On the spontaneous emergence of cell polarity

Steven J Altschuler; Sigurd B Angenent; Yanqin Wang; Lani F Wu

doi:10.1038/nature07119

. Author manuscript; available in PMC: 2009 Mar 14.

Published in final edited form as: Nature. 2008 Aug 14;454(7206):886–889. doi: 10.1038/nature07119

On the spontaneous emergence of cell polarity

Steven J Altschuler ¹, Sigurd B Angenent ², Yanqin Wang ¹, Lani F Wu ¹

PMCID: PMC2562338 NIHMSID: NIHMS63370 PMID: 18704086

Abstract

Diverse cell polarity networks require positive feedback for locally amplifying distributions of signalling molecules at the plasma membrane¹. Additional mechanisms, such as directed transport² or coupled inhibitors³^,⁴, have been proposed to be required for reinforcing a unique axis of polarity. Here, we analyze a simple model of positive feedback, with strong analogy to the classical “stepping stone” model of population genetics⁵, in which a single species of diffusible, membrane-bound signalling molecules can self-recruit from a cytoplasmic pool. We identify an intrinsic stochastic mechanism through which positive feedback alone is sufficient to account for the spontaneous establishment of a single site of polarity. We find that the polarization frequency has an inverse dependence on the number of signalling molecules: the frequency of polarization decreases as the number of molecules becomes large. Experimental observation of polarizing Cdc42 in budding yeast is consistent with this prediction. Our work suggests that positive feedback can work alone or with additional mechanisms to create robust cell polarity.

Many cell types can spontaneously establish and maintain asymmetric distributions of signalling molecules on the plasma membrane¹^,⁶, even in the apparent absence of pre-existing cues⁷^,⁸. Positive feedback circuits, found at the core of diverse biological networks, enable signalling molecules localized at the plasma membrane to initiate processes that further accelerate localized recruitment. These processes allow signalling molecules, such as Cdc42 in budding yeast⁹, mPar3/mPar6 in axons of hippocampal neurons¹⁰, and Rac in canine kidney cells¹¹ and human chemotaxing neutrophils¹², to be concentrated within a defined region of the plasma membrane.

Spontaneous cortical polarization can occur through activation of positive feedback circuits that do or do not depend on mechanisms of directed transport⁸^,¹³^,¹⁴. Transport-dependent circuits make use of cytoskeletal structure to reinforce spatial asymmetries by directing signalling molecules to specified locations on the plasma membrane²^,¹⁵^,¹⁶. However, in the absence of directed transport, it is unclear how positive feedback circuits counteract lateral diffusion to establish and maintain an asymmetric distribution of signalling molecules⁸^,¹⁴. Additional mechanisms have been proposed, such as coupled activation-dependent inhibition³^,⁴^,¹⁶ or depletion of critical signalling components³. Because evidence for these other mechanisms is less generally established, we wondered whether positive feedback alone could generate cell polarization.

The specific details of any given positive feedback circuit can be complex and cell-type specific. For example, signalling molecules can associate to the plasma membrane through docking with transmembrane receptors, binding to phospholipid headgroups, or insertion of prenylated membrane anchors. Additionally, signalling molecules such as GTPases can assume multiple states, which in turn strongly affect properties of their transport. To discover general principles, as with other recent models of polarity¹⁵^,¹⁷^-¹⁹, we abstracted mechanisms common to these circuits and constructed a biologically motivated, yet mathematically tractable model of positive feedback.

Thus, we formulated a model of positive feedback in which a single species of signalling molecules move between non-recruiting, cytoplasmic states, and recruiting, plasma membrane-bound states without mechanisms of directed transport (this model is referred to simply as “positive feedback” in the remainder of the paper). The redistribution of signalling molecules is determined by the rates of four fundamental transport mechanisms (Figure 1; Methods): 1) recruitment (k_fb) of cytoplasmic molecules to the locations of membrane-bound signalling molecules; 2) spontaneous association (k_on) of cytoplasmic molecules to random locations on the plasma membrane; 3) lateral diffusion (D) of molecules along the membrane; and 4) random disassociation (k_off) of signalling molecules from the membrane. The model parameters k_fb, k_on, D, k_off, and total number of signalling molecules N, are readily biologically interpretable, and may be estimated from experimental data (Methods).

A conceptual model of a positive feedback circuit is characterized by five biologically interpretable parameters. Colored arrows indicate four main sources of molecular transport (see Main Text; purple: spontaneous membrane association, or “input” to positive feedback circuit, with rate *k_on*; black: spontaneous membrane disassociation, with rate *k_off*; red: positive feedback, with rate *k_fb*; blue: lateral diffusion, with rate D). Not shown is N, the total number of signalling molecules.

In an initial analysis of the dynamics of this positive feedback circuit, we ignored the spatial distribution of molecules on the membrane. We found that the fraction of total signalling molecules on the membrane can switch from zero to a final equilibrium h_eq with a half time of ∼1/(h_eqk_fb) (Supplemental Materials). This equilibrium fraction h_eq can be estimated simply as 1-k_off/k_fb in the case when k_on is smaller than k_fb (see Supplementary Materials for general formulation), and reflects a balance between the net fluxes of molecules to and from the membrane. Thus, we find that the strength of feedback strongly influences both the switching dynamics and the magnitude of the equilibrium state.

We next considered how signalling molecules are distributed spatially on the plasma membrane. In the case when the number of signalling molecules is large, analysis reveals that all spatial variation eventually vanishes; membrane densities converge exponentially to a spatially homogeneous equilibrium state with membrane fraction h_eq (Methods). Further, perturbations of this state due to spontaneous association events quickly return to equilibrium (Figure 2a; Methods). Thus, no polarity can persist, regardless of the feedback strength k_fb. This suggests that for large numbers of molecules, positive feedback requires additional biological mechanisms to overcome diffusion in order to maintain a persistent switch-like distribution in space. This phenomenon has been studied rigorously for other positive feedback systems, including many motivated by theoretical ecology²⁰.

Numerical simulations reveal an inverse dependence between polarization frequency and large numbers of signalling molecules.

a. Positive feedback alone leads to homogeneous steady-state solutions for large numbers of signalling molecules (Methods). Inset: Evolution of total membrane fraction h(t); theoretical equilibrium value *h_eq* = 0.1.

b. Decreasing numbers of signalling molecules leads to increasing levels of spatial segregation. Shown are kymographs from simulations (vertical-axis: membrane position; horizontal-axis: time). Particle density is rescaled to units of equilibrium fraction (color bar).

c. Small *k_on*/*k_fb* can lead to the establishment of a dominant clan (Text Box 1). Clan color assignment in kymograph is random, and is reset for each molecule after membrane disassociation.

Model parameters are as in Methods.

In contrast, for small numbers of molecules, unexpected model behavior emerges as stochastic fluctuations can drive cellular behaviors from stable equilibrium to dynamic, non-stable states²¹^-²⁴. As the number of signalling molecules decreases, the expected equilibrium fraction of molecules on the membrane remains h_eq, with fluctuations on the order of 1/√N. However, we observed that with decreased numbers of signalling molecules, distinct polarized regions began to emerge spontaneously (Figure 2b). Additionally, we observed that this behavior is strongly affected by the ratio k_on/k_fb of spontaneous membrane association to feedback (Figures 2c, S1). When this ratio is large, frequent association of molecules to random membrane locations results in a spatially homogeneous density (Figure 2c, first and last 10 minutes). When this ratio is small, spontaneous association events are rare and local amplification by positive feedback frequently results in the emergence of spatially isolated clusters of signalling molecules (Figure 2c, middle 40 minutes). We note that simulations were performed using h_eq = 0.1 (Methods), hence the observed polarization behaviour is not due to depletion of the cytosolic pool (Supplementary Materials)³. Thus, these observations predict that decreased numbers of signalling molecules, decreased spontaneous membrane association rate, or increased positive feedback strength can lead to the spontaneous emergence of polarity (Text box 1).

Text Box 1. Intuition for particle clustering.

Intuition for the clustering behavior of positive feedback circuits for small numbers of molecules can be obtained through the simple concept of lineage tracking (Figure 2c). The basic idea is to track the location and source of recruitment for all signalling molecules on the membrane. Signalling molecules that spontaneously associate with the membrane become the founders of new “clans.” Molecules recruited to the membrane join the clans of their recruiters. Molecules that leave the membrane to re-enter the cytoplasmic pool lose their clan identity. A clan that has lost all its members becomes extinct.

Initially each clan has population size one, but over time some clans grow while others become extinct. Hence the number of clans on the membrane decreases between occurrences of spontaneous on-events. At equilibrium the expected total number of signalling molecules at the membrane remains a (non-zero) constant. If the time between spontaneous on-events is sufficiently long, then the number of clans may decrease so much that all molecules on the membrane belong to a single remaining clan. For small diffusion rates, all molecules on the membrane will be located near their most recent common ancestor, resulting in the formation of a single, high-density cluster. (Interestingly, our work bears close resemblance to the well-studied “stepping stone” model⁵ introduced by Kimura to study spatially heterogeneous theories of population genetics and dynamics. Clans in the stepping stone model can be more literally interpreted as collections of individuals sharing a common genetic trait by descent; new mutations create new clans, genetic drift increases or decreases clan sizes, and migration disperses clan members among neighbouring colonies.)

How long does it typically take for a single cluster to form at equilibrium? For relatively small membrane association rates we can deduce that the expected number of clans after waiting any time t is ≈ h_eq·N/(k_offt) (Supplemental Materials). Thus, after a time interval of t≈h_eq·N/k_off, a single clan is expected if no further spontaneous membrane association occurs. Simulations indicate, however, that a single large clan can form even if a small number of spontaneous associations occur; most new clans rapidly become extinct while the dominant clan is forming. What becomes of local cluster formation when diffusion eventually spreads apart the members of a dominant clan? At any given time, lineage tracking can be restarted with all original clan identity “erased” and each membrane-bound molecule regarded as a new clan founder. Then, our analysis shows that after a short time a dominant clan will again emerge. Thus, positive feedback alone, operating on a small number of particles, has the capacity to recurrently generate polarity.

Mathematical analysis of the positive feedback system shows in fact that transitions among clustering behaviors are entirely determined by a simple relationship between the ratio k_on/k_fb and the number of signalling molecules N: a single cluster will certainly form for small k_on/k_fb ≪ N^-2 (Figure 3a, region 1), while no clusters will form for large k_on/k_fb ≫ (N^-1ln N)^1/2 (Figure 3a, region 2). Numerical simulations indicate that the transition occurs for k_on/k_fb ∼ N^-1 (Figure 3a, region 3). Cluster formation is possible for large numbers of molecules, but the larger the value of N, the smaller the value of k_on/k_fb for which clustering actually occurs. This phase plane portrait reveals that polarization is a robust behavior that can be tuned by a small number of parameters.

Dependence of polarization on model parameters.

a. Illustration of phase-plane portrait. Theory reveals robust parameter regimes for cluster formation (Supplemental Materials): 1-polarization; 2-no polarization; 3-transition zone; 4-too few signalling molecules for polarization (Figure S2).

b. Frequency of observing Cdc42 polarization in yeast for increasing numbers of molecules (N). Shown are polarization frequencies estimated from numerical simulations (Top panel) and experimental observation of yeast cells expressing GFP-Cdc42 (Middle panel) (Methods and Figure S3). Green error bars represent standard error from four independent replicates. Bottom panel: Examples of polarized and unpolarized yeast cells. Arrows point to polarized regions; scale bar 1.9 μm.

A clear prediction of this theory is that the probability of spontaneous polarization decreases as the number of signalling molecules becomes large (Figure 3a, follow any horizontal line from left to right). To test whether this trend could be observed experimentally, we made use of an established budding yeast system in which an actin-independent core positive feedback circuit has been shown sufficient to establish cue-independent, spontaneous polarization of the GTPase Cdc42⁸^,⁹^,¹⁴. Mechanistic details of this positive feedback circuit are not well understood; however, several lines of evidence suggest its relevance as an experimental model for our current study: 1) self-recruitment of Cdc42 at the plasma membrane (via its interactions with the guanine nucleotide exchange factor (GEF) Cdc24 and the adaptor protein Bem1) does not require directed transport⁸^,⁹^,¹⁴^,²⁵^,²⁶; 2) polarization is maintained in a state of dynamic equilibrium, with all components rapidly exchanging between the plasma membrane and a significant cytoplasmic pool⁸; and 3) sites of polarization have been observed occasionally to drift and flicker, in phenomenological agreement with our model simulations⁸.

This positive feedback circuit for Cdc42 can be grossly described by the four fluxes in our model. Model parameter values can be estimated from experimental data (Methods), and reflect the combined effects of presumed unmodelled mechanistic interactions, such as required for self-recruitment (e.g. via Bem1and Cdc24⁸^,⁹^,¹⁴) and reversible membrane association (e.g. via guanine nucleotide disassociation inhibitor (GDI)⁸^,²⁷^-²⁹). For varying numbers of signalling molecules, probabilities of polarization can be computed using numerical simulation. As expected from the phase plane diagram (Figure 3a), the probability of spontaneous polarization decreases as the number of signalling molecules becomes large (Figure 3b, top histogram, N > 1000). We note that as the number of signalling molecules becomes very small the overall polarization percentage drops sharply (Fig. 3a region 4, Fig 3b top histogram, N < 500), reflecting more frequent loss of polarity due to spontaneous disassociation of all membrane-bound particles and longer waiting times for re-establishment (Figure S2, Supplementary Materials).

We next assayed polarization of GFP-tagged Cdc42 in yeast cells treated with the drug latrunculin A³⁰ to inactivate an alternative actin-dependent feedback circuit²^,⁸^,¹⁴^,¹⁵ (Methods). To quantify the effects of varying N, cells were binned into different categories based on background-subtracted, total GFP-Cdc42 intensities (Figure S3, Methods). Within each category we computed the fraction of polarized cells (Figure 3b). In general agreement with the model prediction, we observed a significant decrease in the frequency of polarization with increasing Cdc42-GFP expression (Figure 3b, bottom histogram). Notably, this inverse correlation is opposite to what was modelled for transport-dependent positive feedback of constitutively activated Cdc42 in budding yeast. In the case of actomyocin-based feedback, increased concentrations of constitutively activated Cdc42 led to increased frequency of establishing and maintaining distinct sites of cortical polarization²^,¹⁵. Thus, with respect to increasing, large numbers of signalling molecules, models suggest opposite trends in the frequency of polarization for positive feedback circuits that do and do not make use of directed feedback.

The intrinsic ability of positive feedback to polarize spontaneously without requiring additional mechanisms is not apparent from classic deterministic models of pattern formation³^,⁴ (Supplemental Materials, summary). Our work suggests that positive feedback can act alone at the core of complex signalling networks with small numbers of molecules to create and maintain a highly localized cortical distribution required for activation of downstream biological processes.

Methods Summary

Mathematical models

Deterministic and stochastic formulations of the model are given in Methods and Supplemental Materials.

Numerical simulations

Simulations were performed using parameter values estimated from published experiments⁸^,¹⁵ (Methods): D ∼ 1.2 μm²/min; N ∼10³; h_eq ∼ 0.1; k_off ∼ 9 min^-1; k_fb ∼ 10 min^-1; and k_on as indicated in each figure. Numerical simulations were carried out on 1D circles.

Experimental assay

The presence of GFP-Cdc42 polar caps were assayed in yeast as previously described⁸^,¹⁵ (Figure S3). GFP-Cdc42 intensity scores for each cell region were computed using Image J, and probabilities of cap formation computed using custom Matlab software.

Supplementary Material

Supp Mat

NIHMS63370-supplement-Supp_Mat.pdf^{(4.3MB, pdf)}

Acknowledgments

We thank Martin Altschuler, Alex Artyukhin, Tom Kurtz, Claudia Neuhauser, Mike Rosen, Rama Ranganathan, Boris Shraiman, Gürol Süel, and Orion Weiner for their positive feedback. We additionally thank P. Crews for latrunculin A and Rong Li for the yeast strain. This research was supported by an NIH grant (RO1 GM071794), an NSF grant (DMS 0405084), the Endowed Scholars program at UT Southwestern Medical Center, and the Welch Foundation (I-1619, I-1644).

Appendix

Methods

Deterministic formulation of model

The fraction h(t) of all molecules on the membrane changes in time t according to the ordinary differential equation (ODE):

\frac{d h}{d t} = k_{o n} (1 - h) + k_{f b} (1 - h) h - k_{o f f} h .

The density of membrane-bound molecules u(x,t) evolves in time according to the partial differential equation (PDE):

\frac{\partial u}{\partial t} = \frac{1}{2} D \nabla^{2} u + k_{o n} (1 - h) + k_{f b} (1 - h) u - k_{o f f} u,

where ∇² is the diffusion operator on the plasma membrane.

Stochastic formulation of model

In the stochastic model we consider N molecules in a cell, n(t) of which are on the membrane at any given time. The molecules on the membrane have positions x₁(t), ..., x_n_(t)(t), which undergo Brownian motion with diffusion coefficient D. The number of molecules on the membrane can also change randomly in three different ways: 1. A molecule on the membrane may dissociate itself from the membrane (rate constant k_off). 2. A molecule on the membrane may “recruit” some molecule in the cytosol to its location on the membrane (rate constant k_fb). 3. A molecule from the cytosol may spontaneously associate itself to some random location on the membrane (rate constant k_on).

The probability p_n(t) of n molecules appearing on the membrane at time t changes according to the master equation:

\frac{d p_{n} (t)}{d t} = W_{n} - W_{n - 1} where W_{n} = k_{o f f} (n + 1) p_{n + 1} - (k_{o n} + k_{f b} \frac{n}{N}) (N - n) p_{n}

while the numbern(t) of molecules on the membrane evolves by the Markov process determined by the master equation. See the Supplemental Materials for a more detailed description of the stochastic model.

Numerical simulations

While for the theoretical analysis the membrane may be a 1D curve or a 2D surface of arbitrary geometry, numerical simulations were carried out on 1D circles for ease of visualization. Simulation results shown in figures are based on model parameter ranges estimated from experimental measurements of polarized Cdc42 in budding yeast ⁸,¹⁵. For a yeast cell of radius R ∼ 5μm, the plasma membrane diffusion constant of Cdc42 was estimated to be D = 0.001·(2πR)²/min ∼ 1.2 μm²/min (=5·10^-10cm²/sec). We assumed total Cdc42 numbers on the order of N ∼10³ with plasma membrane fraction h_eq ∼ 0.1. FRAP analysis of half-time recovery for polarized Cdc42 (as well as Cdc24, and Bem1) are on the order of ∼ 5 sec ⁸; thus we estimated a off-rate of k_off = (ln 2)/T_1/2 ∼ 9 min^-1. A feedback strength of k_fb = 10 min^-1 was used to obtain an equilibrium plasma membrane fraction h_eq = (1-k_off/k_fb) ∼ 0.1. On-rate values k_on were varied as indicated in each figure legend. Clustering is predicted to occur for k_on ≪ k_fb (h_eq(1-h_eq)N²)^-1 ∼ 10^-4 min^-1, though simulations indicate that clustering can even occur for k_on less than 10^-3 (Figure 3B, Supplemental Materials). Polarization in simulations was determined by whether an interval of 10% of the membrane contained more than 50% of total membrane content; error bars represent standard errors from 4 replicates of 1000 independent simulations (Figure 3B, top panel).

Release assay

Cells from yeast strain RLY1948 (a gift from Dr. Rong Li, Stowers Institute, Kansas City, Kansas), which contain inducible GFP-Cdc42 expression under the Gal1/10 promoter, were grown logarithmically in synthetic complete (SC) medium without methionine⁸,¹⁵. To arrest cells in G1, cells were washed 3 times with autoclaved water, resuspended into YP medium supplemented with 2 mM methionine and 2% Raffinose and distributed into 4 tubes. Cells were then cultured for 150min (tube 1), 135min (tube 2), 120min (tube 3) and 90min (tube 4). To induce expression of GFP-Cdc42, 2% galactose was added to each tube and cells grown for additional 30min (tube 1), 45min (tube 2), 60min (tube 3) and (tube 4). To release from G1 arrest, cells were washed 3 times with water and resuspended in methionine-free SC medium containing 2% glucose plus 150 μM Latrunculin A (LatA). The released cells were seeded in glass bottom microwell dishes (MatTek, Ashlan, MA) and coated with Concanavalin A for 10 min before imaging.

Microscopy

Live cell microscopy was performed on a Nikon epifluorescence microscope (Nikon TE-2000E2) equipped with a 100 Plan Apo objective, a cooled CCD camera (model COOLSNAP HQ; Photometrics Instruments) and a temperature control box set at 30 °C. Image acquisition was performed with Metamorph software (Universal Imaging Corp.) with 1×1 binning. Images were taken in one focal plane at different locations in the dishes. For each condition, images were taken during a 30min period.

Image analysis

Six images from each condition were selected for analysis yielding 145, 243, 211, and 228 individual cells respectively. Using ImageJ, we manually identified cell regions and computed a background-subtracted, averaged GFP-Cdc42 intensity score. The presence of a Cdc42 polar cap in each cell was assessed (Figure S3), and probabilities of cap formation computed using custom Matlab software.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Supp Mat

NIHMS63370-supplement-Supp_Mat.pdf^{(4.3MB, pdf)}

[R1] 1.Drubin DG, Nelson WJ. Cell. 1996;84(3):335. doi: 10.1016/s0092-8674(00)81278-7. [DOI] [PubMed] [Google Scholar]

[R2] 2.Wedlich-Soldner R, Altschuler S, Wu L, et al. Science. 2003;299(5610):1231. doi: 10.1126/science.1080944. [DOI] [PubMed] [Google Scholar]

[R3] 3.Gierer A, Meinhardt H. Kybernetik. 1972;12(1):30. doi: 10.1007/BF00289234. [DOI] [PubMed] [Google Scholar]

[R4] 4.Turing AM. Philosophical Transactions of the Royal Society (part B) 1953;237(12):37. [Google Scholar]

[R5] 5.Kimura M, Weiss GH. Genetics. 1964;49(4):561. doi: 10.1093/genetics/49.4.561. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R6] 6.Ebersbach G, Jacobs-Wagner C. Trends Microbiol. 2007 [Google Scholar]

[R7] 7.Sohrmann M, Peter M. Trends in cell biology. 2003;13(10):526. doi: 10.1016/j.tcb.2003.08.006. [DOI] [PubMed] [Google Scholar]

[R8] 8.Wedlich-Soldner R, Wai SC, Schmidt T, et al. J Cell Biol. 2004;166(6):889. doi: 10.1083/jcb.200405061. [DOI] [PMC free article] [PubMed] [Google Scholar]

[R9] 9.Butty AC, Perrinjaquet N, Petit A, et al. The EMBO journal. 2002;21(7):1565. doi: 10.1093/emboj/21.7.1565. [DOI] [PMC free article] [PubMed] [Google Scholar]

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[R14] 14.Irazoqui JE, Gladfelter AS, Lew DJ. Nature cell biology. 2003;5(12):1062. doi: 10.1038/ncb1068. [DOI] [PubMed] [Google Scholar]

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PERMALINK

On the spontaneous emergence of cell polarity

Steven J Altschuler

Sigurd B Angenent

Yanqin Wang

Lani F Wu

Abstract

Figure 1.

Figure 2.

Text Box 1. Intuition for particle clustering.

Figure 3.

Methods Summary

Mathematical models

Numerical simulations

Experimental assay

Supplementary Material

Acknowledgments

Appendix

Methods

Deterministic formulation of model

Stochastic formulation of model

Numerical simulations

Release assay

Microscopy

Image analysis

Bibliography

Associated Data

Supplementary Materials

ACTIONS

PERMALINK

RESOURCES

Cite

Add to Collections

PERMALINK

On the spontaneous emergence of cell polarity

Steven J Altschuler

Sigurd B Angenent

Yanqin Wang

Lani F Wu

Abstract

Figure 1.

Figure 2.

Text Box 1. Intuition for particle clustering.

Figure 3.

Methods Summary

Mathematical models

Numerical simulations

Experimental assay

Supplementary Material

Acknowledgments

Appendix

Methods

Deterministic formulation of model

Stochastic formulation of model

Numerical simulations

Release assay

Microscopy

Image analysis

Bibliography

Associated Data

Supplementary Materials

ACTIONS

PERMALINK

RESOURCES

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