Self-organization principles of intracellular pattern formation
SSelf-organization principles of intracellular pattern formation
J. Halatek. F. Brauns, and E. Frey ∗ Arnold–Sommerfeld–Center for Theoretical Physics and Center for NanoScience,Department of Physics, Ludwig-Maximilians-Universit¨at M¨unchen,Theresienstraße 37, D-80333 M¨unchen, Germany (Dated: February 21, 2018)
Abstract
Dynamic patterning of specific proteins is essential for the spatiotemporal regulation of manyimportant intracellular processes in procaryotes, eucaryotes, and multicellular organisms. Theemergence of patterns generated by interactions of diffusing proteins is a paradigmatic examplefor self-organization. In this article we review quantitative models for intracellular Min proteinpatterns in
E. coli , Cdc42 polarization in
S. cerevisiae , and the bipolar PAR protein patternsfound in
C. elegans . By analyzing the molecular processes driving these systems we derive atheoretical perspective on general principles underlying self-organized pattern formation. We arguethat intracellular pattern formation is not captured by concepts such as “activators”, “inhibitors”,or “substrate-depletion”. Instead, intracellular pattern formation is based on the redistribution ofproteins by cytosolic diffusion, and the cycling of proteins between distinct conformational states.Therefore, mass-conserving reaction-diffusion equations provide the most appropriate framework tostudy intracellular pattern formation. We conclude that directed transport, e.g. cytosolic diffusionalong an actively maintained cytosolic gradient, is the key process underlying pattern formation.Thus the basic principle of self-organization is the establishment and maintenance of directedtransport by intracellular protein dynamics.
Keywords: self-organization; pattern formation; intracellular patterns; reaction-diffusion; cell polarity;NTPases a r X i v : . [ phy s i c s . b i o - ph ] F e b NTRODUCTION
In biological systems self-organization refers to the emergence of spatial and temporalstructure. Examples include the structure of the genetic code, the structure of proteins, thestructure of membrane and cytoplasm, or those of tissue, and connected neural networks.On each of these levels, interactions resulting from the dynamics and structural comple-mentarities of the system’s constituents bring about the emergence of biological function.Biological systems are the perfect example for the Aristotelian notion that “the whole ismore than the sum of it’s parts”. For centuries this phrase expressed nothing more than avague intuition that some set of organizational principles must underlie the complex phe-nomena we observe around us. Owing to the advances in quantitative biology and theoreticalbiological physics in recent decades, we have begun to understand how biological structureand function originates from fundamental physical principles of self-organization. While weare not yet in a position to define any universal physical principles of self-organization ingeneral, we are now able to identify recurring themes and principles in particular, importantareas like intracellular pattern formation. This will be the main focus of this review article.The generic equilibrium state of any diffusion process is spatially uniform as diffusion re-moves spatial gradients in chemical concentration. Self-organized pattern formation impliesthat this equilibrium can be destabilized, such that an initially uniform system evolves to-wards a non-uniform steady state — a pattern [1, 2]. Historically, the field of self-organizedpattern formation in chemical systems was initiated by Alan Turing in 1952 [1]. In hisseminal article on “
The chemical basis of morphogenesis ”, Turing showed that the interplaybetween molecular diffusion and chemical interactions can give rise to an instability of thespatially uniform state. His general finding was that in a system with multiple reactingcomponents diffusing laterally on different time scales, the diffusive coupling itself can causean instability even if the system is in a stable chemical equilibrium. Turing was the firstto introduce a linear stability analysis for reaction-diffusion systems. To give the reader animpression of the generality of his ideas let us briefly summarise the underlying mathemat-ical concepts: The initial idea is that any random perturbation of a uniform steady statecan be decomposed in Fourier modes ∼ cos( qx ) (Figure 1a). As long as amplitudes aresmall, each of these modes grows or decays exponentially ∼ exp( σ q t ) cos( qx ), depending onthe sign of the growth rate σ q (or more precisely the real parts Re[ σ q ]). By linear stability2nalysis one computes the growth rates for modes with any wavenumber q . Turing foundthat the interaction between chemical reactions and molecular diffusion can give rise tobands of unstable modes with positive growth rates, i.e. situations where some modes withparticular wavelengths are amplified out of a random perturbation (Figure 1a). This givesrise to pattern formation. In the following we will refer to the pattern forming instability as lateral instability since it originates from lateral diffusive coupling.As proof of principle Turing demonstrated this stability analysis for a general reaction-diffusion system with two chemical components, but also discussed (oscillatory) cases withthree components. For 20 years his results received very little attention. It was only in1972 when Segel and Jackson [3] first interpreted Turing’s linear stability analysis of thetwo-component model, while Gierer and Meinhardt [4] in the same year proposed a num-ber of specific two-component models and coined the terms “activator”, “inhibitor”, and“activator–inhibitor mechanism” in this context.Unfortunately, nowadays the terms “activator–inhibitor mechanism” and “Turing insta-bility” are often thought to refer to identical concepts, despite the fact that the “activator–inhibitor mechanism” only represents a particular interpretation of Turing’s proof of princi-ple analysis that is specific to some but not all two-component models. Furthermore, notethat Turing’s general idea of a lateral instability is in fact not even limited to a particularnumber of chemical components. In the literature the “activator–inhibitor mechanism” isusually considered as a combination of “short range activation” and “long range inhibition”or of “local activation” and “lateral inhibition” in order to convey the following heuristicpicture [2, 5, 6]:Consider two chemical components. First – the (short range) activator – enhances itsown production in some autocatalytic fashion such that its concentration can increase expo-nentially. If the diffusion coefficient of this component is small – any concentration peak willonly slowly disperse in the lateral direction. Secondly, the (long range) inhibitor, which isalso produced by the activator, but has a much large diffusion coefficient. Hence, it does notaccumulate locally with the activator but disperses laterally, where it inhibits the action ofthe activator (see Fig. 1b). It is crucial to realize that this mechanisms is merely a heuristicinterpretation of a formal linear stability analysis presented by Turing. A quite commonmisunderstanding in the biological literature is that pattern formation requires an activatorand an inhibitor. This does not in any way follow from the analysis by Turing [1], Segel and3ackson [3], Gierer and Meinhardt [4–6], or any other analysis – activator–inhibitor modelsare simply mathematically idealized examples of pattern forming systems. Moreover, theunderlying interpretation is actually restricted to systems with only two interacting chemicalcomponents. Note that “chemical component” does not refer to a protein species, but to the conformational state of a protein that determines its interactions with specific (conforma-tions of) other proteins. Clearly, protein interaction networks include many conformationalstates – not just two [7].Furthermore, the activator–inhibitor interpretation inextricably links chemical properties(e.g. autocatalytic action) to the diffusibility of the components (e.g. short range activation).However, in the context of intracellular protein pattern formation the general distinction be-tween diffusibilities is that between membrane-bound and cytosolic (conformational) states.Accordingly, membrane-bound protein conformations would have to be considered as acti-vators in the activator–inhibitor picture, and cytosolic protein conformation as inhibitors,respectively. There are many reasons why this picture is not applicable to intracellularprotein dynamics – the most glaring discrepancy is that proteins are not produced auto-catalytically on the membrane, which is the major (implicit) assumption underlying allactivator–inhibitor interpretations. As we will discuss in detail below, intracellular proteinpattern formation is generically independent of protein production and degradation (cf. [8]),and intracellular protein dynamics are generically driven by the cycling of proteins betweenmembrane-bound and cytosolic conformations.Another interpretation of Turing’s mathematical analysis of two-component systems,which appears to take these considerations into account, is the “activator–depletion” model[4, 6] (see Fig. 1c). It differs from the activator–inhibitor model in making a specific choiceof the reaction terms and reinterpreting the rapidly diffusing component (formerly the in-hibitor) as a substrate that is depleted by conversion into the activator. In this interpretationthe autocatalytic production (which increases activator and inhibitor concentrations) is re-placed by an autocatalytic conversion of substrate to activator, which could be understoodas membrane attachment of a cytosolic protein. However, this type of model [4, 6] cruciallydepends on cytosolic production of the substrate and degradation of the activator on themembrane. In particular, concentration minima are not the result of a depleted cytosol (asone might expect intuitively), but arise from the dominance of the activator degradation,which effectively suppresses the autocatalytic conversion process [4, 6] (Fig. 1c). In other4ords, accumulation of cytosolic proteins on the membrane is suppressed by concomitantdegradation of their membrane-bound forms.Obviously, this assumption is highly specific and biologically implausible in terms ofintracellular protein dynamics. The use of a metalanguage with terms like “activator”and “depletion” suggests that these concepts account for intracellular protein dynamicswhere finite cytosolic particle pools play an important role. But the draining of finitereservoirs is not actually the mechanism that drives pattern formation in activator–depletionmodels. Like the activator–inhibitor model the activator–depletion model strictly dependson production and degradation processes to explain pattern formation, and hence, it cannotaccount for pattern formation in mass conserving systems. These issues clearly demonstratethat heuristic interpretations and reinterpretations of specific mathematical models do notgeneralize Turing’s insight in a useful way. Indeed, “activator–inhibitor” and “activator–depletion” models do not even provide a general picture of two-component systems, andtwo-component systems are already a gross simplification of the biological reality. Hence,there is no reason to assume that an intracellular pattern forming system must containactivators and inhibitors, or involve depletion of substrates. In our opinion, the use ofsuch metalanguage to describe the results of quantitative theoretical models does moreharm than good. It suggests that a unifying theoretical understanding is provided by theidealized mathematical models to which intuitive terms like “activator”, “inhibitor”, and“depletion” refer, whereas in reality, little is known about the actual general principles ofactual intracellular pattern formation.Activator–inhibitor models do provide a legitimate phenomenological description of sys-tems based on production (e.g. growth or gene regulation) and degradation, which is appli-cable to some developmental phenomena [9] or vegetation patterns [10, 11]. However, evenin theses cases, it can be argued [10] that such models should be integrated in a more com-plete modelling framework to account for specific details alone, rather than being treated asparadigmatic models that convey the essence of pattern formation in general.In this article, we provide a molecular perspective on intracellular pattern formation,and review the underlying quantitative biological models, without reference to conceptslike “activators”, “inhibitors”, or depleting substrates. Instead, we review in the followingthe specific implementation of pattern forming mechanisms by various protein interactionsystems – i.e. the Min system in Escherischia coli , the Cdc42 system in
Saccharomyces erevisiae , and the PAR system in Caenorhabditis elegans . Based on these systems we willthen extract and discuss recurring principles of the pattern forming dynamics, in particularthe fact that intracellular protein dynamics are based on cycling between different confor-mational states. We conclude that intracellular pattern formation is, in essence, a spatialredistribution process. Cytosolic concentration gradients are the primary means by whichdirected transport is facilitated. The establishment and maintenance of such gradients is thekey principle underlying self-organized pattern formation. The proper theoretical frameworkto study intracellular pattern formation is set by mass-conserving reaction-diffusion systems.We will review recent theoretical advanced in this field [12] at the end of this article.
The Min system in
E. coli
Cell division in
E. coli requires a mechanism that reliably directs the assembly of theZ-ring division machinery (FtsZ) to midcell [13]. How cells solve this task is one of the moststriking examples for intracellular pattern formation: the pole-to-pole Min protein oscillation[14]. In the past two decades this system has been studied extensively both experimentally[15–27] and theoretically [12, 27–31].The Min protein system consists of three proteins, MinD, MinE, and MinC. In its ATPbound form the ATPase MinD associates cooperatively with the cytoplasmic membrane(see Fig. 2a). Membrane-bound MinD forms a complex with MinC, which inhibits Z-ringassembly. Thus, to form a Z-ring at midcell, MinCD complexes must accumulate in thepolar zones of the cell but not at midcell. The dissociation of MinD from the membraneis mediated by its ATPase Activating Protein (AAP) MinE, which is also recruited to themembrane by MinD, forming MinDE complexes. MinE triggers the ATPase activity ofMinD initiating the detachment of both MinD-ADP and MinE. Subsequently, MinD-ADPundergoes nucleotide exchange in the cytosol such that its ability to bind to the membraneis restored (see Fig. 2a).The joint action of MinD and MinE gives rise to oscillatory dynamics: MinD accumulatesat one cell pole, detaches due to the action of MinE, diffuses, and accumulates at the oppositepole. The oscillation period is about 1 min, and during that time almost the entire mass ofMinD and MinE is redistributed through the cytosol from one end of the cell to the otherand back. 6his example nicely illustrates the fact that pattern forming protein dynamics are inessence protein redistribution processes [12, 30]. In other words, the emergent phenomenonis directed transport, and not localized production and degradation (depletion), which serveas the basis of activator–inhibitor (or activator–depletion) models.It was suggested that binding of MinE to the membrane is essential for self-organizedpattern formation [23, 24, 32]. However, theses results were critically debated in the litera-ture [33] and more recent experiments [34] have explicitly confirmed that MinE membranebinding is not required for self-organized pattern formation. Therefore, we will not discussthis process any further.Furthermore, we note that Min protein oscillations are highly regular and thereforeamenable to a deterministic description. Instances where stochastic effects [35] were re-ported turned out to be overexpression artifacts [36] and not an indication for intrinsic noisedue to low copy numbers.A striking lesson to be learned from the study of Min protein dynamics is the dependenceof the pattern forming process on cell geometry [14, 26, 27, 30, 37–39]. The pole-to-poleoscillation in itself is a phenomenon intrinsically tied to the cell’s geometry, which facilitatesthe detection of a specific location in the cell. Over the past two decades a plethora offascinating observations has been made: (i) In filamentous cells, in which cell division isinhibited, the pole-to-pole oscillation develops additional wave nodes showing that the Minoscillation is a standing wave [14]. (ii) Experiments with nearly spherical cells show that, inthe majority of cases, the pattern forming process is able to detect the long axis, even thoughit is much less pronounced than in wild-type, rod-shaped cells [37, 38]. (iii) In mutant cellsthat were grown in nanofabricated chambers of various shapes a broad range of patterns hasbeen observed [26, 27]. In rectangular cells the oscillation can align with the long axis orthe short axis for the same dimensions of the cell. This shows that patterns with distinctsymmetries are stable under the same conditions: Min patterns are multistable.
The Cdc42 system in
S. cerevisiae
Budding yeast (
S. cerevisiae ) cells are spherical and divide asymmetrically by growing adaughter cell from a localized bud. The GTPase Cdc42 spatially coordinates bud formationand growth via its downstream effectors. To that end, Cdc42 must accumulate within a7estricted region of the plasma membrane (a single Cdc42 cluster) [40]. Formation of aCdc42 cluster, i.e. cell polarization, is achieved in a self-organized fashion from a uniforminitial distribution even in the absence of spatial cues (symmetry breaking) [41].Like all other GTPases, Cdc42 switches between an active GTP-bound state, and aninactive GDP-bound state. Both active and inactive Cdc42 forms associate with the plasmamembrane, with Cdc42-GTP having the higher membrane affinity. Furthermore, Cdc42-GDP is preferentially extracted from the membrane by its Guanine Nucleotide DissociationInhibitor (GDI) Rdi1, which enables it to diffuse in the cytoplasm (see Fig. 2b) [42, 43].Switching between GDP- and GTP-bound states is catalyzed by two classes of proteins:Guanine nucleotide Exchange Factors (GEFs) catalyze the replacement of GDP by GTP,switching Cdc42 to its active state; GTPase Activating Proteins (GAPs) enhance the slowintrinsic GTPase activity of Cdc42, i.e. hydrolysis of GTP to GDP [44] (Note that owingto their biochemical role, GAPs are called activating proteins, even though they switchGTPases into their inactive, GDP-bound state. Moreover, these “activating proteins” arein no way related to “activators” in the sense of “activator–inhibitor” models.). Cdc42 inbudding yeast has only one known GEF, Cdc24, and four GAPs: Bem2, Bem3, Rga1, andRga2. Further, a key player of the Cdc42 interaction network is the scaffold protein Bem1which is recruited to the membrane by Cdc42-GTP, and itself recruits the GEF (Cdc24) toform a Bem1–GEF complex (Fig. 2b) [45, 46].Establishment and maintenance of Cdc42 polarization has been shown to rely on twodistinct and independent pathways of Cdc42 transport: (i) vesicle trafficking of vesicle-bound Cdc42 along actin cables which require Cdc42-GTP (via its downstream effectorBni1) [44, 47], and (ii) diffusive transport of GDI bound Cdc42 in the cytosol. CytosolicCdc42 is recruited to the membrane by Bem1–GEF complexes (Fig. 2b) [45, 46, 48]. Eitherof these transport pathways is sufficient for viability, as has been shown by either suppressingvesicle trafficking (by depolymerizing actin) or inhibiting cytosolic diffusion of Cdc42-GDP(by knocking out the GDI Rdi1) [47, 49, 50]. Since both pathways depend on Cdc42-GTP, the pattern formation mechanism in both cases relies on polar activation (nucleotideexchange) of Cdc42 by Bem1–GEF complexes, which are in turn recruited by Cdc42-GTP[48, 51–54]. Various computational models of the Cdc42-Bem1-GEF interaction networkconfirm that a positive feedback loop mediated by Bem1 is able to establish and maintainpolarization [55, 56]. 8eplacing Cdc42 with a constitutively active mutant suppresses GTPase cycling of Cdc42and hence restricts it to membranes [47, 57]. Such mutants show that self-amplified directedvesicle trafficking of Cdc42-GTP provides a viable self-organized polarization mechanismin itself [57, 58]. Because the mutant Cdc42 is locked in its active state, these cells canforego the polar activation of Cdc42 by Bem1–GEF complexes. Conceptual computationalmodels confirm that an actin mediated transport of Cdc42-GTP can in principle maintainpolarity [49, 57, 59–61], although studies of more realistic models show that key details ofthe involved processes – endocytosis, exocytosis, and vesicle trafficking – are still unclear[62, 63].Interestingly, experiments where Bem1 was knocked out (or deprived of its ability torecruit the GEF to active Cdc42) in cells with wild-type Cdc42 revealed that a third polar-ization mechanism must exist, which is independent of both, Bem1 and vesicle trafficking[64, 65]. Furthermore, polarization in the complete absence of Cdc42 transport has also beenobserved [66], hinting at yet another pattern forming mechanism encoded within the inter-action network of Cdc42. How these mechanisms operate independently of Bem1-mediatedfeedback remains an open question that awaits experimental and theoretical analysis.Normal cell division of budding yeast requires the reliable formation of a single bud-site,i.e. a single Cdc42 cluster (polar zone, sometimes also called “polar cap”). Various mutantstrains exhibit initial transient formation of multiple Cdc42 clusters [67, 68], which thencompete for the limited total amount of Cdc42, leading to a “winner-takes-all” scenariowhere only one cluster remains eventually [51, 69, 70].
The PAR system in
C. elegans
So far we have discussed examples for intracellular pattern formation in unicellularprokaryotes (Min) as well in eukaryotes (Cdc42). A well studied instance of intracellularpattern formation in multicellular organisms is the establishment of the anterior-posterioraxis in the
C. elegans zygote [71–74]. The key players here are two groups of PAR proteins:The aPARs, PAR-3, PAR-6, and aPKC (atypical protein kinase C) localize in the anteriorhalf of the cell. The pPARs, PAR-1, PAR-2, and LGL, localize in the posterior half. Inthe wild type, polarity is established upon fertilisation by cortical actomyosin flow orientedtowards the posterior centrosomes, in other words by active transport of pPAR proteins973, 75]. After polarity establishment this flow ceases, but polarity is maintained. In ad-dition, it has been shown that polarity can be established without flow [75]. These resultssuggest that PAR protein polarity in
C. elegans is based on a reaction-diffusion mechanism.The protein dynamics are based on the antagonism between membrane-bound aPAR andpPAR proteins, mediated by mutual phosphorylation which initiates membrane detachmentat the interface between aPAR and pPAR domains near midcell (see Fig. 2c). Thus, PAR-based pattern formation is driven by (mutual) detachment where opposing zones come intocontact, and is therefore quite different than the attachment (recruitment) based systemsdiscussed above.Despite these apparent differences we will argue in the following sections that patternsformation in all three systems is based on the same general principles.
GENERAL BIOPHYSICAL PRINCIPLES OF INTRACELLULAR PATTERN FOR-MATION
Let us take a bird’s eye view and ask: What are the general concepts and recurringthemes that are common to pattern formation in all of these biological systems?In all cases the biological function associated with the respective pattern is mediated bymembrane-bound proteins alone, in other words: intracellular patterns are membrane-boundpatterns (exceptions are discussed further below). Furthermore, the diffusion coefficients ofmembrane-bound proteins are generically at least two orders of magnitude lower than thoseof their cytosolic counterparts, e.g. a typical value for diffusion along a membrane wouldbe between 0 . µm /s and 0 . µm /s , while a typical cytosolic protein has a diffusioncoefficient of about 10 µm /s , e.g. see [66, 76].The key unifying feature of all protein interaction systems is switching between differentprotein states or conformations. The conformation (state) of a protein can change as aconsequence of interactions with other biomolecules (lipids, nucleotides, or other proteins).Likewise, the interactions available to a protein are determined by its conformation. This canbe summarized as the switching paradigm of proteins (Fig. 2d), which is best exemplifiedfor NTPases such as MinD or Cdc42 whose dynamics are in essence driven by deactivationand reactivation through nucleotide exchange. The phosphorylation and dephosphorylationof PAR proteins by kinases and phosphatases, respectively, exemplifies the same principle.10n all these cases, switching is tied to membrane affinity, and thus to the flux of proteinsinto and out of the cytosol.Dynamics based on conformational switching conserve the copy number of the protein.Therefore, intracellular protein dynamics are generically represented by mass-conservingreaction-diffusion systems – and pattern formation in a mass-conserving system can onlybe based on transport (redistribution), it cannot depend on production or degradation ofproteins. In the absence of active transport mechanisms (such as vesicle trafficking) the onlyavailable transport process is molecular diffusion. Given that membrane-bound proteinsbarely diffuse, we can assert that the biophysical role of the cytosol in these systems is thatof a (three-dimensional) ‘transport layer’. Hence, the (functionally relevant) membrane-bound protein pattern must originate from redistribution via the cytosol, i.e. the couplingof membrane detachment in one spatial region of the cell to membrane attachment in anotherregion, through the maintenance of a diffusive flux in the cytosol.However, transport by diffusion eliminates concentration gradients. Hence, if a diffusiveflux is to be maintained, a gradient needs to be sustained. Note that due to fast cytosolicdiffusion, this gradient can be rather shallow and still induce the flux necessary to establishthe pattern (the flux is simply given by the diffusion coefficient times the gradient).Intracellular pattern formation is the localized accumulation of proteins on the membraneby cytosolic redistribution. In this context, self-organization is the emergence of directedtransport that manifests itself as the formation of spatially separated attachment and de-tachment zones, due to the interplay between cytosolic diffusion and protein interactions(reactions).Furthermore, we note that patterns can also be bound to other structures such as thenucleoid [13] or the cytoskeleton [77–79]. In several of these pattern forming systems, thedynamics of a nucleoid-bound, ParA-like ATPase results in different patterns, e.g. midcelllocalization [80, 81] and pole-to-pole oscillations on the nucleoid of a single cargo as wellas equidistant positioning of multiple cargoes [82]. Various mechanisms to explain thesepatterns have been proposed [83], including models that require ParA filament formation[82, 84, 85] and ones which are based on a self-organizing concentration gradient of theATPase along the nucleoid [86–90]. In all these cases the dynamics of patterns are basedon the conformational switching of proteins between the cytosolic and the nucleoid-bound(slowly diffusing) state. For midcell localization, the diffusive flux of the ATPase on the11ucleoid was found to be important [80, 91, 92].Finally, we want to mention pattern forming systems that are based on a preexistingtemplate. The cell geometry itself is such a template, and macromolecules (phospholipids orspecific proteins) can have preferential affinity to accumulate in regions of specific membranecurvature, e.g. the cell poles [93, 94]. However, theoretical analysis of Min protein dynamicsindicates that self-organized pattern formation based on lateral instability is robust againstheterogeneities in the membrane [95].The elementary (i.e. most simple) intracellular pattern is cell polarization: the asym-metric accumulation of proteins in a cell. It lacks any intrinsic length scale and merelyserves to define a specific region of a cell (anterior/posterior domain, budding site). In thefollowing we will use cell polarity patterns as a paradigm to obtain a mechanistic picture ofintracellular pattern formation. Cell polarity: The elementary pattern for intracellular pattern formation
How can one construct a general conceptual model for self-organized intracellular pat-tern formation building on the principles discussed in the preceding section? The proteindynamics are based on cycling of proteins between membrane-bound and cytosolic states,and do not depend on production and degradation of proteins (cf. [8]). Hence, in a steadystate where the protein distribution is spatially uniform, attachment and detachment pro-cesses must be balanced throughout the cell. Lateral instability, e.g. a Turing instability,simply means that small spatial perturbations will be amplified. Let us therefore imaginea perturbation of the density of a membrane-bound species where at some membrane po-sition the protein concentration will be slightly larger than elsewhere on the membrane. Ifthis perturbation is to be amplified, proteins must be transported to the position where themembrane density is (already) highest. For this transport of proteins, we only have cytosolicdiffusion at our disposal. To facilitate a directed transport to a specific position by cytosolicdiffusion, the cytosolic density at this position must itself be at a minimum (Fig. 3). Inorder to reduce the cytosolic density at the position where the membrane density is highest,the balance between attachment and detachment must shift in favour of attachment suchthat protein mass flows from the cytosol to the membrane. Conversely, in the region wherethe membrane density is lowest, the attachment-detachment balance must shift in favour of12etachment, thereby increasing the cytosolic density in this region. Only if changes in den-sity shift attachment–detachment balance in this fashion will the initial perturbation on themembrane be amplified by further attachment and detachment due to cytosolic transport(Fig. 3).To concisely summarize:
Intracellular pattern formation can be understood as the forma-tion of attachment and detachment zones, which are coupled through cytosolic gradients thatfacilitate protein mass redistribution.
In this light, we will now look again at the specific biological systems introduced earlier,and attempt to uncover the basic molecular mechanisms that lead to the formation andmaintenance of attachment and detachment zones.
QUANTITATIVE MODELS FOR INTRACELLULAR PATTERN FORMING SYS-TEMSThe Cdc42 system in
S. cerevisiae
The key interaction that drives self-organized Cdc42 polarization is the recruitment ofGEF by Cdc42, mediated by Bem1, giving rise to mutual recruitment of Cdc42, Bem1and GEF. In a minimal model, the role of Bem1 and GEF can be summarized by aneffective Bem1–GEF complex, which is recruited to Cdc42-GTP on the membrane (Fig. 4a).There, the GEF then recruits more Cdc42 from the cytosol and converts it into the GTP-bound form. As a further simplification, recruitment and nucleotide exchange of Cdc42 byBem1–GEF complexes can be subsumed into a single step. Similarly hydrolysis of GTP(catalyzed by GAPs) and extraction of CDC42 from the membrane can be conflated to asingle membrane dissociation step. Effectively, only active Cdc42 on the membrane andinactive Cdc42 in the cytosol are considered (Fig. 4a).How can this interaction scheme of mutual recruitment establish spatially separated at-tachment and detachment zones for the polarity marker Cdc42? Cdc42-GTP on the mem-brane acts as a “recruitment template” for Bem1–GEF complexes: a zone of high Cdc42-GTP density on the membrane creates an attachment zone for Bem1, which in turn createsan attachment zone for the GEF Cdc24, such that effectively Cdc42-GTP creates an at-tachment zone for Bem1–GEF complexes (cf. panels (1) and (2) in Fig. 4b). A region of13igh Bem1–GEF density on the membrane, in turn, acts as recruitment template for Cdc42,creating a Cdc42 attachment zone, and locally enhances Cdc42 nucleotide exchange leadingto increased local Cdc42-GTP density (cf. panels (1’) and (2’) in Fig. 4b). In the absence ofCdc42-GTP, very little Bem1 attaches to the membrane, such that detachment dominates.Similarly, in the absence of GEF, Cdc42 is dominantly inactive, such that membrane extrac-tion by GDI, i.e. detachment of Cdc42, dominates. Starting from a uniform state, a spatialperturbation of either density will establish a mutual recruitment zone, with Cdc42–GTPsustaining the attachment zone for Bem1–GEF, which in turn maintains the recruitmentand activation zone of Cdc42 (Fig. 4c).Conceptually, cell polarization need not require two protein species that mutually re-cruit each other: conceptual theoretical models for cell-polarity involving only two chemi-cal components (effectively describing a single protein type in two conformational states –membrane-bound and cytosolic) have also been studied [61, 96, 97]. In these models, patterswith multiple density peaks show “winner-takes-all” coarsening dynamics due to competi-tion for the conserved total mass of proteins [96] (cf. the discussion of wavelength selectionand mass-conserving reaction–diffusion systems below).
The Min system in
E. coli
Pole-to-pole oscillations are the result of interactions between MinD and MinE. In Fig. 5,the key phases of the oscillation cycle are shown. Membrane-bound MinD facilitates furtheraccumulation of MinD and MinE on the membrane through recruitment. The recruitmentof MinD shifts the attachment-detachment balance towards further attachment. In contrast,the recruitment of MinE shift this balance towards detachment. The structure of a polarzone is such that MinE is accumulated in the form of MinDE complexes at its rim (sometimescalled the “E-ring”), whereas MinD accumulates at its tip (see Fig. 5 panels (2) and (4)).A theoretical analysis of Min protein dynamics revealed that self-organized Min proteinpattern formation is based on two requirements [30]: The total copy number of MinD mustexceed that of MinE, and the recruitment rate of MinE must be larger than that of MinD.The first condition ensures that all of the MinE can be bound as MinDE complexes onthe membrane (MinE-MinD detachment zone, panels (2) and (4) in Fig. 5) while leavinga fraction of MinD free to initiate and maintain a MinD attachment zone. The second14ondition causes MinE to be trapped immediately upon entry into a polar zone (MinD-MinEattachment zone) and thereby “sequestrated” at the rim, creating a localized MinE-MinDdetachment zone (panels (1) and (3) in Fig. 5). Rebinding of MinE to the MinD-MinEattachment zone at the tip of the polar zone is favoured due to the faster recruitment ofMinE and the fact that MinD is temporarily inactive after detachment. This leads to theprogressive conversion of attachment zones into detachment zones due to the shifting balancein favor of MinDE complexes (detachment).Below we will discuss that the inactivation of MinD upon MinE-stimulated hydrolysisis essential for the establishment of intrinsic length scales and for the dependence of thepattern–forming process on cell geometry (see also Fig. 7).Biochemically the Min system of
E. coli and the Cdc42 system of
S. cerevisiea , areclosely related: both MinD and Cdc42 are NTPases regulated by enzymatic proteins such asNTPase-activating proteins (NAPs), even though the regulation of Cdc42 activity is muchmore complex. As we have argued above, the pattern forming dynamics of both systemsfollow the same underlying physical principle: self-organized spatial separation of attachmentand detachment zones . Indeed, theoretical models have predicted that Min protein dynamicscan also give rise to stationary polar patterns [30]. Conversely, oscillations of Cdc42-markedpolar zones in budding yeast have been observed experimentally [98], while the non-sphericalfission yeast exhibits pole-to-pole oscillations of Cdc42 clusters during the polar growth phase[99]. This raises the question whether there is a common underlying mechanism that unifiesMin protein patterns and Cdc42 polarization at the physical level.
The PAR system in
C. elegans
PAR protein polarization in
C. elegans is based on an antagonism between membrane-bound aPAR and pPAR protein through mutual phosphorylation, cf. [75]. The major differ-ence relative to the above discussed systems is the lack of an evident biochemical mechanismfor the formation of attachment zones, such as recruitment. Instead for
C. elegans , attach-ment zones result from mutual exclusion . In a zone with high aPAR concentration on themembrane only aPAR can attach, as pPARs are immediately phosphorylated. Similarly, inzones with high pPAR membrane concentration only pPAR can attach. At the interface be-tween aPAR and pPAR zones each protein class drives the other off the membrane. Hence,15he interface acts as detachment zone for both aPAR and pPAR, whereas the anterior pole(aPAR dominant) acts as aPAR attachment zone, and the posterior pole (pPAR) acts aspPAR attachment zone. The key to pattern formation is the detachment zone, i.e. themaintenance of the interface. There, aPAR and pPAR domains are actively separated fromeach other, and cycling between the interface and the respective attachment zones maintainsthe bi-polar pattern. Theoretical analysis [75] shows that a stable interface requires the ratesof the antagonistic interactions to be comparable.A very interesting aspect of PAR protein pattern formation is the role of the cortical flow[73, 75]. In the wild type it is used to segregate aPAR zones from pPAR zones, i.e. to formthe respective attachment zones. The polarized state is maintained after the flow ceases,showing that maintenance of the interface is independent of the flow, i.e. it is self-organized.
ADVANCED INTRACELLULAR PATTERN FORMATION: WAVELENGTH SE-LECTION, DEPENDENCE ON CELL GEOMETRY, AND MULTISTABILITY OFPATTERNSPattern formation and length scale selection: the classical picture
Traditionally the phenomenon of self-organized pattern formation in reaction-diffusionsystems has been intrinsically linked to the postulated existence of a characteristic lengthscale [2]. In particular, most authors define a
Turing pattern as a pattern with a character-istic length scale [100]. In our discussion so far such a length scale has only been mentionedin passing as a phenomenon observed in specific
E. coli mutants, and in budding yeast mu-tants as a transient pattern of multiple Cdc42 clusters. Indeed, the theoretical analysis of allquantitative models [30, 55, 71] discussed so far reveals that the existence of such a lengthscale is in no way generic – despite the fact that all patterns emerge from a lateral instabil-ity induced by diffusive coupling, i.e. a
Turing instability . On the contrary, it appears thatthe phenomenon biologists refer to as “the winner takes all” and physicist as “coarsening”is the generic case, cf. [27, 30, 101, 102]. Hence, the generic pattern is a polarized statewith a single concentration maximum and a single concentration minimum for each chemicalcomponent irrespective of the system size.In fact, the question of length scale selection is a highly nontrivial problem, and can only16e addressed in general by numerical simulations. Linear stability analysis (as introduced byTuring) only predicts the length scale of the (transient) pattern that initially forms from theuniform state (see Fig. 1a). This should not be confused with the length scale of the finalpattern. For instance, coarsening dynamics (“the winner takes all”) are generic examplesfor dynamics where the length scale collapses to the system size (or an intermediate lengthscale) regardless of the initially selected length scale [103]. A case where the length scaleis predicted correctly by the linear stability analysis is when the growth of the patternsaturates at small amplitude [2]. Some authors include saturation at small amplitude intheir definition of a Turing pattern [104]. While this definition is mathematically rigorous,it is also very restrictive: For technical mathematical reasons this (supercritical bifurcation,near threshold) case implies that the pattern must vanish if some system parameters areslightly changed. If the pattern does not saturate at a very small amplitude early on, no(reliable) prediction about the final pattern can be made based on the linear stability analysis(except that some non-uniform pattern exists).From the mathematical point of view this specific case (supercritical bifurcation, nearthreshold) is very attractive as it lends itself to analytical calculations [2]. To readers lessinterested in the mathematical details, these points may seem overly technical. However,it is crucial to realize that – from the biological perspective – such technical limitations(a pattern with very small amplitude (weak signal) that is highly sensitive to parameterchanges) imply patterns that are highly fragile (cf. [105]), and will therefore be eliminatedby natural selection. In that light it is not surprising that (robust) quantitative models ofbiological systems, like those presented above, do not meet the constraints of small amplitudeand vicinity to a supercritical bifurcation. Partly because most mathematically motivatedwork on pattern selection is based on the assumption that these constraints are met, verylittle is known about pattern selection in (evolutionarily robust) biological systems.Yet, several key aspects of pattern selection can be inferred from the theoretical analysisof models for actual biological systems. For example, theoretical analysis of Min proteinpatterns [30] showed that standing wave patterns with a finite wavelength emerge if the lat-eral redistribution of Min proteins is canalized , see Fig. 7. In terms of the general principlesdiscussed above, this means that attachment and detachment zones are strongly coupledthrough cytosolic transport. The flux off the membrane in a detachment zone is of the sameorder of magnitude as the flux onto the membrane in the attachment zone. Hence, the frac-17ion of cytosolic proteins remains approximately constant during the redistribution process(Fig. 7a). As we have discussed, attachment and detachment zones are regulated by themembrane kinetics of the specific biochemical model, while transport depends on cytosolickinetics and diffusion. Canalized transfer leads to the emergence of a characteristic sepa-ration distance between attachment and detachment zones which depends in a nontrivialmanner on the system parameters (Fig. 7a). The particular parameter dependence of suchcharacteristic redistribution length scales remains an open question for the Min system, andeven more so for general reaction diffusion systems. However, it was demonstrated that thetotal mass flux due to canalized transfer can be inferred from the linear stability analysisfor the Min model [30]. The flux coupling (cytosolic exchange) between detachment and at-tachment zones is weak, if a cytosolic reservoir is filled and depleted during detachment- andattachment-dominant phases, respectively (see Fig. 7b). It seems intuitive that a redistribu-tion process through a “well-mixed” cytosolic reservoir does not dictate an intrinsic lengthscale for pattern forming dynamics. Moreover, the analysis of quantitative models (such asthe Min model) does provide strong evidence that length scale selection in reaction-diffusionsystems essentially relates to the length scales of directed (“canalized”) transport. However,the precise details underlying the emergence of intrinsic length scales remain unknown.
Cell geometry and pattern selection
In the previous section we discussed why it is important to consider the seemingly tech-nical limitations underlying some mathematical results about pattern formation in orderto correctly understand quantitative biological models for intracellular pattern formation.Besides the question of length scale selection, the effect of system geometry (i.e. cell shape)is in this respect another case in point.At first, linear stability analysis of reaction-diffusion systems was exclusively restrictedto planar geometries such as lines and flat surfaces. Only very recently, the method wasextended to account for (2d) circular geometries where dynamics can take place on theboundary of the circle (membrane) as well as in the bulk (cytosol), and proteins exchangebetween the two domains (membrane-cytosol cycling) [106]. This important first step to-wards quantitative modeling of intracellular protein dynamics was still limited to purelylinear attachment-detachment dynamics (thus excluding the cases of cooperative attach-18ent, recruitment, or antagonistic detachment). Later, linear stability analysis methodswere extended to general attachment-detachment kinetics by for (3d) spherical geometry[55], and for (2d) elliptical geometry [30].The extension to elliptical geometry revealed a very important point [30]: it is in noway generic that patterns align with the long axis of a cell, i.e. there is in general nointrinsic preference for the selection of long axis patterns over short axis patterns. From abiological perspective this is a crucial finding, since proper axis selection is typically linkedto the spatial nature of the biological function mediated by the pattern in the first place.Intuitively, one may expect that axis selection is connected to the “characteristic lengthscale” of the pattern obtained from a linear stability analysis in a planar (flat) geometry:whichever axis length of the cell is closer to this “characteristic length scale” determines theaxis that is selected. The intuition behind this is that the pattern has “to fit” into the cell.So far, however, there is no evidence to support this intuition. On the contrary, it appearsthat the question of axis selection is much more complicated. In a study combining theoryand experiments Wu et al. [26, 27] analyzed the Min protein patterns in rectangular cellgeometries of various sizes and aspect ratios. The experiments found that a broad rangeof patterns (aligned with the long axis or the short axis) can emerge in the same systemgeometries. Hence, intracellular Min protein patterns are multistable, and can conform toa variety of intrinsic length scales instead of one “characteristic” length scale.Theoretical analysis [27] confirmed multistability of Min patterns, and was able to link allobservations to the emergence of an intrinsic length scale for diffusive cytosolic redistribution(“canalized transport”) in the model: the stronger the flux-coupling between attachmentand detachment zones, the stronger was the dependence of the pattern-forming process oncell geometry, and the greater the number of multistable patterns with distinct symmetries(long axis or short axis alignment) observed in a broad range of rectangular cell geometries[27]. This strongly suggest that pattern selection and the influence of cell geometry are –just like wavelength selection – closely tied to the length scale of lateral transport and thestrength of the coupling between attachment and detachment zones. Note that this is avery different picture from the one presented by “activator–inhibitor” models. In the latter,the length scale is set by the degradation and production length scale, e.g. the length scaleover which autocatalytic production of the slowly diffusing component (activator) dominatesover its own degradation. In contrast, for intracellular protein dynamics the essential length19cale appears to be the (mean) distance over which membrane-bound proteins (the slowlydiffusing components) are redistributed in the (fast diffusing) cytosolic state following theirdetachment.We will next discuss how the switch-like behaviour of proteins appears to be essential forthe emergence and regulation of such transport length scales.
The different roles of cytosolic kinetics and membrane kinetics
As we discussed above, the switch-like behaviour of proteins between active and inactivestates is a central paradigm of protein dynamics. In many cases the switch alters theaffinity of proteins for the membrane and can thus be utilized to stimulate attachment ordetachment. In case of the Min system in
E. coli only active MinD-ATP can bind andbe recruited to the membrane. As theoretical analysis [30] has shown, this property isessential for the regulation of intracellular transport and the establishment of “canalizedtransfer”: Since MinD detaches from the membrane in the inactive MinD-ADP form itcannot immediately rebind. Hence, if the timescale of cytosolic reactivation (nucleotideexchange) is sufficiently long, a MinD protein detaching from a polar zone with high MinDmembrane density can leave the polar zone, by diffusion, before being reactivated. In thisway, rebinding of detached MinD to polar zones can be suppressed even if the affinity ofcytosolic MinD-ATP for membrane-bound MinD is very high. In fact, a very high affinity formembrane-bound MinD can serve to promote rebinding of MinD-ATP in new (weak) polarzones. In other words: to promote directed transport of MinD from high (MinD and MinEmembrane) density region to a low density region. In this way, the high density regionbecomes a detachment zone and the low density region an attachment zone. Increasingthe affinity of cytosolic MinD-ATP to membrane-bound MinD (recruitment rate) simplyincreases the attachment in the low density region – but not in the high density regionwhere MinD detached and cannot rebind due to delayed nucleotide exchange. Hence, thecoupling (mass flux) between detachment and attachment zones increases with the MinDrecruitment rate, which leads to “canalized transfer”.A recent study [107] has also shown that the interplay between a membrane affinity switchand cell geometry can lead to an entirely new type of intracellular patterning that is notbased on lateral instabilities (such as the Turing instability) or excitability. In this case the20niform steady state does not become laterally unstable but is replaced by a non-uniformpattern, i.e. it ceases to exist. The experimental observation [108] is that the MinD homologAtMinD from
Arabidopsis thaliana forms a bipolar pattern in ∆MinDE
E. coli cells. Thereis evidence that the ATPase AtMinD can bind (cooperatively) to the membrane in both, itsADP and its ATP form. Assuming that AtMinD detaches from the membrane in the ADPbound form, the theoretical analysis shows that the membrane distribution of AtMinD insteady state is always non-uniform (bi-polar) if, (i) the membrane affinities of AtMinD-ADPand AtMinD-ADP are different, and (ii) the geometry of the cell deviates from a sphericalshape. The mechanism underlying such geometry induced pattern formation is based on the local ratio of membrane surface area to cytosolic volume : in an elliptical cell geometry, aprotein detaching from a cell pole is more likely to re-encounter the membrane in unit timethan a protein that detaches from a site closer to midcell. By utilizing different membraneaffinities (of ADP and ATP states) and cytosolic switching (between these states) a proteinsystem can then establish highly robust bipolar patterns that reflect the cell’s geometry.Again, they key process underlying pattern formation is cytosolic redistribution – in thiscase combined with geometry-dependent reattachment.
MASS-CONSERVING REACTION DIFFUSION MODELS: A NEW PARADIGM?
In the preceding sections we emphasized that mass conservation is the major unifyingproperty of intracellular pattern-forming protein dynamics. Over the past decade, mass-conserving reaction-diffusion systems received considerable attention in the theoretical lit-erature [12, 96, 97, 102, 109–113]. These studies try to answer how mass-conservation affectreaction-diffusion dynamics. But what is the relevance for biological systems?In [97] a mass-conserving model for cell polarity is proposed – where mass-conservationleads to the halting of a propagating wave front. This “pinned” wave represents the polarizedpattern. It has been argued that the corresponding pattern-forming mechanism is not relatedto a Turing instability but instead based on excitability and bistability [97, 112]. A similarline or reasoning is presented in [102]. However, it has been pointed out recently [114, 115]that a Turing instability in the “wave-pinning” model is recovered upon parameter change.Other studies report that mass-conserving reaction diffusion systems are prone to coarsening[96, 109]. However, it remains elusive whether a general relation between mass-conservation21nd coarsening exists.Recently, it has been shown that the general mechanism of pattern formation in mass-conserving reaction diffusion systems is based on the lateral redistribution of the conservedquantities [12]. The total amount of conserved quantities (protein copy number) determinesthe position and stabilities of chemical equilibria. Spatiotemporal redistribution of conservedquantities shifts local chemical equilibria and is generically induced by any lateral instabilitywith unequal diffusion coefficients (such as the Turing instability). The pattern formingdynamics simply follow the movement of local equilibria and the final patterns are scaffoldedby the spatial distribution of local equilibria. This study further demonstrated that “wave-pinning” patterns originate from the same physical processes as Turing instabilities: theredistribution of conserved quantities and the shifting local chemical equilibria. In futureresearch it will be interesting to see how the formation of attachment and detachment zonescan be formalized within the mass-redistribution framework.
SUMMARY AND DISCUSSION
It should be clear by now that activator–inhibitor models do not provide the appropriateconcepts to account for intracellular pattern formation. Rather, the generic key feature ofpattern forming protein system is conformational switching. Proteins cycle between differentstates such as active and inactive, or membrane-bound and cytosolic. It is the switchingof (conformational) states that drives the system and leads to pattern formation, not theproduction and degradation of proteins, which is the basis of any activator–inhibitor oractivator–depletion model. Any dynamics based on the switching (conformational) statesconserved the total copy number. Therefore, the proper models for intracellular proteindynamics are mass-conserving reaction-diffusion systems.In any mass-conserving system pattern formation has to be based on redistribution ofmass. The question to be asked about self-organization in such systems is how redistributioncomes about, i.e. how directed transport is established and maintained.In the context of intracellular protein pattern formation there is a clear functional divisionbetween membrane-bound and cytosolic protein distributions: The biologically (function-ally) relevant pattern is that of the membrane-bound factor(s), while the cytosol acts as atransport medium which facilitates the formation and maintenance of the membrane-bound22attern. The basis of intracellular pattern formation is therefore the emergence of spatiallyseparated attachment and detachment zones and their coupling (transport from detach-ment to attachment zone) through cytosolic gradients. The key design principle for patternforming protein networks is therefore the ability to set up and maintain attachment anddetachment zones in the presence of ongoing protein redistribution.We have discussed how pattern formation in three different biological systems is facili-tated by the formation of attachment and detachment zones. In budding yeast, cell polarityis established by localized accumulation of Cdc42 on the inner face of the plasma mem-brane. Pattern formation is based on the localized formation of mutual attachment zonesfor Cdc42-GTP and Bem1–GEF complexes through mutual recruitment. In
E. coli , pole-to-pole oscillations emerge due to the interactions of MinD and MinE. This process is basedon the formation of attachment zones with high MinD membrane density due to the re-cruitment of MinD and MinE from the cytosol. If the balance in a polar zone is shiftedtowards higher MinE/MinD ratios, an attachment zone becomes a detachment zone. Thesequestration of MinE in detachment zones enables the formation of new attachment zonessome distance away. The conversion of attachment zones to detachment zones and vice versaby the slow shift in the MinE/MinD ratio within a zone is the basis for the oscillation. Theestablishment of the anterior-posterior axis in
C. elegans is based on PAR protein polar-ization. Here, pattern formation originates from the formation of a (mutual) detachmentzone near midcell, and attachment zones exclusive for pPAR or aPAR, respectively, at thetwo cell poles. The establishment and maintenance of this pattern requires that the rates ofboth antagonistic processes are balanced.The question about length scale selection and pattern selection in general is still open.Apart from mathematically idealized cases that do not apply to biological systems no generalstatements about the wavelengths of patterns can be made. However, theoretical studiesof Min protein pattern formation suggest that the emergence and selection of finite wave-lengths is closely tied to the simultaneous formation and diffusive coupling of attachmentand detachment zones. A key step in the regulation of the diffusive coupling between at-tachment and detachment zones is the cytosolic switching between conformation with highand low affinity for the membrane (cytosolic nucleotide exchange). Strikingly, there is evi-dence that such cytosolic switching processes play a key role in mediating the sensitivity ofself-organized pattern formation to cell geometry [27].23any key questions about intracellular pattern formation remain open. In our opin-ion, the focus on concepts based on activator–inhibitor models in the discussion of patternformation phenomena has been a hindrance to progress rather than a help.Intracellular pattern forming protein dynamics are most generally expressed by mass-conserving reaction-diffusion systems. Local equilibria, as recently suggested [12], are apromising candidate for a general and unifying theoretical framework for such systems.To advance our understanding of intracellular protein dynamics a theoretically rigorousanalysis of pattern forming instabilities in mass-conserving reaction-diffusion systems wouldbe highly desirable. In his seminal article Turing presented the general idea of patternforming instabilities in reaction-diffusion systems. Since its publication 65 years ago onlylittle has been learned about the general physical principles underlying the
Turing instability .We expect that a focus on the mass-conserving case will finally enable us to extract somegeneral physical principles of pattern formation systems based on Turing’s lateral instability.
ACKNOWLEDGMENTS
This research was funded by the German Excellence Initiative via the NanoSystems Ini-tiative Munich, and by the Deutsche Forschungsgemeinschaft (DFG) via Project B02 withinSFB 1032 (Nanoagents for SpatioTemporal Control of Molecular and Cellular Reactions),Areas A and C within GRK2062 (Molecular Principles of Synthetic Biology), and ProjectP03 within TRR174 (Spatiotemporal Dynamics of Bacterial Cells). The authors thankS. Bergeler and W. Daalman for feedback on the manuscript. ∗ [email protected][1] Turing AM. The chemical basis of morphogenesis. Philosophical Transactionsof the Royal Society of London Series B, Biological Sciences. 1952;237(641):37–72.DOI: 10.1007/BF02459572.[2] Cross M, Hohenberg P. Pattern formation outside of equilibrium. Reviews of Modern Physics.1993;65(3).[3] Segel La, Jackson JL. Dissipative structure: an explanation and an ecological example.Journal of theoretical biology. 1972;37:545–559. DOI: 10.1016/0022-5193(72)90090-2.
4] Gierer A, Meinhardt H. A theory of biological pattern formation. Kybernetik. 1972dec;12(1):30–9.[5] Meinhardt H, Gierer A. Pattern formation by local self activation and lateral inhibition.BioEssays. 2000;22:753–760.[6] Meinhardt H. Models of Biological Pattern Formation: From Elementary Steps to the Or-ganization of Embryonic Axes. Current Topics in Developmental Biology. 2008;81:1–63.DOI: 10.1016/S0070-2153(07)81001-5.[7] Frey E, Halatek J, Kretschmer S, Schwille P. Protein Pattern Formation. Springer-VerlagGmbH, Heidelberg; 2018. Edited by P. Bassereau and P. C. A. Sens; to be published[arXiv:1801.01365].[8] Shamir M, Bar-On Y, Phillips R, Milo R. SnapShot: Timescales in Cell Biology. Cell. 2016Mar;164(6):1302–1302.e1. DOI: 10.1016/j.cell.2016.02.058.[9] Kondo S, Miura T. Reaction-diffusion model as a framework for understanding biologicalpattern formation. Science. 2010 Sep;329(5999):1616–20. DOI: 10.1126/science.1179047.[10] Tarnita CE, Bonachela JA, Sheffer E, Guyton JA, Coverdale TC, Long RA, et al. A theoret-ical foundation for multi-scale regular vegetation patterns. Nature. 2017 01;541(7637):398–401. DOI: 10.1038/nature20801.[11] Rietkerk M, van de Koppel J. Regular pattern formation in real ecosystems. Trends EcolEvol. 2008 Mar;23(3):169–75. DOI: 10.1016/j.tree.2007.10.013.[12] Halatek J, Frey E. Rethinking pattern formation in reaction–diffusion systems. NaturePhysics. 2018;DOI: 10.1038/s41567-017-0040-5.[13] Lutkenhaus J. Assembly dynamics of the bacterial MinCDE system and spatial regula-tion of the Z ring. Annual Review of Biochemistry. 2007 Jan;76:539–62. DOI: 10.1146/an-nurev.biochem.75.103004.142652.[14] Raskin DM, de Boer Pa. Rapid pole-to-pole oscillation of a protein required for directingdivision to the middle of Escherichia coli. Proceedings of the National Academy of Sciences.1999 Apr;96(9):4971–6.[15] Hu Z, Mukherjee A, Pichoff S, Lutkenhaus J. The MinC component of the division site se-lection system in Escherichia coli interacts with FtsZ to prevent polymerization. Proceedingsof the National Academy of Sciences. 1999 Dec;96(26):14819–14824.
16] Hu Z, Gogol EP, Lutkenhaus J. Dynamic assembly of MinD on phospholipid vesicles regulatedby ATP and MinE. Proceedings of the National Academy of Sciences. 2002 May;99(10):6761–6766.[17] Szeto TH, Rowland SL, Rothfield LI, King GF. Membrane localization of MinD is me-diated by a C-terminal motif that is conserved across eubacteria, archaea, and chloro-plasts. Proceedings of the National Academy of Sciences. 2002 Nov;99(24):15693–15698.DOI: 10.1073/pnas.232590599.[18] Lackner LL, Raskin DM, de Boer PAJ. ATP-dependent interactions between Escherichiacoli Min proteins and the phospholipid membrane in vitro. Journal of Bacteriology. 2003Feb;185(3):735–749. DOI: 10.1128/JB.185.3.735-749.2003.[19] Mileykovskaya E, Fishov I, Fu X, Corbin BD, Margolin W, Dowhan W. Effects of phospho-lipid composition on MinD-membrane interactions in vitro and in vivo. Journal of BiologicalChemistry. 2003 Jun;278(25):22193–22198. DOI: 10.1074/jbc.M302603200.[20] Hu Z, Lutkenhaus J. Topological regulation of cell division in Escherichia coli involves rapidpole to pole oscillation of the division inhibitor MinC under the control of MinD and MinE.Molecular Microbiology. 1999 Oct;34(1):82–90.[21] Hu Z, Lutkenhaus J. Topological regulation of cell division in E. coli. spatiotemporal oscilla-tion of MinD requires stimulation of its ATPase by MinE and phospholipid. Molecular Cell.2001 Jun;7(6):1337–1343.[22] Touhami A, Jericho M, Rutenberg AD. Temperature Dependence of MinD Oscillation inEscherichia coli: Running Hot and Fast. Journal of Bacteriology. 2006 Oct;188(21):7661–7667. DOI: 10.1128/JB.00911-06.[23] Park KT, Wu W, Battaile KP, Lovell S, Holyoak T, Lutkenhaus J. The Min Oscillator UsesMinD-Dependent Conformational Changes in MinE to Spatially Regulate Cytokinesis. Cell.2011;146(3):396–407.[24] Schweizer J, Loose M, Bonny M, Kruse K, M¨onch I, Schwille P. Geometry sensing byself-organized protein patterns. Proceedings of the National Academy of Sciences. 2012Sep;109(38):15283–15288. DOI: 10.1073/pnas.1206953109.[25] Loose M, Fischer-Friedrich E, Ries J, Kruse K, Schwille P. Spatial regulators for bac-terial cell division self-organize into surface waves in vitro. Science (New York, NY).2008;320(5877):789–92. DOI: 10.1126/science.1154413.
26] Wu F, van Schie BGC, Keymer JE, Dekker C. Symmetry and scale orient Min proteinpatterns in shaped bacterial sculptures. Nature Nanotechnology. 2015 Jun;10:719–726.[27] Wu F, Halatek J, Reiter M, Kingma E, Frey E, Dekker C. Multistability and dynamic tran-sitions of intracellular Min protein patterns. Molecular Systems Biology. 2016 jun;12(6):642–653. DOI: 10.15252/MSB.20156724.[28] Huang KC, Meir Y, Wingreen NS. Dynamic structures in Escherichia coli: spontaneousformation of MinE rings and MinD polar zones. Proceedings of the National Academy ofSciences. 2003 oct;100(22):12724–8. DOI: 10.1073/pnas.2135445100.[29] Fange D, Elf J. Noise-induced Min phenotypes in E. coli. Plos Computational Biology. 2006Jun;2(6):e80. DOI: 10.1371/journal.pcbi.0020080.[30] Halatek J, Frey E. Highly canalized MinD transfer and MinE sequestration explainthe origin of robust MinCDE-protein dynamics. Cell reports. 2012 jun;1(6):741–52.DOI: 10.1016/j.celrep.2012.04.005.[31] Hoffmann M, Schwarz US. Oscillations of Min-proteins in micropatterned environ-ments: a three-dimensional particle-based stochastic simulation approach. Soft Matter.2014;10(14):2388–2396. DOI: 10.1039/c3sm52251b.[32] Vecchiarelli AG, Li M, Mizuuchi M, Hwang LC, Seol Y, Neuman KC, et al. Membrane-boundMinDE complex acts as a toggle switch that drives Min oscillation coupled to cytoplasmic de-pletion of MinD. Proceedings of the National Academy of Sciences. 2016 Mar;113(11):E1479–88. DOI: 10.1073/pnas.1600644113.[33] Halatek J, Frey E. Effective 2D model does not account for geometry sensing byself-organized proteins patterns. Proc Natl Acad Sci U S A. 2014 May;111(18):E1817.DOI: 10.1073/pnas.1220971111.[34] Kretschmer S, Zieske K, Schwille P. Large-scale modulation of reconstituted Min proteinpatterns and gradients by defined mutations in MinE’s membrane targeting sequence. PLoSOne. 2017;12(6):e0179582. DOI: 10.1371/journal.pone.0179582.[35] Fischer-Friedrich E, Meacci G, Lutkenhaus J, Chate H, Kruse K. Intra- and intercellularfluctuations in Min-protein dynamics decrease with cell length. Proceedings of the NationalAcademy of Sciences. 2010 Mar;107(14):6134–6139. DOI: 10.1073/pnas.0911708107.[36] Sliusarenko O, Heinritz J, Emonet T, Jacobs-Wagner C. High-throughput, subpixel precisionanalysis of bacterial morphogenesis and intracellular spatio-temporal dynamics. Molecular icrobiology. 2011 Mar;80(3):612–627. DOI: 10.1111/j.1365-2958.2011.07579.x.[37] Corbin BD, Yu XC, Margolin W. Exploring intracellular space: function of the Minsystem in round-shaped Escherichia coli. The EMBO journal. 2002 Apr;21(8):1998–2008.DOI: 10.1093/emboj/21.8.1998.[38] Shih YL, Kawagishi I, Rothfield L. The MreB and Min cytoskeletal-like systems play indepen-dent roles in prokaryotic polar differentiation. Molecular Microbiology. 2005 Oct;58(4):917–928. DOI: 10.1111/j.1365-2958.2005.04841.x.[39] Varma A, Huang KC, Young KD. The Min system as a general cell geometry detection mecha-nism: branch lengths in Y-shaped Escherichia coli cells affect Min oscillation patterns and di-vision dynamics. Journal of Bacteriology. 2008 Mar;190(6):2106–17. DOI: 10.1128/JB.00720-07.[40] Johnson DI. Cdc42: An essential Rho-type GTPase controlling eukaryotic cell polarity.Microbiol Mol Biol Rev. 1999 Mar;63(1):54–105.[41] Chant J, Herskowitz I. Genetic control of bud site selection in yeast by a set of gene productsthat constitute a morphogenetic pathway. Cell. 1991 Jun;65(7):1203–12.[42] Koch G, Tanaka K, Masuda T, Yamochi W, Nonaka H, Takai Y. Association of the Rhofamily small GTP-binding proteins with Rho GDP dissociation inhibitor (Rho GDI) in Sac-charomyces cerevisiae. Oncogene. 1997 Jul;15(4):417–22. DOI: 10.1038/sj.onc.1201194.[43] Johnson JL, Erickson JW, Cerione RA. New insights into how the Rho guanine nucleotidedissociation inhibitor regulates the interaction of Cdc42 with membranes. J Biol Chem. 2009Aug;284(35):23860–71. DOI: 10.1074/jbc.M109.031815.[44] Bi E, Park HO. Cell polarization and cytokinesis in budding yeast. Genetics. 2012Jun;191(2):347–87. DOI: 10.1534/genetics.111.132886.[45] Butty AC, Perrinjaquet N, Petit A, Jaquenoud M, Segall JE, Hofmann K, et al. A positivefeedback loop stabilizes the guanine-nucleotide exchange factor Cdc24 at sites of polarization.EMBO J. 2002 Apr;21(7):1565–76. DOI: 10.1093/emboj/21.7.1565.[46] Bose I, Irazoqui JE, Moskow JJ, Bardes ES, Zyla TR, Lew DJ. Assembly of scaffold-mediatedcomplexes containing Cdc42p, the exchange factor Cdc24p, and the effector Cla4p requiredfor cell cycle-regulated phosphorylation of Cdc24p. J Biol Chem. 2001 Mar;276(10):7176–86.[47] Slaughter BD, Das A, Schwartz JW, Rubinstein B, Li R. Dual modes ofcdc42 recycling fine-tune polarized morphogenesis. Dev Cell. 2009 Dec;17(6):823–35. OI: 10.1016/j.devcel.2009.10.022.[48] Kozubowski L, Saito K, Johnson JM, Howell AS, Zyla TR, Lew DJ. Symmetry-breakingpolarization driven by a Cdc42p GEF-PAK complex. Curr Biol. 2008 Nov;18(22):1719–26.DOI: 10.1016/j.cub.2008.09.060.[49] Freisinger T, Kl¨under B, Johnson J, M¨uller N, Pichler G, Beck G, et al. Establishment of arobust single axis of cell polarity by coupling multiple positive feedback loops. Nat Commun.2013;4:1807. DOI: 10.1038/ncomms2795.[50] Marco E, Wedlich-Soldner R, Li R, Altschuler SJ, Wu LF. Endocytosis optimizes the dynamiclocalization of membrane proteins that regulate cortical polarity. Cell. 2007 Apr;129(2):411–22.[51] Witte K, Strickland D, Glotzer M. Cell cycle entry triggers a switch between two modes ofCdc42 activation during yeast polarization. Elife. 2017 Jul;6. DOI: 10.7554/eLife.26722.[52] Rapali P, Mitteau R, Braun C, Massoni-Laporte A, ¨Unl¨u C, Bataille L, et al. Scaffold-mediated gating of Cdc42 signalling flux. Elife. 2017 Mar;6. DOI: 10.7554/eLife.25257.[53] Woods B, Kuo CC, Wu CF, Zyla TR, Lew DJ. Polarity establishment requires localizedactivation of Cdc42. J Cell Biol. 2015 Oct;211(1):19–26. DOI: 10.1083/jcb.201506108.[54] Irazoqui JE, Gladfelter AS, Lew DJ. Scaffold-mediated symmetry breaking by Cdc42p. NatCell Biol. 2003 Dec;5(12):1062–70. DOI: 10.1038/ncb1068.[55] Kl¨under B, Freisinger T, Wedlich-S¨oldner R, Frey E. GDI-mediated cell polarization inyeast provides precise spatial and temporal control of Cdc42 signaling. PLoS Comput Biol.2013;9(12):e1003396. DOI: 10.1371/journal.pcbi.1003396.[56] Goryachev AB, Pokhilko AV. Dynamics of Cdc42 network embodies a Turing-type mechanism of yeast cell polarity. FEBS Lett. 2008 Apr;582(10):1437–43.DOI: 10.1016/j.febslet.2008.03.029.[57] Wedlich-Soldner R, Altschuler S, Wu L, Li R. Spontaneous cell polarization throughactomyosin-based delivery of the Cdc42 GTPase. Science. 2003 Feb;299(5610):1231–5.DOI: 10.1126/science.1080944.[58] Wedlich-Soldner R, Wai SC, Schmidt T, Li R. Robust cell polarity is a dynamic stateestablished by coupling transport and GTPase signaling. J Cell Biol. 2004 Sep;166(6):889–900. DOI: 10.1083/jcb.200405061.
59] Muller N, Piel M, Calvez V, Voituriez R, Gon¸calves-S´a J, Guo CL, et al. A PredictiveModel for Yeast Cell Polarization in Pheromone Gradients. PLoS Comput Biol. 2016Apr;12(4):e1004795. DOI: 10.1371/journal.pcbi.1004795.[60] Hawkins RJ, B´enichou O, Piel M, Voituriez R. Rebuilding cytoskeleton roads: active-transport-induced polarization of cells. Phys Rev E Stat Nonlin Soft Matter Phys. 2009Oct;80(4 Pt 1):040903. DOI: 10.1103/PhysRevE.80.040903.[61] Altschuler SJ, Angenent SB, Wang Y, Wu LF. On the spontaneous emergence of cell polarity.Nature. 2008 Aug;454(7206):886–9. DOI: 10.1038/nature07119.[62] Layton AT, Savage NS, Howell AS, Carroll SY, Drubin DG, Lew DJ. Modeling vesicle trafficreveals unexpected consequences for Cdc42p-mediated polarity establishment. Curr Biol.2011 Feb;21(3):184–94. DOI: 10.1016/j.cub.2011.01.012.[63] Savage NS, Layton AT, Lew DJ. Mechanistic mathematical model of polarity in yeast. MolBiol Cell. 2012 May;23(10):1998–2013. DOI: 10.1091/mbc.E11-10-0837.[64] Smith SE, Rubinstein B, Mendes Pinto I, Slaughter BD, Unruh JR, Li R. Independence ofsymmetry breaking on Bem1-mediated autocatalytic activation of Cdc42. J Cell Biol. 2013Sep;202(7):1091–106. DOI: 10.1083/jcb.201304180.[65] Laan L, Koschwanez JH, Murray AW. Evolutionary adaptation after crippling cell polariza-tion follows reproducible trajectories. Elife. 2015 Oct;4. DOI: 10.7554/eLife.09638.[66] Bendez´u FO, Vincenzetti V, Vavylonis D, Wyss R, Vogel H, Martin SG. Spontaneous Cdc42polarization independent of GDI-mediated extraction and actin-based trafficking. PLoS Biol.2015 Apr;13(4):e1002097. DOI: 10.1371/journal.pbio.1002097.[67] Caviston JP, Tcheperegine SE, Bi E. Singularity in budding: a role for the evolutionarilyconserved small GTPase Cdc42p. Proc Natl Acad Sci U S A. 2002 Sep;99(19):12185–90.DOI: 10.1073/pnas.182370299.[68] Knaus M, Pelli-Gulli MP, van Drogen F, Springer S, Jaquenoud M, Peter M. Phosphorylationof Bem2p and Bem3p may contribute to local activation of Cdc42p at bud emergence. EMBOJ. 2007 Oct;26(21):4501–13. DOI: 10.1038/sj.emboj.7601873.[69] Howell AS, Savage NS, Johnson SA, Bose I, Wagner AW, Zyla TR, et al. Singularityin polarization: rewiring yeast cells to make two buds. Cell. 2009 Nov;139(4):731–43.DOI: 10.1016/j.cell.2009.10.024.
70] Wu CF, Lew DJ. Beyond symmetry-breaking: competition and negative feedback in GTPaseregulation. Trends Cell Biol. 2013 Oct;23(10):476–83. DOI: 10.1016/j.tcb.2013.05.003.[71] Goehring NW, Trong PK, Bois JS, Chowdhury D, Nicola EM, Hyman AA, et al. Po-larization of PAR proteins by advective triggering of a pattern-forming system. Science.2011;334(Nov):1137–1141. arXiv:1011.1669v3. DOI: 10.1126/science.1208619.[72] Hoege C, Hyman AA. Principles of PAR polarity in Caenorhabditis elegans embryos. NatureReviews Molecular Cell Biology. 2013;14(5):315–322. DOI: 10.1038/nrm3558.[73] Munro E, Nance J, Priess JR. Cortical flows powered by asymmetrical contraction transportPAR proteins to establish and maintain anterior-posterior polarity in the early C. elegansembryo. Developmental Cell. 2004 Sep;7(3):413–424. DOI: 10.1016/j.devcel.2004.08.001.[74] Goehring NW. PAR polarity: From complexity to design principles. Experimental CellResearch. 2014 Nov;328(2):258–266.[75] Goehring NW, Trong PK, Bois JS, Chowdhury D, Nicola EM, Hyman AA, et al. Polar-ization of PAR proteins by advective triggering of a pattern-forming system. Science. 2011Nov;334(6059):1137–41. DOI: 10.1126/science.1208619.[76] Meacci G, Ries J, Fischer-Friedrich E, Kahya N, Schwille P, Kruse K. Mobility of Min-proteinsin Escherichia coli measured by fluorescence correlation spectroscopy. Physical biology. 2006Nov;3(4):255–263. DOI: 10.1088/1478-3975/3/4/003.[77] Graf IR, Frey E. Generic Transport Mechanisms for Molecular Traffic in Cellular Protrusions.Physical Review Letters. 2017 Mar;118(12). DOI: 10.1103/physrevlett.118.128101.[78] Yochelis A, Ebrahim S, Millis B, Cui R, Kachar B, Naoz M, et al. Self-organization of wavesand pulse trains by molecular motors in cellular protrusions. Sci Rep. 2015 Sep;5:13521.DOI: 10.1038/srep13521.[79] Pinkoviezky I, Gov NS. Exclusion and Hierarchy of Time Scales Lead to Spatial Segrega-tion of Molecular Motors in Cellular Protrusions. Phys Rev Lett. 2017 Jan;118(1):018102.DOI: 10.1103/PhysRevLett.118.018102.[80] Schumacher D, Bergeler S, Harms A, Vonck J, Huneke-Vogt S, Frey E, et al. The PomXYZProteins Self-Organize on the Bacterial Nucleoid to Stimulate Cell Division. Dev Cell. 2017May;41(3):299–314.e13. DOI: 10.1016/j.devcel.2017.04.011.[81] Schumacher D, Søgaard-Andersen L. Regulation of Cell Polarity in Motility and Cell Divisionin Myxococcus xanthus. Annu Rev Microbiol. 2017 Sep;71:61–78. DOI: 10.1146/annurev- icro-102215-095415.[82] Ringgaard S, van Zon J, Howard M, Gerdes K. Movement and equipositioning of plasmidsby ParA filament disassembly. Proc Natl Acad Sci U S A. 2009 Nov;106(46):19369–74.DOI: 10.1073/pnas.0908347106.[83] Howard M, Gerdes K. What is the mechanism of ParA-mediated DNA movement? MolMicrobiol. 2010 Oct;78(1):9–12. DOI: 10.1111/j.1365-2958.2010.07316.x.[84] Gerdes K, Howard M, Szardenings F. Pushing and Pulling in Prokaryotic DNA Segregation.Cell. 2010;141(6):927–942. DOI: 10.1016/j.cell.2010.05.033.[85] Banigan EJ, Gelbart MA, Gitai Z, Wingreen NS, Liu AJ. Filament depolymeriza-tion can explain chromosome pulling during bacterial mitosis. PLoS Comput Biol. 2011Sep;7(9):e1002145. DOI: 10.1371/journal.pcbi.1002145.[86] Sugawara T, Kaneko K. Chemophoresis as a driving force for intracellular organization:Theory and application to plasmid partitioning. Biophysics (Nagoya-shi). 2011;7:77–88.DOI: 10.2142/biophysics.7.77.[87] Surovtsev IV, Campos M, Jacobs-Wagner C. DNA-relay mechanism is sufficient to explainParA-dependent intracellular transport and patterning of single and multiple cargos. ProcNatl Acad Sci U S A. 2016 Nov;113(46):E7268–E7276. DOI: 10.1073/pnas.1616118113.[88] Hu L, Vecchiarelli AG, Mizuuchi K, Neuman KC, Liu J. Brownian Ratchet Mechanism forFaithful Segregation of Low-Copy-Number Plasmids. Biophys J. 2017 Apr;112(7):1489–1502.DOI: 10.1016/j.bpj.2017.02.039.[89] Murray SM, Sourjik V. Self-organization and positioning of bacterial protein clusters. NaturePhysics. 2017 Jun;13(10):1006–1013. DOI: 10.1038/nphys4155.[90] Walter JC, Dorignac J, Lorman V, Rech J, Bouet JY, Nollmann M, et al. Surfing on ProteinWaves: Proteophoresis as a Mechanism for Bacterial Genome Partitioning. Physical ReviewLetters. 2017 Jul;119(2). DOI: 10.1103/physrevlett.119.028101.[91] Ietswaart R, Szardenings F, Gerdes K, Howard M. Competing ParA structures space bac-terial plasmids equally over the nucleoid. PLoS Comput Biol. 2014 Dec;10(12):e1004009.DOI: 10.1371/journal.pcbi.1004009.[92] Bergeler S, Frey E. Regulation of Pom cluster dynamics in Myxococcus xanthus .[arXiv:180106133]. 2018;.
93] Huang KC, Ramamurthi KS. Macromolecules that prefer their membranes curvy. MolecularMicrobiology. 2010 Apr;76(4):822–832. DOI: 10.1111/j.1365-2958.2010.07168.x.[94] Laloux G, Jacobs-Wagner C. How do bacteria localize proteins to the cell pole? Journal ofCell Science. 2013 Dec;127(1):11–19. DOI: 10.1242/jcs.138628.[95] Halatek J, Frey E. Highly canalized MinD transfer and MinE sequestration ex-plain the origin of robust MinCDE-protein dynamics. Cell Rep. 2012 Jun;1(6):741–52.DOI: 10.1016/j.celrep.2012.04.005.[96] Otsuji M, Ishihara S, Co C, Kaibuchi K, Mochizuki A, Kuroda S. A mass conserved reaction-diffusion system captures properties of cell polarity. PLoS Comput Biol. 2007 Jun;3(6):e108.DOI: 10.1371/journal.pcbi.0030108.[97] Mori Y, Jilkine A, Edelstein-Keshet L. Wave-pinning and cell polarity from a bistablereaction-diffusion system. Biophysical journal. 2008 may;94(9):3684–97. DOI: 10.1529/bio-physj.107.120824.[98] Howell AS, Jin M, Wu CF, Zyla TR, Elston TC, Lew DJ. Negative feedback enhancesrobustness in the yeast polarity establishment circuit. Cell. 2012 Apr;149(2):322–33.DOI: 10.1016/j.cell.2012.03.012.[99] Das M, Drake T, Wiley DJ, Buchwald P, Vavylonis D, Verde F. Oscillatory dynamicsof Cdc42 GTPase in the control of polarized growth. Science. 2012 Jul;337(6091):239–43.DOI: 10.1126/science.1218377.[100] Yang L, Dolnik M, Zhabotinsky AM, Epstein IR. Turing patterns beyond hexagons andstripes. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2006;16(3):037114.http://aip.scitation.org/doi/pdf/10.1063/1.2214167. DOI: 10.1063/1.2214167.[101] Otsuji M, Ishihara S, Co C, Kaibuchi K, Mochizuki A, Kuroda S. A mass conserved reaction-diffusion system captures properties of cell polarity. PLoS Comput Biol. 2007 Jun;3(6):e108.[102] Semplice M, Veglio A, Naldi G, Serini G, Gamba A. A bistable model of cell polarity. PLoSOne. 2012;7(2):e30977. DOI: 10.1371/journal.pone.0030977.[103] Berry J, Brangwynne C, Haataja MP. Physical Principles of Intracellular Organization viaActive and Passive Phase Transitions. Reports on Progress in Physics. 2018 Jan;.[104] Yang HC, Pon LA. Actin cable dynamics in budding yeast. Proc Natl Acad Sci U S A. 2002Jan;99(2):751–6. DOI: 10.1073/pnas.022462899. + + + + + C o n c . Space autocatalyticproduction productiondominates degradationdominatesslow activator di ff usionfast inhibitor di ff usion Activator-Inhibitor b autocatalyticconversion conversiondominates degradationdominatesfast substrate di ff usionslow activator di ff usion Activator-Depletion c no detachment a timeevolution Space C o n c . band ofunstable modes FIG. 1. Turing’s general linear stability analysis and heuristic ad-hoc interpretations in terms ofthe activator–inhibitor picture based on the production and degradation of reactants. (a) Anyrandom perturbation (black line) of a spatially uniform state (gray line) can be decomposed intoFourier modes. Linear stability analysis yields the growth rates σ q of the amplitude of all modes q . This is represented in the dispersion relation (blue line σ q ). Unstable modes (marked red) growin amplitude an determine the pattern emerging out of the random perturbation during the incip-ient time evolution. (b) The activator–inhibitor model is based on autocatalytic production of aslowly diffusing activator, which in turn stimulates the production of a fast-diffusing inhibitor thatsuppresses autocatalytic activator production. Both activator and inhibitor are subject to degra-dation. The faster diffusion of the inhibitor leads to the formation of an inhibition zone in whichdegradation dominates over activator (and inhibitor) production. (c) In the activator–depletionmodel the inhibitor is replaced by a substrate that is subject to degradation, and autocatalyticactivator production is replaced by the autocatalytic conversion of substrate into activator. Therate of conversion is limited by the available substrate. Heuristically, this conversion could beequated with the attachment of cytosolic proteins to the membrane. However, the reverse process(detachment) is not taken into account. Both substrate and activator are steadily degraded and areproduced at a finite rate. If the activator density is too low, the conversion process is suppressedand the degradation process dominates, as in the activator–inhibitor model. em1-GEF mediatedCdc42 recuritmentand activation Cdc42-GDPGDIBem1 GEF
Cdc42-GTP recruits Bem1-GEFmembranecytosol Cdc42 GTPase cycle
MinD-ATP MinD-ADPMinE mutual detachmentof aPAR and pPAR aPAR pPAR
MinD recruitsitself and MinE MinDE-complexdetachment a c b D T NEFNAPNDPNTP P D PhosphataseKinasePhosphorylatedDephosphorylated d AB Switch regulatorsState AState B
FIG. 2. Biochemical interaction networks of three model systems for self-organized intracellularpattern formation. (a) The Min system of
E. coli [28, 30]. (b) Cdc42 system of
S. Cerevisiae [49, 55] . (c) PAR system of
C. elegans [71]. (d) Switching between two conformal states ofthe proteins involved is a recurring theme in the biochemical networks (a-c). Cycling betweenmembrane bound and cytosolic states is driven by the ATPase/GTPase cycle of MinD and Cdc42respectively, while the PAP-proteins each cycle between different phosphorylation states. In generalwe expect switching between distinct conformal states – catalyzed by “switch regulators” such asNTPase-activating proteins (NAPs), nucleotide exchange factors (NEFs), phophatases, and kinases– to be a core element of biochemical networks that mediate intracellular pattern formation. ttachment zone detachment zone C y t o s o l c o n ce n t r a t i o n random perturbation attachment zonedetachment zone redistribution Return to uniform state redistribution
Increase of membrane density shifts reaction balancein favour of attachement attachment zone detachment zone redistribution membranedi ff usion uniform attachment-detachment balanceMaintained pattern state T i m e e v o l u t i o n Increase of membrane density shifts reaction balancein favour of detachement ab b'c c'
FIG. 3. Linear (in)stability of a uniform initial distribution of proteins. (a) In a uniform steadystate, attachment and detachment must balance everywhere. An external cue or a random per-turbation due to stochastic noise can lead a local increase in membrane density. How the relativebalance of attachment and detachment processes shifts in the region of increased membrane den-sity, determines the stability of the uniform state. If the balance in a region of increased membranedensity shifts in favour of attachment (b), the region becomes an attachment zone leading to afurther increase in membrane density due to redistribution through the cytosol. Hence the spatiallyseparated attachment and detachment zones are maintained, leading the establishment of a pat-tern. If the balance in a region of increased membrane density shifts the attachment-detachmentbalance in favour of detachment (b’), this region becomes a detachment zone, while the region oflower membrane density becomes an attachment zone, such that the system returns to its uniformbalanced state (c’). + mutual recruitmentzonedetachment zone redistribution ++ c y t o s o li c d e n s i t y c y t o s o li c d e n s i t y Bem1-GEFrecruitmentBem1,GEFdetachment redistribution
Cdc42 recruitment,nucleotide exchangeCdc42 detachment redistribution + +
Bem1-GEF template for Cdc42 recruitmentCdc42 template for Bem1-GEF recruitmentBem1-GEF Cdc42-GDP
Mutual recruitment ofBem1-GEF and Cdc42-GTP ab c ' ' FIG. 4. Mutual recruitment of Cdc42 and Bem1–GEF complexes is the core mechanism of Cdc42polarization. (a) The interaction network of Cdc42, Bem1 and GEF (Cdc24) can conceptuallybe simplified to two key processes: Bem1–GEF complexes on the membrane recruit Cdc42-GDPfrom the cytosol, which is followed by immediate nucleotide exchange (conversion to Cdc42-GTP).Reversely, Cdc42-GTP recruits Bem1 to the membrane by, where it immediately recruits Cdc24 toform Bem1–GEF complexes. (b) A local accumulation of Cdc42-GTP thereby acts as recruitmenttemplate for Bem1–GEF complexes (1), creating an attachment zone for Bem1–GEF, while de-tachment of Bem1–GEF dominates in zones of low Cdc42-GTP density (2). Bem1–GEF complexesthen accumulate in their attachment zone. This accumulation acts as a recruitment template forCdc42 (2), creating co-polarized attachment zones of Cdc42 and Bem1–GEF (1’) and (2’). (c)Taken together, the mutual recruitment processes establish and maintain Cdc42-polarization. c y t o s o li c d e n s i t y + redistribution MinDE detachment MinD recruits itselfand MinE + +
MinD self-recruitment MinDE detachmentlocal MinE cycling redistribution ++ MinD self-recruitmentMinDE detachmentlocal MinE cycling redistribution + + redistribution
MinDE detachmentMinD recruits itselfand MinE
FIG. 5. Pole-to-pole oscillations of MinD and MinE: MinD recruits both itself and MinE from thecytosol creating an attachment zone for both proteins (1). As MinDE complexes accumulate in thepolar zone, their detachment begins to dominate over MinD attachment. (2) The old polar zonetraps MinE because it is both an attachment and a detachment zone for MinE, which only cycleslocally, as long as there is free MinD left on the membrane. This allows cytosolic MinD to form anew polar zone at the other end of the cell. MinE trapping ends when all MinD has detached fromthe old polar zone, such that the new polar zone becomes an attachment zone for MinE (3), andthe process starts over at the opposite end of the cell (4). utual detachmentaPAR attachment pPAR attachment redistributionredistribution c y t o s o li c d e n s i t i e s FIG. 6. Mutual antagonism between aPAR and pPar proteins creates a detachment zone at theinterface between aPAR- and pPAR-dominated regions. A region of high aPAR density on themembrane is a detachment zone for pPAR, such that detaching pPARs can only attach to a regionof low aPAR density, and vice versa. A balance of the mutual antagonistic processes is necessary toprevent one protein species from dominating over the other and taking over the whole membrane. imultaneous attachment and detachmentSequential attachment and detachment weak attachmentstrong detachmentcytosolic reservoirstrong attachmentno detachmentcytosolic reservoirLength = L Length = 2 L no attachmentdetachment attachment zonedetachment zonemembranecytosol canalized transferintrinsic transportlength scale weak attachmentstrong detachment cytosolic reservoir strong attachmentno detachment cytosolic reservoirno attachmentdetachment T i m e e v o l u t i o n attachment zone detachment zonecanalized transferdetachment zone canalized transferintrinsic transportlength scale intrinsic transportlength scale ab Cytosolic reservoir does not impose length scaleCanalized transfer determines pattern length scale