Figure 1.
Model of the evolution of resistance in synergistic and antagonistic drug treatments.
(A) Graphical representation of model ODEs describing three bacterial subpopulations: the wild-type sensitive to both drugs (black, Eq. 1), single-drug resistant mutants (blue, Eq. 2), and double-drug resistant mutants (red, Eq. 3). Wild-type and single-drug resistant subpopulations grow with rate (Eq. 4), mutate with rate
and die with antibiotic killing rates
and
, where
and
are the effective drug doses they experience, respectively (Eq. 5). We do not model the growth of the double-drug resistant strain, but simply follow the number of such mutants expected to arise via mutation. (B) The wild-type, single-drug resistant and double-drug resistant mutants experience different effective doses,
, in the multi-drug treatment. The wild-type (black bars) is affected by both the drugs and their interaction, yielding
, where
is the dose of each of the drugs A and B (we assume the two drugs are given at the same dose) and
is the level of their interaction (
, synergistic;
, additive;
, antagonistic). We assume strong resistance, such that resistant mutants are completely unaffected by the drug to which they are resistant; the effective drug dose felt by the single-drug resistant mutant is therefore that of one of the drugs alone,
, and is independent of
(blue bars have a fixed value). Because double-drug resistant mutants are fully resistant to both antibiotics, they feel an effective dose of 0 (red bars). Increased synergy therefore increases killing of the wild-type, but also increases the selective advantage of the single-drug resistant mutants. Antagonistic drug pairs reduce this selective advantage, and can completely eliminate (
) or even invert it (
).
Figure 2.
Choice of drug interaction presents a tradeoff between treatment efficacy and prevention of multi-drug resistance.
Below a critical level of drug interaction (unshaded region, ), treatment efficacy (
, blue) and prevention of multi-drug resistance (
, black) exhibit a tradeoff: increased synergy yields higher efficacy, but at the expense of lower resistance prevention. Above
, however, efficacy plateaus: increasing synergy beyond this ‘synergy ceiling’ fails to improve treatment efficacy, but continues to diminish resistance prevention (shaded region,
).
Figure 3.
The synergy ceiling is determined by clearance of the wild-type population before the single-mutant subpopulation.
(A) Population sizes of the wild-type (, black) and single-drug resistant mutants (
, blue) over treatment courses with levels of interaction below, at or above the critical value
(
). Populations start with sizes
and are killed by antibiotics until they are cleared at times
and
respectively; the overall time of clearance of the infection is simply
(orange markers). The interaction level
affects the order in which the wild-type and the single-drug resistant subpopulations are eliminated: below the synergy ceiling (
, top), the wild-type is eliminated after the single-drug resistant mutant and
; at the synergy ceiling (
, middle), the two populations die simultaneously and
; above the synergy ceiling (
, bottom), the single-drug resistant mutant outlives the wild-type, such that
. Because increasing
increases the wild-type killing rate but has no effect on the single-mutant killing rate, efficacy increases with
below the synergy ceiling (
), but plateaus at and above it (
; vertical dashed line: notice that
is the same both at and above the synergy ceiling). (B) Increased mutation rates,
, give rise to lower
. Inset: treatment efficacy,
, plateaus at lower levels of drug interaction
for higher mutation rates (
, blue;
, orange;
, green; blue and green lines are shifted slightly along y-axis for clarity);
values for each line are indicated by vertical dashed lines, and by circles in the main panel. Orange markers indicate the treatment efficacy achieved for different values of
when
, corresponding to the
values in panel A.
Figure 4.
Prevention of multi-drug resistance by drug antagonism depends on resource competition.
(A) Heat map of instantaneous rates of double-mutant formation, , as a function of single-mutant and total population sizes:
increases with the size of the single-mutant population, and decreases with total population size due to resource competition. Treatment course trajectories for synergistic (
, solid line) and antagonistic (
, dashed line) drug treatments begin with total initial population size
and initial single-mutant population size
(magenta circle), and move toward the origin as the infection is cleared (black circles indicate 20-minute intervals). The different initial slopes of these trajectories (arrows), determined by the relative fitness of the wild-type and single-drug resistant mutants in synergistic versus antagonistic treatments, lead them to different regions of the heat map: synergistic drug pairs quickly kill the wild-type, relieving resource competition before the single-mutant population is killed and leading to a region with high
(solid trajectory goes through red region), while antagonistic pairs kill the single mutants before competition is relieved, leading to a region of low
(dashed trajectory goes through green region). (B) The
over each treatment plotted as a function of time; black circles indicate 40-minute intervals in this panel. (C) Relative ability of these strongly synergistic and antagonistic drug pairs to prevent multi-drug resistance,
, for different initial population sizes (circles). For strong resource competition at the start of treatment (
close to
), antagonistic drug pairs prevent resistance better than synergistic drug pairs (
). For weak competition, however (
significantly less than
), synergistic drug pairs better prevent resistance (
). Artificially turning off wild-type to single-drug resistant mutation during treatment (leaving only the single-mutant population that exists at the onset of treatment) eliminates the advantage of synergy over antagonism at low
(triangles). (D) When initial population size is low and synergy is advantageous, the tradeoff between treatment efficacy and prevention of multi-drug resistance is eliminated, such that maximally synergistic drug pairs yield both the greatest treatment efficacy and greatest prevention of multi-drug resistance (compare panel C to Fig. 2).