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* Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5 Takayama, Ikoma, Nara 630-0192, Japan; and Graduate School of Frontier Biosciences, Osaka University, 1-3 Yamadaoka, Suita, Osaka 565-0871, Japan
Correspondence: Address reprint requests to Shin Ishii, E-mail: ishii{at}is.naist.jp.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODS AND RESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODS AND RESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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,2). This precise construction of neural networks is crucial for carrying out brain functions. Previous studies on axon guidance have provided a wealth of information about guidance factors, their receptors, and cytoplasmic effectors (3–5). A theoretical study that explained the high sensitivity of cellular chemotaxis to molecule gradients, suggested that local activators and global inhibitors amplify a small gradient by a self-enhancing process (6,7). A recent theoretical study showed gradient detection by growth cones using a filopodia-generation mechanism (8). It is still unknown, however, how cellular signaling networks are coupled to growth-cone morphology and gradient detection.
As an axon develops, a growth cone changes its shape by reorganizing actin filaments depending on the type and concentration of external signals (9,10). Rho-family small GTPases, Cdc42, Rac, and RhoA, are important signaling molecules within growth cones, and are well-known regulators of actin polymerization in both nonneuronal cells and neuronal growth cones (11,12). During axon guidance, it is presently thought that: i), external signals (e.g., netrin) are integrated into the intracellular signaling cascade via membrane receptors; ii), these signals interact with GTPases (Cdc42, Rac, and RhoA), which also cross talk with each other; and iii), the resultant signals from the GTPases regulate cytoskeleton dynamics. The resulting phenomena include filopodia elongation mediated by Cdc42 activation, lamellipodia expansion after Rac activation, RhoA-mediated myosin phosphorylation and subsequent filopodia and lamellipodia retraction, and depolymerization of actin filaments, which is inhibited by these three GTPases (Fig. 1 A).
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). In addition, the mechanics of axon guidance and the morphological changes in growth cones are prototypical examples of microscopic molecular interactions resulting in macroscopic biological functions and cellular morphologies. In this study, we first examine the qualitative characteristics of the GTPase cross talk. We show that the GTPase activities can exhibit switching responses as a result of variations in the exogenous guanine nucleotide exchange factors (GEFs). Second, we propose a computational model for the molecular machinery, in which cross talk between Rho-family small GTPases induces growth-cone movement and chemoattractive axon guidance. Our model implies that the GTPase switching response plays a significant role in axon guidance and that local nonlinearity in the GTPase responses, rather than localization of GEF signals, is important for gradient detection. Finally, we explore possible or significant cascades in the GTPase cross talk by the Monte Carlo method and give predictive suggestions about characteristics of the kinetics. |
METHODS AND RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODS AND RESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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,15). These observations can be consolidated into the signaling cascade illustrated in Fig. 1 B in which GTPases can exist in three states: activated, inactivated, and complex. The transition from an inactive state to an active state is mediated by GEFs, whereas GTP hydrolysis is promoted by the activity of GTPase-activating proteins (GAPs) (11,12,16,17,18,19). It is thought that the active form of each of these three G-proteins makes a complex with other proteins, thereby enabling them to act as either a GEF or a GAP. In other words, GTP-bound active GTPases form complexes with other proteins and enzymatically act on other GTPases and/or themselves.
The complex processes and the enzymatic reactions involved in GTPase cross talk can be expressed as the following molecule-molecule interactions and enzymatic reactions:
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is an active GTPase (GTP-GTPase) that binds an effector molecule, B, to form the complex 
A GTPase (G) is transformed into a product (P, a GTP- or GDP-bound GTPase) by the enzyme During the enzymatic reactions, a vast excess of the substrate (
) and steady-state conditions (
) can be commonly assumed. This leads to the additional assumption that the reaction 
is also in a steady state (
) because the amount of the enzyme, 
is much smaller than the amount of 
Thus, the overall rate of the GTPase reactions is dominated by the process 
In the signaling cascade depicted in Fig. 1 B, we can write the ordinary equations for the complex of Cdc42 as follows:
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are the rate constants for the forward and backward reactions, respectively. Using the steady-state conditions of the molecule-molecule reactions (
), the three complex states can be written as 
For the inactive state (Ci), with normalization for the total quantity of Cdc42, 
we obtain 
where 
Similarly, we can write 
with conditions 
for Rac, and 
and 
for RhoA.
The active state of Cdc42 (C) increases with the exchange of GTP for GDP by the inactive state (Ci) and via the backward reactions from the complex states (
and 
), and is decreased by GTP hydrolysis and the synthesizations. Then, the ordinary differential equation for the activation of Cdc42 is
 | (1) |
and 
(normalized concentration), and catalytic constants, k– and k+ (s–1). Similarly, for Rac and RhoA, we obtain:
 | (2) |
 | (3) |
are Michaelis constants (normalized concentration) and 
are catalytic constants; hA and hH are the constant rates of hydrolysis of GTP by activated Rac and RhoA (20). By applying the steady-state conditions, Eqs. 1–3 become:
 | (4) |
 | (5) |
 | (6) |
We examined the characteristics of GTPase cross talk using these three equations.
Characteristics of the kinetics
Although the signals that transduce the external cues to the GTPase network are becoming clear (21), most of the chemical parameters remain unknown. Because many of the reaction coefficients in Fig. 1 B are also unknown, we allocated a number of possible parameter sets to qualitatively analyze the kinetics of these reactions. The inverse values of the dissociation constants (e.g., k1, l2, and m2) were systematically prepared as discrete values ranging from [0.01,0.30], by considering the condition that the levels of the complex states (e.g., 
) should be smaller than the levels of the active state (e.g., C). The values of Michaelis constants (e.g., 
) were from [0.01,0.50] (normalized concentration), which is an appropriate range because the active state variables C, A, and H were normalized to be 1.0. Catalytic constants (e.g., 
and m+) ranged from [0.1,0.9] (s–1), values that approximated experimentally measured constants (22). The concentrations of the three exogenous GEFs (e.g., 
and 
), which are generated downstream of the external cue molecules, were varied from [0.01,0.30], and the concentrations of the corresponding enzymes ranged from [0.003,0.09] if the average activity of a single GTPase was 0.3. According to the Monte Carlo parameter allocation (see the section "Reaction coefficients for oscillation" and Fig. 6), the kinetic responses to various exogenous GEF concentrations could be classified as either oscillatory or convergent.
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) as representatives of the oscillatory responses (Fig. 2). When was increased from zero while the other parameters were fixed (Table 1), the activity of the Rac complex (
) dramatically switched from converging to oscillating in an ultrasensitive manner (Fig. 2 A). This oscillatory activity spontaneously occurred due to the mutual GTPase interactions if exogenous Cdc42-GEF was continuously supplied. The amount of GTPase in a complex state oscillated with a fixed temporal order (Cdc42
RacRhoA) (Fig. 2 B). The temporal order of the activity peaks of Cdc42, Rac, and RhoA did not vary regardless of the applied parameters because this activity stream originated from the signaling pathway, in which Cdc42 activates Rac, Rac activates RhoA, and RhoA inactivates Cdc42. Biochemical oscillations have frequently been observed in the activities of intracellular signaling pathways (23). It is possible that the steady-state conditions assumed in the kinetic model (e.g., ) allow the GTPase activity to oscillate. To test the GTPase oscillations in the absence of the steady-state conditions, we also simulated the GTPase cross talk, using a GENESIS simulator with a Kinetikit interface (24). This simulation included molecule-molecule interactions such as protein syntheses (e.g., PAK/Cdc42 complex). We found that the GTPase cross talk without assuming the steady-state conditions exhibited oscillatory responses (data not shown).
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). It is likely that one of the functions of the growth cone is to control axonal turning and that the activities of intracellular molecules change (switching) to turn the axon toward a different direction.
It is reasonable to assume that the nonlinear and ultrasensitive change (switch) accompanying the oscillation of the local GTPase activity reflects the growth-cone motility. The change of the neurite shape strongly correlates with local GTPase activity, which has a timescale of a few minutes (26). Moreover, the time constants for oscillatory actin dynamics and myosin activity can be estimated to be 
100 s (27), which is in the same range as the timescale of the GTPase enzyme reaction (Fig. 2 B). Considering the fact that GTPases activate LIM kinase, which phosphorylates and inactivates cofilin, an actin-depolymerization factor (28–30), the temporal order of the GTPase activities (Cdc42RacRhoA; Fig. 2 B) regulates the movement of the cytoskeleton as follows: polymerization of actin filaments in filopodia and lamellipodia, actin filament retraction, and an increase in the number of actin monomers due to low GTPase activities. This cytoskeletal cycle can generate an effective expansion of the growth cone. When a forcing oscillation, such as a GTPase oscillation, has a similar period to its responding oscillation, such as the actin cycle, the responding oscillation could be entrained into the forcing oscillation, making that growth-cone expansion stable, because the reaction delay between the GTPase oscillation and the cytoskeletal oscillation may be essentially unchangeable.
We hypothesize that the ultrasensitive switching from a convergent state to an oscillatory state works to switch two qualitatively different modes of elongation during axon guidance: straight elongation and sudden turning. We believe that the growth cone will progress along a straight path while the GTPase activities are in a convergent state, whereas large turns are induced by oscillations of their activities; the ultrasensitivity of the system largely expands the oscillation amplitude, which locally encourages the expansion. One may consider the possibility that the effective expansion is achieved with, for example, low constant RhoA activity and high constant activities of Cdc42 and Rac. In this constant activity regime, however, it is difficult for the growth cone to turn with a large angle due to the absence of the mechanism that governs changes in the direction of expansion. In addition, because we could not find reaction parameters with which the GTPase cross talk showed nonlinear Rac activity in response to an increase of exogenous GEFs, the GTPase switching response between convergence and oscillation is necessary to change the expansion direction (see below).
Winner-takes-direction model
We postulated that the GTPase-activity switch is the threshold mechanism that produces a new growth cone between two filopodia because the oscillations of actin polymerization and filament retraction amplify the expansion of the growth cone (suprathreshold, jumping-expansion mode) more than in the convergent condition (subthreshold, normal expansion mode). The switching response between these expansion modes due to the exogenous GEFs ( for example) is the primary means by which the model growth cone can initiate the turning mechanism (winner-takes-direction model).
The model growth cone we used had six filopodia (M = 6), which are radially and equiangularly distributed with an interval 22.5° (
/8 rad) around the lamellipodia (Fig. 3 A). The length of each filopodium and the radius of each lamellipodium were both set to 1 (normalized length). The model growth cone spatially and temporally integrated external cue molecules and grew in the direction along which its lamellipodia expanded. The growing process of the model growth cone assumed the following two things. First, if all the lamellipodia were in the normal expansion mode, the growth direction and length were defined as the vector summation of the lamellipodia expansions (see "Lamellipodia expansion" below). In this case, the coordinate vector of the center of the growth cone at time 
transfers to 
where 
is the sum of the expansion vectors of the five lamellipodia (see below, Eq. 8). Second, because lamellipodia in the jumping-expansion mode consumed a large fraction of the available actin monomers while depriving other filopodia and lamellipodia of these subunits, the lamellipodium that first reached the jumping-expansion mode, produced a new growth cone while at the same time the old growth cone retrogressed (middle and right panels in Fig. 3 A). This lamellipodium expansion was based on experimental observations (10,31). After either of these two types of axonal progress (normal or jumping), the cone was able to again integrate external cues.
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) in Fig. 2, the expansion rate function of a single lamellipodium in the normal expansion mode is represented as a monotonically increasing function (Fig. 3 B). This function corresponds to the convergent behavior observed for the Rac complex with small values (Fig. 2). Therefore, the expansion rate function is expressed as 
if 
where the parameter 
is the threshold value (=15), which is 150 times larger than the GEF concentration generated by a single cue molecule. 
the integrated concentration of the GEF within the lamellipodium between the ith and the (i + 1)th filopodia (
), is written as:
 | (7) |
is the temporal summation of the amount of the GEF multiplied by the spatial attenuation (
; see "Exogenous GEF") induced by the ith filopodium. The boundary condition is 
This equation models the spatial redistribution of GEF molecules among the four nearest filopodia. In the normal expansion mode, the vector summation, 
is expressed as follows:
 | (8) |
is the expansion rate function (Fig. 3 B) and 
is a unit vector that is parallel to the direction of the lamellipodium between the ith and the (i+1)th filopodia.
For growth-cone production, it is assumed that the amount of GEF (
) in the lamellipodium in the jumping-expansion mode is redistributed by initializing values to 
and 
This redistribution process of GEF molecules restricts the concentration to very low values so that the GTPase activities cannot shift from oscillatory to convergent (the right convergent region in Fig. 2 A).
Exogenous GEF
Guidance cue molecules are captured by receptors on the membrane. The receptors translate the external cue signals into intracellular reactions. The local activation of exogenous GEF in the model growth cone was assumed to exhibit a time-dependent change due to autonomous protein synthesis and degeneration (
(t)). The GEF activity was also attenuated as the growth cone extended (
(D)) because the position of the receptor upstream of this GEF was fixed. These temporal and spatial effects are expressed as
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s and 
(normalized length) (Fig. 3 C). The timescale of GEF activation (
) was estimated from that of the NGF-induced activations of Cdc42 and Rac (26).
Gradient detection
Chemoattractive cue molecules (
) were spatially distributed in a Gaussian form with a center at (0,0) and a standard deviation of 70. The actual number of floating cue molecules is thought to be large (32). Signal detection by moving filopodia and the reactions between membrane receptors and cue molecules are stochastic events. We approximated these situations in our model using stochastically generated molecules of a stationary density.
Resultant growing traces of the model growth cones are shown in Fig. 4. Model growth cones at various starting points successfully detected a chemoattractive gradient (Fig. 4, A–C). Model growth cones starting from (–70,–70) grew with various courses toward a path that was at 45° angle to the positive x axis (Fig. 4 A). This variability in the paths of the growth cones was due to the randomness of both the initial direction the growth cones' migrations and the distribution of the cue molecules. Similarly, model growth cones starting from (70,–70) grew toward a path that was at a 45° angle to the positive x axis (Fig. 4 C). Model growths cone starting from (–50,–50) also grew toward a path that was at a 45° angle to the positive x axis (Fig. 4 B). Additionally, the growth traces fanned out randomly when no cue gradient was supplied (Fig. 4 D).
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= 2), the growth cone turned very frequently and lost efficient guidance. Average turning angles were the largest and smallest in the conditions used in Fig. 4, A and C, respectively. Average turning angle of the growth cones in Fig. 4 B was almost the same as that in Fig. 4 A, whereas the average turning angle of the growth cones in Fig. 4 D was almost zero. Growth cone had the ability of gradient detection even when the threshold value was multiplied by 2/3 (Fig. 4 E). Sole characteristic difference was that the growth cone tended to turn more frequently with a low threshold (dashed line in Fig. 4 F) and to extend along a straight path with a high threshold.
The model growth cones showed zigzag traces (Fig. 4, A–D) and an intermittent motility (Fig. 4 F). Experimental observations supporting our simulation results have been made; developing growth cones exhibit a zigzag trajectory when growing toward a chemoattractant source (25), and stop-and-move behaviors (9).
The model growth cone grew within a restricted distribution range when external molecules were distributed in a specific range (Fig. 5 A). The model growth cone also extended and turned at a densely distributed horizontal band of the external cues (Fig. 5 B). Such restricted or biased distributions of external cue molecules and axons growing along these distributions have been observed (33). Moreover, the model growth cone without a switching function (linear model 1 in Fig. 3 B) could not detect the gradient because all of the lamellipodia remained in the normal expansion mode (Fig. 5 C). Such insensitivity to the gradient was also observed when a linear expansion rate function with a steeper slope (linear model 2 in Fig. 3 B) was used. Even when the spatial diffusion of exogenous GEF was small (
), growth cones without switching did not detect a gradient but extended in straight lines (almost the same as in Fig. 5 C, data not shown), because localized GEF signals generate equal expansion due to linear activation (Fig. 3 B), despite the enhancement of signal localization (weak diffusion). Our results show that by using a switching mechanism for GTPase activities, the model growth cone requires a nonlinear system rather than signal localization to detect an external chemoattractive gradient.
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Rac and Rac Rac (34). The other four pathways have not yet been experimentally confirmed. To clarify the plausibility of the hypothetical signaling cascades drawn in Fig. 1 B, we examined which cascade was significant for the switching responses of all three GTPases by varying the kinetics parameters (Eqs. 4–6). The distributions of the allocation count of a random parameter set for realizing GTPase switching (oscillation and convergence) are shown in Fig. 6. We examined the original kinetics (Fig. 6 A) and the six deficient conditions, each of which lacked one of the six pathways (Fig. 6, B–G). When the distribution was biased to small allocation counts, the absent cascade was not very important for the GTPase switching response. On the contrary, the distributions that were biased to large allocation counts revealed the importance of the absent cascade for switching. The results suggest that GTPase switching response requires four unconfirmed signaling cascades: the inhibition of RhoA by Cdc42 (Fig. 6 D), the activation of RhoA by Rac (Fig. 6 E), and the inhibitions of Cdc42 and Rac by RhoA (Fig. 6, F and G, respectively). We suggest that these chemical pathways should exist if growth cones are able to detect a cue gradient using switching activations of the GTPases. |
DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODS AND RESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). Without the winner-takes-direction function, the model growth cone cannot detect the gradient of guidance cue molecules and its growing trace was a nearly straight line (Fig. 5 C). This result implies that the nonlinear switch produced by the GTPase cross talk is essential for gradient detection, and that the ultrasensitive oscillation is used when growth cones change their growing directions dramatically.
Bacteria, which possess no filopodia, use actin-based motility, but the motility mechanism differs from that of a growth cone (36). Growth cones follow the direction of an external gradient by changing their shape (corresponding to the jumping-expansion mode) and by repeating stop and go behaviors, whereas bacteria detect a gradient by comparing concentrations at two receptor points on a cell and by actively rotating instead of changing their shape. The movements of filopodia precede those of a growth cone, and to offset the small growth-cone body, filopodia work to enhance gradient detection by a scouting mechanism. Conversely, gradient detection by cells that lack filopodia relies on feedback that results from the motility of the cell. These gradient detection methods are essentially the same in that they measure concentration differences of an extracellular molecule. Growth cones, however, have developed an efficient detection mechanism using filopodia but are restricted in that they are always connected to microtubules and cannot move freely.
In the previous theoretical studies (6,7), model cells could detect an external cue gradient by self-amplifying small differences in the gradient and globally inhibiting other parts of the cell. These models explained the sensitivity of cells to the external cue gradient without cell movement or changes in cell morphology. In the previous studies, global inhibitor molecules such as PTEN were modeled to diffuse inside of the whole cell (7) and the external cue gradient was assumed to be spatially smooth and continuous. In our model, on the other hand, we have assumed that the actin polymerization is globally inhibited because the distribution of the actin monomers is locally biased in the model growth cone, i.e., in the area where Cdc42 and Rac GTPases have extremely high activities. In addition, the external cue molecules have been modeled to be discretely distributed, so that the local cue gradient is no longer smooth. Our model growth cone was able to discount the locally rough gradient and used axonal movements to detect the global cue gradient by spatiotemporal integration of the signal. For the GTPase activation, it is possible that Cdc42 and Rac are activated by a feedback loop downstream of PI3K activation (7). In this study, we have attempted to show the possible function of the cross talk between the GTPases in detecting the cue gradient for axon guidance. Moreover, our assumption that the extreme activation of Rac produces a new growth cone may be the molecular foundation underlying recent theoretical studies about the generation of new filopodia (8).
In our model, the concentration of an exogenous GEF () has been introduced as a controlling parameter for axonal guidance. Of the other exogenous GEFs, RhoA-GEF () tends to suppress the GTPase oscillations because active RhoA inhibits the activations of both Cdc42 and Rac. It is likely that such suppression of GTPase oscillation would induce repulsive turning during axon elongation. When the GTPase kinetics are implemented into the model growth cone, six exogenous parameters, a GEF and a GAP for each of the three GTPases, may reproduce the complicated behaviors of axonal elongation. Our study has revealed that the GTPase cross talk shows a nonlinear response to external signals, which could be crucial for nonlinear movements of the growth cone, whereas there is a possibility that another nonlinearity such as protein recruitment by actin filaments at a downstream step (37) works as a switching-like system. Actually, a nonlinear process has been observed upstream (38): transient pulses of phosphatidylinositol 3-phosphate (PI3P), a product of phosphatidylinositol 3-kinase (PI3K). A positive feedback loop including PI3K and GTPases has been also reported (21,39,40).
Using computer simulations, we have shown that the morphological changes of a growth cone (physical behavior), gradient detection by extending axons, and axonal guidance (biological function) can be explained by interactions between activated GTPases (chemodynamic reaction). Although based on a number of assumptions, our approach provides one potential way to integrate and unify molecular interactions and biological phenomena beyond chemodynamics.
Many factors other than G-proteins affect the motility of growth cones in axon guidance. Adhesion molecules such as integrins or cadherins influence the motility of growth cones (41). It has also been observed that protein synthesis is involved in axon guidance (32), and the possibility of protein synthesis at a growth cone has been examined (42). To further clarify the mechanisms underlying axonal guidance, these factors should be included in future work.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMETHODS AND RESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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This work was supported by a Grant-in-Aid for Scientific Research 16014214 and by Special Coordination Funds Promoting Science and Technology (both from the Japanese Ministry of Education, Culture, Sports, Science, and Technology), and by the Inamori Foundation.
Submitted on November 3, 2004; accepted for publication May 2, 2005.
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and hH), thereby becoming GDP-bound proteins. GAPs increase the rate of GTP hydrolysis, whereas GDP for GTP exchange is facilitated by GEFs. Each of the exogenous GEF concentrations (
for inverse constant of dissociation, 
for Michaelis constant, and 
for catalytic constants. The counting process is as follows: 1), Allocate random values in reaction parameters. 2), Calculate the kinetics to draw the GTPase response curve (
). (C) The self-activation of Rac is removed (
). (D) The inhibition of RhoA by Cdc42 is removed (
). (E) The activation of RhoA by Rac is removed (
). (F) The inhibition of Cdc42 by RhoA is removed (
). (G) The inhibition of Rac by RhoA is removed (
).
) (middle), and the activity differences, 
and 
(bottom). The Cdc42-GEF concentrations a, b, and c are the same as those in panel A. Reaction parameters are listed in 
which represents the amount of exogenous GEFs generated downstream of external single cue molecules detected by the ith filopodium, is modeled as a time-dependent function (top). We assume the efficiency is attenuated with growth distance, D. This attenuation rate is expressed as an exponential function, 
(bottom).
