis^r/Linⁱ) populations. The differential equation for the Nis^s/Lin^s population in state 1 (CFU_SS,1) contained second-order killing by nisin and two inhibitory effects of linezolid (Fig. 3; initial condition calculated as for the nisin-plus-amikacin model):

$$\frac{d{\text{CFU}}_{\text{SS},1}}{dt}=\text{REP}·\left(1-{\text{Inh}}_{\text{Rep}}\right)·k_{21}{\text{·CFU}}_{\text{SS},2}-\left(k_{12}·{\text{Inh}}_{k12}+K_{2\text{S}}{\text{·C}}_{\text{Nis}}\right)·{\text{CFU}}_{\text{SS},1}$$ (5)

where REP was defined as above and CFU_ALL was the sum of both states for the three populations of the nisin-plus-linezolid model. Parameters in equations 5 to 9 are explained in Table 1. Linezolid inhibited protein synthesis, and the protein constituent pool (P), with an initial condition of 1.0, was defined as:

$$\frac{d\text{P}}{dt}=k_{\text{prot}}·\left[\left(1-\frac{{\text{C}}_{\text{Lin}}}{{\text{C}}_{\text{Lin}}+{\text{IC}}_{50,\text{Port}}}\right)-\text{P}\right]$$ (6)

k_prot is the turnover rate constant for the protein pool, and IC_50,Prot is the linezolid concentration (C_Lin) required for half-maximal inhibition of protein synthesis. In the absence of linezolid, the protein pool is at its hypothetical baseline of 100%. Depletion of the protein pool by linezolid increases the probability of unsuccessful replication (Inh_Rep):

$${\text{Inh}}_{\text{Rep}}={\text{Imax}}_{\text{Rep}}·\left(1-\text{P}\right)$$ (7)

As described previously (22), an Inh_Rep (equations 5 and 7) of 0.50 results in net stasis of the respective bacterial population and an Inh_Rep of >0.5 results in bacterial killing, since bacteria that replicate unsuccessfully are eliminated (i.e., lost from the system) (Fig. 3). Linezolid additionally inhibited the growth rate (Inh_k12):

$${\text{Inh}}_{k12}=1-\frac{{\text{I}}_{\text{max},k12}·{\text{C}}_{\text{Lin}}^{{\text{Hill}}_{k12}}}{{\text{C}}_{\text{Lin}}^{{\text{Hill}}_{k12}}+{\text{IC}}_{50,k12}^{{\text{Hill}}_{k12}}}$$ (8)

The differential equation for bacteria in state 2 (CFU_SS2) was:

$$\frac{d{\text{CFU}}_{\text{SS},2}}{dt}=k_{12}·{\text{Inh}}_{k12}{\text{·CFU}}_{\text{SS},1}-\left(k_{21}·K_{2\text{S}}{\text{·C}}_{\text{Nis}}\right)·{\text{CFU}}_{\text{SS},2}$$ (9)

The differential equations for state 1 (CFU_IS,1) and state 2 (CFU_IS,2) of the Nisⁱ/Lin^s population were:

$$\frac{d{\text{CFU}}_{\text{IS},1}}{dt}=\text{REP}·\left(1-{\text{Inh}}_{\text{Rep}}\right)·k_{21}{\text{·CFU}}_{\text{IS},2}-\left(k_{12}·{\text{Inh}}_{k12}+K_{2\text{S}}{\text{·C}}_{\text{Nis}}\right)·{\text{CFU}}_{\text{IS},1}$$ (10)

$$\frac{d{\text{CFU}}_{\text{IS},2}}{dt}=k_{12}·{\text{Inh}}_{k12}{\text{·CFU}}_{\text{IS},1}-\left(k_{21}·K_{2\text{S}}{\text{·C}}_{\text{Nis}}\right)·{\text{CFU}}_{\text{IS},2}$$ (11)

The same growth rate constant (k₁₂) was used for the two Lin^s populations (Nis^s/Lin^s and Nisⁱ/Lin^s), and a lower growth rate constant (k_{12_R}) was used for the Nis^r/Linⁱ population to reflect a decreased biofitness. The differential equations for state 1 (CFU_RI,1) and state 2 (CFU_RI,2) of the Nis^r/Linⁱ population contained the inhibitory effect of linezolid on the growth rate (Inh_k12) but lacked the effect on the probability of successful replication (Inh_Rep):

$$\frac{d{\text{CFU}}_{\text{RI},1}}{dt}=\text{REP}·k_{21}{\text{·CFU}}_{\text{RI},2}-\left(k_{12\_\text{R}}·{\text{Inh}}_{k12}+K_{2\text{S}}{\text{·C}}_{\text{Nis}}\right)·{\text{CFU}}_{\text{RI},1}$$ (12)

$$\frac{d{\text{CFU}}_{\text{RI},2}}{dt}=k_{12\_\text{R}}·{\text{Inh}}_{k12}{\text{·CFU}}_{\text{RI},1}-\left(k_{21}·K_{2\text{S}}{\text{·C}}_{\text{Nis}}\right)·{\text{CFU}}_{\text{RI},2}$$ (13)

Observation model.

All log₁₀ viable counts for nisin plus amikacin or nisin plus linezolid were simultaneously fitted using an additive residual error on a log₁₀ scale. For observations below 100 CFU/ml, the number of colonies per plate was directly fitted using a previously developed residual error model (19).

Computation and model selection.

To provide a robust mathematical analysis, model development was performed independently in two software packages (S-ADAPT and NONMEM) which utilize different estimation algorithms. The importance sampling Monte Carlo parametric expectation-maximization method (pmethod = 4) in S-ADAPT (version 1.57 [31]) using SADAPT-TRAN (32, 33) and the first-order conditional estimation method with the interaction option (FOCE+I) in NONMEM VI (level 1.2; using the ADVAN9 subroutine; NONMEM Project Group, Icon Development Solutions, Ellicott City, MD [34]) were applied. Models were evaluated based on the S-ADAPT objective function value (−1× log likelihood), NONMEM objective function value (−2× log likelihood), and a series of standard diagnostic plots as previously described (35–37).

RESULTS

Nisin and amikacin monotherapies.

The MIC of the studied MRSA strain was 16 mg/liter for nisin and 8 mg/liter for amikacin. Both nisin and amikacin displayed (relatively) rapid killing and prevented regrowth over 48 h at the highest tested concentration in monotherapy (Fig. 5A and B). Nisin at 4 and 8 mg/liter yielded 2.3 to 3.5 log₁₀ killing during the first 2 h followed by near-complete regrowth at 24 h.

Fig 5.

Observed and individual fitted viable count profiles from S-ADAPT for nisin (A) and amikacin (B) in monotherapy and sequential (C) and simultaneous (D and E) combinations. Time is the time after start of the experiment. For the sequential combinations, the time axis shows the time after switching treatment to amikacin at 1.75 h after the start of the experiment. Plates with zero colonies are plotted as 0 log₁₀ CFU/ml.

Survival fractions during pretreatment.

Pretreatment with 8 or 32 mg/liter nisin for 1.5 h resulted in 2 to 3 log₁₀ killing (Fig. 5C and Fig. 6C). The modeled survival fraction (Fr_Survive; based on estimated k_2S in Table 1) of the Nis^s populations after 1.5 h pretreatment was <10⁻¹⁸, indicating a complete killing of Nis^s bacteria (calculated from k_2S in Table 1). For the Nisⁱ populations, Fr_Survive was 0.056 (equivalent to 1.25 log₁₀ killing) for 32 mg/liter nisin and 0.49 (equivalent to 0.3 log₁₀ killing) for 8 mg/liter nisin pretreatment. Thus, 32 mg/liter nisin pretreatment killed the Nisⁱ populations to levels below those of the Nis^r populations. For the Nis^r populations, the Fr_Survive values were 0.79 for 32 mg/liter and 0.94 for 8 mg/liter nisin pretreatment, suggesting that these populations were killed by ≤0.1 log₁₀ by either pretreatment. These Fr_Survive results were in good agreement with results from the pilot experiments (results not shown) that optimized the nisin concentrations and duration of pretreatment.

Fig 6.

Observed and individual fitted viable count profiles from S-ADAPT for nisin (A) and linezolid (B) in monotherapy and sequential (C) and simultaneous (D and E) combinations.

Nisin switched to amikacin.

Control arms that switched from nisin pretreatment to no antibiotic showed growth and thus confirmed the viability of the surviving bacteria (results not shown). Nisin control arms that contained a 2-fold-lower nisin concentration compared to that in pretreatment showed stasis followed by growth and thus confirmed the decreased susceptibility of bacteria surviving nisin pretreatment (results not shown).

Against bacteria surviving 32 mg/liter nisin pretreatment, sequential therapy with 16 mg/liter amikacin achieved viable counts below the limit of counting (1.3 log₁₀ CFU/ml equivalent to 1 colony per plate) at 9.5 and 32 h (Fig. 5C). Thus, the surviving bacteria were susceptible to amikacin monotherapy, since amikacin remained active against Nisⁱ and Nis^r bacteria selected via pretreatment.

Simultaneous combination of nisin and amikacin.

Consistent with the assumption that amikacin killed the Nisⁱ and Nis^r populations, the simultaneous combination of 32 mg/liter nisin plus 16 mg/liter amikacin achieved near-eradication from 4 h, and 16 mg/liter nisin plus 16 mg/liter amikacin reduced viable counts to below the limit of counting from 8 h (Fig. 5E).

Modeling nisin and amikacin.

The rate of bacterial killing by nisin over the studied concentration range was satisfactorily described when modeled to be proportional to the nisin concentration, and a more complex function was not necessary to describe the profiles. This second-order killing function for nisin was in agreement with its ability to form pores in bacterial membranes (16, 38). The killing rate constant was 85.5-fold lower for the Nisⁱ and 821-fold lower for the Nis^r populations compared to the Nis^s populations (Table 1). Bacterial killing by amikacin was relatively rapid and adequately described by a Hill function that takes into account the fact that amikacin reaches a maximum effect on bacterial killing at high concentrations and the sigmoidicity of the concentration effect relationship. This choice of a Hill function was in agreement with previous papers on aminoglycosides (39, 40). The estimated maximal killing rate constant for amikacin against the Ami^s populations was 13-fold higher than that of the Ami^r populations (Table 1). Only a very small fraction (10^−7.42 for Nis^r/Ami^r [Table 1]) of the initial inoculum was estimated to be resistant to both nisin and amikacin.

A model with independent (i.e., additive) effects of nisin and amikacin against each of the six populations fitted all viable count profiles well and was in agreement with the observation that amikacin killed the Nisⁱ and Nis^r populations. Thus, a model with subpopulation synergy was suitable to describe the viable counts (Fig. 5C, D, and E).

All 20 viable count profiles with nisin and amikacin monotherapy, sequential and simultaneous combinations (Fig. 5) were successfully comodeled with unbiased and precise fits (r, 0.99; slope, 0.996; intercept, 0.05 log₁₀ CFU/ml for observed versus individually fitted log₁₀ viable counts). For the 1- and 4-mg/liter amikacin curves, the observation at 4 h was overpredicted (Fig. 5B); however, this did not have a large influence, since the overall extent of killing by 1 and 4 mg/liter amikacin was very modest and the 1-h and 8-h samples were reasonably fitted. The observed versus population fitted log₁₀ viable counts (i.e., in the absence of random between curve variability) were also unbiased and reasonably precise (r = 0.94). All parameters for the nisin and amikacin model were estimated with good precision (relative standard errors, ≤23% except for fk_{12_IR} and fk_{12_RR} [Table 1]). Inclusion of an enhanced rate of killing for the simultaneous combination improved neither the objective function nor the curve fits, suggesting a lack of mechanistic synergy. This result on the types of synergy was based on thorough modeling analyses in S-ADAPT and NONMEM which yielded comparable parameter estimates (Table 1) and led to the same conclusion regarding a lack of mechanistic synergy.

Linezolid and nisin.

The MIC was 2 mg/liter for linezolid. At the high inoculum, 32 mg/liter linezolid achieved slow killing by 1 log₁₀ from 8 to 56 h (Fig. 6B), whereas 2 mg/liter linezolid essentially paralleled the growth control. After nisin pretreatment, linezolid yielded mostly static profiles, suggesting that linezolid displayed some activity against the Nisⁱ and Nis^r populations (Fig. 6C). The simultaneous combinations of linezolid with 16 or 32 mg/liter nisin achieved 3 to 6 log₁₀ killing at 48 h (Fig. 6D and E).

Modeling nisin and linezolid.

For nisin and linezolid, all 20 profiles were adequately fit simultaneously (Fig. 6) with an overall r of 0.99, a slope of 1.001, and an intercept of 0.002 log₁₀ CFU/ml for the observed versus individual fitted log₁₀ viable counts and an r of 0.96 for the population fits. Only the 8- and 32-mg/liter curves for the nisin monotherapy and the 2-mg/liter linezolid monotherapy showed slight misfits (Fig. 6A and B) which did not affect the characterization of the type of synergy. In contrast to nisin and amikacin, linezolid did not exhibit fast bacterial killing (Fig. 6B). In agreement with its effect on inhibiting protein synthesis, linezolid prolonged the mean generation time (Inh_k12 in Fig. 3) of all populations. For the Nis^s/Lin^s and Nisⁱ/Lin^s populations, linezolid additionally inhibited the probability of successful replication (Inh_Rep), most likely by affecting the synthesis of proteins that are necessary for replication. Therefore, linezolid (slowly) killed these populations at linezolid concentrations above the IC_50,Prot of 3.92 mg/liter (Table 1) at the high inoculum studied. Linezolid achieved a static response only against the Nis^r/Linⁱ population which was modeled by linezolid prolonging the mean generation time (Inh_k12). Nisin killed this population slowly.

The Nis^r/Linⁱ population was estimated to have a lower growth rate than the Lin^s populations. The estimated log₁₀ mutation frequencies suggested that the initial inoculum comprised a relatively high fraction (10^−4.15) of the Nis^r/Linⁱ population (Table 1). This result indicated that high nisin and linezolid concentrations in combination would be required to achieve more than ∼4.15 log₁₀ killing against this high inoculum.

DISCUSSION

The experimental and modeling strategy developed in the present study was designed to efficiently identify and characterize synergy mechanisms for antibiotic combinations. We proposed to distinguish between two types of synergy. Subpopulation synergy can be greatly beneficial, if one antibiotic kills the resistant population of a second antibiotic and vice versa (Fig. 1). Mechanistic synergy was defined here as a higher rate of killing of a bacterial population under the simultaneous presence of two antibiotics (Fig. 2) compared to the rate of killing of this bacterial population predicted by the independent effects of both antibiotics in monotherapy.

The proposed approach accounted for the limited time available to bacteria to elaborate their tolerance and resistance mechanisms to survive standard, simultaneously dosed antibiotic combinations. Short-term (1.5-h) pretreatment with a rapidly killing first antibiotic (nisin) was utilized to select less-susceptible populations. This approach allows one to test second antibiotics against the entire surviving bacterial population and does not require isolating presumably more stably resistant colonies after growth over 24 h or even longer on agar plates containing the first antibiotics.

Nisin very rapidly killed MRSA at a high inoculum which allowed an efficient selection of the Nisⁱ and Nis^r populations. Amikacin achieved substantial killing against the Nisⁱ and Nis^r populations (Fig. 5B), whereas linezolid yielded only stasis or slow killing (Fig. 6B) against these populations. The Nis^r/Ami^r population had a log₁₀ mutation frequency of −7.42 (Table 1), suggesting that nisin in combination with amikacin can achieve more than 7 log₁₀ killing at high drug concentrations. This was successfully confirmed for both sequential and simultaneously dosed combinations (Fig. 5).

The nisin-plus-amikacin combination leverages subpopulation synergy via nisin killing the Ami^r populations and amikacin killing the Nisⁱ and Nis^r populations (except for the small Nis^r/Ami^r population that was only [slowly] killed by 16 mg/liter amikacin [Fig. 5E]). This sequential dosing strategy was critical, since it provided experimental insights into why eradication was achieved by the simultaneously dosed combination. In immunocompetent patients the small Nis^r/Ami^r population might be eradicated by the immune system (12, 13), as the vast majority of bacteria are rapidly killed by the antibiotic combination.

In contrast, the Nis^r/Linⁱ population had a high mutation frequency of −4.15 log₁₀ (Table 1). This suggested limited promise for the nisin-plus-linezolid combination against this MRSA strain at a high inoculum. However, future studies are required to assess whether the inhibition of protein synthesis and inhibition of bacterial growth by linezolid may postpone the rate of emergence of resistance.

It was critical for the proposed approach to characterize subpopulation synergy against a high total inoculum exceeding the inverse of the mutation frequency of the less-susceptible populations. Standard MIC testing and checkerboard synergy studies in 96-well plates with a low initial inoculum usually utilize ≤5 × 10⁵ CFU/ml (equivalent to ≤1 × 10⁵ CFU for a 200-μl volume per well). Such low inocula have an extremely low probability (here, 0.38%) to carry a single bacterial cell of the Nis^r/Ami^r population that had a log₁₀ mutation frequency of −7.42. Therefore, subpopulation synergy likely cannot be detected in checkerboard synergy studies at low inocula. In contrast, the high inoculum used in this study of ∼2 · 10⁹ CFU per arm (= 20 ml × 10⁸ CFU/ml) had a >99.9999% probability of containing the Nis^r/Ami^r population.

Mathematical modeling and pilot experiments consistently indicated that pretreatment with 8 mg/liter nisin completely killed the Nis^s populations but (essentially) did not kill the Nisⁱ and Nis^r populations. Pretreatment with 32 mg/liter nisin completely killed the Nis^s populations and killed the Nisⁱ populations to viable counts below the Nis^r populations but had very limited effect against the Nis^r populations. This model fitted the viable count profiles well (Fig. 5) based on the assumption of subpopulation synergy and independent killing effects of nisin and amikacin.

Inclusion of a mechanistic synergy term with an enhanced rate of killing of one or multiple bacterial populations for the simultaneously dosed combination was not needed for nisin and amikacin. In general, an enhanced rate of killing due to the simultaneous presence of two antibiotics (i.e., mechanistic synergy) may be difficult to estimate for antibiotic combinations with subpopulation synergy, since eradication is already achieved by the first antibiotic killing the resistant population of the second antibiotic and vice versa. Mechanistic synergy is easier to identify and estimate, if one antibiotic has no (or limited) effect in monotherapy, such as for a β-lactam/β-lactamase inhibitor combination or for combinations with an efflux pump inhibitor.

The present study also showed that the development of life cycle growth models with multiple (here, up to six) bacterial populations is robust and the model parameters were well estimated in two software packages utilizing different estimation algorithms. The proposed model for nisin employed a second-order killing function similar to previously developed models for colistin (19), a peptide antibiotic against Gram-negative pathogens. Published models for aminoglycosides against Pseudomonas aeruginosa and Staphylococcus aureus utilized Hill-type killing functions in agreement with the present combination model (39, 40). Consistent with previously developed models for linezolid against Enterococcus faecalis (41) and MRSA (18), the present model included an effect of linezolid prolonging the mean generation time and inhibiting the probability of successful replication due to linezolid inhibiting protein synthesis. The population analysis in S-ADAPT, in addition to the analysis in NONMEM, demonstrated that the model structure is sufficiently complex to capture all viable count profiles well, as shown by the r ≥ 0.99 for the individual fitted versus observed log₁₀ CFU/ml, and it suggested there is a small between-curve variability that is reflected by variability in the parameter estimates.

In summary, nisin achieved subpopulation synergy with amikacin but not with linezolid against a high inoculum of MRSA. Additional mechanistic synergy was not needed to simultaneously model the time course of all viable counts for nisin and amikacin. The proposed experimental and modeling strategy is valuable to predict the potential benefits of combination therapies, as shown here with nisin as a probe drug. The proposed strategy can efficiently identify and quantify subpopulation synergy and explain why simultaneously dosed combinations yield eradication. Short-term (1.5 h) pretreatment with a rapidly killing first antibiotic allows this approach to minimize the impact of upregulation of tolerance and resistance mechanisms. This may be important, as bacteria have limited time to elaborate their tolerance and resistance mechanisms to survive standard, simultaneously dosed antibiotic combinations in patients. Therefore, this sequential dosing strategy may be beneficial for future studies on combinations with established and new antibiotics.