Chemotaxis is a dynamic cellular process, comprised of direction sensing, polarization and locomotion, that leads to the directed movement of eukaryotic cells along extracellular gradients. As a primary step in the response of an individual cell to a spatial stimulus, direction sensing has attracted numerous theoretical treatments aimed at explaining experimental observations in a variety of cell types. Here we propose a new model of direction sensing based on experiments using Dictyostelium discoideum (Dicty). The model is built around a reaction-diffusion-translocation system that involves three main component processes: a signal detection step based on G-protein-coupled receptors (GPCR) for cyclic AMP (cAMP), a transduction step based on a heterotrimetic G protein Gα2 βγ, and an activation step of a monomeric G-protein Ras. The model can predict the experimentally-observed response of cells treated with latrunculin A, which removes feedback from downstream processes, under a variety of stimulus protocols. We show that G α 2 β γ cycling modulated by Ric8, a nonreceptor guanine exchange factor for G α 2 in Dicty, drives multiple phases of Ras activation and leads to direction sensing and signal amplification in cAMP gradients. The model predicts that both G α 2 and Gβγ are essential for direction sensing, in that membrane-localized G α 2 *, the activated GTP-bearing form of G α 2, leads to asymmetrical recruitment of RasGEF and Ric8, while globally-diffusing Gβγ mediates their activation. We show that the predicted response at the level of Ras activation encodes sufficient ‘memory’ to eliminate the ‘back-of-the wave’ problem, and the effects of diffusion and cell shape on direction sensing are also investigated. In contrast with existing LEGI models of chemotaxis, the results do not require a disparity between the diffusion coefficients of the Ras activator GEF and the Ras inhibitor GAP. Since the signal pathways we study are highly conserved between Dicty and mammalian leukocytes, the model can serve as a generic one for direction sensing.
In light of the restriction to LatA-treated cells, the backbone of the chemotactic pathway activated in response to changes in extracellular cAMP is Δ cAMP →Δ GPCR occupation →Δ Gαβγ activation →Δ Ras activity. We describe this pathway in terms of three modules: the GPCR surface receptors cAR1-4, G α 2 β γ and Ras, as illustrated in Fig 1.
Next we investigate how cells respond to a linear cAMP gradient along the x-axis, which we define as follows. C (x, y, z) = Δ C 10 · (x - x r) + C r where C (x, y, z) is the cAMP concentration on the membrane at (x, y, z) ∈ S 5 2 (a sphere of radius 5), ΔC ≡ Cf − Cr, and subscripts f and r denote the points (5,0, 0) (the ‘front’) and (-5,0, 0) (the ‘rear’). In the context of Dicty aggregation, the ‘back-of-the-wave’ problem refers to the fact that cells do not turn to follow the cAMP gradient after the wave has passed, despite the fact that the spatial gradient reverses as the wave passes over a cell [15,62]. This requires some level of persistence of ‘orientation’ of a cell, but there is as yet no agreed-upon mechanistic solution for this problem, since polarization and other factors may play a role [63]. Under uniform stimuli, cells are said to show rectification if there is an asymmetry in the amplitude and evolution of the response to a step increase in cAMP compared with the response following removal of the stimulus [21]. To test whether the proposed network exhibits rectification in this sense, we apply a uniform stimulus of various concentrations for 60 seconds and then remove it, as was done experimentally in fully aggregation-competent cells [21]. Fig 20 (left and center) show the simulation and the experimental results, resp. In both cases the concentration of cAMP is increased from 0 M to the concentrations indicated for 60 seconds (green shaded area), followed by a decrease to 0 M, and in both cases one sees a much larger and faster change in RBD following application of the stimulus than on removal. We also applied the same stimuli as used above to gα-null cells and ric8-null cells. Results given in the Supporting Information show that Ric8 plays a significant role in the rectification, as will also be seen later in the traveling wave analysis. Some insight into this behavior can be gained from simple models of excitation and adaptation, such as the cartoon description defined by the system of equations d y 1 d t = S (t) - (y 1 + y 2) t e, d y 2 d t = S (t) - y 2 t a. (1) Here S (t) represents the signal and the magnitudes of te and ta reflect the time scale for excitation and adaptation, resp. , and one see that y1 adapts perfectly to a constant stimulus whereas y2 compensates for the stimulus. However, the temporal responses to increasing and decreasing stimuli are symmetric, and therefore such a simple model cannot explain the observed response. Nakajima et al. [21] suggest that a single-layered incoherent feedforward circuit with zero-order ultrasensitivity [64] is necessary to generate rectification, but our model does not include an ultrasensitive circuit. Instead, rectification is induced solely by the balanced regulation of RasGEF and RasGAP activity. The ratio of RasGEF* to RasGAP* increases 2–4 fold very rapidly in response to a step increase in the cAMP concentration, but when the stimulus is removed this ratio does not drop significantly, as shown in the right panel of Fig 20. Thus Ras activation persists because the ratio equilibrates rapidly while the absolute levels of the factors decrease more slowly. To study how cells would respond in wave-like spatially-graded stimuli, we first generate a simple trianglular wave that approximates a natural cAMP wave. Let W (x, y, z, t) denote the cAMP concentration at (x, y, z) of the cell at time t, and specify it as W (x, y, z, t) = 0,0 + 350 k ≤ t ≤ x + 5 v + 350 k 10 (t - x + 5 v - 350 k), x + 5 v + 350 k < t ≤ x + 5 v + 100 + 350 k - 10 (t - x + 5 v - 350 k) + 2000, x + 5 v + 100 + 350 k < t ≤ x + 5 v + 200 + 350 k 0, x + 5 v + 200 + 350 k < t ≤ 350 (1 + k), where v is the wave speed and −5 ≤ x, y, z ≤ 5, k = 0,1, ⋯. This wave resembles a natural wave when we choose the natural wave speed v = 5μm/s, as shown in Fig 21. The wave length is 1000μm, and at the natural speed any point on a cell is subject to an increasing stimulus for 100 sec on the upstroke of the wave and a decreasing stimulus for 100 sec on the downstroke. As shown in Fig 22, Ras is activated everywhere as the wave passes, but Ras activation is delayed about 1 sec in the rear half (Fig 22 -right) for a wave traveling at the natural wave speed. Ras activation is higher at the front of the cell than at the rear throughout passage of the wave, thereby providing persistent directionality in Ras activation and the potential for persistent orientation as the wave passes. It should be emphasized that we are simulating the rounded LatA-treated cells that have no intrinsic polarity, which suggests that polarity is not necessary for the persistence of direction sensing at the natural wave speed, even at the level of Ras activity. By comparing Figs 20 and 22, one sees a similar pattern in Ras activation. In fact, due to the rectification characteristic observed in uniform stimuli, Ras* activity does not drop significantly in a wave, and therefore the front is able to maintain a higher Ras*. To determine whether the cell is able to respond after the first wave passes, we applied the same wave for three periods, and one sees in Fig 23 that the cell responses are almost identical for three successive passages of a wave. It is also known that wave speeds affect the spatial pattern of Ras activity over a cell [21], in that Ras is activated uniformly for a fast wave, and activated at both the wavefront and waveback for slow waves. To test the effects of the wave speed, we apply a fast wave (50μm/s) and a slow wave (0. 5μm/s) to the rounded LatA-treated cells. The results are shown in Fig 24. At a wave speed of 50μm/s, Ras activation is uniform along the cell periphery, as is observed in the experiments, but at 0. 5μm/s we see a significant Ras reactivation at the rear of the cell and the Ras* distribution reverses at the back of the wave. In order to demonstrate the effect of wave speed on rectification more clearly, we plot the time course of Ras activation at the front-most and rear-most points of the cell in Fig 25. At a wave speed of 0. 1μm/s, Ras is reactivated at the rear of the cell when the back of the wave passes over the rear. As the wave speed increases, the reactivation at the rear becomes weaker, and at the normal wave speed of 5μm/s persistent directionality is well-preserved. Of course, when a fast wave passes over the cell, Ras activation is almost spatially uniform. As was pointed out earlier, Ric8 plays an essential role in rectification under uniform stimuli, and to further emphasize that the back of the wave problem is closely connected with the disparity in the response to increasing vs. decreasing stimuli, we applied the same wave used previously to a ric8-null cell. The Ras* activity is shown in Fig 26, where one sees that the persistence of directional information is essentially lost. It is not surprising to see that Ras* at the front becomes smaller than the rear, which indicates a reversal in the Ras* distribution, further reinforcing the importance of the asymmetric response to increasing vs decreasing stimuli in solving the back of the wave problem. Clearly there is a trade-off between the persistence of directionality in Ras activation and the ability of cells to respond to new gradients. To investigate whether the Ric8-induced rectification has an adverse effect on reorientation in response to a reversed gradient, we subject cells in a 0–100 nM gradient to reversals to increasingly weaker gradients. In each case we keep the mean concentration experienced by the cell fixed to eliminate the mean concentration effect (see. Fig 14). For an equally strong reverse gradient (100–0 nM), the directional persistence of Ras* is reversed within 100 seconds of gradient reversal, as shown in Fig 27. The spatial profile also indicates that Ras* distribution is strongly reversed after switching to equally strong reversed gradients, (Fig 27 –center and right). It is observed in Dicty that all cells (20/20) reversed their direction of migration under this protocol [22]. For intermediate gradients (75–25 nM), Ras* is slightly reversed (Fig 28 –left) in the same time window (0–200 s). The spatial plot of Ras* indicates spatial oscillations along the cell periphery at almost the end of the time window t = 180 s, (see Fig. M in S1 Text) suggesting spatio-temporal complexity in Ras* redistribution. Consistently, experiments show that a fraction of the cells (5/17) did not reverse their migration direction. For weak gradients (60–40 nM) a difference in Ras activation is still maintained at the end of the time window (t = 200 s) (Fig 28 (right) ), consistent with the observation that that all cells continued moving in their original direction in this case [22]. These simulations suggest that Ric8-induced rectification does not harm cells’ reorientation in response to large amplitude reversals of the gradient, but it delays the reorientation in a weak reversed gradient.
Chemotaxis is a dynamic spatio-temporal process that involves direction sensing, polarization, and cell movement, and direction sensing is the first essential step in this process, becuase it defines the cell’s compass. A growing body of evidence suggests that Ras is an ideal candidate within the chemotactic signalling cascade to play an essential role in direction sensing [31,68]. In this article, we developed a novel modular model of direction sensing at the level of Ras activation. The model incorporates biochemical interactions in Dicty and captures many aspects of its response. The model consists of the cAMP receptor, the G-protein G α 2 β γ, and a Ras GTPase module in which both adaptation and amplification occur. Utilizing a rounded cell pretreated by LatA as was done in experiments, we investigated Ras activation patterns in various cAMP stimuli. Simulations of this model give insights into how the signal transduction network determines Ras activation characteristics in wild type cells, how an altered network in mutant cells changes Ras activation, and how the spatial profile and persistence of Ras activation can lead to directional persistence. We proposed an experimentally-based kinetic model of G α 2 β γ signaling in which the intact G α 2 β γ and the Gβγ subunit can cycle between the membrane and the cytosol, while the G α 2 subunit remains membrane-bound. Moreover, G α 2 can be reactivated by the only known (to date) GEF for G α 2, Ric8. The regulation of Ric8 is not well-defined, but we assume that it is also cycles between the cytosol and the membrane, and that its recruitment to the membrane is promoted by G α 2 *. The model replicates the persistent Gαβγ dissociation in the presence of cAMP, and also demonstrates that Gβγ and G α 2 * are produced in a dose-dependent manner. Interestingly, the model reveals that G α 2 exhibits dose-dependent kinetic diversities. The variety of G α 2 dynamics revealed here may have important implications in direction sensing because in neutrophils G α 2-GDP accumulates at the leading edge and is involved in regulating directionality [74], although it has not been demonstrated that Ric8 is involved there. Adaptation of Ras activity is controlled by a balance between RasGEF and RasGAP, both of which can cycle between the membrane and the cytosol. This component of the network involves incoherent feed-forward, and becuase both can cycle betweenmembrane and cytosol, can give rise to spatial asymmetry in Ras activation. Both RasGEF and RasGAP are activated at the membrane by free Gβγ, but the translocation of RasGEF from the cytosol is enhanced by G α 2 *. The proposed translocation-activation topology is able to capture the dose-dependent Ras activation and various patterns such rectification and refractoriness under uniform stimuli. It also predicts that imperfect adaptation is inevitable in wild type cells due to the asymmetrical translocation of RasGEF. Takeda et al. [28] proposed an incoherent feedforward activation model to explain adaptation of Ras activity in which RasGEF is assumed to be confined to the membrane and RasGAP diffuses in the cytosol. In our model, both RasGEF and RasGAP can diffuse in the cytosol at equal rates, and both can be recruited to the membrane and activated by Gβγ. Direction sensing, biphasic Ras activation and signal amplification are achieved by complex interactions between the modules. The incoherent-feedforward-activation by globally-diffusing Gβγ contributes to a transient activation along the entire cell perimeter. The activation at the front of the cell (facing the higher cAMP concentration) is initially faster and stronger due to the cAMP gradient, but it provides no symmetry breaking or signal amplification since diffusion eliminates the initial Gβγ concentration gradient. This means that Gβγ does not reflect the external stimulus gradient and provides no basis for direction sensing in LatA-treated cells, although it is essential for RasGEF and RasGAP activation. It is the Ric8 regulated, membrane-bound G α 2 * that determines the symmetry breaking and signal amplification. G α 2 * creates an asymmetrical recruitment of RasGEF in a cAMP gradient, which in turn induces asymmetrical RasGEF activation, providing a basis for symmetry breaking. More importantly, Ric8 recruitment to the membrane is elevated by G α 2 *, while activated Ric8 reactivates G α 2, forming a positive feedback loop. In addition, faster G α 2 β γ reassociation at the back of the cell due to less reactivation of G α 2 there induces faster G α 2 β γ cycling. Since G α 2 β γ diffuses in the cytosol, this provides a potential redistribution of G α 2 β γ from the back to the front, which in turn results in more G α 2 * at the front, thereby forming another positive feedback loop. These two positive feedback loops generate the symmetry breaking and signal amplification of Ras activation in a cAMP gradient. We also studied cell responses to gα2 and ric8 mutations extensively. It is predicted in numerical simulations that in the presence of uniform stimuli, adaptation of Ras activity is perfect and the maximum cytosolic RBD depletion is reduced in gα2-null cells. In a cAMP gradient, gα2-null cells fail to sense directions and there is only an initial transient Ras activation. Adaptation of Ras activity is still imperfect in ric8-null cells, but the magnitude of imperfectness is reduced as compared with wild type cells. Moreover, simulations suggest that ric8-null cells fail to sense direction when they are exposed to a shallow gradient or a steep gradient with high mean concentration, highlight the importance of Ric8 in regulating Ras activation. In contrast to LEGI-type models, the global diffusing Gβγ does not act as an inhibitor directly in our model—instead, it induces both activation and inhibition by activating RasGEF and RasGAP respectively. Gβγ also serves as a ‘global’ activator for the pool of RasGEF* and as a ‘global’ inhibitor by creating a uniform inhibition pool of RasGAP*. Asymmetry in their localization at the membrane arises from the fact that membrane-bound G α 2 * recruits RasGEF from the cytosol, thereby creating an asymmetrical pool of RasGEF*. Hence, our model can be regarded as a local-global transitions of both excitation and inhibition with a delayed local sequestrations of excitation model, in the sense that initially both activation and inhibition go through a local-global transition due to diffusion of Gβγ while a delayed localized translocation by G α 2 * contributes to a local excitation. Direction sensing is results from the Gβγ- mediated, G α 2-Ric8 dependent signal transduction network. Although the model is based on cAMP induced Ras activation in Dicty, GPCR-mediated Ras activation is highly conserved between Dicty and mammalian leukocytes [8]. GEF translocation via interaction with an upstream GTP-bound G protein is a principle conserved in evolution [47] and Gα’s role in GPCR-mediated signalling has been emphasized in other systems [50,75] and in drug discovery [76]. Therefore, our model could serve as a generic framework for GPCR mediated Ras activation in other systems and suggest new experiments in those systems.
We first formulate the reaction-diffusion system of signal transduction in general terms and then list the specific equations for the model. Consider a bounded three dimensional domain Ω ⊂ R3 representing a cell, and denote ∂Ω as the plasma membrane. Then the reaction diffusion equation for a cytosolic species A is ∂ C ∂ t = ∇ · (D ∇ C) + ∑ i s i r i, (2) in which C = C (t, x) represents the concentration of A at time t at x ∈ Ω and D is the diffusion coefficient of A. The summation is a reaction term indicating A participates in cytosolic reactions which either depletes it or produces it. The ith reaction produces si molecules of A, or consumes −si > 0 molecules of A with a reaction rate ri = ri (t, x). In the signal transduction network considered in this article, si = 0,1. The boundary conditions involve reactions on the boundary and binding and release of molecules at the membrane. We assume that the volume density C (the concentration in the cytosol) for A has the units μM and that the surface density (the concentration on the membrane), Cm, has the units #/μm2. We also assume that the binding reactions at the membrane take place within a layer of thickness δ (nm) at the membrane. Then the net flux to the boundary, which can be positive or negative, can be written as - n → · D ∇ C = - D ∂ C ∂ n = k + · δ · C - k - · C m ≡ j + - j -, (3) where n → is the exterior unit normal to ∂Ω, k± are the on and off rate of binding to the membrane, and κ = 602 relates the units of volume density and surface density scaled by Avogadro’s constant. For the membrane form of species A we have the translocation-reaction-diffusion equation, ∂ C m ∂ t = ∇ · (D m ∇ C) + κ (j + - j -) + ∑ s m i r m i, (4) where Cm = Cm (t, x) denotes the concentration on the membrane and Dm is the surface diffusion coefficient [77,78]. The first term represents the diffusion on the membrane, which we ignore throughout, and the second represents transolcation between cytosol and membrane, which could be absent if A is confined on the membrane, such as Ras, Ras*. There may also be conservation laws for certain substances. If the substances are confined to the membrane we write ∫ ∂ Ω ∑ i = 1 n A n d S = A t o t, (5) where Ais are the concentrations of different forms and Atot represents the total amount in the cell. If the substances are present both in the cytosol and on the membrane, we write ∫ Ω ∑ i = 1 k A i c d x + ∫ ∂ Ω ∑ j = 1 n A j m d S = A t o t, (6) where Ais are the concentrations of different forms in the cytosol and A i m s are the concentrations of different forms on the membrane. We are now ready to assemble the system of equations that constitute the full kinetic model in a given geometry Ω. We have to account for 6 cytosolic species in the system Gαβγ, c, Gβγ, c, RasGEFc, RasGAPc, Ric8c and RBDc. The evolution can be described by a system of diffusion-translocation equations ∂ G α β γ, c ∂ t = ∇ ⋅ (D G α β γ, c ∇ G α β γ) ∂ G β γ, c ∂ t = ∇ ⋅ (D G β γ, c ∇ G β γ) ∂ R a s G E F c ∂ t = ∇ ⋅ (D R a s G E F c ∇ R a s G E F c) ∂ R a s G A P c ∂ t = ∇ ⋅ (D R a s G A P c ∇ R a s G A P c) ∂ R i c 8 c ∂ t = ∇ ⋅ (D R i c 8 c ∇ R i c 8 c) ∂ R B D c ∂ t = ∇ ⋅ (D R B D c ∇ R B D c) with the following conditions on ∂Ω, D G α β γ, c ∂ G α β γ, c ∂ n = j 1 D G β γ, c ∂ G β γ, c ∂ n = j 2 D R a s G E F c ∂ R a s G E F c ∂ n = j 5 − j 6 D R a s G A P c ∂ R a s G A P c ∂ n = j 7 D R i c 8 c ∂ R i c 8 c ∂ n = j 3 − j 4 D R B D c ∂ R B D c ∂ n = j 8 − j 9 The species that evolve on the membrane are: R*, Gαβγ, m, Gβγ, m, G α *, Gα, Ric8m, Ric8*, RasGEFm, RasGAPm, RasGEF*, RasGAP*, Ras, Ras* and RBDm. The evolution equations for these are given by ∂ R * ∂ t = r 1 ∂ G α β γ, m ∂ t = − κ j 1 − r 2 + r 7 ∂ G β γ, m ∂ t = − κ j 2 + r 2 − r 7 ∂ G α * ∂ t = r 2 − r 3 + r 5 ∂ G α ∂ t = r 3 − r 5 − r 7 ∂ R i c 8 m ∂ t = − κ j 3 + κ j 4 − r 4 + r 6 ∂ R i c 8 * ∂ t = r 4 − r 6 ∂ R a s G E F m ∂ t = − κ j 5 + κ j 6 − r 8 + r 9 ∂ R a s G A P m ∂ t = − κ j 7 − r 10 + r 11 ∂ R a s G E F * ∂ t = r 8 − r 9 ∂ R a s G A P * ∂ t = r 10 − r 11 ∂ R a s * ∂ t = r 12 − r 13 + r 14 − r 15 ∂ R a s ∂ t = − r 12 + r 13 − r 14 − r 15 ∂ R B D m ∂ t = − κ j 8 + κ j 9 The following conservation laws are also imposed: ∫ ∂ Ω (R + R *) d s = R t, (7) where Rt is the total amount of receptors. ∫ Ω G α, c + G β γ, c + G α β γ, c + G α * d x + ∫ ∂ Ω G α + G β γ, m + G α β γ, m d s = G α β γ t, (8) where G α β γ t is the total amount of heterotrimetric G protein, indicating the cell does not produce additional heterotrimetric G protein. ∫ Ω R a s G E F c d x + ∫ ∂ Ω R a s G E F m + R a s G E F * d s = R a s G E F t. (9) Similarly, for RasGAP ∫ Ω R a s G A P c d x + ∫ ∂ Ω R a s G A P m + R a s G A P * d s = R a s G A P t. (10) For Ras, we have ∫ ∂ Ω R a s + R a s * d s = R a s t. (11) The parameters involved in the Receptor module are taken from the literature. We estimated the parameters in the heterotrimeric G protein module from steady state analysis (SSA) of the spatially lumped model averaged from the spatially distributed model. The parameters in the Ras module are also estimated from SSA and time dynamics of Ras activation. The detailed estimation scheme is described in the supporting information (see section Parameter estimation in S1 text). We summarize the parameters in Table 3.
Many eukaryotic cells, including Dictyostelium discoideum (Dicty), neutrophils and other cells of the immune system, can detect and reliably orient themselves in chemoattractant gradients. In Dicty, signal detection and transduction involves a G-protein-coupled receptor (GPCR) through which extracellular cAMP signals are transduced into Ras activation via an intermediate heterotrimeric G-protein (G α 2 β γ). Ras activation is the first polarized response to cAMP gradients in Dicty. Recent work has revealed mutiple new characteristics of Ras activation in Dicty, thereby providing new insights into direction sensing mechanisms and pointing to the need for new models of chemotaxis. Here we propose a novel reaction-diffusion model of Ras activation based on three major components: one involving the GPCR, one centered on G α 2 β γ, and one involving the monomeric G protein Ras. In contrast to existing local excitation, global inhibition (LEGI) models of direction sensing, in which a fast-responding but slowly-diffusing activator and a slow-acting rapidly diffusing inhibitor set up an internal gradient of activity, our model is based on equal diffusion coefficients for all cytosolic species, and the unbalanced local sequestration of some species leads to gradient sensing and amplification. We show that Ric8-modulated G α 2 β γ cycling between the cytosol and membrane can account for many of the observed responses in Dicty, including imperfect adaptation, multiple phases of Ras activity in a cAMP gradient, rectified directional sensing, and a solution to the back-of-the-wave problem.
cell motility
signaling networks
g-protein signaling
network analysis
cellular structures and organelles
g protein coupled receptors
cytosol
computer and information sciences
proteins
ras signaling
transmembrane receptors
cell membranes
chemotaxis
biochemistry
signal transduction
cell biology
biology and life sciences
cell signaling
A Model for Direction Sensing in Dictyostelium discoideum: Ras Activity and Symmetry Breaking Driven by a Gβγ-Mediated, Gα2-Ric8 -- Dependent Signal Transduction Network