Coarse-grained computational framework for simulating cell polarization
We sought to enumerate all possible simple molecular regulatory networks that could yield polarization to better understand the design principles governing this fundamental self-organizing behavior (). In natural networks, core network motifs may be obscured by evolutionary history and pleiotropic function (Ma et al., 2006
Coarse-grained computational model for cell polarization
We developed a computational framework simple enough to enable efficient screening of a large number of network architectures, yet complex enough to represent the essential spatial behavior of polarization. We decided to explore the full space of two-node networks. The two nodes represent two molecular species – a “polarity marker” species whose distribution we measure, and a second, “regulatory” species that has the potential to alter the behavior of the polarizing species (). The full circuit space is defined by different combinations of regulatory links between the two nodes (either positive, negative, or no link), as well as self-regulation (positive, negative, or no feedback), yielding a total of 81 possible network architectures.
We implemented a coarse-grained model in which the plasma membrane is represented by a one-dimensional circular lattice (of size 100) surrounding a “cytosolic” pool of the two molecules (). Many examples of cell polarization involve localization of molecules to the plasma membrane. Thus, we defined reactions to be simple membrane-binding and dissociation reactions () and polarization as the asymmetric distribution of the polarity marker along the membrane.
We used a stochastic algorithm to simulate binding and dissociation events (Gillespie, 1977
). Thus, at lattice location i, a signaling component X bound to the membrane with rate kXbind
(i) or a membrane-bound component X dissociated into the cytosol with rate kXdissoc
(i). Regulatory interactions due to membrane-bound molecules at neighboring positions in the lattice act by modifying the binding and dissociation rates, kXbind
(i) and kXdissoc
(i) respectively. Lateral diffusion rates for each component within the membrane were also defined as parameters in the model (DX
, ). We treated the cytosol as a well-mixed reservoir and simulate only lateral diffusion on the membrane, assuming that membrane binding and dissociation occur on a much slower timescale than cytosolic diffusion.
In natural polarization circuits, local regulatory interactions between molecules can occur through mechanisms such as enzymatic reactions, physical recruitment, and cytoskeletal transport. We chose a generalizable, abstract representation of local regulatory interactions that modeled the overall effect of regulation rather than its implementation. The regulatory links in our model affected the local apparent membrane affinities of the signaling components by changing basal binding and dissociation reaction rates (see Extended Experimental Procedures for full equations). A positive regulatory link from node X to Y (RX→Y > 0) increased the binding rate of Y, kYbind(i), and decreased the dissociation rate of Y, kYdissoc(i), as a function of the local concentration of membrane-bound X at lattice location i. This effectively increased the local concentration of membrane-bound Y in the vicinity of membrane-bound X, leading to “positive feedback” when Y=X or “cross activation” when Y ≠ X. The reverse was true for negative, “inhibitory” links.
We enumerated all network topologies by combinatorially varying the possible regulatory links between two network nodes (). Note that a network topology only encodes the types of regulatory interactions between the nodes, whereas the magnitudes of these regulatory interactions are specified as parameters.
Scoring polarization and searching parameter space for robust networks
After each simulation reached steady state, a polarization score (P) was calculated as the normalized magnitude of the vector sum of each of the membrane-bound polarity markers (, see Extended Experimental Procedures). A non-polarized cell with a random distribution of membrane-bound polarity markers resulted in many randomly oriented vectors and thus, low P (, top). In a polarized cell with a cluster of membrane-bound polarity markers, many aligned vectors resulted in a high P (, bottom).
To test the robustness of each network topology, we sampled the performance of the network topology with 10,000 parameter sets (). Each network had at most eight associated parameters - the strengths of the four network regulatory links, the lateral membrane diffusion rates of the two molecular species, and the concentrations of the two molecular species. We scored each network topology by its polarization robustness, Q, the fraction of parameter sets that polarized with P > 0.6 (Dassow et al., 2000
). More robust topologies are most likely to emerge through a semi-random process such as evolution and are also likely the easiest targets for engineering of polarization networks.
Identification of three minimal network motifs for cell polarization
Within the complete set of 81 network topologies, 33 (or 41%) were able to polarize with a robustness of Q>0.0005 (). The performance of these 33 candidate topologies consistently stood out above background, even when we altered the basal conditions of the simulations (Figure S1A
). Further, to confirm that our parameter sampling was sufficient, we tested a subset of representative topologies using five times larger sample size (50,000 parameter sets) and found that robustness values did not change significantly (Figure S1B
Three minimal motifs drive robust self-organizing cell polarization
We hypothesized that some of the candidate topologies were degenerate, i.e. they contain regulatory links that are either extraneous or detrimental to performance. We compared the Q value of each network topology with the Q values of all its “ancestor” topologies consisting of one fewer regulatory links (Figure S2A
). If a network topology was more robust than all of its ancestors, then all of the regulatory links were indeed essential and we considered it to be a new core network topology. If a network topology failed to display increased robustness in comparison to at least one of its ancestors, then some of its regulatory links were unnecessary elaborations, and the topology was considered degenerate. After this analysis, only 8 out of the 33 candidate topologies were classified as core network topologies (), all of which achieve polarization over similar ranges of time scales depending on parameters (Figure S2B
Within the core network topologies, we observed three recurring minimal motifs: positive feedback on the polarizing molecule, either direct (topology 23) or indirect (topology 44); an inhibitor with positive feedback (topology 69); and mutual inhibition (topology 65). All of the eight core topologies that emerged from our analysis were found to contain one or more of these three minimal motifs (). Note that for simplicity, direct and indirect positive feedback are considered to be different implementations of the same minimal motif of positive feedback. But as subsequent analyses will show, these two implementations in fact display distinct robustness behaviors.
Minimal motifs only achieve polarization within constrained regions of parameter space; combination motifs are more robust
Since all of the identified core network topologies could be constructed using one or more of these three minimal motifs (), we asked if there were functional differences between minimal motifs and topologies that contained motif combinations. We explored how distinct topologies performed when biological parameters, such as component concentrations, diffusion constants, and regulation strengths, were varied.
We first investigated each topology’s robustness to variation in concentrations of signaling components. For each of the three minimal motifs, we performed finer sampling (100,000 parameter sets), binned parameter sets by concentrations, and calculated each motif’s robustness Q to variation in the remaining parameters (regulation strengths and diffusion constants). We visualized the robustness landscape as a heatmap in two-dimensional concentration space ().
Combining minimal motifs increases robustness to variation in component concentrations and lateral diffusion rates
The positive feedback minimal motif can achieve polarization, but only within a limited region of concentration space (). For direct positive feedback, polarization requires a limiting concentration of the polarizing species -- less than the number of total binding sites on the membrane. This observation is consistent with previous studies in which a network combining self-activation with limiting concentrations (a form of “local activation and global inhibition”) was capable of both polarization and pattern formation (Gierer and Meinhardt, 1972
; Meinhardt and Gierer, 2000
; Altschuler et al., 2008
). Intuitively, a network with positive feedback and an excess concentration of molecules fails to robustly polarize because the molecules simply promote each other’s binding, leading to symmetric saturation of the membrane. Our analysis also shows that indirect positive feedback networks (involving a secondary activator) are even more constrained in concentration space (). Because the feedback loop involves both of the network nodes, both red and blue molecules must be present in limiting concentrations to yield asymmetric membrane binding.
The inhibitor with positive feedback motif displays distinct robustness constraints (). The inhibitor must be present in limiting concentrations, and an excess concentration of the polarity marker is required. The inhibitor’s positive feedback combined with a limiting concentration effectively implements asymmetric clustering of the inhibitor on the membrane. The asymmetric spatial organization of the inhibitor then restricts the localization of the polarity marker, resulting in polarization.
The third minimal motif we observed was mutual inhibition between the two nodes. Although cross-antagonism between two components (a double negative feedback loop) can in some contexts be considered to be equivalent to positive feedback (Meinhardt and Gierer, 2000
), our results indicate that in this spatial context, this motif is actually highly distinct in behavior (). Specifically, mutual inhibition can drive polarization in the presence of excess concentrations of both signaling components, and limiting concentrations actually hinder its ability to self-organize polarization.
Thus, although each minimal motif can generate cell polarization, it is only able to do so in limited regions of concentration space. Combining motifs into more complex topologies, however, increases robustness to concentration variations. For example, coupling mutual inhibition to an inhibitor with positive feedback enables polarization in a larger fraction of concentration space () when compared to either of the minimal motifs alone. Combining all three motifs resulted in the most robust self-organizing polarization topology ().
A similar increased robustness of combinatorial networks was also observed with respect to variation of other parameters such as diffusion constants and regulatory strengths. In “diffusion space” where lateral diffusion rates of the molecular species were varied () and in “regulation space” where regulatory link strengths were varied (Figure S3
), combinatorial networks drove polarization with higher tolerance to parameter variations. In all three situations, a topology containing all three minimal motifs is the most robust ().
Engineering synthetic PIP3 polarization in yeast: Building new regulatory interactions using combinations of modular localization and catalytic domains
Our computational analysis defines the landscape of possible polarization circuits and provides a guide for the design of new polarization circuits. To test these findings, we attempted to construct synthetic polarization circuits and to systematically probe the in vivo requirements for spatial self-organization.
We chose the membrane-associated phospholipid species, phosphatidylinositol (3,4,5)-trisphosphate (PIP3
) as a novel polarity marker in S. cerevisiae
(). Although PIP3
is an important polarization marker in higher eukaryotic cells, PIP3
is not normally present in budding yeast (Dove et al., 1997
; Rodríguez-Escudero et al., 2005
). The total amount of PIP3
in yeast can be controlled by expressing lipid kinases and phosphatases from higher eukaryotes – PI3 kinase, which converts PIP2
, and PTEN, the lipid phosphatase that converts PIP3
(Rodríguez-Escudero et al., 2005
). Thus variants of these enzymes could be engineered to act as specific regulatory links controlling PIP3
generation and degradation, if we could find a way to spatially target their activities. The spatial distribution of PIP3
can also be easily tracked with an in vivo
reporter (the PH domain of Akt).
Construction of a synthetic polarization system using PIP3, a phospholipid not normally found in S. cerevisiae
To build specific spatially-controlled links in PIP3
regulatory circuits, we fused the catalytic domains from PI3K or PTEN to different localization domains (, and S4
). For example, to generate a positive feedback regulatory link for PIP3
, we fused a PIP3
-binding domain (PHAkt
) to the PI3K catalytic domain (p110α). This fusion protein should produce more PIP3
at a location that already has PIP3
(i.e. “IF PIP3
, THEN make more PIP3
”). Using this strategy, fusion proteins made from combinations of localization and catalytic domains can be used to create diverse spatially-controlled regulatory links in a PIP3
A fusion protein will only function as a conditional regulatory link if its catalytic function depends on its localization. We demonstrated that the PI3K catalytic domain (p110α) does not produce PIP3
at the membrane unless properly localized. Expression of the catalytic domain alone does not lead to membrane localization of a PIP3
-2xGFP (). However, when PI3K is targeted to the plasma membrane (where PIP2
is present), PIP3
is produced. Similar localization dependent function is observed for the PTEN phosphatase catalytic domain (Figure S4
To create a second regulatory node to serve as an inhibitor of the polarizing molecule, we utilized the endogenous GTPase protein Cdc42 as an opposing landmark. Cdc42, which exists in two distinct states (one active GTP-bound and one inactive GDP-bound), is an ideal inhibiting regulatory node. Active Cdc42 localizes to the membrane in a highly polarized manner via a combination of catalytic- and actin transport-mediated positive feedback () (Kozubowski et al., 2008
). We created synthetic mutually inhibitory links between activated Cdc42 (Cdc42*) and PIP3
as follows. By fusing a Cdc42* binding domain (from the protein Gic2) to the PTEN phosphatase domain, we encoded the regulatory link: “IF Cdc42* THEN dephosphorylate PIP3
”. Conversely, by fusing a PIP3
binding domain (PHAkt
) to a Cdc42 GAP domain (inactivates Cdc42*) from the protein Rga1, we could encode the following regulatory link: “IF PIP3
, THEN inactivate Cdc42*”.
We generated a set of chimeric enzymes corresponding to the regulatory links in the coarse-grained model (). By selectively transforming combinations of enzymes from this set of links into yeast, we re-created a subset of the topologies identified through our computational analysis. We could alter the strength of individual links in the circuit by altering the strength of the constitutive promoter used to express each synthetic fusion protein.
To detect the distribution of PIP3
in cells containing these synthetic circuits, we expressed a reporter protein consisting of a fluorescent protein fused to a domain that binds PIP3
-2xGFP). To simplify the automated analysis of polarization, we pretreated the cells with the drug Latrunculin A (LatA), which disrupts actin polymerization, thus preventing budding, resulting in cells which maintained their shape over a longer period of time (hours). LatA also disrupts the actin transport based positive feedback on Cdc42* (Ayscough et al., 1997
) but leaves the Bem1/Cdc24 protein-based positive feedback loop intact (Ziman et al., 1993
; Wedlich-Soldner et al., 2003
; Kozubowski et al., 2008
We classified cells bearing these synthetic circuits into three phenotypes (): 1) no visible PIP3
(and therefore no PIP3
pole), 2) PIP3
observed throughout the plasma membrane (no pole), or 3) a concentrated region of PIP3
pole). We used a simple metric for the polarity score: the ratio of the maximum to the mean cell-edge intensity minus one (see Figure S4B
for more detail). The fraction of PHAkt
-GFP recruited to the membrane (indicating PIP3
production) was also calculated and termed the “production score.” Cells with a low average production score (<0.3) were classified as “no PIP3
.” Those cells with a higher production score were divided into “polarized” (maximum observed polarity score>0.5) and “PIP3
Synthetic circuit that includes positive feedback and mutual inhibition can generate artificial PIP3 poles in living cells
Our model predicted the combination of all three minimal motifs to be the most robust to variations in component concentration and diffusion rates. We implemented this three-motif combination circuit by expressing all of the components of our enzymatic toolkit in one strain: PIP3 positive feedback (PI3K-PHAkt), dephosphorylation of PIP3 in response to Cdc42* (CRIBGic2-PTEN), and deactivation of Cdc42* in response to PIP3 (PHAkt-GAPRga1). This circuit combined synthetic positive feedback on PIP3 with mutual inhibition between PIP3 and Cdc42* (Cdc42* also has native positive feedback regulation). This combinatorial circuit was expected to be the most robustly performing network and thus the easiest network to implement without parameter fine-tuning.
As predicted, many (65%) of the cells expressing these three synthetic signaling proteins exhibit strong PIP3
poles (). Two-dimensional time-lapse images ( and S5
) show that the poles are relatively stable, lasting for tens of minutes. A three-dimensional reconstruction of one of these synthetic PIP3
poles is shown in Movie S1
and . Analysis of cells that have not been treated with LatA shows PIP3
poles are located roughly opposed to the bud, where Cdc42* is localized ( and S6A
). More detailed three-dimensional reconstructions of confocal time courses show that PIP3
polarization is, overall, highly persistent. In some cases, stable PIP3
polarization is observed for more than 45 minutes (Movie S2A
; 50 minutes in Movie S2D
). The PIP3
poles, however, are also dynamic, appearing/disappearing or dividing/fusing on the minute timescale as well as moving rapidly throughout the plasma membrane (Movie S2
). Thus, two-dimensional time analysis probably underestimates pole persistence, as poles move out of the plane of focus.
Experimental design of synthetic PIP3 polarization networks
For comparison, we also constructed an identical circuit lacking the mutual inhibitory links (PIP3
positive feedback only). To assess the frequency of polarization in both types of circuits, we analyzed ~70 cells for each circuit and measured their polarity scores. The cells containing the three-motif circuit showed a distribution with high polarity scores; cells containing only the PIP3
positive feedback motif were all clustered at low polarity scores (). Cells with positive feedback alone occasionally had weak poles (polarity score≥0.5), but cells with the full circuit had stronger poles with a much higher frequency. Only 5% of cells with positive feedback alone exhibited poles, with most having PIP3
everywhere on the plasma membrane. In cells with the engineered three-motif circuit, 65% of cells had PIP3
poles, many of which were much stronger than any seen with positive feedback alone. In addition, polarization in the three-motif circuit persisted longer than polarization in cells expressing positive feedback alone (Figure S6B
Requirements for synthetic polarization: Analysis of circuit variants underscores the importance of combinatorial motifs
We explored the circuit requirements for PIP3 polarization, perturbing individual circuit links in our designed networks by either omitting them or altering the expression levels of the equivalent fusion protein. As described in the previous section, cells expressing only the PIP3 positive feedback loop (PI3K-PHAkt) from a medium strength promoter (pCyc1) did not show significant polarization. However, our model suggested that this class of circuit could generate polarization but would only do so in limiting concentration regimes (). A small amount of PI3K-PHAkt could create an initial quantity of PIP3 stochastically, as the enzyme encounters its substrate PI(4,5)P2 incidentally (without recruitment). This initial PIP3 would be amplified by positive feedback as PI3K-PHAkt is recruited and would phosphorylate nearby lipids. This positive feedback loop could generate transient PIP3 polarization. If the concentration of the positive feedback node (PI3K-PHAkt) is too high, however, PIP3 is likely to overtake the membrane symmetrically. We varied the expression level of the PIP3 positive feedback fusion protein alone using different constitutively active promoters (). In this designed circuit, PIP3 and the positive feedback fusion protein (PI3K-PHAkt) can be considered to be a single virtual node, where the concentration of the fusion protein has the potential to be limiting.
Network topologies with motif combinations produce PIP3 polarization more frequently, demonstrating their increased robustness
We observed that cells with lowest expression of the PIP3
positive feedback regulatory link alone (weak promoter: pIno4s) showed higher frequencies of polarization; at the lowest concentration of positive feedback, one in four cells polarized (). As the expression level of PI3K-PHAkt
increased, fewer cells displayed polarization and instead showed more uniform PIP3
over their plasma membrane. Thus, the positive feedback only circuit can yield polarization but is highly sensitive to enzyme concentration, consistent with theoretical predictions () and previous work (Altschuler et al., 2008
). In kinetic experiments, when positive feedback alone (pGal10-PI3K-PHAkt
expression) is rapidly induced, PIP3
polarization can be transiently observed before PIP3
becomes distributed throughout the plasma membrane (Figure S6C
Next we dissected which interactions allowed the full three-motif circuit () to produce robust PIP3
poles. We constructed a series of circuit variants in which individual regulatory link proteins were omitted and determined the percentage of PIP3
-polarizing cells. Removal of the PIP3
positive feedback link resulted in loss of polarization (). Conversely, induction of PIP3
positive feedback in cells constitutively expressing the mutual inhibition regulatory links resulted in the rapid induction of strong polarization (Figure S6D
). Moreover, expression of the mutual inhibitory links with a version of PI3K that is either cytoplasmically or plasma membrane targeted (but not positive feedback regulated) also failed to yield polarization ( and S7B
). Thus, despite the intrinsically strong polarization of Cdc42*, the mutual inhibition circuit between PIP3
and Cdc42* alone is not sufficient to confer robust PIP3
polarization ( and S7B
Based on this link deletion analysis, positive feedback on PIP3 and cross-inhibition from Cdc42* to PIP3 seem to be the most critical links in the network, consistent with our computational results. The PIP3 to Cdc42* inhibitory link appears less critical than predicted by modeling, perhaps because Cdc42* is endogenously a strong pole which may not require the additional spatial sharpening of this inhibitory link. The combinatorial two- or three-motif circuit that balances self-propagation with competition appears to be a circuit that is optimized for robust performance, rather than an overly complex solution to a simple biological problem.