A symmetry breaking mechanism for epithelial cell polarization
In multicellular organisms, epithelial cells form layers separating compartments responsible for different physiological functions. At the early stage of epithelial layer formation, each cell of an aggregate defines an inner and an outer side by breaking the symmetry of its initial state, in a process known as epithelial polarization. By integrating recent biochemical and biophysical data with stochastic simulations of the relevant reaction-diffusion system we provide evidence that epithelial cell polarization is a chemical phase separation process induced by a local bistability in the signaling network at the level of the cell membrane. The early symmetry breaking event triggering phase separation is induced by adhesion-dependent mechanical forces localized in the point of convergence of cell surfaces when a threshold number of confluent cells is reached. The generality of the emerging phase separation scenario is likely common to many processes of cell polarity formation.
The development of epithelial tissues (e.g., kidney tubules, respiratory and gastrointestinal tracts, etc.) results from complex morphogenetic processes implying the arrangement of cells in layers organized along specific directional axes Comer and Parent (2007); Martin-Belmonte et al. (2007). Epithelial cells are endowed with a self-polarization mechanism defining an ‘inner’ and ‘outer’ side, which is mandatory to allow organs to exert their vital functions. In a well established in vitro cell system, which recapitulates the in vivo morphogenesis, after a single epithelial cell is seeded in a three-dimensional gel (Fig. 1a), cell division begins, and a multicellular aggregate arises Comer and Parent (2007). The cells in the aggregate are bound each other (through cadherin molecules, Fig. 1b) and to an extracellular matrix (through integrin molecules, Fig. 1c). When the cell number reaches 5-6 units, an inner cavity, named lumen, is spontaneously opened Wang et al. (1990) (Fig. 1a,b). Afterwards, cells develop a top (called apical) and a bottom (basolateral) side (Fig. 1d) having different chemical features, while cell-cell and cell-matrix contacts only persist in the basolateral region. Finally, the border between apical and basolateral sides is sealed by ring-shaped tight junction proteins, which spontaneously find their functional position (Fig. 1d) and prevent intermixing of chemical components between the apical and the basolateral membranes, as well as the outpouring of liquids from the lumen. The full polarization process has a complex nature and involves different factors and stages. However, recent experiments have determined its master regulator (see Bryant and Mostov (2008); Mellman and Nelson (2008) and ref.s therein): intracellular asymmetry and lumen opening are controlled by PIP2/3 phospholipids (see Fig.2) and their interaction with PTEN/PI3K enzymes which induce PIP2/3 segregation to opposite poles, while the PAR complex further stabilizes axial polarity (see the review Comer and Parent (2007) and ref.s below). Even in such an in vitro system, however, the mechanism whereby the cell original spatial symmetry is spontaneously broken and polarization develops remains mysterious Comer and Parent (2007).
To describe cell differentiation, polarization and signal localization mechanisms, stochastic reaction-diffusion van Kampen (2007) and coupled kinetic rate equations have been widely used for intracellular signaling, gene regulation and autocatalytic reaction systems (see e.g. Ramanathan et al. (2005); Thattai and van Oudenaarden (2002); Paulsson et al. (2000) and ref.s therein). In this context, here we use reaction-diffusion equations to model the PIP2/3 master regulator in order to address some open questions Comer and Parent (2007), namely the mechanisms for: i) the lumen site choice; ii) its opening; iii) the control of its final size; iv) the localization of tight junctions. After delineating the chemical reactions involved in the process, by simulation of its master equation we show that self-polarization can be understood in terms of statistical physics concepts as a symmetry breaking mechanism driven by the chemical regulatory network. We finally interpret the simulation results in terms of a simple mean-field model.
Recent experiments have shown that PIP2 and PIP3 for a module that acts as a master regulator controlling all signaling pathways and cytoskeletal dynamics required for epithelial cell polarization Bryant and Mostov (2008); Mellman and Nelson (2008). PIP2/PIP3 levels are regulated by the counteracting enzymes PI3K and PTEN, which respectively catalyze the switch of PIP2 to PIP3 and vice versa Comer and Parent (2007); Kölsch et al. (2008) (Fig. 2). The phospholipids (PIP2/3) are stably localized in the inner face of the cell membrane where they diffuse. The enzymes (PI3K/PTEN) diffuse instead in the cell volume, where they are present in limiting amounts, and become active upon association with membrane spanning proteins or lipids. PTEN levels in the membrane are controlled by its binding to PIP2, thus realizing a positive feedback loop (see Fig. 2). PI3K levels in the membrane are controlled by its binding to cell-cell adhesive receptors (cadherins) and cell matrix adhesive receptors (integrins, schematically indicated by C/M in Fig. 2) Watton and Downward (1999). To bind PI3K, cadherins must be activated (Cad in Fig. 1b and 2) by engagement with cadherins of a neighboring cell (named C/C in the diagram of Fig. 2) PI3K is active only when associated to either activated cadherins or integrins. Since PIP3 stabilizes the activated form Cad Yap and Kovacs (2003), these interactions create a positive PIP3-PI3K feedback loop, mediated by the existence of cell-cell contacts (Fig. 2).
Before polarization, cadherins and integrins are activated along the whole membrane and PIP3 uniformly prevails on PIP2 determining a stable PIP3-rich phase over the whole membrane. A local depletion of PIP3-PI3K can be created if a large enough membrane area with disrupted cell-cell links is formed, thereby breaking the PIP3-PI3K feedback loop (Fig. 2) and originating a germ of a PIP2-rich phase (Fig. 1b and 2). Then, the PIP2-PTEN feedback loop may locally prevail, inducing a PIP2 and PIP3 surface compartmentalization that splits the cell membrane in two regions, or phases Landau and Lifshitz (1980), characterized by different chemical concentrations of the signaling molecules.
Several mechanisms have been proposed Bryant and Mostov (2008) to explain the initiation of the polarization process, which is typically found in high curvature membrane regions, especially in correspondence of multiple cell-cell contacts (cellular vertex) Martin-Belmonte et al. (2007) away from the extracellular matrix (Fig. 1). We observe here that cell-cell contacts can be broken Kroschewski (2004) as soon as the adhesion energy (i.e. the energy per unit area needed to break cadherin contacts) becomes comparable to the elastic energy stored in the cell membrane in high curvature regions, such as at the confluence of several cells (Fig. 1a,b). Since the elastic energy per unit area stored in the membrane is Landau and Lifshitz (1986); Fournier (2007); Simson et al. (1998)
where is the membrane bending rigidity and the local curvature radius Fournier (2007), the condition allows to estimate the critical curvature radius where cell-cell contacts start being disrupted as Simson et al. (1998)
The critical value can be easily estimated. In eukaryotic cells, the typical adhesion energy of cadherin contacts is Simson et al. (1998), while the typical bending rigidity is Simson et al. (1998), giving 111It is worth observing here that since the values and used for the estimate have been measured on live cells Simson et al. (1998), they take effectively into account the mechanical contributions from both the lipid bilayer and the cortical cytoskeleton.
According to the above estimate, when the number of cells in the initial aggregate increases up to the 5-6 cell stage, at the cell convergence points (see Fig. 1b) the membrane curvature increases as well and, especially in areas not in contact with the extracellular matrix, cadherin bridges are subject to forces that can disrupt links. By such a mechanism, a local opening of cell-cell contacts breaks the PIP3-PI3K feedback loop and induces a local unbalance towards PIP2 formation (Fig. 2) and a germ of the PIP2-rich phase can be ushered in (Fig. 1b).
In Sect. III we show that only germs of the PIP2-rich phase larger than a treshold radius actually survive and grow. This fact suggests that although the uniform PIP3-rich phase is not the more stable state for the signaling network, a polarized state characterized by the coexistence of the PIP3-rich and the PIP2-rich phase may be reached only by overcoming a barrier in a suitably defined effective energy (see Sect. IV for a detailed discussion of this point). Therefore, we are faced with the following physical picture: if elastic forces due to high membrane curvature in the region of cell convergence (cellular vertex) trigger disruption of cadherin links in a region of size larger than , the PIP2-rich patch grows favoring further breaking of cadherin links (Fig. 2) and the formation of a lumen. Thus, for the process of lumen formation to start, it is necessary that the local curvature radius in the cellular vertex satisfies
The growth of the PIP2-patch and lumen slows down and eventually comes to a stop as soon as cytosolic PTEN is depleted. This way, at the end of the process the cell reaches a stable polarized state characterized by the coexistence of the PIP2-rich and the PIP3-rich phase Ferraro et al. (2008); Gamba et al. (2007), and a lumen coinciding with the PIP2-rich phase is formed.
In this Section we investigate on quantitative grounds the above described scenario of polarization.
Since the chemical reaction and diffusion processes are intrisically noisy, we simulate the corresponding dynamics by a stochastic algorithm, using realistic values for reaction and diffusion rates. We can check this way that noise alone is not sufficient here to overcome the energy barrier separating the uniform and polarized state in observational times, if an initial PIP2 seed of size larger than is not created by an external interaction.
We represent the plasmamembrane by a lattice of (mostly hexagonal) sites of area on a sphere surface with radius (Fig. 3). Each site is populated by a number of molecules of the chemical factors and their dynamics is described by standard master equations. For instance, the PIP2 PIP3 process is described by
where is the probability to have at time a number of type molecules at a given site (say, .) The list of relevant reactions with their corresponding rates is given in Table 1 222 The values for processes involving cadherins are educated guesses since no precise data are available. One order of magnitude changes in these values do not result however in appreciable modifications of the system dynamics. . PIP2/3 diffusion is described by random jumps of a molecule from site to its neighboring site with rate , where is phospholipid diffusivity Gamba et al. (2005). Since the diffusivity of cytosolic enzymes (PI3K/PTEN) is much larger than that of membrane pospholipids Gamba et al. (2005), their distribution in the cytosol is treated as uniform.
For the simulations we use a variation of Gillespie algorithm Gillespie (1977) taking into account the spatial non-uniformity of the system. At time zero, a random number is generated to determine the next reaction or elementary diffusion process to occur, with a probability proportional to the corresponding factor from Table 1. Then, time is advanced as a Poisson process with rate again determined by the factors. These steps are repeated iteratively until the desired simulation time is reached.
|0 min||4 min||10 min|
We suppose that a circular PIP2-rich patch of radius is initially formed in the sea of the PIP3-rich phase 333Where not otherwise stated, the following experimentally realistic values for the initial concentration are used: (according to Vanhaesebroeck et al. (2001)), (according to Carpenter et al. (1990) and Duguay et al. (2003), respectively), (molecules/cell). For the PTEN concentration, we assumed the same order of magnitude as in Gamba et al. (2005) and studied in Fig. 5 the system behaviour on varying the PTEN molecules number. The initial PIP3-rich phase is 98% PIP3, 2% PIP2. and investigate its dynamics to check whether a stable polarization state is attained (Fig. 3). Fig. 4 shows the time evolution of circular patches of different initial radii . Patches smaller than a threshold radius are dissolved by diffusion and thermal processes, and do not impair the stability of the uniform PIP3-rich phase. Conversely, patches larger than grow in time triggering the separation of the cell surface in a PIP2-rich and a PIP3-rich region and eventually reach an equilibrium (Fig. 3). Notably, the threshold radius derived from the above calculation is consistent with the previously independently derived value for the adhesion radius, . The two phases are divided by an interface of characteristic width Gamba et al. (2007), where is of the order of the catalytic constants of the two catalytic reactions of Table 1 (first two rows).
The kinetic of this heterogeneous nucleation process can be understood in terms of non-equilibrium, reaction-diffusion stochastic dynamics. In reaction-diffusion systems instabilities are often produced by Turing’s mechanism Turing (1952). Here we find that pattern formation starting from a locally stable homogeneous state is triggered by a local perturbation by a nucleation center of size larger than a critical size Bray (1994); Schöll (2000); Gamba et al. (2007, 2009a).
Fig. 5 shows that the equilibrium size of the PIP2-rich patch, and therefore of the lumen, is controlled by the number of PI3K and PTEN molecules. In the absence of any limiting mechanism, the growth of the PIP2-rich patch would in fact lead to a PIP2-rich phase completely invading the cell surface. However, due to the coupling to a finite PTEN and PI3K reservoir, the system self-tunes to a phase-coexistence state and the process stops when the PIP2-rich patch reaches the equilibrium size Gamba et al. (2005, 2007); Ferraro et al. (2008). Interestingly, the fact that the size of the PIP2-rich patch, and consequently of the lumen, is controlled by the precise number of PTEN molecules, is in qualitative agreement with the observation that deletion of a single PTEN allele can interfere with the polarization process Gassama-Diagne et al. (2006).
The observation in the present scenario of a comparatively large threshold radius, of the order of one tenth of the cell size, suggests the existence of a correspondingly large barrier of effective energy dividing the uniform state from the phase-separated one. This prevents thermal and chemical noise from triggering spontaneous symmetry breaking and lumen formation. However, an external mechanical action creating a sizeable PIP2-rich patch, due to the presence of localized regions of high membrane-curvature, can overcome the barrier and start polarization 444Interestingly, the role of mechanical forces has been suggested also in other settings of tissue morphogenesis Shraiman (2005).. Our picture also explains tight junction localization. Experimental data show that the stable binding of tight junction proteins to the membrane requires both a protein complex named PAR3-PAR6, which is localized in the PIP2-rich phase by a chain of reactions, and cell-cell contacts, which are maintained only in the PIP3-rich phase Margolis and Borg (2005). The spontaneous aggregation of tight junctions is thus constrained by a biochemical logical AND to take place only on the ring-shaped boundary separating the PIP2-rich from the PIP3-rich phase.
In this Section we show that the results of the simulations can be conveniently interpreted in terms of an effective mean-field model, following the approach detailed in Refs. Gamba et al. (2007, 2009a).
Fast diffusion of PI3K and PTEN enzymes in the cytosol, and the conservation law allow to effectively describe the state of the cell membrane in terms of the configuration of the single-component concentration field Gamba et al. (2007, 2009a):
The resulting effective equation for can be set in the simple Landau-Ginzburg form:
complemented by an integral costraint expressing the coupling of the concentration field to the reservoir of free cytsolic enzymes (see Ref.s Gamba et al. (2007, 2009a) and Supplementary Information in Gamba et al. (2005).) In Eq. (7), is the diffusivity of lipids on the cell membrane, is an effective potential, and is a stochastic term taking into account the effect of thermal and chemical noise.
The mean-field effective potential can be easily derived, via a quadratic approximation, from the stochastic model described in Sect. III under the assumption that the cytosolic PI3K, PTEN and Cad fields are in approximate equilibrium with the membrane PIP2 and PIP3 fields, and therefore “slaved” to the field Gamba et al. (2007, 2009a):
The terms in the r.h.s. of Eq. (8) describe respectively conversion of PIP3 into PIP2 due to the action of PTEN, and conversion of PIP2 into PIP3 due to the action of PI3K activated by cadherins (Fig. 2). The quadratic terms encode respectively the PIP2 PTEN and the PIP3CadPI3K feedback loops (Fig. 2). In particular, when cadherin links are broken.
In a wide region of parameter space around the realistic parameter values from Table 1, the effective potential is bistable (Fig. 6; for a detailed description of the bistability region, see Ref. Gamba et al. (2009b)).
The two potential wells in Fig. 6 correspond to a stable PIP2-rich and a metastable PIP3-rich phase, separated by an energy barrier .
The mean-field model (7,8) and the bistability of the effective potential provide an interpretation to the simulation results, showing that the stable polarized state characterized by the coexistence of PIP2 and PIP3 in complementary regions is separated from the metastable PIP3-rich phase by an effective energy barrier . According to the theory of Landau-Ginzburg equation, PIP2-rich seeds larger than a critical value are bound to expand in the PIP3-rich sea with a velocity proportional to Bray (1994).
The cytosolic concentrations , , and appearing in (8) may be expressed as integrals of the concentration field Gamba et al. (2009a, b). The resulting global coupling has the effect of driving dynamically the cell membrane towards an equilibrium polarized state where the PIP2-rich and PIP3-rich phases coexist: the growth of the PIP2-rich phase “eats up” free PTEN molecules from the cytosol, decreasing until phase coexistence is reached Gamba et al. (2009a, b). This process may be understood via a simple physical analogy with the non-equilibrium process taking place during the liquid-vapor transition in a sealed vessel: there, the rise of the vapor pressure (which in our analogy corresponds to the number of cytosolic enzymes) provides a negative feedback, slowing down the growth of the vapor phase and eventually leading the system to a state of phase coexistence. The main difference between the two systems is that in the liquid-vapor transition a local conservation law holds for the particle field, while in the growth of signaling domains on the cell membrane the field satisfies only an approximate global constraint Gamba et al. (2009a) encoded in the integral expressions for the coefficients appearing in (8,9,10).
It is worth observing here that while spontaneous polarization in eukaryotic directional sensing Gamba et al. (2005) can be described in terms of an homogeneous nucleation process, whereby seeds of a PIP3-rich phase are created by thermal and chemical noise in the sea of the PIP2-rich phase and grow by a coarsening process Gamba et al. (2007, 2009a), the present picture of epithelial polarization reminds instead heterogeneous nucleation, i.e. a situation where the effective potential barrier is so high that spontaneous nucleation does not occur in typical observational times, and needs to be triggered by the introduction of a large enough nucleation germ.
We have shown that a simple symmetry breaking mechanism, informed with the recently discovered biochemical and biophysical details of the system, accounts for a wealth of morphogenetic processes in epithelial polarization. The model makes specific predictions on the dependence of the threshold radius, lumen size, tight junction positioning and width, on the biochemical system parameters. Our results shed light on the role of PTEN as a tumor suppressor protein, whose expression levels are known to be critical to prevent the onset of cancer. In particular, our models predicts that by decreasing the number of PTEN molecules, the lumen size should decrease, and for very low PTEN levels no lumen at all should form. The experimental validation of the model could be performed by genetic manipulation of the amounts or activity of cadherins, PI3K and PTEN.
We have also shown that curvature induced forces are a very plausible candidate for triggering the symmetry breaking process at the right time. This could be verified experimentally by trying to induce localized formation of a growing PIP2 patch and lumen by mechanically breaking adhesion bonds in localized regions of the membrane of epithelial cells surrounded by extracellular matrix. Under these conditions, our model predicts that only breaking adhesion bonds in regions larger than should induce the formation of a growing patch of the PIP2-rich phase, while smaller PIP2-rich patches, induced by breaking adhesive bonds on smaller regions, should shrink spontaneously.
Interestingly, the bistable PI3K-PTEN module here described plays also a key role in chemotaxis, where PI3K is initially activated by chemotactic receptors (see the review Iglesias and Devreotes (2008) and ref.s therein) rather than by adhesive receptors. While experimental evidences and our results suggest that epithelial polarization is induced by a nucleation center of the PIP2-rich phase generated by mechanical forces, the polarization of migrating cells is likely to be triggered by spontaneous fluctuations in PIP3/PI3K levels Gamba et al. (2007).
regulates chemotactic polarization Gamba et al. (2005, 2007) and
orientation Toyoshima et al. (2007).
The similarity underlying the mechanisms in these very different
aspects of cell life hints to the possibility that phase separation
phenomena might have a general role in the
cell Gamba et al. (2005, 2007) and in its nucleus Nicodemi and Prisco (2007); Nicodemi et al. (2008).
The principles emerging here could explain in a universal way the deep
analogies observed in a variety of cellular processes involving spatial
polarity formation Comer and Parent (2007).
We thank G. Boffetta for discussions, hospitality at ISAC-CNR and access to computational facilities, and S. Vegetti for suggestions. This work was partially supported by Telethon-Italy GGP04127, AIRC, MIUR (PRIN 2007BMZ8WA), Regione Piemonte, PRESTO, Fondazione CRT, Ministero della Salute.
- Comer and Parent (2007) F. Comer and C. Parent, Cell 128, 239 (2007).
- Martin-Belmonte et al. (2007) F. Martin-Belmonte, A. Gassama, A. Datta, W. Yu, U. Rescher, V. Gerke, and K. Mostov, Cell 128, 383 (2007).
- Wang et al. (1990) A. Z. Wang, G. K. Ojakian, and W. J. Nelson, J Cell Sci 95 ( Pt 1), 137 (1990).
- Bryant and Mostov (2008) D. M. Bryant and K. E. Mostov, Nat Rev Mol Cell Biol 9, 887 (2008).
- Mellman and Nelson (2008) I. Mellman and W. J. Nelson, Nat Rev Mol Cell Biol 9, 833 (2008).
- van Kampen (2007) N. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, 2007), 3rd ed.
- Ramanathan et al. (2005) S. Ramanathan, P. B. Detwiler, A. M. Sengupta, and B. I. Shraiman, Biophys J 88, 3063 (2005).
- Thattai and van Oudenaarden (2002) M. Thattai and A. van Oudenaarden, Biophys J 82, 2943 (2002).
- Paulsson et al. (2000) J. Paulsson, O. G. Berg, and M. Ehrenberg, Proc Natl Acad Sci U S A 97, 7148 (2000).
- Kölsch et al. (2008) V. Kölsch, P. G. Charest, and R. A. Firtel, J Cell Sci 121, 551 (2008).
- Watton and Downward (1999) S. J. Watton and J. Downward, Curr Biol 9, 433 (1999).
- Yap and Kovacs (2003) A. S. Yap and E. M. Kovacs, J Cell Biol 160, 11 (2003).
- Landau and Lifshitz (1980) L. Landau and E. Lifshitz, Course of Theoretical Physics, vol. 5 (Butterworth-Heinemann, 1980).
- Kroschewski (2004) R. Kroschewski, News Physiol Sci 19, 61 (2004).
- Landau and Lifshitz (1986) L. Landau and E. Lifshitz, Course of Theoretical Physics, vol. 7 (Butterworth-Heinemann, 1986).
- Fournier (2007) J. Fournier, Soft Matter 3, 883 (2007).
- Simson et al. (1998) R. Simson, E. Wallraff, J. Faix, J. Niewöhner, G. Gerisch, and E. Sackmann, Biophys J 74, 514 (1998).
- Ferraro et al. (2008) T. Ferraro, A. de Candia, A. Gamba, and A. Coniglio, Europh. Lett. 83, 50009 (2008).
- Gamba et al. (2007) A. Gamba, I. Kolokolov, V. Lebedev, and G. Ortenzi, Phys Rev Lett 99, 158101 (2007).
- Gamba et al. (2005) A. Gamba, A. de Candia, S. D. Talia, A. Coniglio, F. Bussolino, and G. Serini, Proc Natl Acad Sci U S A 102, 16927 (2005).
- Gillespie (1977) D. T. Gillespie, The Journal of Physical Chemistry 81, 2340 (1977).
- Turing (1952) A. M. Turing, Phil Trans R Soc Lond B 237, 37 (1952).
- Bray (1994) A. Bray, Adv. Phys. 43, 357 (1994).
- Schöll (2000) E. Schöll, Stochastic Processes in Physics, Chemistry, and Biology (Springer, 2000), pp. 437–451.
- Gamba et al. (2009a) A. Gamba, I. Kolokolov, V. Lebedev, and G. Ortenzi, Journal of Statistical Mechanics: Theory and Experiment 2009, P02019 (2009a).
- Gassama-Diagne et al. (2006) A. Gassama-Diagne, W. Yu, M. ter Beest, F. Martin-Belmonte, A. Kierbel, J. Engel, and K. Mostov, Nat Cell Biol 8, 963 (2006).
- Margolis and Borg (2005) B. Margolis and J.-P. Borg, J Cell Sci 118, 5157 (2005).
- Gamba et al. (2009b) A. Gamba, G. Naldi, M. Semplice, G. Serini, and A. Veglio, in preparation (2009b).
- Iglesias and Devreotes (2008) P. A. Iglesias and P. N. Devreotes, Curr Opin Cell Biol 20, 35 (2008).
- Toyoshima et al. (2007) F. Toyoshima, S. Matsumura, H. Morimoto, M. Mitsushima, and E. Nishida, Dev Cell 13, 796 (2007).
- Nicodemi and Prisco (2007) M. Nicodemi and A. Prisco, Phys. Rev. Lett. 98, 108104 (2007).
- Nicodemi et al. (2008) M. Nicodemi, B. Panning, and A. Prisco, Genetics 179, 717 (2008).
- Vanhaesebroeck et al. (2001) B. Vanhaesebroeck, S. J. Leevers, K. Ahmadi, J. Timms, R. Katso, P. C. Driscoll, R. Woscholski, P. J. Parker, and M. D. Waterfield, Annu Rev Biochem 70, 535 (2001).
- Carpenter et al. (1990) C. L. Carpenter, B. C. Duckworth, K. R. Auger, B. Cohen, B. S. Schaffhausen, and L. C. Cantley, J Biol Chem 265, 19704 (1990).
- Duguay et al. (2003) D. Duguay, R. A. Foty, and M. S. Steinberg, Dev Biol 253, 309 (2003).
- Shraiman (2005) B. I. Shraiman, Proc Natl Acad Sci U S A 102, 3318 (2005).