Pattern formation inEscherichia coli: A model for thepole-to-pole oscillations of Min proteins and thelocalization of the division siteHans Meinhardt*†and Piet A. J. de Boer‡*Max-Planck-Institut fu¨ r Entwicklungsbiologie, Spemannstrasse 35, D-72076 Tu¨ bingen, Germany; and‡Department of Molecular Biology and Microbiology,Case Western Reserve University, School of Medicine, 10900 Euclid Avenue, Cleveland, OH 44106-4960Edited by Lucy Shapiro, Stanford University School of Medicine, Stanford, CA, and approved October 4, 2001 (received for review May 1, 2001)Proper cell division requires an accurate definition of the divisionplane. In bacteria, this plane is determined by a polymeric ring of theFtsZ protein. The site of Z ring assembly in turn is controlled by the Minsystem, which suppresses FtsZ polymerization at noncentral mem-brane sites. The Min proteins in Escherichia coli undergo a highlydynamic localization cycle, during which they oscillate between themembrane of both cell halves. By using computer simulations weshow that Min protein dynamics can be described accurately by usingthe following assumptions: (i) the MinD ATPase self-assembles on themembrane and recruits both MinC, an inhibitor of Z ring formation,and MinE, a protein required for MinC兾MinD oscillation, (ii) a localaccumulation of MinE is generated by a pattern formation reactionthat is based on local self-enhancement and a long range antagonisticeffect, and (iii) it displaces MinD from the membrane causing its ownlocal destabilization and shift toward higher MinD concentrations.This local destabilization results in a wave of high MinE concentrationtraveling from the cell center to a pole, where it disappears. MinDreassembles on the membrane of the other cell half and attracts a newaccumulation of MinE, causing a wave-like disassembly of MinDagain. The result is a pole-to-pole oscillation of MinC兾D. On timeaverage, MinC concentration is highest at the poles, forcing FtsZassembly to the center. The mechanism is self-organizing and doesnot require any other hypothetical topological determinant.bacteria 兩 cell division 兩 polar pattern 兩 FtsZ 兩 center findingHow a bacterium finds its center to localize the division ma-chinery is a long standing question (1, 2). The preparation fordivision starts with the assembly of a polymeric ring of the tubulin-like GTPase FtsZ just underneath the cytoplasmic membrane (Zring). Recruitment of other division factors to this structure cul-minates in the septal ring organelle, which mediates cell wallinvagination in the exact plane of the initial Z ring (3, 4). Placementof the Z ring is regulated negatively by two autonomous but partiallyredundant systems (5). The best understood is the Min system,which directs Z-ring assembly toward midcell by blocking theprocess at noncentral membrane sites. The other is called ‘‘nucleoidocclusion,’’ which refers to the observation that nucleoids somehowinterfere with FtsZ assembly in their direct vicinity (6, 7). Themechanism of nucleoid occlusion is obscure, and the phenomenonis not considered in detail here.In Escherichia coli, the Min center-finding system is based onhighly dynamic behavior of the MinC, D, and E proteins in vivo.MinC inhibits FtsZ polymerization (8), and its activity is regulatedby MinD and MinE through modulation of its cellular location. Inwild-type cells virtually all of MinC and MinD and a fraction ofMinE assemble on the membrane in the shape of a test tubecovering the membrane from one pole up to approximately midcell.In contrast, the majority of MinE accumulates at the rim of this tubein the shape of a ring (the E ring). The rim of the MinC兾D tube andassociated E ring move from a central position to the cell pole untilboth the tube and ring vanish. Meanwhile, a new MinC兾D tube andassociated E ring form in the opposite cell half and the processrepeats, resulting in a pole-to-pole oscillation cycle of the divisioninhibitor. A full cycle takes ⬇50 s (9–16).MinC binds to and colocalizes with MinD but is itself notinvolved in the oscillation mechanism (11, 12, 17). Oscillationrequires both MinD and MinE, however. These two proteins alsointeract and modulate each other’s behavior (10, 13–16, 18).MinD is an ATPase that accumulates on the cytoplasmic side ofthe membrane to where it recruits both MinC and MinE (9–12, 19).In the absence of MinD, both MinC and MinE remain in thecytoplasm. A lack of MinC at the membrane results in a MinC⫺phenotype where cells frequently produce minicells because ofinappropriate assembly of Z rings near cell poles. In the absence ofMinE, MinD (and hence MinC) is distributed evenly over the entiremembrane. As a result, Z-ring assembly is blocked at any site, andcells form long nonseptate filaments (Sep⫺).In this paper we develop a theory addressing two outstandingquestions raised by the oscillatory behavior of the Min proteinsin E. coli. What is the purpose of an oscillating division inhibitionsystem, and how might such oscillation be accomplished? Wesuggested previously that oscillation of a division inhibitor mightfunction as a center-finding tool, because on time average theconcentration of the inhibitor would be expected to be lowest atthe cell center (10, 11, 15). Here we show that pole-to-poleoscillation can be explained by assuming that MinD and MinEundergo coupled pattern-forming reactions based on local self-enhancement and long range antagonistic effects. The minimummolecular machinery for such oscillation, the localization of asignal to the cell center, and the actual implementation in E. coliare worked out. As demonstrated by computer simulations, thesystem is self-organizing, does not require any prelocalizeddeterminants, and returns to its normal mode after separationinto two daughter cells. The model accurately describes theknown behavior of the Min proteins under various conditionsand suggests particular functions for the known components.Molecular Interactions That Enable the Formation of StablePatterns: Short Range Self-Enhancement and Long RangeInhibitionThe possibility of generating patterns by the interaction of twosubstances with different diffusion rates was discovered by Turing(20). However, a different spread of two interacting substances doesnot guarantee pattern-forming capabilities. In fact, only a veryrestricted class of interactions allows pattern formation, specificallythose in which a local self-enhancing reaction is coupled