Isolated cell behavior drives the evolution of antibiotic resistance

Bacterial antibiotic resistance is typically quantified by the minimum inhibitory concentration (MIC), which is defined as the minimal concentration of antibiotic that inhibits bacterial growth starting from a standard cell density. However, when antibiotic resistance is mediated by degradation, the collective inactivation of antibiotic by the bacterial population can cause the measured MIC to depend strongly on the initial cell density. In cases where this inoculum effect is strong, the relationship between MIC and bacterial fitness in the antibiotic is not well defined. Here, we demonstrate that the resistance of a single, isolated cell—which we call the single-cell MIC (scMIC)—provides a superior metric for quantifying antibiotic resistance. Unlike the MIC, we find that the scMIC predicts the direction of selection and also specifies the antibiotic concentration at which selection begins to favor new mutants. Understanding the cooperative nature of bacterial growth in antibiotics is therefore essential in predicting the evolution of antibiotic resistance.

= −Cn(t)(a out (t) − a in (t)) − y(t) Vmaxaout(t) aout(t)+K M dn(t) dt = γ g n(t)(1 − n(t) nmax ), a in < scM IC Dh5α γ d n(t), a in ≥ scM IC Dh5α , : whenever antibiotic concentration exceeds the scMIC of DH5a, any change in a cell density n(t) is due to the cell death, therefore the density of lysed cells y(t) increases by the same amount as by which the cell density n(t) decreases; there is no change in the density of lysed cells y(t) when periplasmic concentration of antibiotic a in (t) is low enough for cells to divide. The last equation describes the balance between the inux of antibiotic from the environment to periplasm of the cell and enzymatic inactivation of antibiotic in the periplasm [6].
The OD units for cell density correspond to OD600 -absorbance or optical density at 600 nm wave length light of 1 cm-wide sample of the cell culture. The OD of 1 corresponds to 4 · 10 8 CFU per ml. For competition experiments, we use the following system of equations: The system of equations 2 is essentially 1 for two dierent cell types with densities n (1) (t) and n (2) (t), periplasmic antibiotic concentrations a (1) in (t) and a (2) in (t), densities of lysed cells y (1) (t) and y (2) (t). These two cell types share external environment which can be seen in the system of equations in two ways: a out (t) is the same for both cell types and logistic growth part ensures that the carrying capacity of nutrients is shared by two types evenly (γ g n (1)

Solutions in various limits
In order to get some intuition about model prediction for the inoculum eect curve of a single strain, we will consider the limits of small and large initial antibiotic concentration relative to the K M of the enzyme. In section 1.2.1, we will derive the expression for the duration of the death phase of the bacterial growth for which a out (t = 0) = M IC. In sections 1.2.2 and 1.2.3, we will derive the expression for the inoculum eect curve in the limits of low and high initial antibiotic concentrations a out (t = 0) respectively.

Death and growth time
In our model, the cells either grow exponentially or die exponentially. Assuming that initial cell density is n 0 and nal cell density at t 20 = 20 hours is n f (which is xed in the MIC experiment), we can write the following system of linear equations on the time intervals when the culture dies t death and the time interval when the culture grows t growth : with the solution for t death Next, while a out > scM IC (the same as a in > scM IC Dh5α ), Therefore, For the parameter values that we have, we can ignore the exponent at the right-hand side and logarithmic dependence on n 0 : Thus, at low initial antibiotic concentrations (M IC K M ), MIC increases exponentially with the increase of n 0 .

a out (t = 0) K M
Typical scM IC ≤ K M , so the initial antibiotic concentration is much higher than scM IC and the cells initially die. In order for initial antibiotic concentration to be an M IC, the death phase should be signicantly long (otherwise, the regrowth will happen faster than in 20 hours). This fact together with an observation that a in ≤ a out allows us to disregard the antibiotic hydrolysis inside the cell and only consider hydrolysis outside: note that we assume a out K M for all t, which is not the case when antibiotic is almost completely hydrolyzed.
Generality of scMIC: growth rate as a function of internal antibiotic concentration 1 MODEL Whenever a out becomes comparable to K M , a out starts to be broken exponentially in time. However, in the limit of high enough a out , this time of exponential hydrolysis will be much smaller than the time of linear hydrolysis 13.
Plugging in 13 a 0 = M IC, a(t = t death ) = scM IC, Ignoring the logarithmic term, Thus, at high initial antibiotic concentrations (M IC K M ), MIC increases linearly with the initial cell density n 0 .

Generality of scMIC: growth rate as a function of internal antibiotic concentration
In the derivations above, we assume that the growth rate is a step function of antibiotic concentration: γ(a out ) = γ g for a out < scM IC and γ(a out ) = −γ d for a out > scM IC. However, γ(a out ) can be any weakly decreasing function. The concept of scM IC is general and useful for various functions γ(a out ). The following statements hold as long as resistance mechanism is cooperative: a. The general scaling of M IC * as a function of initial cell density are independent of the exact functional form of γ(a out ): 1. M IC * scales exponentially with the initial cell density when smaller than K M ; 2. M IC * scales linearly with the initial cell density when larger than K M .
b. scM IC is well-dened because the inoculum eect curve asymptotically approaches a limit at small initial cell densities.

Small cell densities
The term small (initial) cell densities that we are using corresponds to the dilute conditions, when the cooperative part of the resistance is very weak. From equation11, the dilution condition is as follows: which leads to which under conservative assumptions (see 1.5) of K M = 10µg/ml, V max = 10 4 µg/ml per hour per OD results in the following condition: n 0 2 · 10 4 cells/ml   While C may seem to be another parameter to the inoculum curves t, it has a constraint that it should be the same for several inoculum eect curves. Thus, every inoculum eect curve except for one has two free parameters in their model ts.

The model with the enzyme degradation
While the simple model presented in 1.1 explains qualitatively the behavior of the system at low antibiotic concentrations, it fails to explain some properties of the system at high antibiotic concentrations. There are two major discrepancies: a. Inoculum eect curve, high cell densities and high M IC * s. The data points are not only always lower than the model prediction, but also suggest dierent scaling of M IC * as a function of initial cell density than the model. b. Competition data, high antibiotic concentrations. The data suggests that there is a second peak of selection for the more resistant strain at high antibiotic concentration, while the simple model suggests that above the scM IC of the more resistance strain, selection level relaxes to some level with no dips or peaks.
The discrepancies above happen in dierent experiments under similar conditions -at high initial antibiotic concentrations. This is why it might be the case that they happen for the same reason.
We have considered several ways in which our model can be modied, out of which introducing beta-lactamase degradation turned out to be the most promising one.
The enzyme degradation may happen on its own and because of the reversible substrate-induced inactivation [1]. In the model below, we make two assumptions: 6 a. Dierent enzymes have dierent degradation rate in the absence of antibiotic.
b. The degradation rate of an enzyme is a linear function of antibiotic concentration -the higher antibiotic concentration, the higher the degradation rate.
Generally, as long as inhibition changes V max as a function of antibiotic concentration, the scaling of the inoculum eect curve at high antibiotic concentrations should become sublinear. That means that if the inhibition is accounted for it takes longer to inactivate the antibiotic to the level of scM IC than without inhibition and the eect of inhibition is larger at high antibiotic concentrations, which makes the selection increase the second time at high antibiotic concentrations.
Given two observations above, we constructed a model, which incorporates the degradation rate of beta-lactamase, linearly proportional to the cefotaxime concentration.
where α the enzyme degradation rate per unit of antibiotic concentration. Figure

Sequencing summary
Below is the summary of the mutations observed in the end of the evolution experiment (Fig 1c   and S4). greater than the scMIC of the initial strain is shown. Dierent colors correspond to dierent strains evolved. The data presented is the same as in Fig. 1c and S4.