The drug interaction profile, <i>i</i>(<i>θ)</i>, as defined in Materials and Methods.

<p>The drug interaction profile is closely related to the two ‘checkerboard’ diagrams shown in (a) and (c). In a checkerboard, the concentration of both drugs is given on the <i>x</i> and <i>y</i> axes, bacterial growth inhibition (or population density or some other fitness measure) is then plotted on the <i>z</i> axis. The contour of all concentrations that reduce this measure by half is an <i>isobole</i> here denoted <i>IC</i>₅₀ and figures (a) and (c) show two checkerboard plots viewed from above. Basal concentrations of both drugs that achieve the same inhibitory effect in this illustration are <i>D</i>₅₀ and <i>E</i>₅₀, <i>θ</i> then parameterises the equidosage line between these two values. The fitness measure evaluated along this line is shown in (b) and (d) and we define the degree of interaction based on this curve, this is <i>i</i>(<i>θ</i>). We say the interaction is <i>synergistic</i> when the drug proportion that minimises <i>i</i>(<i>θ</i>) satisfies 0<<i>θ</i><1 as in (b), we denote the resulting value by <i>θ</i>_syn. In (d) we observe <i>θ</i>_syn = 0 or <i>θ</i>_syn = 1, in this case the drugs are said to be <i>antagonistic</i> as <i>i</i>(<i>θ</i>) is maximised by some drug combination and minimised by the monotherapies.</p

Smile-frown transition: a verbal argument and a toy mathematical model.

<p>(a) Synergistic drugs suppress drug-susceptible sub-populations (yellow cells) more than single-drug therapies however, this eliminates competitors of the drug-resistant red cells who grow more rapidly than the yellow cells would have done at weaker synergies. Thus greater synergy can increase population densities. (b) Solving <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1001540#pbio.1001540.e001" target="_blank">Equation 1a</a>–<a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1001540#pbio.1001540.e002" target="_blank">b</a> and plotting population density against drug proportion shows that a short-term synergistic combination (blue) can maximise densities later (red). The red dots show the path of the optimal combination, note this idealised model is symmetric about <i>θ</i> = 1/2 but empirical data will not be. (c and d) The densities of drug-susceptible cells (<i>S</i> on the vertical axis in (c)) and resistants (<i>R</i> on the vertical axis in (d)) are shown at different times where, again, the blue line denotes a treatment of short duration and the red line denotes a longer treatment. The arrow in (c) represents the loss of <i>S</i> that occurs because of the drug whereas the arrow in (d) represents the analogous gain in <i>R</i>. For longer treatments the latter more than compensates for the former and by summing the red and blue lines in (b) and (c), respectively, we obtain the red and blue curves showing population density, Δ = <i>S</i>+<i>R</i>, in (a).</p

Drug checkerboards and isobolograms.

<p>(a) Empirical dose-response checkerboards show population density data on the <i>z</i>-axis versus drug concentration on the x and y-axes. This data was obtained by culturing <i>E. coli</i> sampled from the highly synergistic 50-50 environment at days one and five (the treatment with 4.8 µg/ml ERY and 0.08 µg/ml DOX), it corroborates the known synergism on day 1 and indicates the appearance of a more complex interaction by day 5. Note, 50% inhibition relative to the zero-drug control population is indicated by white blocks; (right) the 70% isobole is highlighted as a green line, indicating an interaction where one drug appears to suppress the other. (b) Isoboles (lines of equal inhibitory effect) are shown based on a numerical filter of the data from (a) (the fitting algorithm and code are described in <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1001540#pbio.1001540-DErrico1" target="_blank">[50]</a>). Black lines correspond to isoboles in intervals of 10% inhibition, the darkest red areas illustrate increasing drug concentrations with inhibition towards 100%, the darkest blue areas denote inhibition closest to 0%. The white region denotes 50% inhibition.</p

Drug interaction profiles are dynamic.

<p>(a) The degree of interaction, <i>I</i>(<i>T</i>) defined in <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1001540#s4" target="_blank">Materials and Methods</a>, is shown at different times from 12 h to 108 h: <i>I</i>(<i>T</i>) is negative for <i>T</i>≤24 h denoting synergy, but is positive for all <i>T</i>≥36 h denoting antagonism (vertical bars are s.e., 19 replicates). (b) A finer interaction measure than that used in (a), the degree of interaction obtained using the <i>α</i>-test defined in <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1001540#s4" target="_blank">Materials and Methods</a> produces a locus of drug interaction measures as a function of time. Consistent with (a), this measure changes sign, indicating a change of interaction near 30 h (note: −<i>α</i> is plotted). (c) The smile-frown transition resembles a phase transition when applying the <i>α</i>-test to <i>i</i>(<i>θ</i>,<i>T</i>) derived from MC4100 density data: the grey line shows the optimal drug combination that minimises <i>i</i>(<i>θ</i>,<i>T</i>), the red line shows the maximising combination. As the drug interaction profile ‘inverts’, the short-term optimal therapy shifts over a very short period to become the worst therapy beyond approximately 30 h. (The <i>y</i>-axis varies from <i>θ</i> = 0 (denoting an ERY-monotherapy) to <i>θ</i> = 1 (for a DOX-monotherapy), s.e. is shown as a pair of dashed lines.)</p

The smile-frown transition in empirical data and modelled bacterial densities.

<p>Shown are empirical and modelled bacterial densities (dots and lines, respectively) for 16 different drug proportions, denoted by <i>θ</i> and ranging from 0 (denoting ERY) to 1 (denoting DOX) on the horizontal axis. Population densities, measured as optical densities, are plotted against drug proportion are shown here in a panel of six time points with each blue and red datum 24 hours apart. The data was obtained using <i>E. coli</i> K12 (MC4100) challenged by erythromycin and doxycycline. The smile-frown transition described in the text occurs near 30 h at which point drug synergism is replaced by an antagonism. The model assumes multi-drug efflux is the only resistance mechanism and interpolates the discrete dataset to produce a series of continuous interaction profiles, as shown.</p

The stronger the synergy on day 1, the stronger the antagonism on day 2, both in models and data.

<p>(a) A theoretical model trained on prior predicts that the difference between 18 h synergy and 42 h antagonism will be greater at greater doses. (The prior training data is included in one of the panes and basal dosages are given within each pane.) (b) Predicted changes in interaction are shown as blue points that were determined using <i>α</i> values from the simulations in (a) above. Alongside these are the analogous <i>α</i> values from data in (c) which are black, the dashed line is the linear regression from (c). (c) The correlation between day 1 synergy and day 2 antagonism measured empirically at different basal dosages can be seen in this linear regression showing the interaction measure <i>α</i> at 12 h versus <i>α</i> at 48 h (see <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1001540#s4" target="_blank">Materials and Methods</a> for the definition of <i>α</i>; horizontal and vertical lines are s.e.). Labels denote the level of growth inhibition, 40, 80, 90 and 95%, observed at 18 h relative to a zero-drug control for each of four basal drug dosages.</p

Coverage plots highlight the suspected duplication: a 2× increase in coverage suggests a gene duplication.

<p>A 315 Kb region of the <i>E. coli</i> K12 (MC4100) genome contains the <i>acrAB</i> operon and is highlighted in red. The region was not duplicated for treatments with no antibiotic (‘No drug’), it was duplicated for monotherapies (both ‘ERY’ and ‘DOX’) but was duplicated most often for combination treatments with the greatest synergy (‘50-50’). The outer ring (black line) indicates genome position, grey blocks encompass the different replicates of each treatment (replicates are marked with an alphabetic label) and the reddest regions are most likely to have been duplicated.</p

Overview of single nucleotide polymorphisms in the genomes of <i>E. coli</i> K12 (MC4100) that evolved within five days in erythromycin, doxycycline treatments or in a 50-50 combination of both.

<p>The number of polymorphic sites indicates how many independent nucleotide positions in the gene carry a SNP in at least one replicate, the frequency reflects the number of replicates where a polymorphism in the gene was found. The table only shows SNPs unique to the three treatments.</p

Several antibiotic-binding and resistance genes are found in the 315 Kb genomic region duplicated most frequently in the 50-50 combination treatment, including the following genes and their annotations.

<p>Several antibiotic-binding and resistance genes are found in the 315 Kb genomic region duplicated most frequently in the 50-50 combination treatment, including the following genes and their annotations.</p

The greater the synergy, the more rapid adaptation is to treatment.

<p>This illustrates an entirely expected aspect of our data that corroborates a previous finding <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1001540#pbio.1001540-Hegreness1" target="_blank">[14]</a> on differences in rates of adaptation between antibiotic treatments using different drug pairs: selection for resistance is greater when treatments are more synergistic. The figure shows that our data also supports this idea (degree of interaction defined in <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1001540#s4" target="_blank">Materials and Methods</a>; rate of adaptation is defined in Section 4 of <a href="http://www.plosbiology.org/article/info:doi/10.1371/journal.pbio.1001540#pbio.1001540.s001" target="_blank">Text S1</a>).</p