Introduction

The ability of eukaryotic cells to process nonisotropic extracellular stimuli lies at the heart of many aspects of cellular behavior including directed cell growth, movement, and cell-cell communication [1–4]. When an external stimulus is localized to a discrete patch of membrane, for example, during contact-dependent cell-cell communication, the physical recruitment of signaling components into receptor-associated multimolecular complexes provides a straightforward mechanism for establishing the appropriate directionality of intracellular responses. Under these circumstances, it is readily apparent how the activation of excitatory components at response onset and of inhibitory components during negative regulation (sometimes called adaptation) is constrained to a specific region of the cell.

In contrast, during chemotaxis (directed movement along a chemical gradient) cells display a very strong intracellular biochemical polarization even though the external stimulus may surround the entire cell and differ by only a few percent in concentration at different points along the membrane. Here, the principle of physical locality invoked in the case of cell-cell contact does not seem to provide an obvious mechanism for the translation of small extracellular directional clues into an almost digital biochemical, cytoskeletal, and morphological polarization of the responding cell. The striking capacity of cells to reliably sense and respond to minute gradients has led to intense experimental analysis at the cell biological level as well as to development of quantitative mathematical models of the chemosensing phenomenon.

Many of the core components of the underlying signaling network have been elucidated in several cell types, especially neutrophils and the amoeba of Dictyostelium discoidium [5–7]. A central role is played by the phospholipid phosphatidylinositol-(3,4,5)P₃ (PIP₃), the product of the action of the lipid kinase phosphoinositide 3-kinase (PI3K) on the relatively abundant membrane component phosphatidylinositol-(4,5)P₂ (PIP₂). The local concentration of PIP₃ is determined by the competing actions of PI3K and the phosphatase PTEN (phosphatase and tensin homolog deleted on Chromosome 10) that dephosphorylates PIP₃ at the 3′ position to produce PIP₂ [8]. PIP₃ functions as a membrane anchor for signaling proteins with pleckstrin-homology (PH) domains [9]. Such proteins include several that regulate the polymerization of actin, which, in turn, is needed for the extension of pseudopods/lamellipodia that allow cells to crawl [10]. For this reason, localized PIP₃ accumulation is frequently used as a read-out of polarization in response to chemoattractant gradients. In spite of this progress in defining the main molecular players and their interactions, however, we do not yet fully understand how eukaryotic cells are able to amplify the primary receptor signals induced by shallow extracellular gradients of chemoattractants into steep intracellular gradients of signaling molecules like PIP₃. Nor is it known precisely how the chemosensing signaling machinery adapts to attractant stimulations or what the connection is between the mechanisms that lead to amplification and those that lead to adaptation.

The dominant hypothesis embodied in existing conceptual and computational models for how cells achieve polarization in the face of shallow chemoattractant gradients is termed “local excitation – global inhibition” [6]. Meinhardt and Gierer [11,12] in particular showed that combinations of localized positive feedback and long-range inhibition could potentially produce the biochemical inhomogeneity encountered during chemotactic responses, and variations on this theme appear in the other models in this class [13,14]. The main idea is that polarization results from a combination of local activation processes and one (or more) globally acting, signal-induced inhibitor(s) tuned to be just strong enough to suppress the local responses everywhere except in those regions of the cell where the extracellular concentration of the stimulus is highest. The globally acting inhibitor, whose level is determined by the overall occupancy of the chemoattractant receptors, indirectly plays the role of a messenger that provides cell-wide information about the average extracellular chemoattractant concentration. This average is the standard used by the cell to determine which side is experiencing “high” and which side “low” relative chemoattractant concentrations. Adaptation, in this context, is the failure of local signals to significantly rise above the average and overcome the induced level of global inhibition.

While these modeling efforts have had a major influence on the way investigators view the molecular basis for chemosensing, the field still lacks a model that reflects to the greatest extent possible the detailed biochemistry revealed by bench experiments and whose predictions have been widely validated by experimental tests at the single cell level. For example, in Dictyostelium, stimulation with homogeneous fields of the chemoattractant cyclic adenosine monophosphate (cAMP) produces a fast rise in PIP₃ around the entire cell circumference, followed by a rapid return to (or below) prestimulus level [6,15]. PTEN shows the inverse pattern, leaving the membrane upon exposure to cAMP and then returning during the adaptation phase [7]. However, the concentration of membrane-bound PTEN is still considerably below its prestimulus level at the time when the PIP₃ concentration peaks and starts to decay. Given the current belief that in these cells, PI3K is the only generator of PIP₃ and PTEN is the most important enzyme metabolizing PIP₃, the point at which PIP₃'s concentration in the membrane peaks must mark the moment at which the activities of PI3K and PTEN are temporarily in equilibrium. The cAMP signal must therefore rapidly activate a component that negatively regulates PI3K so that its activity is balanced by the lower than resting level of PTEN present at the membrane at this point in time. Such a negative regulatory component is not explicitly included in existing models of Dictyostelium chemosensing.

In part, the absence of such specific components in previous computational treatments arises from a tendency to use abstract modules in models of complex biological signaling networks whenever the details are incompletely understood. Although such approaches are in part guided by the principle of parsimony, another, more practical reason for the introduction of abstract modules is that the literature mining involved in the creation of complex biological signaling models and their translation into computational representations can be quite difficult.

Here we report a two-pronged approach to the development of a more complete and explicit model of Dictyostelium chemosensing. First, at the biochemical level, we started with a basic chemosensing model containing only the principal molecular players incorporated into the current paradigm in the field. We then tried to fill in “logical” blanks (such as the negative regulator of PI3K discussed above) so as to be able to reproduce the experimentally reported dynamical behaviors of PIP₃, PTEN, and PI3K upon exposure of a morphologically unpolarized Dictyostelium cell to chemoattractant exposure [7,16]. Our specific aim was to construct a model that did not contain abstract “black-box” modules but instead used specific molecular components and interactions reported in the experimental literature to provide necessary functions (positive feedback, negative regulation, amplification).

Second, to help achieve this goal of molecular detail, we utilized a new software approach called Simmune that was created to facilitate the development and simulation of realistic, and therefore frequently quite complex, cell-biological models. Originally developed because of an interest in simulating immune responses, Simmune has no attributes that are unique to immunology and it is applicable to the simulation of any cell-biological system. The software suite allows molecular reactions to be defined directly at the level of interactions between molecular binding sites, using simple graphical representations of molecules and molecular complexes. This permits expert biologists to construct and simulate complex models without direct involvement in the underlying mathematics.

The signaling network that we developed through literature mining and the application of Simmune shares some properties with the abstract model developed by Meinhardt [12], but it goes further by providing an explicit molecular definition of the feedback module and the inhibitory regulator postulated in this earlier work. Our model also includes the action of a second, slower inhibitory component that is reminiscent of Meinhard's “poisoning” element. This element was introduced to prevent cells from locking into the direction of given stimulus and becoming blind to subsequent changes in direction and strength of the external signal. However, the present scheme differs in important aspects from this previous proposal and from the “local excitation–global inhibition” models suggested by Devreotes and coworkers [14] because the locally acting feedback mechanism and the distribution of the inhibitory component (PTEN) are now coupled.

Simulations performed with our model predicted previously unrecognized biphasic spatiotemporal changes in PIP₃ and PTEN localization and concentration in Dictyostelium cells responding to cAMP gradients. These predictions were confirmed across a range of gradient conditions by single-cell imaging studies, leading us to propose a substantial modification to the standard “local excitation–global inhibition” model, with locally acting negative feedback now seen as playing a key role in controlling the development of chemoattractant-induced biochemical polarity.

Results

In constructing a new detailed model of Dictyostelium cAMP chemosensing, we needed to ensure that the major features of the cellular response to chemoattractant exposure would be present in the simulated behavior of a cell. First, when exposed to a uniform concentration of attractant, the model should show a transient membrane response that rapidly reverts to the resting state (global adaptation). Second, when exposed to a shallow gradient of attractant, the simulated cell should generate a much steeper intracellular biochemical gradient (amplification). The starting point for the construction of a model with these features was the well-established biochemical scheme described in [8] and [17] (Figure 1). Ligand binding to the cAMP receptor leads to activation of the associated Gαβγ, yielding Gα and Gβγ. These effectors, in turn, promote the activation of Ras and the membrane recruitment and activation of PI3K, along with a G protein–dependent translocation of PTEN from the membrane to the cytosol. The combination of PI3K activation and a loss of membrane-bound PTEN results in a rapid increase of PIP₃ upon stimulation of the cells with cAMP (Figure 1).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 1. Standard Model of the Extracellular and Intracellular Distribution of Key Components of Dictyostelium Chemotactic Signaling

(A) In an unstimulated cell PTEN is homogeneously distributed at the membrane. The cell membrane contains very little PIP₃.

(B) Stimulation of the cell leads to the membrane recruitment and activation of PI3K, as indicated by the arrows (1) leading from inactive, mainly cytosolic PI3K (yellow) to membrane-proximal, active PI3K (orange). Activated PI3K transforms PIP₂ into PIP₃. PTEN is deactivated following cAMP stimulation and leaves the membrane. This process is indicated by arrows (2) connecting active PTEN (dark green) and the mainly cytosolic inactive PTEN (light green). Regulatory processes lead to reactivation of PTEN (3). Differences in the speed and degree of cAMP receptor ligation between front and back of the cell lead to preferential accumulation of PTEN at the back of the cell. As a result, the front experiences a higher concentration of PI3K and a lower concentration of PTEN than the back and accumulates PIP₃. Note: To emphasize the changes in PIP₃ content, the amount of PIP₃ relative to that of PIP₂ has been overstated. Even after cAMP stimulation, the actual amount of PIP₂ will be much higher than that of PIP₃.

https://doi.org/10.1371/journal.pcbi.0020082.g001

Development of a Refined Signaling Model of Chemosensing Based on Quantitative Live Cell Imaging and Literature Mining

Fusion molecules containing both a PIP₃-specific PH domain and a fluorescent protein domain have been produced to monitor the dynamics of the distribution of PIP₃ under various modes of Dictyostelium exposure to chemoattractant [1,18,19]. When stimulated with a homogeneous field of cAMP, previously unpolarized cells respond with a rapid, transient increase of PIP₃ around their entire perimeter [6,20]. In contrast, upon exposure to gradients of chemoattractant, the plasma membrane concentration of PIP₃ increases throughout the cell initially and then decays everywhere to below the pre-stimulus level, except for the side of the cell exposed to the higher chemoattractant concentration [6,15]. Using a fluorescent fusion of PTEN (GFP [green fluorescent protein]-PTEN) [7,21], similar imaging studies have been used to show that, following a homogeneous cAMP stimulus, much of the pool of PTEN translocates from the membrane to the cytosol within a few seconds and then slowly returns to the membrane [7].

Activation of membrane-recruited PI3K and translocation of PTEN to the cytosol are the two “excitatory” mechanisms that lead to the initial increase in membrane PIP₃ revealed by PH-GFP relocalization (Figure 1) and qualitative diagrams of chemosensory signaling frequently focus on these excitatory signaling events [17]. Simulations using a model incorporating only these mechanisms would, however, show the concentration of membrane PIP₃ to increase upon introduction of cAMP without being followed by either the experimentally reported rapid decay in the concentration of this membrane phospholipid or the relocalization of PTEN to the membrane. Therefore, the cells must contain inhibitory components that quench PI3K signaling and also reverse those changes that cause PTEN to dissociate from the membrane, allowing it to return to its pre-stimulus distribution (adaptation). As already noted, a comparison of the dynamic changes in the concentrations of PIP₃ and membrane-bound PTEN upon exposure of Dictyostelium cells to a homogeneous field of cAMP allowed us to conclude that PI3K activity must be rapidly negatively regulated to account for the observed post-stimulus changes in membrane PIP₃ levels.

In seeking to account for this regulation of PI3K activity, we took advantage of our previous findings that the α and the βγ subunits of the G proteins remained dissociated as long as chemoattractant was present, establishing that there was ongoing signaling through the excitatory pathways [15]. The inhibitory components thus do not simply shut off receptor signaling but actively compete with ongoing excitatory mechanisms. For the PI3K signaling branch, we therefore included stimulus-induced activation and membrane recruitment of a phosphatase that deactivates PI3K (Figure 2A, module 1) and of a Ras GAP [22] that suppresses Ras activity after an initial spike [16] (Figure 2A, module 2). Additionally, we assumed that the amount of free Gβγ contributing to activation of PI3K is controlled by a blocking element, because the dynamics of the system did not appear to permit re-association with Gα to dampen Gβγ effector function quickly enough (Figure 2A, module 3). Receptor-associated kinase (RAK) was subsequently reported to function in mammalian cells in a manner consistent with this assumption [23].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 2. PI3K and PTEN Regulatory Modules

(A) Elements controlling the activity of PI3K and upstream components. In addition to the basic, “excitatory,” signaling elements like the cAMP receptor, Gβγ, and Ras, we introduced further elements controlling the activity of PI3K and upstream components. “PI3Ktp” (module 1) stands for a tyrosine phosphatase that deactivates PI3K. This phosphatase becomes enzymatically activated and is recruited to the membrane after interaction with Gβγ. “RasGAP” (module 2) translocates to the membrane and deactivates Ras after activation by Gβγ. RAK blocks Gβγ, Gα, and the receptor, thereby reducing all signals (module 3).

(B) Elements controlling the activity and localization of PTEN. In our model, PTEN is phosphorylated by a Src-like kinase, here simply called “Src” (module 1). Src is activated by Gα and deactivated by Csk, which in turn is recruited by phosphoPaxillin (“pPaxillin”) (module 2). SHP2, which is membrane recruited by pGab1 bound to PIP₃, dephosphorylates pPaxillin (module 3), thereby leading to increased activation of PTEN.

https://doi.org/10.1371/journal.pcbi.0020082.g002

We also needed to add to the model a set of specific biochemical processes regulating PTEN localization within the cell, in particular a link from activation of the sensing receptor to the dissociation of PTEN from its resting location at the plasma membrane and a negative regulatory limb that reverses this active induction of membrane dislocation. We hypothesized that the molecular change causing PTEN to dissociate from the membrane is a phosphorylation event that in mammalian cells is mediated by a Src-like kinase [24] (Figure 2B, module 1). Reversal of the process that leads to release of PTEN from the plasma membrane thus requires control of the activity of this kinase. We adopted the pathway reported by Ren et al. [25], according to which Src activity is negatively controlled by Csk, which is recruited to the plasma membrane by phosphopaxillin (Figure 2B, module 2). The phosphorylation of paxillin in turn is controlled by the tyrosine phosphatase SHP2, which is brought into proximity of the membrane-bound paxillin through the PIP₃-binding adaptor Gab1 (Figure 2B, module 3). Figure S1 shows the full network of enzymatic interactions. Figures S2 and S3 focus on those branches of the network that control PI3K and PTEN activity, respectively.

Simmune Permits Construction of Detailed, Biology-Based Models That Avoid “Black-Box” Abstractions

In order to construct the model signaling network outlined above, with regulatory modules that account for the time-dependent return of the chemosensing apparatus to baseline after a stimulus (adaptation) and that could be tested for whether the model was also adequate to account for intracellular amplification, we incorporated molecules and mechanisms that are well-documented in the literature whenever possible, though in some cases we had to speculate about components and interactions (see also Text S1). Text S2 provides a detailed analysis of the resultant chemosensing signaling network in terms of modules with specific functionalities (“signal transduction module,” “adaptation module,” and “gradient amplification module”) and a discussion of the behavior of the network upon modifications of these modules.

This approach of incorporating molecular detail in regulatory pathways that have been incompletely defined in the experimental literature differs from the more frequently encountered abstract methods of filling these gaps that appear in many theoretical treatments of this topic. Often, limitations in experimental data and also in computational tools have led modelers to utilize conceptual signaling modules (“black boxes”) with the right input-output behavior, to achieve the desired overall behavior of a signaling network. While this approach reduces the effort needed to construct the differential equation equivalent of the signaling network and may provide insights about general dynamic properties of the simulated system, it cannot be used to investigate the behavior of specific molecular mechanisms and frequently fails to incorporate essential regulatory dynamics into the resultant model.

The software suite Simmune was developed to overcome many of the difficulties associated with the creation of detailed, biologically realistic, quantitative models suitable for simulation. It uses the familiar steps of dialog box entry and menu selection to allow the user to readily define molecule types as well as to specify the number and properties of their binding and/or enzymatic sites (Figure 3). This same interface also allows the investigator to define whether a molecule is membrane bound, in which case its movement will be limited to intramembranous diffusion, and to note which part of a transmembrane molecule is in the extracellular environment and which inside the cell, restricting the other molecular species with which these topologically distinct domains can interact during a simulation. Symbolic graphical representations of the defined molecule types appear on the screen and interactions between these entities can be specified in Simmune by simple mouse clicking and dragging to draw connections between the potential binding sites of the reaction partners.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. Defining Quantitative Interactions between Molecular Binding Sites Using Simmune (Screenshot)

The screenshot of Simmune's modeling interface shows the graphical representations of PI3K and PIP₂ and the binding sites through which they interact, as well as the sites of membrane attachment. The turquoise circle around the upper binding site of PI3K identifies it as an enzymatically active site. The dotted line represents the possibility of a binding interaction between these two molecules. Selecting the interaction by clicking on the handle on the dotted line allows entry of the relevant binding parameters.

https://doi.org/10.1371/journal.pcbi.0020082.g003

By this simple process of defining binary interactions between the binding sites of reacting species (which closely follows the way biologists traditionally think about signaling networks), the user provides Simmune with the information needed to determine which molecular complexes are possible in the system, for example, a complex consisting of the cAMP receptor and an associated Gαβγ heterotrimer. The program then automatically constructs the complete network of complexes from the component definition input, collects the reactions in which each complex participates, and calculates the contributions of these reactions to the rate of change of the concentration of each complex (which depends on the reaction rates and the—typically changing—concentrations of the reaction partners). Given a set of initial conditions, Simmune can then calculate the time course of changes in the concentrations of all molecular complexes in the model after a stimulus is applied. The software also permits the investigator to both visualize and obtain a dynamic quantitative readout of the subcellular location (Figure 4) and binding states of all the molecular complexes and of the flux through the network during the simulated signaling process (Figure S1, Video S1).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 4. Comparison of the Simulated Activities of PI3K, Membrane-Bound PTEN, and the Resulting Behavior of PIP₃ (Composite Screenshot)

Stimulation of a cell in a 2:1 cAMP gradient (mean concentration 500 nmol) leads to a rapid 3-fold increase in the membrane proximal activity of PI3K (green) and to a loss of membrane-bound PTEN (blue; tracked as GFP-PTEN in real cells). This results in a rapid accumulation of PIP₃ (red; reported by the location of PH-GFP in real cells). Subsequently, the PI3K activity is strongly quenched by the recruitment of regulatory components to the membrane and falls below its prestimulus level in less than 20 s. PTEN returns more slowly to the membrane. During the phase of downregulation of PI3K activity and reattachment of PTEN to the membrane, the concentration of PIP₃ decays. In the front of the cell (which experiences a high cAMP concentration), membrane-associated PTEN only returns to a fraction of its prestimulus level and then enters a second phase of decline. After approximately 50 s, the low level of membrane-bound PTEN that is reached in the front of the cell allows PIP₃ to increase again, even though the amount of active PI3K in this region is modest. In the back of the cell (low cAMP concentration), membrane-bound PTEN increases beyond its prestimulus level, resulting in a decrease of PIP₃ below its resting state concentration. The circular inset shows a two-dimensional representation of the dynamics of membrane-bound PTEN and PIP₃ in different regions of the three-dimensional simulation of a cell.

https://doi.org/10.1371/journal.pcbi.0020082.g004

Simulation Predictions versus Experimental Findings

Computational models are best evaluated by examining their ability to make correct predictions how a biological system will behave under conditions for which experimental data are not yet available and that were not used to establish the model's core parameters. In the present case, this meant using responses to homogeneous chemoattractant fields to establish parameters that were then used by Simmune to simulate the response of Dictyostelium not only to other (untested) homogeneous concentrations of cAMP but, more important, to gradients of cAMP that were not used for creating the model.

To measure quantitatively the behavior of PIP₃ and PTEN during the chemosensing response of Dictyostelium, experiments were performed by single-cell microscopy of cells in a one-well chamber reacting to microinjector-delivered cAMP stimulation (see [15] for details). We simulated these experiments using Simmune by placing simulated cells with a given initial composition of signaling molecules into three-dimensional extracellular compartments (representing the culture well). The model was then run for a period of time to allow the simulated cells to achieve a biochemical steady-state in the absence of cAMP stimulation The fluctuations in shape and PIP₃ distribution seen even in the absence of cAMP exposure using amoeboid cells with actin-based morphological prepolarization [26] were avoided in the present study by using lactrunculin to inhibit actin polymerization, thus allowing the result of this computational equilibration process to reflect the biological situation before stimulation with cAMP. After equilibration the simulated cells were exposed to in silico cAMP gradients equivalent to the actual chemoattractant conditions experienced by the real cells under experimental conditions. Simmune then calculated the spatiotemporal response of the concentrations of components like PIP₃, activated PI3K, and membrane-bound or cytosolic PTEN. The software furthermore allowed us to save the time-dependent behavior of molecular concentrations and flows through the signaling pathways of the simulated cell to data files for visualization using automatically generated diagrams of the cell's reaction network (Figure S1, Video S1).

First, we examined in this manner the responses of simulated cells to varying concentrations of homogeneously applied cAMP. In these simulations, the time T_max post-cAMP exposure at which PIP₃ reached its peak decreased with increasing concentration of the applied chemoattractant. After having adjusted the model parameters in such a way that T_max for one concentration of cAMP (25 nmol) in the simulation matched the experimentally observed value, simulations quantitatively predicted the correct concentration dependence of T_max for 10-fold higher and 10-fold lower concentrations of cAMP in latrunculin-treated (morphologically nonpolarized) Dictyostelium cells expressing the PIP₃-binding chimeric molecule PH_crac-GFP (Figure S4; [1,18,19]). This established that the model was able to accurately predict cellular behavior under conditions not used to establish the specific parameter values employed for the simulations.

We next examined the model's performance when used to simulate a cell's more complex response following exposure to extracellular cAMP gradients. The simulated cells showed the previously reported transient PIP₃ elevation along their entire perimeter [6,20], which then completely decayed everywhere except for the “front” side of the cell that was exposed to the highest concentration of cAMP. Also, simulated PTEN translocated from the cytoplasmic membrane to the cytosol and reassociated with the membrane in the posterior of the cell as expected [7]. However, the PIP₃ response at the “front” side of the cell showed an unanticipated behavior in these simulations: after an initial peak, it decayed transiently and then recovered to go through a second phase of increase, followed by a second, typically higher, peak. Previous experimental studies had not reported this type of biphasic behavior of PIP₃ at the front of a cell [6], so we sought to determine whether the results indicated a flaw in the model or whether the model parameters were “incorrect” and could be adjusted to make the dip in between the two peaks shallower, yet preserve the cells' quantitatively correct behavior upon exposure to homogeneous stimuli. Although for gradients with low (less than 10% of the K_d for the binding of cAMP to the receptor) absolute cAMP concentrations we could find a parameter set yielding a monotonic response, this was not possible to achieve with any reasonable parameter set when the concentration of cAMP was increased.

We then examined in detail the time-dependent changes in PIP₃, active PI3K, and membrane-bound as well as phosphorylated cytosolic PTEN in the different regions of the cell during a simulated gradient response, using a reasonable parameter set consistent with the accurate simulation of homogeneous field behavior. This analysis indicated that the biphasic PIP₃ kinetics arose from a difference between the speed of signal-induced negative regulation of PI3K activity and that of PTEN membrane reattachment (Figure 4). In accordance with previously published data [27], the simulation showed PI3K activity to be downregulated very rapidly (within several seconds) after the onset of stimulation (local adaptation), while the translocation phase of PTEN took considerably longer, typically 1 min [7]. Because the level of PIP₃ is determined by the balance between local PI3K activity and local availability of membrane-bound PTEN, the simulated cell's front experienced two phases of PIP₃ accumulation. During the first phase PI3K is highly activated and PTEN has just begun its dissociation from the membrane. This phase ends with the rapid decrease of PI3K activity due to the operation of the regulatory pathways noted above and a partial return of PTEN to the membrane. During the second phase, the imbalance in Gα-induced Src-like kinase activity between the front and the back of the cell leads to a gradual loss of membrane-bound PTEN in the front and accumulation of PTEN in the back. With a strongly decreased level of PTEN at the front of the cell, the residual activity of PI3K now can induce a second, slower increase of PIP₃ at the cell's anterior. Figure S5 illustrates this process that translates (even small) differences in receptor occupancy between front and back of the cell into steeper intracellular gradients of PIP₃ and PTEN (amplification).

Following the development of new experimental techniques for rapid exposure of Dictyostelium cells to a defined cAMP gradient [15], we found that the predicted biphasic PIP₃ response at the front of the cell corresponded to the actual behavior of the gradient sensing system, as did the concentration dependence of the extent of the decline in PIP₃ levels after the initial peak (Figure 5). Furthermore, the simulation predicted that the second increase would happen later and take longer to reach its peak if the absolute concentration of the chemoattractant was increased while maintaining the relative concentration difference between front and back of the cell. We also found this prediction to be correct (Figure 5). Adjusting the model parameters so that the dip between the first and second peak of PIP₃ corresponded quantitatively to the experimentally determined time for a single chemoattractant concentration, the simulation correctly predicted the altered slope for the second increase and for the position in time of the second PIP₃ peak when using 10-fold higher or lower absolute concentrations of cAMP (Figure 5). The model was thus able to predict with great accuracy in both time and space the PIP₃ response of Dictyostelium cells over a several-log range of chemoattractant concentration.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 5. Correspondence in Time and Space between the Predicted and Measured Changes in PIP₃ at the Front and Back of Cells Exposed cAMP Gradients

Experimental data from exposure of Dictyostelium to a 2:1 gradient with a mean cAMP concentration of 100 nmol were used to adjust model parameters. The other two responses are predictions of the model. (A), (B), and (C) are simulated responses. (D), (E), and (F) are experimental measurements, using PH-GFP to monitor PIP₃ levels in single cells exposed to gradients with a mean cAMP concentration of 1 μmol, 100 nmol, and 10 nmol. See Figure S6 for details on the full dataset of experimental replicates.

https://doi.org/10.1371/journal.pcbi.0020082.g005

To test further the predictive power of the model, we used cells expressing GFP-PTEN to examine whether the simulated spatiotemporal properties of PTEN relocation corresponded to those measured experimentally. At the cell's front, GFP-PTEN showed the triphasic behavior we expected, involving a decline, a temporary increase, and then a substantial decay. The local peak of PTEN occurred at the predicted time point and the ensuing decay correlated well with the second increase of PH_crac-GFP at the front of an adjacent cell exposed to the same gradient. As predicted, with progressively lower concentrations of cAMP, the temporary increase of PTEN seen with a high-concentration stimulus first turned into a shallow plateau and then vanished (Figures 6, S6, and S7; see also Text S3).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 6. Correspondence in Time and Space between the Predicted and Measured Changes in Membrane-Bound PTEN at the Front and Back of the Cells Exposed to cAMP Gradients

These dose-dependent dynamics of PTEN were produced by the simulation after the dynamics of PIP₃ for one cAMP concentration (100 nM) had been used to adjust model parameters. (A), (B), and (C) are simulated responses. (D), (E), and (F) are experimental measurements based on GFP-PTEN analysis in single cells exposed to gradients with a mean cAMP concentration of 1 μmol, 100 nmol, and 10 nmol. See Figure S7 for details on the full dataset of experimental replicates.

https://doi.org/10.1371/journal.pcbi.0020082.g006

Discussion

Here we describe the construction and evaluation of a new detailed model of the chemotactic signaling response of Dictyostelium. We tested the model's explanatory and predictive capabilities in a close interplay between computer simulations and laboratory analysis using quantitative single-cell microscopy. The simulations correctly predicted the biochemical behavior of the cells under conditions that had never before been experimentally investigated in detail and led us to propose specific substantial changes to the existing model of chemosensory signaling in this organism, modifications that are needed to reconcile theory with experimental observations.

The model and the simulations reported here illustrate how signal-induced activation and membrane-recruitment of components controlling the availability and/or activity of Gβγ, Ras, PI3K, and PTEN can induce rapid adaptation of the level of PIP₃ and of the localization of PTEN following homogeneous stimuli with chemoattractant. Interestingly, these same mechanisms lead to a polarized distribution of PIP₃ in gradients of cAMP. In contrast to the predominant view in the field, our studies suggest that the intracellular amplification of the applied chemotactic gradient is the consequence of the local, rather than global, feedback-regulation of excitation and adaptation. Upon stimulation of the cell, the product (PIP₃) of the activation of an excitatory component (PI3K) reinforces the processes leading to deactivation of an inhibitory component (PTEN), thereby allowing for a rapid, pronounced response. Following this first response, adaptation is achieved through a strong, locally controlled and signal-dependent suppression of the excitatory component which, in turn, decreases the deactivation of the inhibitory component, i.e., allows for reactivation of the inhibition. In a gradient, however, the higher concentration of PIP₃ at the side of the cell facing the higher chemoattractant concentration (“front”) supports the activation of more PTEN-deactivating components than at the opposite side (“back”) of the cell. This leads to a net transfer of PTEN from the front to the back, which in turn decreases the concentration of PIP₃ in the back, accelerating that side's transition into the adaptation phase. In our model, polarization of a cell in a chemoattractant gradient is thus caused by the (diffusive) communication between the sides of the cell with high or low chemoattractant concentration, respectively, interfering with the local adaptation processes.

The recruitment of Ras GAP through PIP₃ [22] and the multiple functions of receptor-associated kinase, which in addition to phosphorylating the receptor binds to Gα and blocks Gβγ from acting on Ras and PI3K [23], are well-documented negative feedback mechanisms in G protein–coupled receptor signal transduction. Our model suggests the action of two additional regulatory components: a protein phosphatase that deactivates PI3K and a lipid phosphatase (SHIP [SH2-containing inositol 5′-phosphatase]) that is recruited to the front of the cell to control the level of PIP₃ when the local concentration of active PTEN is low. These elements act on two different time scales. The phosphatase that deactivates PI3K is rapidly activated to control PIP₃ levels during early adaptation of the network. SHIP is more slowly recruited to those regions of the membrane that have lost PTEN. An abstract slow second inhibitory (“poisoning”) element had been previously suggested by Meinhardt [12] to prevent the cells from losing their sensitivity to gradient changes and we can speculate that this hypothetical element corresponds to SHIP in our more specific scheme. Biochemical studies are now required to see if the various components we have incorporated into our extended biochemical model from mammalian studies have molecular analogs in Dictyostelium with the anticipated functions.

As noted before (and discussed in detail in Text S1), we think that it is helpful to use modeling to speculate about the role of the different components in a signaling network not in terms of abstract modules but in terms of interacting molecules. Instead of postulating, e.g., a “PTEN-translocation-control-module,” we asked which molecules have been observed to play a role in the control of the membrane attachment of PTEN and how they exert this control. Simmune in particular allows the facile construction and simulation of such biochemically specific models because components represent concrete signaling molecules. This allows simulated responses to be analyzed in molecular detail and the predicted behavior of these molecules to be directly compared to results obtained by traditional laboratory methods such as immunoprecipitation, immunoblotting, flow cytometry, and confocal microscopy. As we discuss in Text S2, previous, more abstract models with PTEN-control modules neither suggested concrete molecular equivalents of those modules nor were they able to generate the correct behavior of PTEN or PIP₃ in response to chemoreceptor engagement.

Not all aspects of Dictyostelium chemotactic behavior are fully captured in even the extended model we have implemented here. While the simulations arising from the present model were quite accurate in their spatiotemporal details under diverse conditions, it is very likely that details of the regulatory mechanisms included in the present model are incorrect or that important additional components are absent. An indication that this is the case comes from observations made while experimenting with models containing alternative pathway configurations or very different parameter sets (see Text S2). In certain situations, residual PI3K inhibitors persisted at the former high-concentration side of the cell membrane once the stimulating gradient was removed. In these simulations, the persistence of these inhibitors then led to a temporarily inverted biochemical response during secondary exposure to a chemotactic gradient, a behavior that has recently been observed experimentally (XX, MMS, and TJ, manuscript in preparation). The specific model we describe here does not permit a single parameter set to give a simulation output that reproduces in detail this inversion behavior, while also being in full quantitative agreement with the primary polarization kinetics reported in this paper across a wide range of cAMP concentrations. These recent findings suggest that additional elements will need to be added to the circuits controlling local PIP₃ levels to generate a more robust molecular model.

This issue of persistence of inhibitory components at the leading edge after cAMP removal is closely related to the key question of what mechanisms keep a cell responsive to changes in the externally applied gradient. Dictyostelium cells that have fully differentiated into the chemotactically competent stage behave like neutrophils and turn their “body” instead of changing the orientation of their internal biochemical polarization when the extracellular gradient changes direction. During earlier differentiation stages, however, Dictyostelium reacts to such changes by reorienting its internal PIP₃-dependent polarity (C. Parent, personal communication). Cells simulated with our current model react to changes in the direction of externally applied cAMP gradient in a manner similar to these immature amoeba, with a reorientation of the internal PIP₃ gradient. However, the simulated reorientation is only partial and occurs more slowly than that observed directly in live cells (XX, unpublished data), again pointing to the need for a reworking of the mechanisms that control the activity of PI3K at the leading edge of a cell exposed to a gradient.

Many experimental biologists see no immediate need for measuring in a highly quantitative manner either intracellular molecular concentrations, their stimulus-induced changes, or the corresponding reaction rates. One factor contributing to this viewpoint is the limited availability of user-friendly computational tools that would allow investigators to make direct use of such data for detailed simulations that could enhance their biological understanding. Simmune, the software we developed and used here to investigate the mechanisms behind Dictyostelium's chemosensing capabilities, enables biologists lacking advanced mathematical or computer coding expertise to conduct the type of simulations that make the pursuit of quantitative experimental data a more useful undertaking. The software provides a simple interface for model building and parameter entry, provisions for automated construction of complex molecular interaction networks from the input of binary molecular interactions, visual output of the spatial distribution and number of specific molecules/molecular complexes in model cells during a simulation, and the capacity to view at varying levels of resolution the time-dependent changes in component concentrations or higher-order molecular complexes within a signaling pathway. It also permits creating models across varying scales of biological resolution, from intracellular molecular networks to individual cell behavior to the activity of groups of simulated cells.

Quantitative modeling is just beginning its foray into the cell biology of intracellular signaling, and measurements of the temporal behavior of the concentrations of most of the relevant molecules are still missing. We hope that Simmune will help encourage investigators in diverse fields of biological research to fill these gaps by providing them with an easily mastered yet powerful tool for translating biological knowledge and data into models and simulations that can enhance understanding of a complex biological system, as we show here for chemosensing in Dictyostelium.

Materials and Methods

Supporting Information

Acknowledgments

The authors wish to thank David Margulies, Ronald Schwartz, Carole Parent, Alan Kimmel, William Paul, Zvi Grossman, and Frederick Klauschen for helpful discussions. MMS is especially grateful to Gerhard Mack for his guidance and advice during the development of Simmune [43].