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* Institute of Applied Mathematics and Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada; and Center for Cell Dynamics, Friday Harbor Labs, University of Washington, Friday Harbor, Washington
Correspondence: Address reprint requests to A. T. Dawes, E-mail: atdawes{at}u.washington.edu.
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ABSTRACT |
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INTRODUCTION |
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). Cells can detect gradients as shallow as 1–2% (2–5), by transducing and amplifying external signals into clear, persistent intracellular maps. These internal maps correspond to differences in levels of enzymes, lipids, and proteins at an emergent front and back of the cell. Notable among the first to redistribute are the kinase PI3K (front), and phosphatase PTEN (back), followed by phosphoinositides, and small GTPases of the Rho protein family (1,6–8). The interactions and crosstalk of these signaling components at many levels form an important organizing principle, and the subject of this article. A key aspect of the downstream effect of these signaling components is their effect on the actin cytoskeleton. By regulating the initiation of new growth sites (i.e., by nucleation of new actin filament barbed ends), and by releasing inhibition of growth at some sites (i.e., by inhibition of barbed end capping), these regulatory pathways lead to the directional protrusion of the cytoskeleton, formation of a leading edge, and eventual initiation of cell motion (Fig. 1 b).
Aside from amplification of weak stimuli, chemotactic cells display a panoply of characteristic behaviors. The term "amplification" denotes the fact that internal gradients are macroscopic, with similar magnitudes in response to strong or weak stimuli (4,9). The term "adaptation" denotes the fact that some cells (e.g., Dictyostelium) return to rest after transient responses to spatially uniform chemoattractant distributions (see Fig. 3 in (10)). Other cells (e.g., neutrophils) randomly choose a direction and initiate directed motion (11,12). Normal motile cells move up gradients of attractants, but remain sensitive to new or changing stimuli from other directions (6,13). Finally, some cell types appear to initiate directed movement in the absence of spatial cues (14). In this article, we explore how detection of an external stimulus can lead to directed movement.
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). Some difficulties with the current state of knowledge is that these diagrams are difficult to comprehend, some key components are missing or not well-characterized, the plethora of those included sometimes obscures which are central and which peripheral, and the static diagrams cannot indicate how these components work together in real time. In this article, our aim is first, to investigate three important "modules" that act in concert to initiate cell motility, second, to understand their essential dynamical functions, and third, to show how they work together in real time to produce cellular polarization and initiate motility (both normally, and in knockout or mutant cells). By a "module", we mean a set of interrelated proteins or lipids that can be identified as a unit with specific dynamical functions (amplifying, switching, filtering, etc.). Based on our experience, we focus on the Rho family of small GTPases, the phosphoinositides, and the actin cytoskeleton, as shown in Fig. 1, a and b. We explore their components, crosstalk, and interactions based on the experimental literature.
Previous theoretical work has addressed cell polarization and motility phenomena. Some studies focus on putative activators, inhibitors, etc., with overall appropriate dynamics (15–18). Many rely on a local excitation/global inhibition (LEGI) module to produce specific behaviors such as gradient amplification and adaptation (9,19). Others have investigated signaling networks (20–25). Excellent recent reviews of both theoretical and experimental approaches can be found in the literature (26,27). We compare our model to others in Table 1 (listing components parts included and indicating behaviors that each model could account for). To our knowledge, ours is the first attempt to link together the above three biochemical modules in a model for the polarization and initiation of cell motility. In this work, our guiding principles have been to base assumptions, where possible, on what is known, to assume the simplest mechanisms where knowledge is lacking, and to explore hypotheses for parts of the system that are uncertain. We restrict attention to a one-dimensional "motile cell" (see Geometry of the Model) to enable us to understand the dynamical behavior in the simplest possible geometry, before attempting to move to more computationally challenging or intensive two- or three-dimensional versions (but see (28), for initial steps in simulating a two-dimensional moving cell).
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BIOLOGICAL BACKGROUND |
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Regulation of actin polymerization in motile cells
To initiate motility, cells reorganize their actin cytoskeleton into a thin protruding sheet (the lamellipod), with filaments oriented toward the membrane (29–31), impinging on and pushing the leading edge forward. In motile cells, active polymerization occurs almost exclusively at the leading edge (29,32–34). The extension of actin filaments (at their fast growing "barbed ends") is tightly regulated by many factors that nucleate, cap, and depolymerize them (35,36). Arp2/3 is essential for cell motility (37). It is activated close to the membrane by WASp or N-WASp (34,38,39), associates with, and causes, side-branching off actin filaments, thereby nucleating new growing barbed ends (40,41). Arp2/3 is incorporated into the actin network, and is recycled when filaments depolymerize, at the back of the lamellipod. The literature on theoretical approaches to actin cytoskeletal growth is extensive, and includes Mogilner and Oster (42,43), Carlsson (44), Mogilner and Edelstein-Keshet (45), Carlsson et al. (46), and Rubinstein et al. (47).
Rho proteins
The best-studied members of the Rho subfamily, Cdc42Hs, Rac1, and RhoA (henceforth Cdc42, Rac, and Rho) are expressed by many cell types (e.g., fibroblasts, neutrophils, neurons) and are crucial for cell motility (48–50), and organization of the actin cytoskeleton (51,52). In a resting cell, the Rho proteins are evenly distributed, but when a cell is stimulated by a spatially graded signal, active Rho proteins (bound to the membrane and GTP) reorganize into spatially graded distributions (Cdc42 and Rac high at the leading edge and Rho high at the rear) (13,53–56). Inactive (GDP bound) Rho proteins transit between the membrane and cytosol, where they diffuse more rapidly. Interconversion of these forms is accelerated by the activating guanine-nucleotide exchange factors (GEFs) and inactivating proteins (GAPs) (57–60). Rho family proteins interact with one another via crosstalk, although the detailed mechanism of that crosstalk is not yet known. It has been demonstrated in many cell types that Cdc42 activates Rac and Rac activates Rho (61–63). Whether mutual Rac-Rho inhibition (51,63–65) or mutual Cdc42-Rho inhibition (62,66) are the rule, is less clear. In theoretical work, Sakumura et al. (23) analyzed several variants. In our recent work (28,67,68) we adopted the crosstalk proposed by Giniger (66) (see also Fig. 1 a), with mutual inhibition of Cdc42 and Rho. We investigate the effect of inhibitory arrow 9 in Fig. 1 a with our numerical experiments in Cdc42 Spatially Excludes PTEN by Inhibiting Activation of Rho. A well-recognized actin-related role of the Rho family GTPases includes the following:
,69,70). In the models developed here, we incorporate Cdc42 activation of Arp2/3 (arrow 6 of Fig. 1 a) and investigate the role Cdc42 plays in gradient sensing and polarization.
Phosphoinositides
Excellent reviews of the structure, function, and interconversions of these membrane lipids are given in the literature (71–74). In this article, we focus on PI(4)P, PI(4,5)P, and PI(3,4,5)P (henceforth PIP1, PIP2, and PIP3, respectively). The kinases PI5K and PI3K add phosphates to the 5- and 3-positions, respectively, while the phosphatase PTEN removes phosphates from the 3-position. Together, these inter-convert phosphoinositides (PIs), as shown in Fig. 1 a. PI3K, PTEN, and PI5K can diffuse in the cytoplasm. When a cell is exposed to a chemoattractant gradient, PTEN is released from the membrane at the front of the cell, allowing PI3K to associate with the membrane. (PTEN remains bound to the sides and the back of the stimulated cell.) This spatial redistribution of PI3K and PTEN causes PIP3 to be elevated at the leading edge (75–77). PIs diffuse on the membrane at the same rate or faster than Rho proteins. We use the two biological facts:
,69). We include this effect in our model (arrow 7 in Fig. 1 a). ) and Rohatgi et al. (79), PIP2, (not Cdc42) is required to activate WASp and N-WASp, which then activate Arp2/3. In the presence of both PIP2 and Cdc42, activation of Arp2/3 is enhanced (80), suggesting that Cdc42 amplifies this effect (arrow 6 in Fig. 1 a).
Interconnection of the modules
It has been shown experimentally that PIP3 is required for the activation of Cdc42 and Rac (4,8,53,81). Indeed, using RNA interference, it was determined that PIP3 activates both Cdc42-specific (PIX
, (82)) and Rac-specific (P-REX1, (83)) GEFs, as well as shared GEFs such as Vav2 and Vav3 (84). Based on this, we explore the hypothesis that:
Evidence that Rac enhances the activity of the kinases PI5K and PI3K comes from several sources. In platelets and neutrophils, it was shown that Rac can directly activate PI5K (85,86). Observations of the fluorescent distribution of PIP3 and active Rac in neutrophils demonstrated that Rac (and not Cdc42) is required to enhance activity of PI3K (53). Experiments in neuron-like cells suggested that Cdc42 and Rac can interact with the GEFs Vav2, Vav3 to enhance PI3K activity (84), and further, in vitro, active Rac and Cdc42 can bind directly to PI3K (87–89). Based on this evidence, we assume that:
In epithelial cells and neutrophils, Rho activates Rho kinase, which directly binds and activates PTEN by phosphorylation (82,90). Based on this we assume that
Moreover, we show that this fact has important consequences for the spatial exclusion of PTEN from areas with high concentration of PI3K and Cdc42 (see Cdc42 Spatially Excludes PTEN by Inhibiting Activation of Rho).
We consider feedback from the actin cytoskeleton to its upstream signaling components. Blocking actin polymerization in motile cells results in a loss of asymmetry in the PIs, but not in the Rho proteins. When cells are exposed to the actin-sequestering latrunculin, and then to chemoattractant, their PIP3 level increases transiently, while Rac is persistently elevated at the leading edge (53). (Latrunculin treatment halts polymerization and prevents the cell from initiating movement.) Cells treated with jasplakinolide (which halts actin filament turnover), stop moving within 1 min and lose PIP3 at their membrane (4). This leads us to explore the hypothesis that:
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QUESTIONS WE ADDRESS |
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)? (c) What prevents PI gradients from dissipating by diffusion along the membrane, i.e., what preserves that gradient (91)? (d) If such internal gradients are persistent, how does a cell turn in response to a new distinctly oriented signal, i.e., what prevents the internal gradient from becoming "frozen" (15,92)? ,12,93)? ,4,5,12)? (b) Why do neutrophils exposed to a uniform concentration of chemoattractant polarize in a random direction and move (14,93)? )? ,53)? ,94)? ,91,95)? ,90)? )? We present our model in the next section and then describe detailed numerical experiments that shed light on these questions.
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MODEL OF PIs, RHO PROTEIN, AND ACTIN DYNAMICS |
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,68).) The key and important distinction between membrane-versus-cytosolic forms is the rate of transverse diffusion along the one-dimensional axis of the model, and this is preserved by appropriate choices of diffusion coefficients. With this geometry, neither membrane nor cytosolic compartments are homogeneous along the one-dimensional domain, but since diffusion is much faster in the cytoplasm, gradients along the domain are much shallower in the cytoplasm.
Basic scheme
As a general rule, we restrict variables to known biochemical entities, and avoid including hypothetical inhibitors or activators. Activation/inactivation of a given substance, X (in a well-mixed system), subject to the influence of Y are described by the basic scheme
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 | (1b) |
In Eq. 1a, the notation IX denotes a rate of activation (e.g., dependent on feedback from Y), whereas dX is the rate of inactivation (or turnover) of X. In Eq. 1b, we assume a linear activation, i.e., that a fractional increase of Y above some constant basal level, Yb, leads to a proportional increase in the activity of X. (As discussed later, we also tested Michaelis-Menten activation rates, and found qualitatively similar results.)
Spatial terms in Eq. 1a include diffusion and transport. The speed of the one-dimensional moving-cell depends on the cytoskeleton-mediated protrusion forces. We assume that both cytoplasmic and membrane-associated small molecules diffuse in this domain, and that they are also transported by a bulk flow, vbulk, when the cell edge protrudes forward.
Model of actin dynamics
The model of actin dynamics keeps track of actin filament density (F), density of growing barbed ends (B), and concentration of activated Arp2/3 (A) available to nucleate new barbed ends. Filaments are assumed to be essentially immobile due to crosslinking and attachment to adhesion sites (retrograde flow is here ignored). The growth of barbed ends deposits new filament density, and filaments turn over at some average constant rate, 
. The motion of the cell is modeled as protrusion-limited, with barbed ends pushing the membrane at the leading edge (x = xedge). We omit adhesion-contraction mechanics (but see (96)). We approximate the effect of barbed ends on protrusion speed at the leading edge, with a thermal ratchet relationship (42,43,97). In a previous article, we explored the relationship between Arp2/3-mediated branching of actin filaments and cell speed resulting from the formation of new barbed ends at the membrane (98). Here, we are concerned with the regulation of Arp2/3 activation that creates the surge of barbed ends in response to a stimulus.
Based on biological facts B1 and B3, Arp2/3 activation depends on Cdc42 (C) and PIP2 (P2). However, a fully linear relationship implies that even small elevations in PIP2 induce cell motion, which is unrealistic. Hence, we assumed that activation occurs once PIP2 exceeds some threshold,
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 | (3) |
Model of Rho protein dynamics
The Rho protein module forms the central component of the regulatory pathway in our model. Our treatment of this module is related to the previous work by Sakumura et al. (23) with the following essential differences: based on the experimental observations of spatial segregation of these proteins and their mutual crosstalk, we predict that this module is a switch between multiple equilibria (rather than an oscillator, as in (23)). We model crosstalk through GEF-mediated activation (rather than GAP-mediated inhibition): the actual mechanism is not yet known definitively. Inactive Rho proteins distribute to the cytosol where diffusion is much more rapid. This means that they can rapidly transmit "global information" (in the sense of (1,26,93), see also Discussion, this article). We use the simplest crosstalk scheme (66), to describe this module.
In our previous treatment of the Rho protein module (28,67,68), we showed that a minimal module that contains the essential features described above can be constructed from the following basic scheme: Crosstalk between the Rho proteins has to be of the type that allows for multiple coexisting steady states (i.e., high Cdc42 with low Rho and vice versa), as in the literature (23,66). Mutual inhibition between Cdc42 and Rho is a requirement of this scheme, and so is nonlinearity higher than Michaelian kinetics (i.e., some degree of cooperativity). These assumptions are essential for ensuring the existence of multiple steady states needed to account for observations. Finally, the rapidly diffusing inactive forms of the Rho proteins are important for stabilizing the polarized wavefront, and preventing one or another homogeneous steady state from sweeping through and taking over the whole domain (68). The basic scheme we arrived at has the form
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 | (4b) |
active, Gi = Ci, Ri, i, inactive forms of Cdc42, Rac, and Rho, respectively, and dG an inactivation rate. The activation rate, IG, is some function of the form
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. Based on assumed GEF-mediated crosstalk, QC decreases with Rho (), Q decreases with Cdc42 (C), QR increases with C, and Q increases with Rac (R). (Based on B4, QC and QR increase with PIP3 (P3), a feature not explored in our previous work.) See Appendix A for details.
The spatial terms in Eqs. 4 and 5 include advection and diffusion. This leads to six equations for the three active and three inactive forms of these proteins. The motivation for specific choices of activation functions and other details are described in the literature (28,67,68). As shown in these background articles, with these assumptions, the Rho module dynamics are consistent with multiple distinct equilibria for a large range of parameter values. The module accounts for spatial polarization in response to graded or noisy inputs (e.g., in IC), producing a stable polarized distribution. Furthermore, in the preliminary two-dimensional cell motility simulation (28), we have shown that, once polarized, this module retains sensitivity to stimuli along new directions.
Model of PI dynamics
We do not model the full temporal dynamics of PI3K, PTEN, etc. Rather, we use equations based on the general strategy of Eq. 1, and Rho protein effects (B5, B6) to formulate a quasi-steady-state (QSS) assumption for these. This leads to a general form for the level 
= PTEN, PI3K, PI5K of the kinase/phosphatase that is then incorporated into the PI dynamics. To simulate the observed spatial asymmetry of PI3K and PTEN in response to a spatially graded external signal, we impose a gradient in the parameter I (or k) across the one-dimensional domain (see Appendix A for details). Eqs. 13 tracks the dynamics of PIP1, PIP2, and PIP3 across the cell. These forms are interconverted, under the influences of PTEN, PI3K, and PI5K, mentioned above. Further, based on B5, Rac enhances the conversion of PIP1 to PIP2 (via PI5K) and the conversion of PIP2 to PIP3 via PI3K. Based on B6, Rho enhances the conversion of PIP3 to PIP2 via PTEN. We assume all PIs diffuse in the membrane at the same rate, DP, and undergo bulk convection as described above.
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ESTIMATING PARAMETERS AND SIMULATING THE MODEL |
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) and Schneider and Haugh (100). To deal with this discrepancy we studied the behavior of the model for a wide range (several orders of magnitude) of values of DP, as described in Robustness of the Model.
The full model equations (Eqs. 6, 8, 9, and 13 based on the pathways of Fig. 1 a) were simulated in a one-dimensional domain of length L, where x = xedge and x = xedge – L represent the two edges of the cell (eventually, the front and back, but initially not so specified). In some cases, we studied chemical polarization in the static domain xedge – L 
x xedge to investigate dynamics upstream of motility. Where motility was simulated, we allowed xedge(t) to evolve with time. Figures show some results in stationary lab-frame coordinates (e.g., Fig. 3), and others in a coordinate system moving with the cell (e.g., Fig. 4).
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Polarization was examined by plotting the distributions of signaling components and actin across the one-dimensional domain (e.g., Fig. 4). Motility was assayed by relating the number of actin filament barbed ends at x = xedge to protrusion speed in the positive x direction. (We did not simulate motility of left-moving cells.) Details of the simulations and of the protocol used to run specific tests are provided in Appendix C.
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RESULTS |
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The profiles of internal signaling components and actin cytoskeleton are shown in a coordinate system moving with the protruding membrane in Fig. 4. By this time (t = 100 s), the magnitudes of the steady-state spatial profiles are independent of the stimulus strength. PIP2 and PIP3 are highest at the front of the cell (left column, Fig. 4), consistent with observations (2,8,91). Note that the scales for phosphoinositides in Fig. 4 differ by two orders of magnitude: PIP3 occurs at very low levels relative to other forms (consistent with (9,71)). Cdc42, Rac, and Rho (center column, Fig. 4) also segregate spatially, with Cdc42 and Rac high at the front, and Rho high at the back. This is consistent with fluorescence imaging in live motile cells (13,54,55,101). Similar profiles of these proteins were obtained by us previously in Jilkine (67) and Jilkine et al. (68).
The profiles of actin filament density, barbed ends, and Arp2/3 are shown on the left column of the same figure. Arp2/3 is highest at the leading edge, followed by a peak of barbed ends and further back, a peak of actin filament density. Similar profiles that we previously obtained in an actin-only one-dimensional model (98) were compared to experimental profiles observed by (32,34,39). Related results for actin with Rho proteins were also discussed in Dawes (102) and in a two-dimensional motile cell (28).
As shown by this "control" simulation, the model produces appropriate one-dimensional spatial profiles of signaling components, and initiates protrusive motility in response to a strong graded stimulus. Thus, the interactions shown in Fig. 1 (the default model) suffice to account for basic phenomena to be explained (see Q1(a)). The persistence of polarization and motion is in agreement with the literature (4,12,93), and addresses question Q2.
Niggli (91) noted that the mechanism for formation of a steep gradient of PIP3 was uncertain, and that the localization of active Rac was also unclear. We can understand both phenomena from the scheme of Fig. 1: A weak stimulus that sets up a shallow gradient in the PIs feeds onto the Rho protein module, tripping the GTPase-switch and setting up a spatially segregated profile of these Rho proteins. The asymmetric gradient of the Rho proteins in turn feeds back onto the PI module, reinforcing and amplifying that PI gradient. When PIP3 is thereby concentrated at the front, it further reinforces and elevates the activity of Rac (arrow 2, Fig. 1). This set of interactions then maintains the asymmetric profile of the PIs after the stimulus is removed. This result addresses question Q1(b).
Before carrying out numerical experiments, we first investigated robustness of the default model to parameter values, kinetic terms, and types of stimuli. We afterwards modified the default model by one or another numerical experiments to investigate the effects of knockouts and mutants, or to understand the roles of specific pathway components or interactions.
Robustness of the model
The behavior described above constitutes a default behavior that was obtained in one realization of the model. We asked whether this behavior was robust to variations in the values of parameters, choices of kinetic terms, and applied stimuli.
We varied each kinetic parameter by 10% and found the same qualitative behavior (see Appendix B for further details). Many parameters could be varied on a much greater range. For example, we varied the rate of PI diffusion (Dp) in the range of 0.5–5 µm2/s and found no qualitative effect and little quantitative effect on the model. Thus the model cell is robust to parameter variations, with one exception: lowering the Hill coefficient in the Rho protein module leads to loss of bistability and polarization.
We explored whether assumed linearity versus nonlinearity of terms in the model creates artifacts in behavior. As previously discussed, nonlinearity in the Rho module is essential for the type of spatial bistability observed experimentally. Changing all linear terms in the PI and Rho protein equations to saturating (Michaelis-Menten) kinetics makes the slope of the internal steady-state gradients shallower, but does not qualitatively change the behavior of the model. In both linear and Michaelian cases, the model cell polarizes persistently in response to a graded stimulus.
The response of the model was also tested using stimuli superimposed on various baseline levels. (See protocol discussed in Appendix C.) We found that the steady-state response of the cell was identical. Based on these findings, we adopted the default model, parameter values, and stimuli as the basic protocol, from which further numerical experiments were done.
Weak stimuli cause a delay in initiation of motion
To determine the response of the model-cell and its sensitivity to weak graded stimuli, we varied the steepness, strength, and duration of graded stimuli (see protocol in Graded Stimuli). Stimuli applied for a short period of time (Fig. 3 b) resulted in directed motion, but only after some delay (a similar result was found for weak stimuli). A graded stimulus was applied for periods of time ranging from 0.01 s to the full length of the simulation. Membrane speed was computed as the simulation progressed, and plotted. The onset of motion is shown in Fig. 3 b. Stimuli applied for a short period of time (for example 0.1 s) cause directed motion but only after a time delay of 
15 s. Longer or stronger stimuli organized the polarization and motion more rapidly, and caused a slight overshoot in the membrane speed before steady-state motion was established. (See also the overshoot previously described in Fig. 3 a.) Similar results are seen for shallower (i.e., weaker) graded stimuli applied for the same period of time (not shown).
These simulation results suggest that the pathways of Fig. 1 a suffice to account for the cell's response to weak or short-lived stimuli, addressing question Q3(a). However, we also found that the weaker the stimuli, the longer the time lag. The delay stems from the time taken to assemble the internal map. Our model is constructed so that this occurs even for very weak or short-lived stimuli, but the weaker/shorter the stimulus, the longer it takes the cell to polarize. This delay in initiation of motion (for a reasonable range of stimulus strengths) is one of our testable predictions. In a real cell, there are likely further mechanisms (not included in our model) to prevent polarization in response to extremely weak stimuli.
Activation of Cdc42, not Rac, by PIP3 required for proper gradient sensing
As noted in B4, PIP3 activates both Cdc42 and Rac by enhancing the activity of specific GEFs (82–84). However, the relative importance or roles of these feedbacks has not previously been examined. We investigated this question (Q4) by selectively abolishing either arrows 1 or 2 in Fig. 1 a, and simulating the model as previously described.
We first cut only arrow 2 of Fig. 1 a, from PIP3 to Rac. We obtained proper spatial localization of PIs and Rho proteins, with high Cdc42, Rac, PIP2, and PIP3 at the front (Fig. 5, solid lines). Once the cell polarizes, it begins moving to the right (the leading edge is formed in the area of the cell with high Cdc42, Rac, PIP2, and PIP3).
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) that directional sensing is not obligatory for motility: mutants lacking the pathway represented by arrow 1 would still polarize and move, but may not correctly detect and align with the gradient. (This addresses aspects of question Q5.) The reverse polarization can be understood in the context of Fig. 1: in a control-cell, a graded signal in PIs activates Cdc42, which both elevates Rac and depresses Rho at the front. In the mutant-cell lacking arrow 1, Cdc42 is bypassed: the PIs activate Rac, which activates Rho at the front. Due to mutual (Cdc42-Rho) inhibition, Rho takes over the front, Cdc42 is relegated to the back, and the polarity is reversed. GTPase and PI segregation can still occur, but Cdc42 cannot establish dominance at the front. Thus, the cell could still develop a polarity and move, but may not detect the gradient correctly or move in the correct direction.
Finally, we cut both arrows 1 and 2. The graded stimulus was applied to Cdc42 by imposing a linear gradient in the activation rate, Ic (since gradients in PIs no longer affect the Rho proteins). In this case (results not shown), the cell takes longer to polarize and initiate movement and the steady-state speed is much slower compared to the full model. This stems from a lack of positive feedback from the PIs to the Rho proteins. The Rho module polarizes first, and then, through arrows 3, 4, and 5, leads to PI polarization. The polarization and movement of the model cell are persistent.
We can also understand other observations in the context of these results. According to the literature (91,95,103), Rac activation depends somewhat on activation of PI3K. Moreover, Wang et al. (4,53) noted that if PI3K were entirely inhibited, both cell polarity and cell motility would be blocked. Since PI3K activation creates PIP3, which, in turn, feeds forward to the Rho module, this observation makes sense, and addresses question Q6. It is consistent with arrows 1 and 2 in Fig. 1, both of which lead to enhanced activation of Rac.
Further, we can put our numerical results into the context of experiments demonstrating that cells lacking active Cdc42 are able to move but cannot properly detect the direction of the gradient or the source of the stimulus (53,94) (see Q7). Absence of Cdc42 is analogous to severing arrows 1 and 9 of Fig. 1 a. First, this destroys the spatial polarizability of the Rho protein module, which depends on the double negative feedbacks of Cdc42 and Rho, and second, it prevents the spatial PI asymmetry from properly biasing the Rho proteins. This means that an external gradient may lead to internal restructuring, but not to the correct corresponding internal gradient. In summary, these results indicate that PIP3 activation of Cdc42 is required for proper polarization, not PIP3 activation of Rac.
Feedback from Rac to PI5K or PI3K required to maintain PI asymmetry
It has been shown experimentally (see B5) that Rac directly interacts with and activates PI5K (85,110) and is important for PI3K activity (53,91,95). To explore the relative importance of Rac feedback onto PI5K and PI3K (see Q8), we conducted numerical tests in which arrows 3 and 4 in Fig. 1 a were cut (see protocol in Dissecting the Pathways). As above, a graded stimulus (100% for 10 s) was applied after 20 s and the final data was collected at 100 s.
We first removed both feedbacks. This resulted in complete loss of polarity in PIP3 (but not Rho proteins) as soon as the graded stimulus was removed. In this case, PIP2 did not build up sufficiently to activate Arp2/3, and not enough barbed ends were nucleated to initiate movement. As shown in Fig. 6, restoring either feedback (Rac activation of PI5K or PI3K) leads to an asymmetric distribution of PIP3. When Rac enhances only PI3K (i.e., cut arrow 3), there is no asymmetric distribution of PIP1 or PIP2, only of PIP3, while Rac activation of PI5K (i.e., cutting arrow 4) leads to asymmetric distributions of all three PIs.
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Cdc42 spatially excludes PTEN by inhibiting activation of Rho
While PI3K is essential for gradient sensing, the role of PTEN (and even its localization) is still controversial. In many motile cell types, PTEN establishes a spatial gradient reciprocal to that of PI3K. (However, there have been reports that PTEN is strictly cytosolic in migrating neutrophils; e.g., see (104).) It has been suggested that PTEN and active Cdc42 do not spatially colocalize (82,90), but the underlying reason for this has not been clear (see Q9). In this section we discuss a possible reason for this spatial exclusion. (Note, however, that the model cell is able to properly detect a gradient and initiate movement even when PTEN activity is constant everywhere on the domain; not shown.) In our default model runs, this phenomenon is observed indirectly: The activity of PTEN is assumed to depend positively on the activity of Rho. However, as previously discussed, mutual exclusion of Cdc42 and Rho is essential in our Rho protein module for spatial polarization to occur. (New evidence for an inhibitory role of Rho appears in Ohta et al. (65).) This inhibitory interaction then also implies mutual exclusion of Cdc42 and PTEN in the model predictions.
We simulated a variant of the model in which arrow 9 of Fig. 1 a was cut (i.e., omitting the inhibitory effect of Cdc42 in the dynamics of Rho). Using the usual protocol (see Dissecting the Pathways), we ran the model to its steady state and determined resulting profiles.
Whereas in simulations of the full model (Eqs. 8, 9, and 13), PTEN activity is low in areas where active Cdc42 is high, if Cdc42 does not inhibit Rho, the stable spatial asymmetry of the PIs and small G proteins is lost and PTEN activity is higher than baseline everywhere in the cell (not shown). These results suggest that the spatial exclusion of PTEN from areas with active Cdc42 may be due to the inhibition of Rho by Cdc42, addressing question Q9. Since Rho has been shown experimentally to activate PTEN (82,90), PTEN activity will be high only in those areas where active Rho is high and Cdc42 low.
This sheds light on experimental observations that Cdc42 activation plays a role in excluding PTEN from leukocyte protrusions (1,105). Ridley (1) postulated the existence of a "network of feedback loops" that conspire to produce and sustain motile cell polarity. We have shown here that the loops in Fig. 1 a suffice to explain this result, but that when loop 9 is cut, this spatial exclusion of PTEN is not observed. In a related comment, Niggli (91) remarked that it is not clear what mechanisms are responsible for retaining the gradient in PI molecules that forms in response to graded stimuli, i.e., what prevents these molecules from diffusing laterally along the membrane (see Q1(c)). While our model does not address the two- or three-dimensional localization, we have shown that the polarization of the Rho module in turn reinforces and maintains the polarity of the lipids, preventing the smearing of the PIs and loss of that internal gradient. Finally, these results are in concert with the statement in Kimmel and Parent (105) that PI3K-PTEN pathways are essential in chemotaxis, and that elevation of PIP3 in the front of the cell results from their reciprocal distributions. Our predictions here can be tested experimentally by microinjecting resting cells with active Cdc42 and observing the resulting spatial distribution of Cdc42, Rho, and PTEN.
Competing random stimuli cause the cell to polarize and initiate movement
We investigated the response of the one-dimensional model cell to a 10-s spatially irregular, but temporally fixed input, representing a superposition of competing random stimuli (protocol in Competing (Random) Stimuli).
Without PIs, Rho proteins can form multiple domains of high Cdc42
We first examined the model's response in the absence of the PIs (i.e., with only Rho proteins and actin). To do so, we modified the model so that PIP2 activation of Arp2/3 (arrow 6 in Fig. 1) is replaced by Cdc42 activation of Arp2/3, and PIP2 suppression of capping (arrow 7) is replaced by Rac inhibition of capping to preserve downstream effects on the cytoskeleton (as in (28)). We then ran the simulation with the stimulus applied directly to the Cdc42 activation rate, IC.
If the input is graded, no multiple peaks occur. Further, for weak random stimuli, transient multiple peaks coalesce into one, exactly as reported by Wong et al. (56). A strong random stimulus, however, leads to multiple stable domains with high Cdc42 and Rac, and reciprocal Rho as shown in Fig. 7. The profile of barbed ends, Arp2/3, and filament density reflect the spatial profile of the Rho proteins, i.e., produce a distribution of growing barbed ends that is inappropriate for efficient motility. In our one-dimensional model, this means that the cell slows down (speed not shown), since not as many barbed ends localize to the leading edge. In a more realistic two-dimensional version, it is easier to interpret multiple peaks of Cdc42 as multiple sites where nascent competing lamellipodia would form. Indeed, the appearance of multiple lamellipodia was noted by us in two-dimensional simulations of the motile cell in which PIs were not yet included (28). We can understand these results by noting that barbed ends do not persist outside areas with high levels of active Rac due to rapid capping. Barbed ends are nucleated in areas within the domain with high Cdc42 and Rac, but even if they grow toward the leading edge, they are mostly eliminated by capping before they reach the cell edge. This simulation suggests that in the absence of the PIs, a variety of putative Rho protein distributions can be manifested in response to competing or contradictory signals (see Q10).
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ABSTRACT
INTRODUCTION
and activation by 
. Arrows are based on the following references: 1, (82
