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Diffusionlimited phase separation in eukaryotic chemotaxis

Edited by H. Eugene Stanley, Boston University, Boston, MA (received for review May 12, 2005)
Abstract
The ability of cells to sense spatial gradients of chemoattractant factors governs the development of complex eukaryotic organisms. Cells exposed to shallow chemoattractant gradients respond with strong accumulation of the enzyme phosphatidylinositol 3kinase (PI3K) and its D3phosphoinositide product (PIP_{3}) on the plasma membrane side exposed to the highest chemoattractant concentration, whereas PIP_{3}degrading enzyme PTEN and its product PIP_{2} localize in a complementary pattern. Such an early symmetrybreaking event is a mandatory step for directed cell movement elicited by chemoattractants, but its physical origin is still mysterious. Here, we propose that directional sensing is the consequence of a phaseordering process mediated by phosphoinositide diffusion and driven by the distribution of chemotactic signal. By studying a realistic reaction–diffusion lattice model that describes PI3K and PTEN enzymatic activity, recruitment to the plasma membrane, and diffusion of their phosphoinositide products, we show that the effective enzyme–enzyme interaction induced by catalysis and diffusion introduces an instability of the system toward phase separation for realistic values of physical parameters. In this framework, large reversible amplification of shallow chemotactic gradients, selective localization of chemical factors, macroscopic response timescales, and spontaneous polarization arise naturally. The model is robust with respect to orderofmagnitude variations of the parameters.
The general picture emerging from the analysis of chemotaxis in several different eukaryotic cell types indicates that, in the process of directional sensing, a shallow extracellular gradient of chemoattractant is translated into an equally shallow gradient of receptor activation (1) that in turn elicits the recruitment of the cytosolic enzyme phosphatidylinositol 3kinase (PI3K) to the plasma membrane, where it phosphorylates PIP_{2} into the D3phosphoinositide product of PI3K, PIP_{3}. However, phosphoinositide distribution does not simply mirror the receptor activation gradient, but rather a strong and sharp separation in PIP_{2} and PIP_{3}rich phases arises (1), realizing a powerful and efficient amplification of the external chemotactic signal. PIP_{3} acts as a docking site for plekstrin homologydomaincontaining effector proteins that induce cell polarization, i.e., the generation of biochemically defined cell anterior and posterior sides, regulate cytoskeletal dynamics (2), and eventually cell motion (3). Cell polarization can be decoupled from directional sensing by the use of inhibitors of actin polymerization so that cells are immobilized but respond with the same signal amplification of untreated cells (4). The action of PI3K is counteracted by the phosphatase PTEN that dephosphorylates PIP_{3} into PIP_{2} (1). PTEN localization at the cell membrane depends on the binding to PIP_{2} of its first 16 Nterminal amino acids (5). Because PI3K and PTEN constitute a pair of enzymes with counteracting biochemical activities, it has been conjectured that in chemotacting cells mutual regulation between the two enzymes could be responsible for their localization into complementary regions of the cell surface.
In physical terms, the process of directional sensing shows the characteristic phenomenology of phase separation (6). However, it is not clear which mechanism could be responsible for it. In known physical models, such as binary alloys, phase separation is the consequence of some kind of interaction among the constituents of a system, which can favor their segregation in separated phases (7). In this work we show that, even in the absence of direct enzyme–enzyme or phosphoinositide–phosphoinositide interactions, catalysis and phosphoinositide diffusion mediate an effective interaction among enzymes, which is sufficient to drive the system toward phase separation.
Materials and Methods
A semiregular tessellation of the sphere composed of 10,230 hexagonal and 12 pentagonal sites was used to represent the plasma membrane. On each site, the number of molecules of any given type was represented as an integer variable. The probability rate for a reaction or diffusion step of type s (s = 1,..., R) to take place at site i (i = 1,..., N_{s} ) was decomposed in the product of a global (site independent) rate f_{s} and a local (site dependent) probability . Global rates were functions of the global number of available reactants, whereas local probabilities were functions of their local concentrations, following from the expressions given in Tables 1 and 2 and the normalization conditions . Reaction and diffusion processes were performed stochastically according to the following algorithm (see Data Set 1, which is published as supporting information on the PNAS web site) (8 and 9). At each time step, the reaction or diffusion step of type s was chosen with probability . Then the site i where the reaction or diffusion step had to take place was chosen with probability . The reaction or diffusion step was then performed. The tables of global rates and local probabilities were updated to take into account the variation in the values of the local and global values of available reactants. The tables of local probabilities were ordered with the use of a bisection algorithm to speed up the choice of the next reaction–diffusion site i. Time was advanced as a Poisson process with mean . The simulations were performed on a 99 dualprocessor nodes Beowulf cluster [2800 MHz Athlon processors (AMD, Sunnyvale, CA) with 4 Gbyte of memory each] using slackware linux (Slackware Linux, Brentwood, CA) and the gnu c compiler (http://gcc.gnu.org). Simulation results were represented graphically by using matlab (Mathworks, Natick, MA).
Results and Discussion
We simulated the kinetics of the network of chemical reactions that represents the ubiquitous biochemical backbone of the directional sensing module and contains the following: (i) binding of PI3K to activated membrane receptors, (ii) binding of PTEN to PIP_{2}, (iii) catalytic activity of PI3K and PTEN, and (iv) phosphoinositide diffusion within the plasma membrane. Because the chemical system is characterized by extremely low concentrations of chemical factors [0–50 nM for enzymes (10) and 0.1–1 μM for phosphoinositides (11)] and evolution takes place out of equilibrium, we used a stochastic approach (8, 9). Indeed, rare, large fluctuations are likely to be relevant for kinetics in the presence of unstable or metastable states. This assumption is also consistent with the observation that cell response to chemotactic stimuli has a stochastic character (12). Simulated reactions and diffusion processes are summarized as follows:

PI3K(cytosol) + Rec(i) ↔ PI3K·Rec(i)

PTEN(cytosol) + PIP_{2}(i) ↔ PTEN·PIP_{2}(i)

PI3K·Rec(i) + PIP_{2}(i) → PI3K·Rec(i) + PIP_{3}(i)

PTEN·PIP_{2}(i) + PIP_{3}(i) → PTEN·PIP_{2}(i) + PIP_{2}(i)

PIP_{2}(i) → PIP_{2}(j)

PIP_{3}(i) → PIP_{3}(j)
Index i represents a generic plasma membrane site and j one of its nearest neighbors. We have simulated the chemical kinetics on the inner face of the plasma membrane as a twodimensional lattice gas coupled to an unstructured cytoplasmic reservoir. On each site i of the lattice we have assumed the presence of a number of molecules of the kth species. Chemical reactions and diffusion have been simulated as random processes with intensities proportional to kinetic reaction and diffusion rates (see Tables 1 and 2).
The probability of performing a reaction on a given site is taken to be proportional to realistic kinetic reaction rates and local reactant concentrations (Tables 1 and 2).
Stochastic Simulation. The plasma membrane is represented as a sphere of radius R = 10 μm partitioned in N _{s} = 10,242 computational sites forming a honeycomb lattice with 12 pentagonal defects. Chemical concentrations are represented as integer variables giving the number of molecules of the kth species present on site i. Reaction–diffusion kinetics is simulated according to Gillespie's method (8) generalized to the case of an anisotropic environment. Catalytic processes are described by Michaelis–Menten kinetics. The density of activated receptors is proportional to extracellular chemoattractant concentration. The probability of diffusion from a computational site to a neighboring one is assumed proportional to the difference in local concentrations, according to Fick's law. Time is advanced as a Poisson process. To provide complete reproducibility, the C code of the simulation algorithm is included as Data Set 1.
The density ρ of activated receptors controls PI3K recruiting to the plasma membrane, thus playing the role of a chemical potential (13). Although receptor activation is directly controlled in the experiments through the extracellular concentration of chemoattractant stimulus, it is difficult to control experimentally [PI3K]. We have therefore set [PI3K] to a realistic fixed value (Table 2) and varied ρ in the range 0–100 nM. Diffusion controls the behavior of the system in an obvious way, because for high values it tends to mix chemical species, but in conjunction with catalytic activity it also exerts an ordering action, transferring information between neighboring uniformly populated patches of the plasma membrane. Diffusion is therefore a second parameter, which strongly influences the system's dynamic and stationary state. We have considered D values in the range 0–5 μm^{2}/s.
Order Parameter. Because PTEN localizes on PIP_{2}rich regions of the plasma membrane, phase separation can be observed either at the enzyme or phosphoinositide level. In real cells, phosphoinositide clusters trigger actin polymerization and motility. For this reason we have studied symmetry breaking in phosphoinositide distribution. The physics of phaseseparation dynamics has been thoroughly studied (14–18) and is known to give rise to a variety of effects such as selforganization, pattern formation, and pattern selection in many physical–chemical systems (refs. 6, 19, and 20; for a general reference on pattern formation in reaction–diffusion systems, see ref. 21). The degree of order of a chemical mixture undergoing phase separation can be quantified by means of an order parameter, i.e., a dimensionless observable assuming the value 0 in the symmetric, mixed state and a value of order 1 in the symmetrybroken state in which chemical species are separated (13). A convenient order parameter measuring the degree of phase separation of the phosphoinositide mixture is Binder's cumulant (9, 22) where φ = φ _{i} = [PIP_{3}] _{i}  [PIP_{2}] _{i} is a difference of local concentrations on site i, and 〈... 〉 denotes average over many different random realizations. Binder's cumulant measures the distance of the probability distribution of φ from a Gaussian distribution. When phosphoinositides are mixed, fluctuations around the uniform average value are Gaussian, and g tends to zero, whereas when phosphoinositides separate into distinct clusters (Fig. 1), the probability distribution of φ is characterized by two distinct peaks, and g becomes of order 1 (Fig. 1 d and e ). Maximal observed values of g for phaseseparating systems are between 0.4 and 0.8, corresponding to the fact that the PIP_{2} and PIP_{3} distributions partially overlap.
Spontaneous phase symmetry breaking leads to the formation of PIP_{2}, PIP_{3}rich clusters of different size. Cluster sizes can be characterized by harmonic analysis. For each realization, the fluctuations δφ = φ  〈φ 〉 of the φ field can be expanded in spherical harmonics (23) as where u is a unit vector identifying a point on the spherical surface. Let us consider the twopoint correlation functions where 〈... 〉_{iso} denotes average over ensembles (7) and over the action of the sphere rotation group, , and P_{l} are Legendre polynomials (23). A measure of the average cluster size is πR/2 〈l 〉, where . In particular, if most of the weight is concentrated on the lth harmonic component, average phosphoinositide clusters extend over the characteristic length πR/2l.
When receptor activation ρ = ρ _{i} = [Rec] _{i} is not uniformly distributed we are interested in its correlation with localized PIP_{2}, PIP_{3} clusters, which is measured by the components of the covariance matrix c _{ρρ} = 〈(ρ 〈ρ 〉)^{2} 〉, c _{ρφ} = 〈(ρ 〈ρ 〉)(φ 〈φ 〉) 〉, c _{φφ} = 〈(φ 〈φ 〉)^{2} 〉 and by the correlation coefficient (24).
Dynamic Phase Diagram. We have run ≈10 random realizations of the system for ≈400 (ρ_{R}, D) pairs with 1 μM < ρ < 100 μM, 0.1 μm^{2}·s^{1} < D < 5 μm^{2}·s^{1} for 2 h of simulated time [≈1 Tflop (trillion floating point operations) per realization]. For each (ρ, D) pair we have computed g as a function of time by performing surface and ensemble average at fixed time intervals. For each random realization we have started from a stationary homogeneous PTEN, PIP_{2} distribution (10 s of simulated time evolution were enough to reach stationarity). To measure automatically phaseseparation events and characteristic phaseseparation times, we have computed the 10min running average of g to cut off rapid fluctuations and selected the moment when it reached the threshold 0.4 and did not fluctuate below that value during the following 30 min of simulated time. Patterns observed at t = 2 h are then quite close to stationarity. Similarly, we have measured the harmonic components C_{l} . In this case we have identified the phaseseparation time with the moment when the weight of the 10min running average of a single harmonic component surpassed the 80% of the total weight and did not fluctuate below that value for the following 30 min of simulation time.
We let the system evolve to stationarity, in the absence of receptor activation, obtaining a homogeneous PTEN and PIP_{2} distribution at the plasma membrane, identical to that observed in unstimulated cells. At time t = 0 receptor activation is switched on; either activated receptors are isotropically distributed or the isotropic distribution is perturbed with a linear term producing a 5% difference in activated receptor density between the North and the South poles.
In the isotropic case, we found that in a wide region of parameter space, the chemical network presents an instability with respect to phase separation (25, 26), i.e., the homogeneous phosphoinositide mixture realized soon after receptor activation is unstable and tends to decay into spatially separated PIP_{2} and PIP_{3}rich phases (Fig. 1 a–d ). Characteristic times for phase separation vary from the order of a minute to that of an hour, depending on receptor activation (Fig. 1e ). The dynamic behavior and stationary state of the system strongly depend on the values of two key parameters: the concentration ρ of activated receptors and the diffusivity D.
In the case of anisotropic stimulation, orientation of PIP_{2} and PIP_{3} patches clearly correlates with signal anisotropy (Fig. 1 g–i ), and, compared with isotropic stimulation, phase separation takes place in a larger region of parameter space and in times that can be shorter by one order of magnitude (Fig. 1j ).
Average phaseseparation times are plotted in Fig. 2a , where red areas correspond to nonphaseseparating systems. In the deep blue area, phase separation takes place in <5 min, whereas close to the boundary of the broken symmetry region phase separation can take times of the order of an hour (Fig. 1e ). Average cluster sizes at stationarity are plotted in Fig. 2b . In the red region, cluster sizes are of the order of the size of the system, corresponding to the formation of pairs of complementary PIP_{2} and PIP_{3} patches (Fig. 1). For diffusivities <0.1 μm^{2}/s the diffusionmediated interaction is unable to establish correlations on lengths of the order of the size of the system, and one observes the formation of clusters of separated phases of size much smaller than the size of the system (Fig. 2c ). For diffusivities >2 μm^{2}/s the tendency to phase separation is contrasted by the disordering action of phosphoinositide diffusion (Fig. 2 a–c ).
Average phaseseparation times for the anisotropic case are plotted in Fig. 2d . By comparing with the isotropic case (Fig. 2a ), it appears that there is a large region of parameter space where phase separation is not observed with isotropic stimulation, while a 5% anisotropic modulation of activated receptor density triggers a fast phaseseparation process. Cluster sizes (Fig. 2e ) are in the average larger in the anisotropic case than in the isotropic case (Fig. 2a ). Fig. 2f shows that the orientation of PIP_{2} and PIP_{3} patches is strongly correlated with signal anisotropy. Therefore, anisotropy has two main effects: on one hand, it triggers the phaseseparation process on shorter timescales and in a wider region of parameter space; on the other hand, its direction breaks the system's rotational symmetry and selects the stationary state of the system. For receptor activation ρ < 20 nM, the few PI3K enzymes bound to the plasma membrane are not sufficient to create extended PIP_{3}rich regions; however, small intermittent PIP_{3} clusters are still observed (Fig. 3a ).
Simulating a gradient in receptor activation similar to the one imposed in experimental assays, we are able (Fig. 4) to reproduce the input–output relationship observed on Dictyostelium (4). It is also worth noting that the characteristic timescales for phase separation emerging from our dynamical simulations, which have been performed by using realistic reaction and diffusion rates, are in agreement with the observed ones (4).
Physical Picture. The transition from a phaseseparating to a phasemixing regime results from a competition between the ordering effect of the interactions and the disordering effect of molecular diffusivity. The frontier between these two regimes varies continuously as a function of parameters. Importantly, we found that the overall phaseseparation picture is robust with respect to parameter perturbations, because it persists even for concentrations and reaction rates differing from those of Table 2 by one order of magnitude. Moreover, both in isotropic and anisotropic conditions, signal amplification is completely reversible. Switching off receptor activation abolishes phase separation, delocalizes PI3K from the plasma membrane to the cytosol, and brings the system back to the quiescent state (see Movie 1, which is published as supporting information on the PNAS web site).
Thus, phosphoinositide diffusion is directly responsible for establishing correlations between neighboring sites, leading the system to a phaseseparation instability. Although large diffusivity has a mixing effect, intermediate diffusivity cooperates with catalysis to order the system on large scales. Physically, this process can be understood as follows. Receptor activation shifts the chemical potential for PI3K, which is thus recruited to the plasma membrane. PI3K catalytic activity produces PIP_{3} molecules from the initial PIP_{2} sea. Initially, the two phosphoinositide species are well mixed. Fluctuations in PIP_{2} and PIP_{3} concentrations are, however, enhanced by preferential binding of PTEN to its own diffusing phosphoinositide product, PIP_{2}. Binding of a PTEN molecule to a cell membrane site induces a localized transformation of PIP_{3} into PIP_{2}, resulting in higher probability of binding other PTEN molecules at neighboring sites. Such a positive feedback loop not only amplifies the inhibitory PTEN signal, but via phosphoinositide diffusion it also establishes spatiotemporal correlations strong enough to drive the system toward spontaneous phase separation. The time needed by the system to fall into the more stable, phaseseparated phase, however, can be a long one if the symmetric, unbroken phase is metastable. In that case, a small anisotropic perturbation in the pattern of receptor activation can be enormously amplified by the system instability.
It is worth noting that, when the system phase separates, the final size of the clusters is limited only by the size of the system and the availability of cytosolic enzymes. In an infinite system, clusters would grow indefinitely. In a finite system instead, as long as PTEN molecules are recruited on the PIP_{2} patch, they are no longer available to compete with PI3K on the residual PIP_{3} patch, which is therefore stabilized. The net effect is that the cluster size saturates at a stationary value of the order of the size of the system.
Dimensional considerations suggest that the size of the patches should grow proportionally to , 1/k being the characteristic enzyme association–dissociation timescale, for low diffusivity values, and saturate to the size of the system for intermediate diffusivities. Simulations confirm the dimensional estimate and show that for higher diffusivities the species are mixed up and the size of clusters drops abruptly (Fig. 5).
The properties of the real reaction and diffusion processes described in Results and Discussion and Table 1 are better understood through the study of a onedimensional model derived from them under simplifying assumptions (see Supporting Text and Fig. 6, which are published as supporting information on the PNAS web site), where a more complete analysis is possible. The simplified model presents a parameter region where multiple stable equilibria are possible, and stochasticity in the number of membranebound enzymes can trigger a transition from a less stable to a stabler state. The transition takes place through the formation of a small region of the stabler phase in the sea of the less stable one. Although small regions of the stabler phase are wiped off by diffusion, larger regions propagate with finite velocity in the less stable phase and would eventually take over the whole system if the number of available cytosolic enzymes was infinite. Because, however, the number of cytosolic enzymes is limited, the process slows down and eventually stops with the formation of a stationary front separating a PTENrich and a PI3Krich region. The critical size of the nucleating region is determined by the relative strength of two intrinsically dynamic quantities: diffusivity and the velocity of front propagation.
Conclusions
Our results provide a simple physical clue to the enigmatic behavior observed in eukaryotic cells. As we have shown, there is a large region of parameter space where the cell can be insensitive to uniform stimulation over very large times but responsive to slight anisotropies in receptor activation in times of the order of minutes (Fig. 1 and Movie 1). Accordingly, by simulating shallow gradients of chemoattractant, we observed PIP_{3} patches accumulating with high probability on the side of the plasma membrane with higher concentration of activated receptors, thus resulting into a large amplification of the chemotactic signal (Figs. 1 and 4). Moreover, we identified an intermediate region of parameters, where phase separation under isotropic stimulation is observed on average in a long but finite time. In this case, one would predict that on long timescales cells undergo spontaneous polarization in random directions and that the number of polarized cells grows with time. Intriguingly, this peculiar motile behavior is known as chemokinesis and is observed in cellmotility experiments when cells are exposed to chemoattractants in the absence of a gradient (27). An additional consequence of the tendency to phase separation is that for low values of receptor activation small intermittent clusters should form, because diffusion cannot establish a correlation between toodistant phosphoinositide patches and the number of enzymes producing the patches is too low. Indeed, intermittent phosphoinositide clusters are observed in our simulations (Fig. 3a ), and their formation has been recently described in Dictyostelium cells exposed to very low cAMP concentrations (28). It appears that D3phosphoinositide patches serve as a spatial cue for pseudopod formation, which enhances the sensitivity and amplitude of chemotactic movement. Furthermore, cells migrating within tissues, such as neurons or immune system cells, may encounter multiple chemoattractant signals in complex spatial and temporal patterns that can potentially direct their path. Notably, cell polarization induced by multiple chemoattractant sources has been observed in vitro (4), and this experimental situation can be mimicked in silico by simulating the receptor activation pattern produced by multiple chemoattractant sources (Fig. 3b ).
In summary, the phaseseparation scenario provides a simple and unified framework to different aspects of directed cell motility, such as large amplification of slight signal anisotropies, insensitivity to uniform stimulation, appearance of isolated and transient phosphoinositide patches, and stochastic cell polarization. It provides a link between known microscopic and macroscopic timescales. Finally, it unifies apparently conflicting aspects that previous modeling efforts (29, 30) could not satisfactorily reconcile (31), such as insensitivity to absolute stimulation values, large amplification of shallow chemotactic gradients, reversibility of phase separation, robustness with respect to parameter perturbations, stochastic character of cell response, use of realistic biochemical parameters, and space–time scales.
One of the characteristic features of our model is that it brings together into a unified picture several seemingly disconnected phenomena, such as response to anisotropic chemotactic stimulation, stochastic polarization, and small cluster formation for low activation levels. It would be useful to perform systematic quantitative observations on stochastic cell polarization under uniform stimulating conditions, both at high and low activation leves. Because nucleation can be arguably modeled as a Poisson process, in the presence of isotropic stimulation one should observe a number of stochastically polarized cells growing with time, with a rate comparable with the model predictions.
Acknowledgments
We thank I. Kolokolov, V. Lebedev (Landau Institute), G. Ortenzi, and L. Rondoni (Turin Polythecnic) for useful discussions. We thank A. Giorgilli, L. Marsella, G. Naldi, and the Departments of Mathematics of the Universities of Milano and Milano Bicocca for kind hospitality and help with computational resources. This work was supported by Ministero dell'Istruzione, dell'Università e della Ricerca–Progetti di Rilevanza Nationale 2003 (to A.G.), Telethon Italy Grant GGP04127 (to G.S.), European Union Network Grant MRTNCT2003504712, Ministero dell'Istruzione, dell'Università e della Ricerca–Progetti di Rileranza Nationale 2004, Ministero dell'Istruzione, dell'Università e della Ricerca–Fondo per gli Investimenti della Ricerca di Base 2001, Centro Regionale di Competenza–Analisi e Monitoraggio del Rischio Ambientale, Istituto Nazionale di Fisica Della MateriaProgramma Quadro per la Competitività e l'lnnovazione, Istituto Nazionale di Fisica NucleareFB11 (to A.d.C. and A.C.), Sixth Framework Programme of European Union Contract LSHMCT2003503254, and the Associazione Italiana per la Ricerca sul Cancro (to F.B.).
Footnotes

↵ † To whom correspondence may be addressed. Email: gamba{at}polito.it or guido.serini{at}ircc.it.

Author contributions: A.G., A.d.C., S.D.T., A.C., F.B., and G.S. designed research, performed research, analyzed data, and wrote the paper.

Conflict of interest statement: No conflicts declared.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviations: PI3K, phosphatidylinositol 3kinase; PIP_{3}, D3phosphoinositide product of PI3K.
 Copyright © 2005, The National Academy of Sciences
References
 ↵
 ↵

↵
Ridley, A. J., Schwartz, M. A., Burridge, K., Firtel, R. A., Ginsberg, M. H., Borisy, G., Parsons, J. T. & Horwitz, A. R. (2003) Science 302 , 17041709. pmid:14657486

↵
Janetopoulos, C., Ma, L., Devreotes, P. N. & Iglesias, P. A. (2004) Proc. Natl. Acad. Sci. USA 101 , 89518956. pmid:15184679

↵
Iijima, M., Huang, Y. E., Luo, H. R., Vazquez, F. & Devreotes, P. N. (2004) J. Biol. Chem. 279 , 1660616613. pmid:14764604

↵
Seul, M. & Andelman, D. (1995) Science 267 , 476483.

↵
Stanley, H. E. (1987) Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, Oxford).
 ↵

↵
Binder, K. & Heermann, D. W. (2002) Monte Carlo Simulations in Statistical Physics: An Introduction (Springer, Berlin).

↵
Carpenter, C. L., Duckworth, B. C., Auger, K. R., Cohen, B., Schaffhausen, B. S. & Cantley, L. C. (1990) J. Biol. Chem. 265 , 1970419711. pmid:2174051
 ↵
 ↵

↵
Landau, L. & Lifshitz, E. (1980) Statistical Mechanics (Butterworth–Heinemann, London), Part 1.

↵
Langer, J. (1992) in Solids Far From Equilibrium, ed. Godrèche, C. (Cambridge Univ. Press, Cambridge, U.K.).

Bray, A. (1995) Adv. Phys. 45 , 357459.

Wagner, C. (1961) Z. Elektrochem. 65 , 581591.
 ↵
 ↵

↵
Shraiman, B. & Bensimon, D. (1985) Phys. Scripta T9 , 123125.
 ↵

↵
Binder, K. (1981) Z. Phys. 43 , 119140.

↵
Abramowitz, M. & Stegun, I. (1964) Handbook of Mathematical Functions: With Formulas, Graph, and Mathematical Tables (U.S. Government Printing Office, Washington, DC).

↵
Cowan, G. (1998) Statistical Data Analysis (Oxford Univ. Press, Oxford).

↵
Gunton, J., San Miguel, M. & Sahni, P. (1983) in Phase Transitions and Critical Phenomena, eds. Domb, C. & Lebowitz, J. L. (Academic, Berlin), Vol. 8, pp. 267466.

↵
Lifshitz, E. & Pitaevskii, L. (1981) Physical Kinetics (Pergamon, Oxford).
 ↵

↵
Postma, M., Roelofs, J., Goedhart, J., Gadella, T. W., Visser, A. J. & Van Haastert, P. J. (2003) Mol. Biol. Cell 14 , 50195027. pmid:14595105
 ↵
 ↵

↵
Devreotes, P. & Janetopoulos, C. (2003) J. Biol. Chem. 278 , 2044520448. pmid:12672811

↵
Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K. & Walter, P. (2002) Molecular Biology of the Cell (Garland Science, New York).

↵
Shen, B. Q., Lee, D. Y., Gerber, H. P., Keyt, B. A., Ferrara, N. & Zioncheck, T. F. (1998) J. Biol. Chem. 273 , 2997929985. pmid:9792718

↵
Fujiwara, T., Ritchie, K., Murakoshi, H., Jacobson, K. & Kusumi, A. (2002) J. Cell Biol. 157 , 10711081. pmid:12058021
 ↵

↵
Adam, G. & Delbruck, M. (1968) in Structural Chemistry and Molecular Biology, eds. Rich, A. & Davidson, N. (Freeman, San Francisco), pp. 198215.

↵
Panayotou, G., Gish, G., End, P., Truong, O., Gout, I., Dhand, R., Fry, M. J., Hiles, I., Pawson, T. & Waterfield, M. D. (1993) Mol. Cell. Biol. 13 , 35673576. pmid:8388538