Predicting the dynamics of bacterial growth inhibition by ribosome-targeting antibiotics

Understanding how antibiotics inhibit bacteria can help to reduce antibiotic use and hence avoid antimicrobial resistance—yet few theoretical models exist for bacterial growth inhibition by a clinically relevant antibiotic treatment regimen. In particular, in the clinic, antibiotic treatment is time-dependent. Here, we use a theoretical model, previously applied to steady-state bacterial growth, to predict the dynamical response of a bacterial cell to a time-dependent dose of ribosome-targeting antibiotic. Our results depend strongly on whether the antibiotic shows reversible transport and/or low-affinity ribosome binding (‘low-affinity antibiotic’) or, in contrast, irreversible transport and/or high affinity ribosome binding (‘high-affinity antibiotic’). For low-affinity antibiotics, our model predicts that growth inhibition depends on the duration of the antibiotic pulse, and can show a transient period of very fast growth following removal of the antibiotic. For high-affinity antibiotics, growth inhibition depends on peak dosage rather than dose duration, and the model predicts a pronounced post-antibiotic effect, due to hysteresis, in which growth can be suppressed for long times after the antibiotic dose has ended. These predictions are experimentally testable and may be of clinical significance.


Appendix A. Bifurcation points of the model
As we show in figure 2, the model described by equations (1)-(3) may have different numbers of stationary points (stable and unstable fixed points), depending on the parameter values. Changes in the number and character of the fixed points of the model occur at critical parameter values, and are known as bifurcation points. Figure A1 illustrates these bifurcation points in more detail. Here we use a parameter set intermediate between the low and high-affinity cases studied in the rest of the paper: We vary the parameter k off and plot the fixed points of the model as a function of a P ex in . Figure A1(a) shows results with = k 1000 off − h 1 , for which there is only a single fixed point. In contrast, for a smaller value of = k 100 off − h 1 , as shown in figure A1(b), for some values of a P ex in the model has one fixed point (which is stable), and for other values of a P ex in there are three fixed points, two of which are stable and one unstable. The regime in which there are three fixed points is bistable (shaded) and is bounded by two bifurcation points, labelled * a ex,low and * a ex,high in figure A1(b), where the number of fixed points changes. The upper bifurcation point * a ex,high is associated with a steep decrease in the growth rate λ, since at this bifurcation point the upper stable fixed point is lost and λ drops to the lower fixed point. This critical value * a ex,high is very close to (though not exactly equal to) the IC 50 .
The bifurcation points can be calculated by noting that the fixed points of the model dynamics are given by the roots of equation (7). The number of rootsand thus the number of fixed points-is determined by the discriminant of equation (7): if the discriminant is positive there are three roots, otherwise, there is only one root. Thus the zeros of the discriminant mark the bifurcation points. Since equation (7) is a cubic equation in λ, it may be written as λ λ λ . The zeros of the discriminant can be computed numerically. Figure A1(c) shows the results of such a computation: here the bifurcation points * a ex,low and * a ex,high are plotted as a function of k off . Since the discriminant itself is cubic in a ex it may have either one or three zeros; those at positive a ex correspond to * a ex,low and * a ex,high

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. For low values of k off there are two bifurcation points as in figure A1(b), while for high values of k off there is no bifurcation, as in figure A1(a).
We can also obtain an analytical estimate for the upper bifurcation point = * a a ex ex,high , which corresponds to the antibiotic concentration at which the model predicts a threshold drop in growth rate. To this end, we rewrite equation (7) in the form (note that we have multiplied equation (7) by a factor of 4). Interestingly, this equation depends only on the combination a P ex in rather than on a ex and P in independently. This is why we have used the parameter combination a P ex in in figure A1; it also implies that the critical values * a ex scale as ∼ * , and k P off out not too large, the second term in equation (A.1) can be neglected and we arrive at the quadratic equation The zero of the discriminant is then at figure A1(c) as the blue dashed line. Figure A1(c) shows that for small values of k off this provides a very good estimate for the upper bifurcation point * a ex,high . Thus, µ = * a 11.62 ex M − h 1 /P in is a good estimate for the threshold antibiotic concentration and the IC 50 for high-affinity antibiotics. Remarkably, this approximation does not explicitly depend on P out , k off or k on . For large values of k off or P out , however, this approximation does not hold anymore, since the prefactor k P 1 off out /( ) in equation (A.1) decreases the importance of the first term relative to the second term.

Appendix B. Analytical calculation of inhibition time for a high-affinity antibiotic using the adiabatic approximation
Incorporating expression (6) for the ribosome synthesis rate into the dynamical equations (1)-(3), our model can be expressed as: ) and λ is a function of r u via equation (4). Making the adiabatic approximation, i.e. setting = a 0 , and using equation (4) Here, the system has regimes with one fixed point and a bistable regime with three fixed points (shaded area)-two of which are stable (solid lines) and one unstable (dashed blue line). The bistable regime is bounded by the critical values * a ex,low and * a ex,high . (c) The critical points * a ex,low and * a ex,high , computed from the discriminant of equation (7) as detailed in the text and plotted as function of k off . The bistable regime is shaded. The blue dashed line marks the analytical estimate λ ∆r 4 0 / for the upper critical point * a ex,high , which is valid for κ k t on and λ ∆ P k a P r out off ex in / . Note that since equation (7) is symmetric under exchange of k off and P out , * a P ex in as a function of P out would look exactly the same for Since we are dealing with a high-affinity antibiotic, we also set = k 0 off . This allows us to express the model as an equation in one variable only (the growth rate λ t ( )): Returning to equation (B.7), we can integrate the trajectory λ t ( ) to predict the time T c required for λ to reach a predefined threshold λ c : This integral can be solved using the substitution κ κ λ where we have defined = + κ C k P a 1 k on in ex t on . This gives the following result: . Otherwise, the denominator becomes zero and T c diverges. This would be the case if the system is at the upper stable fixed point of the dynamics, such that the growth rate is not significantly decreased upon exposure to the antibiotic.
This analysis also allows us to understand the origin of the very slow inhibition dynamics for the high-affinity antibiotic, for values of a ex just above the bifurcation point, as shown in figure 4(a). Figure B1 shows the rate of change of the growth rate, λ, plotted as a function of λ, as predicted by equation (B.7), for the highaffinity parameter set. Figure B1(a) shows results for = × a 0.95 ex IC 50 (just below the bifurcation point): the fixed points correspond to zeroes of λ and the stable one is indicated by the arrow (there is of course also another stable fixed point at very small λ, but this is lost in the quadratic approximation of equation (B.7)). Figure B1(b) shows equivalent results for a slightly higher antibiotic concentration, = × a 1.05 ex IC 50 , just above the bifurcation point. Here the fixed points are lost, but the rate of change of λ still comes close to zero, implying that the speed of inhibition by the anti- 1.05 ex IC 50 (just above the bifurcation point). 15 We expect this approximation to be valid close to the upper fixed points, whose bifurcation we are concerned with here. Close to the lower stable fixed point, where λ 0 → , the approximation may not hold.
biotic will be very slow. This slow dynamics close to the bifurcation point can be thought of as a 'bottleneck' in the inhibition trajectory.