Fig 5. Simulated optimal drug sequences constrain resistance on long timescales and outperform simple collateral sensitivity cycles.

(A) Discretized collateral sensitivity or resistance C_d∈{−2,−1,0,1,2} for a selection of six drugs. For each selecting drug, the heat map shows the level of cross-resistance or sensitivity (C_d) to each testing drug (the subscript d indicates the profiles are discretized) for n_r = 4 independently evolved populations. See Fig 1 for original (nondiscretized) data. (B) Average level of resistance (〈R(t)〉) to the applied drug for policies with γ = 0 (red), γ = 0.7 (black), γ = 0.78 (magenta), and γ = 0.99 (blue). Resistance to each drug is characterized by 11 discrete levels arbitrarily labeled with integer values from −1 (least resistant) to 9 (most resistant). At time 0, the population starts in the second lowest resistance level (0) for all drugs. Symbols (circles, triangles, squares) are the means of 10³ independent simulations of the MDP, with error bars ± SEM. Solid lines are numerical calculations using exact Markov chain calculations (see Materials and methods). Light red line, long-term optimal policy (γ = 0.99) calculated using the data in (A) but with collateral sensitivity values set to 0. Black shaded line, randomly cycled drugs (±SEM). (C) The time-dependent probability P(Drug) of choosing each of the six drugs when the optimal policy (γ = 0.99) is used. Inset, steady-state distribution P_ss (Drug). (D) The probability P(Resist) of the population exhibiting a particular level of resistance to the applied drug when the optimal policy (γ = 0.99) is used. Inset, steady-state distribution P_ss (Drug). (E) Steady-state joint probability distribution P(current drug, next drug) for consecutive time steps when the optimal policy (γ = 0.99) is used. (F) Average level of resistance (〈R(t)〉) to the applied drug for collateral sensitivity cycles of 2 (dark green, LZD-RIF), 3 (pink, AMP-RIF-LZD), or 4 (dark green, AMP-RIF-TGC-LZD) drugs are compared with MDP policies with γ = 0 (short-term, red) and γ = 0.99 (long-term, blue). For visualizing the results of the collateral sensitivity cycles, which give rise to periodic behavior with large amplitude, the curves show a moving time average (window size 10 steps), but the original curves are shown transparently in the background. Data underlying this figure can be found in S1 Data. AMP, ampicillin; DAP, daptomycin; FOF, fosfomycin; LZD, linezolid; MDP, Markov decision process; RIF, rifampicin; TGC, tigecycline.