INTRODUCTION
There is much interest in achieving a quantitative understanding of the underlying signal transduction networks that regulate biological processes, an area referred to as "Systems Biology." Eukaryotic chemotaxis-the directed migration of eukaryotic cells in response to gradients of external chemicals-is an excellent candidate for such a detailed theoretical and experimental study for various reasons. First, it incorporates basic biological elements such as motility and the sensing and response to external signals. second, it is implicated in many physiological processes including wound healing, tumor metastasis, and development.
Experimentally, single-cell chemotaxis is studied in a diverse number of cells (1-3). Because chemotaxis is a complex process, many experimental and modeling efforts have focused on the first stage of this process, known as gradient sensing: the process by which cells convert information about the chemical concentration of their surroundings into an internal signal to guide their motion (4-10). In many systems, the gradient sensing process is functional in cells where the actin cytoskeleton has been impaired (2,11,12). Valuable insights into gradient sensing have been obtained by modeling the process in this simpler setting (13,14). Other phenomenological efforts focus on the description of random motility, biased random walk and persistent chemotaxis (15).
Additional features need to be understood to make a transition from gradient sensing to chemotaxis and cell migration, including a detailed study of the interaction of gradient sensing with the actin cytoskeleton leading to motility. Here, we focus on a crucial feature of chemotaxing cells: polarity. Though it has long been recognized that cell polarity is of critical importance in migration (16), there are few systematic experimental or theoretical studies of cell polarity in this context. In fact, the term "cell polarity" is used to mean somewhat different things by different researchers. We understand cell polarity to be the localization of various signaling components to opposite ends of the cell in a persistent fashion, with any attendant morphological change (4,5). This definition of polarity is appropriate for our purpose, because the formation of a persistent front and back is of vital importance for efficient chemotaxis. Polarity as a phenomenon, that is, the establishment of an axis, is a topic that cuts across many biological processes, and is of relevance in development, cell growth and division besides cell migration (17,18).
In chemotaxing cells, polarization occurs at a timescale of one to a few minutes after exposure to an external chemoattractant gradient. However, cells can also become polarized without an externally imposed gradient (19,20). For instance, neutrophils polarize (in an apparently random direction) when stimulated by a spatially homogeneous dose of chemoattractant. Dictyostelium cells become highly polarized in the course of their development (5). Thus, in these systems, there is an intrinsic capability for polarization that does not need an externally imposed gradient Different assumptions regarding how this intrinsic polarization may be exploited for chemotaxis have been made, either explicitly or implicitly (9,21).
There are many questions regarding this polarization process: how does it come about in the absence of externally imposed chemical gradients? How, if at all, does the cell exploit this capability to polarize, when subject to a chemical gradient? What are the factors that control the intrinsic cell polarity and how do receptor signals interact with them? Are there any differences in the timescales of these processes? What are the roles of nonlinear dynamic phenomena and self-organization in these processes?
Here, we develop a modeling framework to address various aspects of the polarization process, including the interaction between intrinsic polarization, such as the one induced during development in Dictyostelitim cells, and that driven by an externally applied chemoattractant gradient. We employ simplified models, motivated by experimental data on various pathways involved in gradient sensing and motility. We study the implications of various scenarios regarding lhe interaction. This analysis provides us with insight into how external signals and intrinsic processes may interact, and makes various testable predictions that do not require full knowledge of the biochemical entities involved. It also brings into focus the possible role of nonlinear dynamic transitions in the polarization process.
The modeling framework
We develop a modeling framework to investigate the relationship between intrinsic and externally induced polarity; see Fig. 1 A. We consider several possibilities for the underlying mechanisms in a way thai suggests experiments that can distinguish between them, thus enabling us to study polarity from a systematic perspective and focus on several key questions.
A polarized chemotaxing Dictyostelium cell has different components localized near either end. For instance, P13K, PI(3,4,5)P^sub 3^, PI(3,4)P^sub 2^, F-actin, PAK1, and Rac are found preferentially near the front, whereas PTEN, Rho, ACA, and myosin-II are found near the rear of the cell (21-23). The G-protein coupled receptors are present mostly uniformly along the cell surface, though they apparently exhibit a slightly nonuniform distribution in strongly polarized cells (24,25). We focus on biochemical entities that are early in the polarization pathways, many of which are also involved in gradient sensing.
Much of the recent biochemical focus related to gradient sensing has been on phosphoinositide lipids (PI(3,4,5)P^sub 3^ and PI(3,4)P^sub 2^) and the enzymes (PI3K and PTEN) that regulate them (26-28). Further regulation, related directly or indirectly to the actin cytoskeleton, contributes to a sharp localization of these lipids at the leading edge (12). The amplified production of these lipids relative to unstimulated cells arising from the dual contribution of P13K and PTEN has been previously modeled (13,29). A model describing this dual regulation and relating it to adaptation to homogeneous stimulation has been previously presented (14).
To study the interaction of externally induced and intrinsic polarity, we employ qualitatively simplified, rather than detailed biochemical models, because many biochemical details that are most pertinent to this problem are not known (e.g., details regarding receptor regulation of P13K. PTEN, regulation of G^sub α^ and G^sub βγ^ proteins). The simplified models are motivated by known aspects of the biochemical pathways.
Our model includes signaling pathways corresponding to both receptor-mediated and intrinsic polarity: these regulate common biochemical components leading to motility. We first discuss various elements of the model and then present the underlying variables and equations.
Receptor-mediated signaling
Our modeling framework incorporates two types of receptorregulated pathways: global and local (Fig. 1 B). Local pathways are those where the extent of regulation of downstream components depends on the degree of local occupancy of receptors. In contrast, global pathways offer downstream regulation, dependent on receptor occupancy averaged over the cell periphery and involve components that are highly diffusible. Both local and global pathways can regulate downstream processes in either excitory or inhibitory capacities. The combination of local and global pathways is able to account for gradient sensing and polarization consistent with adaptation to spatially homogeneous signals (30).
Intrinsic polarity
How polarity arises independently of externally imposed gradients remains unclear. For example, in Dictyostelium cells, a change occurs during the developmental process (5). Cells that are 4 h in the developmental process are, at best, weakly polarized: they do not have a clear and persistent separation of front and back. Though these cells are motile, they extend pseudopods in more-or-less random directions. In contrast, cells 7 h into development have sharp and persistent localization of various components al both ends of the cell.
The transition from unpolarized to a strongly polarized state may result from some kind of symmetry-breaking (31). It is possible to build models that give rise to a nonuniform steady svate, representative of a persistent front and back, based on symmetry breaking (32-35). However, the symmetry breaking may not necessarily be induced by, or related to the chemotactic pathway. Moreover, there are different ways in which symmetry might break to give rise to polarity cues. It is possible that landmarks for polarity cues are established earlier in development (spatial symmetry breaking), and the polarity pathways are activated by another temporal signal later in development, as in Saccharomyces cerevisiae (20). To keep our approach as general as possible, we assume that the intrinsic polarity pathways are controlled by signals with concentration profiles that are localized at the front (F) and back (B). Below, we will consider how these signals may interact with receptor-mediated signaling.
Model domain
Because most of the important reactions regulating chemotaxis take place on the cell membrane and cell cortex, we assume that all the elements of our model reside there. The spatial coordinate, s, corresponds to the arc-length of the membrane (see Fig. 1 B), We do not make use of cytosolic dynamics explicitly; these can also be incorporated in a simple manner by including an additional compartment that has transport to and from the membrane. Periodic boundary conditions are invoked. Though polarization involves a change of shape from an essentially circular cell to a more elongated cell (see Appendix A), we assume that this shape change is a downstream effect of the polarization of the various signaling components; possible feedback effects of shape change on the concentration profiles of the components will not be dealt with here. When we employ global entities, we assume that diffusion is through the cell membrane. This provides essentially the same kind of global regulation as the case where diffusion is cytosolic, and is sufficient for our purpose. In general, the time taken to recruit a species from the cytosol to the cell membrane through diffusion or any other transport limitations can be accounted for by a suitable choice of the corresponding rate constants.
One-dimensional gradients
As chemotaxis and polarization occur in response to relatively simple concentration fields, such as those with variation in mainly one direction, we concentrate on this case in detail. We assume a two-dimensional circular cell in the plane subject to a concentration field varying only in one direction. This one-dimensional variation allows us to address issues relating to the motility without having to consider the complexity associated with motility in higher dimensions. In this case, the cell either remains stationary or moves to either the left or right, depending on the net signal from the motility apparatus (described below). For cells whose shapes are surfaces of revolution, if the angular coordinate of an element of the surface is θ, the symmetry of the cell shape and external concentration field implies that, for the receptor occupancy R, R(θ) - R(-θ) for-π ≤ θ ≤ π. The symmetry condition implies that dR/dθ - 0 at θ = 0, π; equivalently, dR/ds - 0 at the front and back of the cell.
Turning mechanism
The directional regulator signal, S, is related to the receptor signal by a local excitation, global inhibition mechanism (30); see Eq. 3 below. This mechanism determines a condition for the cell to turn that incorporates contributions from both the local concentration at the front, and the spatial average of the concentration around the cell. In our onedimensional model, we make the cell reverse orientation instantaneously (i.e., R(s) is replaced by R(1/2 - s)) if the signal T* at the current "front" of the cell falls below a threshold value, 7^sub cr^. The T* value is reset to its basal level whenever the cell turns. This mechanism allows for a cell to change direction abruptly, without necessarily rearranging its intrinsic polarity.
Thus, a cell moving in the direction opposite the direction of a strong enough gradient will eventually change its direction either by creating a new pseudopod at the current "back" of the cell (reorganization of polarity), or by turning abruptly based on this criterion. The rate constants k^sub t^ and k^sub -t^ quantify the sensitivity and tendency of the cell to turn: a cell may take either a relatively long (small k^sub t^, k^sub -t^ relative to other rate constants) or short (large k^sub t^ and k^sub -t^) time to decide to turn sharply. It is also possible to choose parameters so that the tendency of the cell to turn sharply in reasonable gradients is suppressed (this is done by simply choosing a low enough value for T^sub cr^), in which case the only option for a cell to change direction is to reorganize its polarity. Thus, the incorporation of this module allows the model to exhibit a variety of responses to changing spatial and/or temporal signals, by variation of parameters.
Model variables and equations
All equations are nondimensionalized using appropriate time (1 s), spatial (5
As our main focus in this study is on the relation between intrinsic and receptor-mediated polarization pathways, we use simplified models most relevant to this investigation. We omit a number of other features that may be relevant in actual cells, including static non linearities and thresholds in the pathways, oscillatory effects involved in cell motility signaling, a realistic description of the pseudopod, as well as shape change.
RESULTS
We first consider the response of the receptor-mediated pathway under the assumption that the cell is not intrinsically polarized (Eqs. 1-11 above, corresponding to α = β = 0 in the equations below; see also Fig, 2). The steady-state response, which is a function only of the external concentration field, depends on the relative gradient as a consequence of signaling through a combination of local and global pathways (12,30,39). The frontness components are above their basal values in the front, and below them at the back; the opposite holds for the backness signals.
We also consider the effect of the intrinsic polarity pathway (Eqs. 12-15) when no external gradient is present (see Fig. 3). The intrinsic polarity components (IF*, IB*) become increasingly pronounced with the progression of intrinsic polarity (Fig. 3 A) and this affects all the downstream components (see Fig. 3, B and C).
We now examine different scenarios describing the interaction between the receptor-regulated and intrinsic pathways.
Parallel regulation by receptor and intrinsic cues
The parameters α and β describe the effect of intrinsic polarity on the frontness and backness components. The relative values of these parameters and those of the receptorcontrolled pathways (k^sub p^, k^sub p^, etc.) determine the relative strengths of the two pathways in establishing a steady response; higher values denote a stronger contribution of the intrinsic polarity.
We first assume that cell is motionless. However, unlike cells that are immobilized through actin inhibitors (12,40), we assume that all signaling components are intact. We consider a scenario where this intact cell is anchored to the surface. This anchoring could be achieved by engineering the microenvironment of cultured cells causing them to adhere to the surface (41). In our model, an anchored cell implies that it is subject to a static gradient and that it is incapable of turning (sharply); that is, the turning mechanism has no effect on the cell.
Whenever the external gradient is coaligned with the intrinsic polarity, the net result is that the external signal reinforces the intrinsic polarity and leads to stronger polarization than is possible from either (see Fig. 4, B and C). Note that for a fixed external gradient, the contribution of the receptor-controlled pathways need not necessarily be dominant. When an external gradient is imposed in a direction opposite to the intrinsic polarity, the two pathways work against each other (Fig. 4, D and E). The counteractive effects of the receplor controlled and intrinsic pathways are particularly acute when the external gradient cannot overcome the intrinsic polarity, which happens when the intrinsic polarity is strong (Fig. 4 E). In this case, the resulting steadystate profiles of F^sub 1^* and B^sub 2^* indicate a polarity opposite to the direction of the external gradient. An analytical description of parallel regulation by receptor-mediated and intrinsic cues is presented in Appendix A.
We now assume that the cell is not anchored to the surface, but is free to move and turn. This movement is assumed to be sufficiently slow so that the cell experiences an essentially static signal in the timescales of interest. Also, note that if the external gradient is in the same direction as the intrinsic polarity, the turning signal is not triggered and the effect is purely additive, so that the results are identical to those of the anchored cell (Fig. 4, B and C).
Different results arise, however, when the direction of the applied gradient is opposite to the cell's intrinsic polarity. If the intrinsic polarity is weak, then it is counteracted by the external gradient leading to net polarization in the direction of the applied gradient. Eventually, a stronger protrusive force develops in the rear-F*^sub m^ is greatest at s = ± 1/2 in Fig. 4 D-causing the cell to change its direction of motion by the gradual reorganization of polarity.
If the cell's intrinsic polarity is sufficiently strong, then the receptor-mediated pathways regulate the downstream pathways and attempt to reorganize its polarity. In this case, the turning mechanism induces the cell to turn sharply (flip); see Appendix A. This has the effect of aligning the intrinsic polarity with the external gradient. At this point, the net effect is reinforcement of the intrinsic polarity by the external gradient because, after turning, the profile is the same as when the external gradient's direction coincided with the intrinsic polarity (Fig. 4 C). This is true even though, initially, the external gradient was opposite to the intrinsic polarity.
For intermediate levels of intrinsic cell polarity, which effect-turning or protrusion at the old rear-dominates will depend on the relative timescales for meeting the turning threshold and for developing strong net protrusive force (correlated with concentration of F^sub m^*) at the back of the cell. Clearly, an increase in the intrinsic polarity works in favor of turning, as opposed to the reorganization of polarity, because the external signal has to counteract the existing inhibitory effect at the old rear of the cell. For a sufficiently polarized cell in a weak opposing gradient, turning sharply is its only option for chemotaxis. On the other hand, if the capability to turn is removed (as in an anchored cell), then lhe cell relies entirely on the reorganization of polarity by the gradient to change direction. In this case, a purely one-dimensional gradient of insufficient strength to counteract the intrinsic polarity will not be able to elicit a change in direction.
Inhibition of intrinsic pathways by receptor pathways
In various contexts in polarity generation, it is assumed that external signals inhibit intrinsic cues (17,20). In our setting, we examine the possibility that receptor-mediated pathways inhibit intrinsic pathways while imposing their own contribution to downstream signaling components,
Direct local inhibition
From the expressions for IB* and IF* above, we note that if k^sub ib^, k^sub if^ » k^sub c^, k^sub -c^, and k^sub e^ and k^sub -e^, then the equations for R^sub Fin^ and R^sub Bin^ enforce a state of weak intrinsic polarity. This is true under both the homogeneous stimulation (b - 0), irrespective of strength (in which case R^sub Fin^ and R^sub Bin^ do not depend on a), and in the gradients. Thus, for such a complex inhibition mechanism to work in gradients, we require that k^sub ib^ and k^sub if^ be of the same order as k^sub c^, k^sub -c^, k^sub e^, and k^sub -e^.
Because of the tendency to suppress the intrinsic polarity pathways, this inhibitory mechanism would tend to reduce, or delay the tendency of a polarized cell to turn sharply in this case, when compared to the case of no inhibition. Of course, it remains to be seen if such an inhibitory mechanism exists at all. Therefore, rather than vary the effect of such an inhibition, experiments that could point to the presence of such an inhibition would prove more useful. These effects could be detected by modifying receptor-controlled pathways of downstream components, such as PI3K. Thus, inhibiting the last component in the PI3K recruitment pathway starting from the receptor (which is not part of the intrinsic pathway), and subjecting a cell with some degree of intrinsic polarization to a gradient, would give useful information. If there was genuinely an inhibitory mechanism as described above, a gradient would affect the PI3K localization (which results from intrinsic pathways), even though the direct pathway leading from the receptor has been disrupted. Such experiments, however, require more biochemical knowledge regarding the receptor regulation of PI3K than is currently known.
In conclusion, it is certainly possible that inhibitory pathways emanating from the receptor could impact upon intrinsic pathways. However, the presence or absence of such pathways may not be easily directly discerned. Inhibitory pathways that are either local or global (but not both) could be detected by experiments that focus on the effect of a homogeneous stimulus on intrinsic polarity components. However, if the inhibition were to involve local and global pathways, and itself adapt to homogeneous inputs, the only way of proving its presence would be to perform experiments in gradients, by comparing regular cells and cells where the direct receptor controlled pathway was disrupted.
CONCLUSIONS AND DISCUSSION
Much of the recent attention in eukaryotic chemotaxis at the single-cell level has been on the process of gradient sensing. To connect gradient sensing to chemotaxis and migration in these cells, a number of important issues need to be addressed. Among these is a thorough understanding of the relation and interaction between gradient sensing and cell polarization, and the role of the actin cytoskelelon therein.
We have focused on some core issues related to polarity, the process by which different signaling components localize at opposite ends of the cell persistently, along with any attendant morphological change (5,16,26,42-44). This definition includes the ability of the cell to polarize in the absence of externally imposed gradients. We described a modeling framework to analyze how this inherent polarity can be reconciled with that induced by externally imposed chemoattractant gradients.
One example of an intrinsic polarization process is that which occurs during Dictyostelium development: cells 7 h into the developmental cycle are strongly polarized (36,45). Our modeling framework deals with different ways in which such an intrinsic polarization process may be exploited by the chemotactic pathways. Thus, we formulated our framework to deal with the relationship between the intrinsic and chemotactically induced polarity processes, and addressed different questions regarding their interaction. While our modeling framework was constructed in a specific setting, its insights are relevant to other systems/situations involving competing polarity mechanisms.
Our modeling framework includes only the most important aspects of signaling pathways and information flow in the actual system, relevant to the issues at hand. This allowed us to deal with the main questions of interest and relate some core hypotheses to implications in a transparent manner. Our framework was qualitatively simplified rather than detailed biochemical since many relevant biochemical details, such as the dynamics and regulation of the Gα and the Gβγ proteins, as well as the regulation of Raps, PI3K, and PTEN by the receptor, are still under experimental investigation. We also did not incorporate change in morphology, feedforward nonlinearities and thresholds in signaling, a realistic description of a pseudopod, or oscillatory effects in motility signaling. These aspects will be dealt with in future studies.
The model was formulated on a membrane of a twodimensional representation of a cell exposed to a one-dimensional external gradient that is able to induce chemotaxis and polarization. This restriction allows us to treat motility in a simple manner. Different aspects of motility in the one-dimensional setting have been previously studied (46,47). We also incorporated, in a phenomenologic al manner, a mechanism describing the cells' ability to turn sharply when faced with a changing and/or unsuitable gradient-a behavior that has been observed in both strongly polarized neutrophils and Dictyostelium cells (36). This was accomplished by the incorporation of a "turning module." Different parameters in this module allow for the possibility of sharp turning, and also for its abolition. We modeled the initiation and progression of intrinsic polarity, as controlled by some polarity cues that were described phenomenologically.
We considered different scenarios regarding the interaction of intrinsic and externally induced polarity pathways. In the first case, we assumed that these pathways act in parallel and regulate common downstream components (Fig. 4). The implications of this possibility were most transparent when the intrinsic pathway dynamics are much slower than those of the receptor-controlled pathways. Stimulation of a cell, anchored to the surface but otherwise intact, led to additive effects: if the gradient was in the same direction as the intrinsic polarity, stronger polarity ensues. In contrast, a gradient in the opposite direction acted to counteract the intrinsic polarity. Thus, cells that have sufficiently strong intrinsic polarization do not reorganize their polarity in response to weak external gradients, so that the net resulting polarity was opposite to that of the external gradient. In contrast, motile cells with sufficiently strong intrinsic polarization changed direction when exposed to a gradient in the opposite direction as a result of the competition between the tendencies of the cell to reorganize its polarity and then to turn sharply. For strongly (intrinsic) polarized cells, turning dominates because of the greater time taken to reorganize polarity.
The parallel action of externally induced and intrinsic pathways has important implications for chemotaxis. It suggests that cells with sufficiently strong intrinsic polarity can respond to relatively weak gradients only by reorienting themselves appropriately and not by the reorganization of their polarity. This is also relevant lo the cell's response to multiple sources/chemotactic cues: the history of the cell's exposure to these cues and the location of these cues relative to the front of the cell is crucial in determining the cell's response. Thus, the nature of the response is different from that in immobilized celts (39) and weakly polarized cells.
We also considered an implicit assumption made in different contexts that extrinsic polarity effects (receptormediated pathways in this context) suppress intrinsic ones (Fig. 5). The simplest scenario examined was one involving direct local inhibition of intrinsic pathways. This assumption implies that homogeneous stimulation of intrinsically polarized cells leads to a reduction in the concentration of intrinsic polarity components and, hence, overall polarity, and that the extent of reduction of polarity depended on the degree of stimulation. We also considered the possibility that the suppression could occur downstream of an adaptation mechanism, in which case, spatially homogeneous stimulation would have minimal effect in suppressing intrinsic pathways. In this case, depending on the manner of the suppression, the net polarity in a gradient emerges from the combination of external and intrinsic pathways in a nontriviat way. Experiments performed with gradients imposed in the same and opposite direction as the intrinsic polarity could test the presence/absence of such suppressive pathways.
While feedforward nonlinearities, feedback, and other interactions affect signal propagation in the polarity pathways, the inclusion of these effects does not alter our main conclusions. This is because the critical issue remains as to how the intrinsic and receptor-mediated signals are coupled.
Finally, it is worth considering the possible role of nonlinear dynamic transitions in polarization and chemotaxis (see Appendix B). Other models of eukaryotic gradient sensing employ nonlinear dynamic transitions to describe the origin of polarization in homogeneous stimulation, and the same nonlinear dynamic transition is at the core of the amplification effects in gradient sensing (9,10,32). In Dictyostelitim, the basal state is always one in which the cell is moving, even if the cells are weakly polarized. Weakly polarized cells move by extending pseudopods in apparently random directions. The homogeneous stimulation of these cells does not lead to strong persistent polarization but instead results in a degree of polarization that is essentially the same as before stimulation. Thus, unlike the scenario described by the aforementioned models, we do not have a situation where homogeneous stimulation necessarily orchestrates a nonlinear dynamic transition leading to a strongly polarized state. We note that it is possible that homogeneous stimulation could actually regulate the signaling system so that it transiently passes through a parameter regime that supports multiple asymptotic states, but we know of no corroborating experimental evidence yet.
We investigated whether nonlinear dynamic transitions could occur in the propagation of polarization. As demonstrated in Appendix B, a spatially varying receptor signal is able to induce multiple steady states as a result of the interplay between nonlinearities and heterogeneity, and activate a transition, Wilh upstream regulation of this mechanism arising from a combination of local and global pathways, it is possible for such a nonlinear transition to be involved in signal propagation either at the front or at the back of the cell. While some pattern-forming process may be responsible for the creation of intrinsic polarization, we see from this article that it is entirely possible that this process is not directly exploited by the chemotactic pathways.
Our results demonstrate the need for systematic experimental investigations contrasting the relative effects of intrinsic and external pathways on both frontness and backness components. This entails performing experiments with the same imposed gradients on cells at different stages of their developmental state, for example. Systematically varying the external gradient and measuring the response of the cells is also important. It is also important to work with static external gradients. Gradients established in microfluidic devices may be especially useful here (48,49). The clearest way to address various related issues is to perform experiments on cells that are anchored, thus preventing or minimizing motility without impairing the actin cytoskeleton. Environments where the cells are made to adhere strongly to the surface and/or changing surface properties to minimize movement could prove particularly useful (41). Investigating the response of such cells to homogeneous increases and decreases in receptor stimulation provides further important information. These experimental settings would allow for a clearer investigation of the roles of intrinsic and receptor-mediated pathways than experiments performed with transient external signals and moving cells. Such controlled experiments would provide invaluable information on how the response of migrating cells depends on both signal detection and their intrinsic state. Experimentally checking for the presence of a nonlinear dynamic transition is more difficult, especially if homogeneous stimulation does not yield useful information. The signature of a nonlinear transition would be a discontinuous response as the gradient is varied.
For the most part, we have sidestepped the issue of what processes may be involved in generating the intrinsic polarity cues. This would involve symmetry breaking, but the crucial issue is related to the exact stage where the symmetry breaks, and whether this is at all related to chemotaxis. Employing a concrete model for symmetry breaking, similar to that of Narang (32) to describe the generation of intrinsic cues, does not alter our main conclusions.
Polarity generation is a complex and subtle process. In this article, we have constructed a simplified model as a first step to address this complex problem. Nevertheless, several issues remain to be addressed. For example, how the intrinsic polarity cues generated, and whether they depend on an intact actin cytoskeleton. Thus, the role of adding actin inhibitors to cells at different stages of development (and in general, in different stages of intrinsic polarity) needs to be studied systematically. In our model, we have assumed that we are working with cells with an intact cytoskeleton in which the developmental process or any other progression of intrinsic polarity is unimpaired.
An additional aspect that deserves special attention is the origin of an apparent random motility in essentially unpolarized cells. It appears that there is an intrinsic process that is responsible for this seemingly random pseudopod generation, and this is functional even in cells where the receptor is not expressed (50). However, this process is overridden by gradients in weakly polarized cells. Recent experiments have suggested that small levels of chemoattractant induce random pseudopod extension leading to random cell motility (51). However, this cannot explain how cells lacking the receptor extend pseudopods randomly (50). We further note that even if a low level of chemoattractant were to cause symmetry breaking, resulting in the transition from a completely immobile cell to a mobile cell with pseudopod extension, it does not result in a strong and persistent polarity as produced, for example, during development. In our work, we assume that there is always a small basal amount of chemoattractant.
A systematic experimental and theoretical investigation of the interaction between chemotactic signals and intrinsic polarity is of critical importance in understanding chemotaxis. The extent to which this differs between eukaryotes would shed light on how and to what extent cells might employ this intrinsic capacity for chemotaxis. Finally, this also provides an example of interaction of different cues in polarization, which could have analogs in developing and other biological systems.
Source: Biophysical Journal
