Bistable Bacterial Growth Rate in Response to Antibiotics with Low Membrane Permeability Johan Elf, 1,2, * Karin Nilsson, 1 Tanel Tenson, 3 and Ma ˚ns Ehrenberg 1,† 1 Dept. Cell and Molecular Biology, BMC, Husargatan 3, Box 596, Uppsala University, Sweden 2 Present affiliation: Dept. Chemistry and Chemical Biology, 12 Oxford Street, Harvard University, 02138 Cambridge, Massachusetts, USA 3 Institute of Technology, University of Tartu, Estonia (Received 6 July 2006; published 19 December 2006) We demonstrate that growth rate bistability for bacterial cells growing exponentially at a fixed external antibiotic concentration can emerge when the cell wall permeability for the drug is low and the growth rate sensitivity to the intracellular drug concentration is high. Under such conditions, an initially high growth rate can remain high, due to dilution of the intracellular drug concentration by rapid cell volume increase, while an initially low growth rate can remain low, due to slow cell volume increase and insignificant drug dilution. Our findings have implications for the testing of novel antibiotics on growing bacterial strains. DOI: 10.1103/PhysRevLett.97.258104 PACS numbers: 87.17.Ee, 87.17.Aa Bacterial infection is a major cause of human suffering and death. Therefore, design of new antibiotics and devel- opment of more efficient ways to deliver already existing antibiotics are mandatory. Antibiotics with low membrane permeability have been regarded as clinically less interest- ing, although they often are very efficient when they have reached their intracellular targets. The current Letter ad- dresses this class of antibiotics and how its members could become clinically more useful by taking into account some striking relations between bacterial growth and intracellu- lar drug concentration. We develop a general dynamic model for the intracellular concentration of these antibi- otics and show that the bacterial growth rate will be bi- stable in response to the antibiotic if the membrane permeability is sufficiently small and the intracellular re- sponse to the antibiotic is sufficiently sensitive. The total concentration, a, of an antibiotic in a bacterial cell depends on its net flow over the membrane, J mem , and its dilution by cell volume growth, J gr . When the drug transport into and out from the cell is passive [3], we model the time evolution of a by the ordinary differential equa- tion da dt ca ex a fr |{z} Jmem a |{z} Jgr (1) The inflow is proportional to the external antibiotic concentration a ex , and the outflow is proportional to the free intracellular concentration of the antibiotic, a fr , which is a function of the total intracellular concentration a: a fr a fr a. The constant of proportionality, c, is the permeability coefficient of the cell wall multiplied by the cell surface [3] divided by the cell volume and will for simplicity be referred to as the ‘‘permeability.’’ [An ex- pression that is mathematically equivalent to (1) can be obtained when there is active transport by an unsaturated efflux pump system [1,2], leading to different permeability for the in-flow and the out-flow: J mem c 1 a ex c 2 a fr c 2 c 1 c 1 2 a ex a fr . Here, c 2 replaces c, and c 1 c 1 2 is a scale factor for the external concentration a ex in Eq. (1).] Since the growth rate is a function of the concentration of bound drug, a-a fr , it follows that also is a function of the total intracellular drug concentration a: a. We will show that biologically motivated constraints on the functions a fr a and a in Eq. (1) in conjunction with a sufficientlylow permeability c lead to bistability for a range of external antibiotic concentrations a ex . Our analysis deals with stationary states during exponential growth and the approach to such stationary states, but the treatment can in a straight-forward manner be extended to cases in which the growth rate varies with external conditions or in which different bacteria experience different growth conditions, like in biofilms [4]. We will first make the natural assumption that the real valued and continuous function a fr a fr a is smaller than the total intracellular concentration a, i.e., a fr a, and that both the free and bound intracellular drug concentrations increase with increasing a, so that da fr =da a 0 fr 1. We further assume that a fr is concave in a (a 00 fr 0), since a 0 fr is expected to increase as the target binding sites become saturated. When a !1, all binding sites are occupied so that a 0 fr ! 1. Some functions, a fr a, that fulfill the criteria are illustrated in Fig. 1(a) and the corresponding convex membrane flow functions J mem a ca ex a fr a in Fig. 1(b). Here, J mem 0 ca ex , the slope of J mem a is determined by ca 0 fr , and J mem a is zero when a fr a a ex . The exponential growth rate a is assumed to be a continuous, finite, positive function [Fig. 1(c)] that de- creases monotonically with increasing a, such that the dilution flow J gr a < 1 and J 0 gr 0 in the limit a ! 1. We will, finally, assume that dJ gr =da < 0 in some interval of a, implying that here J 0 gr decreases more rap- idly than 1=a with increasing a. If, to give an example, the growth rate can be modeled by a Hill function, then J gr a ak 1 =1 a=K H m , and the requirement is that m> 1, meaning that in this special case a 0 in the PRL 97, 258104 (2006) PHYSICAL REVIEW LETTERS week ending 22 DECEMBER 2006 0031-9007= 06=97(25)=258104(4) 258104-1 2006 The American Physical Society