Abstract
Fisher suggested advantageous genes would spread through populations as a wave so we sought genetic waves in evolving populations, as follows. By fusing a fluorescent marker to a drug efflux protein (AcrB) whose expression provides Escherichia coli with resistance to some antibiotics, we quantified the evolution and spread of drug-resistant E. coli through spacetime using image analysis and quantitative PCR. As is done in hospitals routinely, we exposed the bacterium to a gradient of antibiotic in a ‘disk diffusion’ drug susceptibility test that we videoed. The videos show complex spatio-genomic patterns redolent of, yet more complex than, Fisher’s predictions whereby a decelerating wave front of advantageous genes colonises towards the antibiotic source, forming bullseye patterns en route and leaving a wave back of bacterial sub-populations expressing AcrB at decreasing levels away from the drug source. qPCR data show that E. coli sited at rapidly-adapting spatial hotspots gain 2 additional copies of acr, the operon that encodes AcrB, within 24h and imaging data show resistant sub-populations thrive most near the antibiotic source due to non-monotone relationships between inhibition due to antibiotic and distance from the source. In the spirit of Fisher, we provide an explicitly spatial nonlinear diffusion equation that exhibits these properties too. Finally, linear diffusion theory quantifies how the spatial extent of bacterial killing scales with increases in antibiotic dosage, predicting that microbes can survive chemotherapies that have been escalated to 250× the clinical dosage if the antibiotic is diffusion-limited.
Introduction
Quantifying Fisher’s idea1 that an advantageous gene moves through space as a diffusive wave is difficult for populations of large organisms. It requires spatiotemporal measurements of a single gene under positive selection, sampled repeatedly from those organisms in situ, on an evolutionary timescale with minimal, non-destructive interference from the observer. So, to meet these requirements we turn to single-cell organisms.
We created an observational rig for bacterial populations using a bespoke time-lapse camera and used it to seek Fisher waves in an antibiotic disk diffusion assay that is routinely used to determine the antibiotic susceptibility of bacteria. We videoed a bacterial population expressing a green fluorescent protein (GFP) that had been physically fused to an antibiotic efflux mechanism (protein AcrB) in Escherichia coli and by culturing the latter in an antibiotic gradient photographed at regular intervals in appropriate light conditions, we used image analyses, corroborated by qPCR, to determine where high AcrB-expressing antibiotic efflux mutants were at all times. This method quantified an increase in antibiotic resistance that occurred within 24h in this clinical assay. Image data show this was due to a 3-fold increase in efflux pumps per cell appearing within complex spatiotemporal patterns.
An important question for in vivo treatments that we sought to quantify using an in vitro approach is how does an antibiotic become too spatially diffuse, and so achieve too low a concentration, to inhibit bacteria? Moreover, antibiotics exhibit gradients in vivo2 so how does dose escalation mitigate this effect? Indeed, human infections are treated at high dosages that exceed clinical breakpoints3–8 yet aggressive chemotherapies can fail,9,10 whether pathogens have pre-existing resistance or else develop resistance during chemotherapy.11–17 So, here we take a diffusion-theoretic approach to quantify features of the ubiquitous disk diffusion antibiotic susceptibility assay, paying attention to how very high, super-clinical dosing increases bacterial killing. We revisit this classical problem in microbiology18 and quantifying bacterial ‘zones of inhibition’ (ZoI) mathematically, thus relating gains in inhibition to dosage increases, providing new quantitative expressions that relate dose to ZoI that are consistent with our imaging data.
Results
Part I: Waves and bullseyes from antibiotics: predicting spatial structure from ecological diffusion theory
Intriguingly, patterns are often visible in antibiotic diffusion assays.19–21 To explain them, we turn to an ecological genetics model written in the spirit of Fisher - the CARS equation: carbohydrate-antibiotic-susceptible-resistant - that captures fundamental features of bacterial growth and antibiotic diffusion:
Here, drug-resistant and susceptible bacteria (R and S, respectively) colonise agar containing a carbohydrate at concentration C. This is taken into the cell at rate u(C) and converted into biomass with efficiency c, where all cells have the same half saturation coefficient for resource uptake (κ), uptake rate (u(C)) and maximal uptake rate (ν). Thus G(C) = c · u(C) represents bacterial growth rate, where u(C) = νC/(κ + C) is carbohydrate uptake rate and σ and σ′ are diffusion coefficients of small molecules and bacteria, respectively. μ is a gain of resistance mutation rate and D an antibiotic-dependent death rate and α is the antibiotic uptake rate.
In the absence of antibiotic, so that A ≡ 0 when t = 0, CARS is a spatial logistic equation of the type studied by Fisher. To see this, define the number of cells, N, by N = S + R. Then define the total system mass as N + cC and note that the latter is constant through time if A = 0, call this constant C0. Adding (1c) to (1d) yields which is a Fisher equation with a vast mathematical history that is known to support waves.22
Bullseye patterns
CARS predicts the formation of spatial patterns when antibiotics are deployed (Figure 1A) because it generates moving bi-modal distributions of bacterial densities formed from waves whereby local population density maxima occur both near to the antibiotic source and far from it, thus creating ‘bullseyes’. Intuition behind the bullseyes is this: high antibiotic dosages kill (or inhibit) so many cells that they create spatial regions abundant in nutrients close to the drug source. So, counterintuitively, the resistant cells that are able to grow in those high-nutrient, low-competition regions can grow more quickly than drug-susceptible cells situated much further from the antibiotic where nutrients are lower per cell because competitor densities are higher there.
This intuition (formalised in Supplementary §1) is borne out by imaging data extracted from photographs of disk diffusion assays (Supplementary §2) where E. coli K12 (AG100) was exposed to gradients of doxycycline (Figure 1B), creating bullseye patterns of population densities. Our explanation for this is known as competitive release which exacerbates resistance because resistant mutants, namely ones that can survive high antibiotic dosages, are granted access to more nutrients than would have been the case if the drug were not present.
The theory of competitive release predicts (Supplementary §1) that the bullseye can form from different genomes which exhibit their fastest growth rates at different distances from the drug source, thus one genotype could be largely responsible for one bullseye ring. To test this idea, we sought spatiotemporal data on population densities of a genetic mutation and so we observed E. coli K12(eTB108) in a disk diffusion assay using doxycycline (Supplementary §2). Strain eTB108 has GFP physically fused to AcrB and it amplifies the efflux operon acr that encodes subunits of the multi-drug efflux pump AcrAB-TolC23,24 under doxycycline stress because this drug is a substrate of the pump. By photographing the diffusion assay through green filters and using quantitative PCR to determine the number of copies of acr per chromosome to corroborate the fluorescence images, we sought to track the spatial locations of acr amplification mutants across the agar plate.
Bullseye rings of population densities formed during this assay (Figure 2A). A ring was first measured 160 pixels (NB: 13.2 pixels per mm throughout) from the drug source at 8h, which subsequently transits with diminishing wavespeed (Figure 2B) towards the antibiotic. qPCR data show acr per genome was amplified within 24h and acr copy number correlates positively with distance to the doxycycline source at 24h (Figure 2C; linear regression p < 0.002, F ≈ 19.6, AICc ≈ 16.2, quadratic regression p ≈ 0.0026, F ≈ 12.4, AICc ≈ 18.1). However, the spatial distribution of acr per genome had changed quantitatively by 48h and a ‘hotspot’ formed whereby acr copy number was now maximised some intermediate distance from the drug, as indicated by a quadratic regression being a more likely descriptor of the data than a linear regression (linear regression p ≈ 0.05, F ≈ 4.7, AICc ≈ 27.0; quadratic regression with unimodal geometry p ≈ 0.0012, F ≈ 12.5, AICc ≈ 22.1). Thus, later populations harbour mutants with 2 mean copies of acr per genome (and 3 copies were observed in 1 biological replicate) whereas earlier populations have fewer operons (c.f. Figure 2C and D).
Genomic waves
Seeking empirical evidence of waves, we examined relative fluorescence image data (green light intensity per pixel divided by white light per pixel) as a proxy for the spatial distribution of the per-cell abundance of AcrB (Figure 3A shows a positive nonlinear correlation). These data indicate (Figure 3B) an expanding population front at the edge of the Zol situated approximately 80 pixels from the drug source at 24h where qPCR and GFP imaging data both indicate between 2 and 3 copies of acr per genome. Imaging data indicate a spatial gradient of up to 3 AcrB relative protein units per cell where cells have 1 AcrB relative unit per cell at those spatial locations sited as far as possible from the drug source (Figure 3B and C).
The wave front exhibits a bi-phasic wave speed (Figure 2B) that approaches the antibiotic source at two different rates that slow but do not stop (Figure 2B, linear regression: 0.019 pixels per h ± 0.012, slope ± 95% confidence interval, p ≪ 0.001). The faster initial phase (before 16h) is likely driven by the carbon gradient-mediated growth of drug sensitive, wild-type cells with 1 acr per genome. After 24h, the emergence of higher acr copy-number variants likely forms the second, slower phase of colonisation towards the drug source as mutants overcome both the increase in antibiotic dose at those locations and the large fitness costs of carrying acr am plifications.24 Finally, we highlight a wave-back behind the front, away from the drug source, where AcrB per cell increases towards the antibiotic source (Figure 3C), up to approxim ately 3 units relative to 1 for the wild-type.
Part II: how zone of inhibition size scales with dose
We are now interested in how the size of the zones of bacterial clearance depend on the antibiotic dosage supplied. The above theory and data indicate subtle nonlinear dependancies that we henceforth ignore in the expectation that linear diffusion theory might be sufficient to answer this classical question. Indeed, much can be gleaned from basic dimensional considerations alone. For instance, spatially-extended chemotherapy should follow a law of diminishing returns because as antibiotics diffuse through 3D-space, killing as they go, the ZoI radius should depend on the 1/3 (cube) root of the dose applied: to double the ZoI volume, 8 times more antibiotic must be supplied. This number is larger if the antibiotic degrades as it diffuses, as we now show.
Diffusion-kill theory
We now present the simplest possible theory of spatially-extended antibiotic killing that we call diffusion-kill theory. Assume a spatially extended bacterial population encounters an antibiotic diffusing isotropically from a point source at rate σ. Suppose antibiotic is supplied at the centre of a spatial domain at concentration Ac and that the distribution of antibiotic, A, is described by the fundamental solution of the diffusion equation in n-dimensions (n ≤ 3). This is At = σ(Axx + Ayy + Azz), where x, y and z are spatial coordinates, thus:
Here, r is distance from the antibiotic source so that r2 is either x2, x2 + y2 or x2 + y2 + z2 for n = 1, 2 and 3 respectively and A(r, t) represents a spatially-normal distribution that expands out from the source as a decaying, spherical wave. Experiments correspond to n = 2 and n = 3 where the former approximates an agar plate where molecules cannot penetrate far into the agar and the latter approxim ates a region of homogeneous tissue.
Suppose a threshold concentration exists above which the antibiotic kills cells, call this threshold Ad (c.f. the minimal bactericidal concentration (MBC)). The ZoI is described by the set of coordinates at distance r from the antibiotic source that satisfy
We can solve inequality (3) (Supplement §3) which yields a constant depending on spatial dimension and Ad, call it C, whereby killing occurs at those distances, r, for which
This formulae are expected from dimensional considerations but the calculations generalises to the situation where the drug decays: in 2 dimensions we find constants C1, C2 and C3 (Supplement §3) such that
Testing diffusion-kill theory
We tested (5) by imaging disk diffusion assays, as follows. We excised a circular section of agar from a plate, replacing it with agar supplemented with identical nutrients but with added antibiotic at a defined dose (Methods). We photographed the latter (Figure S3) to track bacterial inhibition, taking photographs at regular intervals to produce data on the dynamics of the sizes of ZoIs.
The resulting data agree with theory: ZoIs at up to 256×MIC follow the diminishing returns law (4) although the model (5) better fits data for penicillin (Figures 4A, S6) though not for doxycycline (Figures 4B). Prior theory (25, equation 1),18 predicts ZoI geometries of the form r ≤ C1 ln(Ac/C2))1/2 which capture data well too, though not as well as diffusion kill theory (Figures 4A and B; more replicates in Supplement §3). However, prior theory makes a non-physical prediction that the ZoI boundary (where r = C1 ln(Ac/C2))1/2) has a logarithmic singularity that grows without bound as the antibiotic concentration reduces. Our theory corrects this: exponentiating (5) predicts r → 0 as Ac → 0, meaning the ZoI disappears when there is no drug.
Discussion
Theory and data highlight a fundamentally different bacterial dose response to antibiotics in spatial assays compared with non-spatial assays conducted in mixed liquid environments. Notwithstanding the Eagle effect26 whereby dose-response data can be non-monotone in liquid cultures, methods for determining antibiotic dose responses generally predict that bacterial growth declines with increasing antibiotic dose. Here, in contrast, spatial structures form when antibiotics diffuse from a source so that bacteria can grow fastest in a mid-dose regime, leading to patterned growth within waves and rings of colonisation moving towards the antibiotic source at slowing rates.
One mechanism supporting non-monotone spatial dose responses is competitive release: antibiotic gradients create nutrient gradients, thus cells that can grow in zones rich in antibiotic (i.e. resistant cells) access more nutrients than cells far from the antibiotic source. This creates conditions whereby the most rapid population growth can arise close to (and not far from) the antibiotic source because growth benefits provided by the nutrients there outweigh any costs of being resistant. Here, this dynamic promotes resistance by producing cells with 2 additional copies of the acr efflux operon within 24h in a mid-dose region. We tracked these mutants in real-time by combining qPCR with image analysis of GFP-labelled AcrB protein and observed gradients of acr amplifications in a wave structure that qualitatively resembles Fisher’s predictions,1 albeit with additional complexities, like biphasic wave speeds and bullseye rings.
Antibiotics form gradients in the body 2 so predictions on resistance progression in vitro can only have limited predictive power in vivo. Our data is consistent with this idea in the sense that spatial and non-spatial assays behave differently: here, the evolution to 3-fold acr amplifications observed within 24h in a spatial antibiotic gradient dosed at 128× MIC compares unfavourably against the mere 4×MIC needed in a shaken liquid environment to prevent detectable bacterial growth, therefore preventing acr amplifications, following 7 rounds of antibiotic treatment (Figure 4C).
In conclusion, fluorescence photography and imaging data extracted from a ubiquitous clinical assay show a 250-fold increase in antibiotic concentration increases bacterial clearance by much less than 1cm on an agar plate. Thus, without active transport mechanisms to a bacterial site or else spatial mixing to prevent gradients forming, diffusion can prevent antibiotics from achieving the high concentrations needed to kill bacteria. Drug diffusion can therefore promote resistance by providing a smooth path that leads from the lowest to the highest dosages over short spatial scales.
Methods
Media and strains
We used strains E. coli K-12 MG1655 and TB108. Strain TB108 derives from strain MG1655 in which a superfolder GFP locus was inserted in-frame and downstream of acrB. As a consequence, the GFP is attached to the cytoplasm atic c-terminal end of AcrB after translation, with an additional eight amino acids long polylinker peptide between AcrB and GFP. The fluorescence tag led to a partial malfunction of the pump AcrA B-TolC, as determined by the MIC of the strain TB108 to erythromycin, that was repaired (increased back to wild-type) by propagating TB108 in M9 media supplemented with 0.2% glucose (w/v), 0.1% casamino acids and 10μg/mL of erythromycin for seven days. After this step, the MIC of the evolved TB108 strain (now denoted eTB108) was identical to the wild-type MG1655. By using erythromycin we sought to prevent mutations that pre-adapt eTB108, that we use used to the track abundance of GFP-AcrB, to doxycycline and penicillin. The same M9 media, but with no antibiotic, was used for Petri dish assays by adding 6g/L of agar powder (semi-solid agar). To embed the microbes in semi-solid media, 1mL of overnight culture in liquid media was added to 99m L of semi-solid media prior to use.
Spatially-extended diffusion assays
For spatial diffusion plates we prepared two sets of semi-solid media, one being inoculated from an overnight culture as described above, and the other containing doxycycline equivalent to from 1 to 300 times the minimum inhibitory concentration (1×MIC = 0.227 ± 0.007μg/m L, mean ± s.e.m. with n = 8). To our theoretical assumptions, a volume of 20mL from inoculated semi-solid media was used to fill the Petri dish and we later removed a circular section, at the centre of the Petri dish, we then refilled this with semi-solid media containing the drug (1.3m L used). The resulting concentration of nutrient was therefore uniform at the start of this assay whereas that for the drug was not because the drug source lay at the centre of the plate.
Quantitative polymerase chain reaction (qPCR)
We first removed a section of agar of 2.5×5×5mm, approximately, at different distances from the source of antibiotic and degraded the agar using agarase (ThermoScientific #EO0461) as per manufacturer instructions but with additional incubation time. We then extracted DNA from each fragment using the DNA extraction kit ‘GeneJET’ (ThermoScientific #K0729), quantified DNA yield using ‘Qubit’ and diluted accordingly to harmonise DNA content. The kit ‘Luminaris Color Probe Low ROX’ (ThermoScientific #0342) was used as per manufacturer instructions to quantify the abundance of the loci acrB and rob, the latter being a housekeeping gene of reference, sited outside the genomic region known to be amplified under our experimental conditions. The amplification efficiency was ≈ 100% with the primers and probes described in Table 1 and, based on the resulting calibration curves (Figure S7), we used 25ng of material from the agar samples.
Image analysis pipeline
To track culture growth we used a custom-built plate imaging system (Figure S3) equipped with a digital camera (Canon EOS 1100D). To detect GFP-AcrB, we used high intensity LEDs with an effective excitation bandwith of 475-490nm (Comar #475GY25 and #490IK25) and filtered emission bandwiths in the 530-560nm range (Comar #530GY50, #560IK50). We implemented an algorithm in MATLAB to read the resulting time lapse data, with photographs taken every 2h for three days, thus determining dynamic, spatial dose-responses. After a baseline correction to harmonise readings from subsequent photographs, we generated a dose-response profile measuring the pixel intensity in a 16-bit scale (0-255 for each red, green and blue channel) across a line connecting the centre of the zone of inhibition to the edge of the Petri dish, and rotated this line every 15° to generate technical replicates.
Data analysis
To measure the effect of drug dose on microbial growth, we fitted m athem atical models to population density and growth data to determine dose-response profiles (Figure S4A). The following logistic model was used to model bacterial density timeseries at different distances from the drug source:
Here B0 is an estimated experimental blank due to reading an empty microtitre plate, K is the culture density during stationary phase or carrying capacity, A is a composite parameter that includes the initial cell density (inoculum), r the growth rate per cell per hour and λ the duration of the lag phase in hours. Curves for GFP-AcrB were reconstructed (Figure S4B) by fitting the following model to timeseries data
The constant and logistic terms here are analogous to those in the above population growth model and the additional exponential term Ce−dt models the potential for down-regulation or degradation of GFP-AcrB, at rate d.
1 Supplementary Information: Growth rate is maximal an intermediate distance from an antibiotic source
One can use the competitive release concept to explain bullseye patterns quite directly. To do so, modify the basic Monod growth law from the main text and represent bacterial growth rate (G) as where G is reduced by a function of the antiMicrobiol, A, by an inhibition function, f, that satisfies 0 ≤ f(A) ≤ 1. Assuming A and C diffuse, we can predict how G changes through space assuming two things: the drug decays and carbohydrates create the conditions that exhibit competitive release at very high drug dosages. For this, write the steady-state solution of the decay-diffusion equation At = σ(Axx + Ayy) − dA, as A(r) = Acϕ(r), where d is the antibiotic degradation rate and Ac is the antibiotic supply concentration. Assume the free diffusion of carbohydrate away from the antibiotic source occurs due to the use of high antibiotic dosages, so the steady solution for C is analogous: C(r) = C0ϕ(r), where C0 is the carbon supplied that is initially uniformly distributed on agar at t = 0.
If one could show these assumptions imply the maximal bacterial growth rate occurs at intermediate spatial locations sited away from the antibiotic source and if the intracellular antibiotic concentration for two sub-populations were different, for instance if one were to express more antibiotic efflux pumps than another, then their spatial location of optimal growth would be different too. Thus bullseyes would form from the growth rings of the different sub-populations.
So, to demonstrate this, take growth rate defined in (6) and suppose that the spatial distribution of A is given by the steady-state solution of the diffusion equation, so A = Acϕ(r) where ϕ(r) is a strictly decreasing function. So ϕ(r) satisfies, for this example, (although our argument cam be made much more general than this single model). Now assume that where the drug source is, doses to cells are so high so they are rapidly killed due to the high dose, thus a carbon ‘haven’ is formed which begins to supply carbon to surrounding cells by diffusion. Thus, C is also the steady-solution of a diffusion equation and C = C0ϕ(r). In the following, a dash (′) and a subscript (typically r) will both denote derivatives, as is standard notation.
Growth rate is maximal where ∂G/∂r = 0 and lnG = ln v + ln (f(A)) + ln (C) − ln (κ + C), so that
Using f(A) = 1/(1 + pA) for the sake of definiteness, we find f′(A) = −p/(1 + pA)2 so that f′(A)/f(A) = −p/(1 + pA) and therefore
The assumed monotonic property of ϕ (assuming the drugs decay away from the source, which is consistent with solutions of diffusion equations) now means that we are seeking the values of r for which where ϕ = ϕ(r). Simplifying this, we arrive at the following fixed point equation for ϕ = ϕ(r):
Elementary sign and growth considerations of the RHS of this equation (which is quadratic in ϕ) show it always has a unique solution, ϕ*, so that the point of maximal growth rate in the spatial domain is the value r = r* for which ϕ(r) = ϕ*.
Now note that if a population of cells has two different responses to antibiotics whereby each cell is associated with a different internal drug concentration, then this is represented in this model by having A take on lower values for the more resistant cells. For instance, having more drug efflux pumps would reduce A when is modelled here by scaling the value of Ac by a constant between 0 and 1, call this λ: A = λAcϕ(r) is the internal concentration of antibiotic of a cell at location r where λ is lower for cells with more efflux pumps. Thus the equation for maximal growth rate becomes
Now, as the resistance phenotype becomes increasingly significant, represented by A reducing towards 0, the form of these equations makes the value of ϕ* become larger and so r* becomes inreasingly smaller because ϕ(r) is a decreasing function. It is feasible that the value of ϕ* becomes so large that r* approaches 0 and so the fastest growth rate of some highly resistant strains could be achieved very close to the drug source.
Continuing this argument, because of the efflux from highly resistance cells, it could arise that there is a local increase in drug concentration within some highly drug-susceptible cells above the value Ac. In this case, λ would be bigger than 1 and, as this value grows, the solution ϕ* heads towards zero and so r* could grow towards being a point very far from the drug source.
To get a sense of how the resulting spatial distributions of population density might appear, consider the mathematical solutions provided by the above analysis. So, first we need solutions o f the radially symmetric diffusion equation where x and y coordinates are replaced by the distance, r, from the drug source and the angular coordinate is removed: , such that A(r) → 0 as r → ∞ and A(r = 0) = Ac, or Arr + r−1 Ar −Ad/σ = 0. As the solution of this equation is a Bessel function that requires lengthy techniques to derive that are likely to obscure our point, we illustrate the 1-dimensional version ofthiscalculation in Figure S1 where A(r) is an exponential function A(r) = Ac exp(−r(d/σ)1/2).
When we compare the theoretical Figure S1 against the empirical equivalent of the same concept, we find Figure S2 which has been determined for the total population, not for a sub-population. Thus growth rate is maximised some intermediate distance from the drug in data, as the theory claims.
2 Supplementary Information: a fluorescence and white-light photography rig
We constructed a fluorescence photography rig to take images and videos in laboratory conditions. This is based on a standard camera, LEDs used to excite bacterial samples on agar plates in white light and at specific wavelengths, and light filters that are rotated into place by a servo to read different protein emission wavelengths. A heating mat is placed below the camera on a mounting point to control agar temperature.
Antibiotic disk diffusion assays are illustrated in Figure S3B which shows a circular piece of agar (in black) into which high dosage antibiotic has been impregnated which diffuses outwards to create a zone of inhibition on a bacterial lawn. We can measure this lawn either in white light to determine population density, or else in a fluorescent wavelength if wish to determine the level of expression of a GFP-tagged gene throughout space by that population. Dividing white light measurements by GFP measurements gives a proxy for local expression levels per cell through space. Figure S4 shows two exemplars of these spatiotemporal datasets.
NB
Figure S5 shows that there is a positive, albeit nonlinear and saturating correlation between plated cell counts and white light intensity as measured using a camera. Thus white light intensity can be used as a proxy for cell densities provided the latter is not too high.
3 Supplementary Information: How zone of inhibition size scales with antibiotic dose
In this section, we use inequality (3) to predict the effect of dose escalation on the size of the zone of growth inhibition. By calculus, differentiate the expression for A(r, t) from the main text with respect to t, set this derivative to zero and solve it for t. This shows that the maximal drug dose is reached when time, t, satisfies t = t* = 2r2/(4nσ), at this time . Placing this expression in (3), antibiotics inhibit growth provided , where C denotes a dimension-dependent constant. Setting n = 1, 2 and n = 3 and re-arranging, respectively, we determine the radii of the zones of inhibition: microbes do not grow if they are placed at distances, r, from the drug source if where C is a placeholder for constants of proportionality.
We now generalise this calculation to the more realistic case where the drug decays. We again need solutions of the inequality (3) but now where A(t, r) is a solution of the isotropic decay-diffusion equation which, with decay rate d, in 3-d reads:
The solution of this is the solution (2) from the main text multiplied by a decaying exponential, at rate d:
By calculus, we differentiate a(r, t) with respect to t, set this derivative to zero and solve it for t:
Thus, a(t, r) > Ad for some t > 0 if ln a(t*, r) > ln Ad, or where c = 4d/σn2, C1 = d/2σn, C0 = n ln(2πσn/d)/2. Given zone of inhibition radius data on from agar plate diffusion assays, where n = 2, this provides a 3-parameter nonlinear regression model to fit against data of the form (re-stating (5) from the main text) where A is the antibiotic supply concentration, r is the zone of inhibition radius and p1, p2 and p3 are unknown parameters to be determined from data.
Dimensional arguments from main text suggests the use of the 2-parameter regression model for the same ZoI data where we expect p1 to be close to the value 2 for diffusion assays on agar plates. This may not be true in practise because agar plates are not 2-d objects, but anisotropic, 3-d objects which are much wider than they are high.
We tested the ZoI regression models defined by equations (4), (9), (10) and one used already in the literature18,25,27 defined as against ZoI data. The result of applying these 4 regressions can be seen in Figure S6.
NB
Throughout, the equation f(A, r; p) = 0 determines the ZoI boundary as a function of dose (A) and this observation is used to determined the resulting model parameters p.
Figure S6 shows that the four different possible regressions for the same phenomenon that we propose all capture data, having adjusted R2 values above 0.88. However, while equation (9) predicts the ZoI radius at the MIC dose, equation (11) makes an unusual and non-physical prediction whereby as dosage heads towards zero, the radius of the ZoI diverges to infinity (see Figure S6). This paradoxical behaviour (of very good fitting at high dose and non-physical behaviour at low dose) can be seen in Figure S6D.
4 Supplementary Methods
Methodological data for determining quantitative PCR amplification efficiency are shown in Figure S7.