A Proposed Kinetic Pathway
The model was based on the reaction scheme shown in . A few steps are labeled, such as the synthesis of axin and β-catenin, the degradation of axin, the axin-independent (basal) and axin-dependent degradation of β-catenin, as well as the critical cycle involved in the phosphorylation of β-catenin for degradation (Destruction Core Cycle). The output is the formation of the β-catenin/T-cell factor (TCF) complex and the input is the Wnt signal. Although many proteins interact with the Wnt pathway, we have focused only on core components known to be necessary for mediating a Wnt signal in most contexts. These core proteins include GSK3β, protein phosphatase 2A (PP2A), β-catenin, APC, axin, Dsh, TCF, and Wnt. The reactions incorporated into our model include protein synthesis/degradation, protein phosphorylation/dephosphorylation, and the assembly/disassembly of protein complexes (, solid arrows). Reactions mediated by proteins that activate a process are represented with broken arrows: (1) activation of Dsh by Wnt (step 1), (2) activation of the release of GSK3β from APC/axin/GSK3β by Dsh (step 3), and (3) activation of APC-dependent axin degradation (step 15). The reactions and components in blue are concerned with additional features of the pathway, as discussed below.
Reaction Scheme for Wnt Signaling
The centerpiece of the model is the formation of the unstable core complexes involved in β-catenin phosphorylation and subsequent destruction. In addition to β-catenin, this set of complexes contains GSK3β and the scaffold proteins APC and axin. The complexes assemble in several steps: (1) binding of axin to APC (forward reaction of step 7); (2) binding of GSK3β (forward reaction of step 6); (3) phosphorylation of axin and APC by GSK3β (step 4). Dephosphorylation of the core complex (step 5) is mediated by PP2A. The first step in β-catenin degradation is its binding to APC*/axin*/GSK3β (step 8), after which it is phosphorylated by GSK3β (step 9) and released from the complex (step 10). Our model assumes that the phosphorylation of β-catenin by GSK3β is negligible in the absence of axin. Indeed, recent work indicates that axin stimulates the phosphorylation of β-catenin by GSK3β at least 24,000-fold (Dajani et al. 2003
). Free, phosphorylated β-catenin is rapidly polyubiquitinated and degraded by the SCF complex and the proteasome, respectively (step 11).
The dynamic properties of the model, such as the flux through the pathway, are also affected by binding of β-catenin to other partners, such as TCF (step 16) and free APC (step 17). In special cases (high axin concentrations), the flux through the system is affected by the binding of axin to GSK3β (step 19) as well as β-catenin (step 18). We have previously shown experimentally that TCF reduces the rate of β-catenin degradation (Lee et al. 2001
). Turnover of β-catenin (steps 11, 12, and 13) and axin (steps 14 and 15) are included in our model, but since biochemical experiments in Xenopus
egg extracts indicate that the turnover of GSK3β, Dsh, and TCF is relatively slow (no detectable degradation after 3 h at room temperature; unpublished data), the synthesis and degradation of these proteins are not explicitly modeled. The activation of the pathway in vivo, which results in the stabilization of β-catenin, is initiated by binding of Wnt ligands to Frizzled receptors and the subsequent transition of Dsh from its inactive form (Dshi
) to its active form (Dsha
). Since these events are still poorly defined, both processes have been combined in step 1. Interaction of Dsha
with the nonphosphorylated complex APC/axin/GSK3β (step 3) activates the release of GSK3β . This latter process requires the activity of the GBP/Frat (not shown on our diagram). Deactivation of Dsha
occurs through an as-yet-unidentified mechanism (step 2).
The mathematical analysis is based on a series of balance equations that describe the concentrations and complexes of proteins in the Wnt pathway, as depicted in . The set of variables and the set of 15 differential equations we obtained are given in Table S1
, and Dataset S1
, respectively (Equations [A-1]–[A-15]). Stimulation of the pathway by Wnt is described by a time-dependent function, Wnt(t)
. Since Dsh, TCF, and GSK3β are degraded very slowly, we assume that their concentrations remain constant throughout the timecourse of a Wnt signaling event. The conservation equations for Dsh, TCF, and GSK3β are as follows:
Symbols with the superscript "0" denote total concentrations.
Since the concentration of axin is very low (see below) compared to the concentration of GSK3β, we replaced Equation (3) with the simple relationship GSK3β0
. Similarly, we omitted the concentration of complexes containing axin in the conservation relationship for APC, which leads to the following equation:
We will, however, take into account the contribution of axin-containing complexes for GSK3β and APC conservation equations when we later consider the effect of large increases in axin concentration.
The simplest possible equation was chosen to describe the kinetics of each individual reaction. Synthesis of β-catenin and axin are described by constant rates νi
. Unimolecular reactions are assumed to be irreversible and are described by linear rate equations νi
, where ki
denotes the first-order rate constant and Xj
denotes the concentration of the reactants. Reversible binding steps (double-headed arrows in ) are described by the equation νi
), where Xj
denote the concentrations of the binding partners and (Xj.Yl
) the concentration of their complex. The Dsh-mediated release of GSK3β from the destruction complex is described by an irreversible reaction that is bimolecular in the concentrations of Dsh and the degradation complex. The model is simplified by assuming that the reversible binding steps between axin, β-catenin, APC, and TCF are very fast, such that the corresponding protein complexes are in rapid equilibrium, so that only the dissociation constants Ki
are considered in the kinetic description of these steps. The conservation equations and the binding equilibria reduce the number of independent dynamic variables. Accordingly, the original set of 15 differential equations is transformed into a set of only seven ordinary differential equations coupled to four conservation equations and four relationships for binding equilibria. For a detailed mathematical description of the model and the procedure for reducing the number of systems variables, see Dataset S1
Experimental Evaluation of the Reference and Stimulated States
We define the reference state as the absence of Wnt signaling (Wnt
= 0). In this unstimulated stationary state, Dsh is inactive and does not affect the degradation complex. β-Catenin concentration is kept low by continuous phosphorylation and degradation. The reference state can be characterized by the special values for its rate constants, its equilibrium constants, and its conservation quantities. If one can obtain values for all of these system parameters, the model equations should allow for a straightforward calculation of the variables in the reference state. Currently, we have experimental data for many of these parameters (see below). For the remaining system parameters that were not directly measured, we were able to derive numbers based on experimental data of steady-state concentrations and fluxes. A number of parameters were set such that the results of the model were in agreement with previous experimental data, specifically with the experimentally determined rate of β-catenin degradation (Salic et al. 2000
; Lee et al. 2001
). Finally, a few parameters had to be estimated; the constraint used was that the resulting model should be compatible with the steady-state and flux values. lists the numeric values of all of the input quantities of the model. These quantities are either specific parameters, such as dissociation constants, or systemic properties, such as steady-state concentrations or fluxes, from which the other parameters have been derived. Both types of input quantities include experimental data as well as estimated values. The specific numerical values affect the model to differing degrees. In a later section, we analyze the effects of changing the values of the parameters around their reference numbers. The types of input data used for our modeling can be divided into five groups.
Numeric Values of Input Quantities of the Model for the Reference State
The first group of input data contains both total concentrations (Dsh0, APC0, TCF0, and GSK3β0) and steady-state concentrations (Axin0, β-catenin0, β-catenin*). The total concentrations of Dsh, TCF, GSK3β, axin, β-catenin, and APC in Xenopus egg extract were determined experimentally using Western blot analysis by comparing the intensity of the signal to that of known amounts of recombinant protein. The concentration of phosphorylated β-catenin w as estimated because we have not been able to directly determine its level in extracts. However, we estimate that this value is small compared to that of nonphosphorylated β-catenin for the following reasons: (1) Addition of axin to Xenopus extracts dramatically increases the rate of β-catenin degradation. Since the role of axin is to promote phosphorylation of β-catenin, which is subsequently degraded, this would suggest that normally a significant pool of β-catenin exists in the nonphosphorylated form. (2) Western blot analysis of Xenopus extracts demonstrates that only a small percentage (<10%) of total β-catenin can be detected as migrating with a slower mobility, which likely represents the phosphorylated form of β-catenin.
The second group of input data was experimentally obtained from measurements of rates of dissociation of protein complexes. Binding constants were calculated based on the assumption that association rates approached that of the diffusion limits for a typical protein in an aqueous solution. The ratio K17
= 10 of the dissociation constants characterizing the binding of β-catenin to APC and APC*/axin*/GSK3β, respectively, is based on previous experimental results demonstrating that β-catenin has a 10-fold lower affinity for nonphosphorylated compared to phosphorylated APC (Salic et al. 2000
). In addition, we have shown experimentally (unpublished data; see Materials and Methods) that phosphorylated β-catenin dissociates from axin more rapidly (reaction 10) than nonphosphorylated β-catenin. Once phosphorylated, β-catenin will thus dissociate from the axin complex and undergo polyubiquitination and proteolysis.
The third group of input data consists of the two concentration ratios in the Destruction Core Cycle for complexes that either contain or lack β-catenin. The concentration ratio for the complexes that lack β-catenin is represented by the ratio of its phosphorylated versus nonphosphorylated forms and reflects the relative activities of its kinase(s) and phosphatase(s), respectively. By contrast, the concentration ratio of the two β-catenin-containing degradation complexes represents the relative activities of β-catenin phosphorylation and the rate of release of phosphorylated β-catenin from the complex. These parameters were chosen rather arbitrarily to indicate roughly equal kinase and phosphatase activities and yielded realistic values for the overall fluxes, given the known concentrations and kinetic rate constants.
The fourth group of data includes the steady state flux ν11
for the degradation of β-catenin via the Wnt pathway and the flux ratio ν13 / ν11
describing the extent to which β-catenin is degraded via non-Wnt mechanisms (e.g., via Siah-1 and presenilin [Liu et al. 2001
; Matsuzawa and Reed 2001
; Kang et al. 2002
]). We have now measured this Wnt pathway–independent degradation in Xenopus
extracts (see Materials and Methods; value shown in ).
The final group of input data consists of the characteristic time constant (τ) of selected processes. This is the time it takes for the concentration to drop to 1/e of its initial value. These characteristic times include τK.P
) for the kinase/phosphatase cycle that mediates phosphorylation/dephosphorylation of both APC and axin in the degradation complex (steps 4 and 5), τGSK3β.ass
= 1 /
) for the binding equilibrium of GSK3β with the APC/axin complex (step 6), and τax.deg
for axin degradation (step 15). Values for the rate of axin degradation were determined directly from experiments performed in Xenopus
egg extracts (unpublished data). Experiments to determine the rate of APC and axin dephosphorylation (τK.P
≈ 2.5 min) were performed using in vitro 32
P-labeled recombinant APC and axin. Radiolabeled proteins were added to Xenopus
egg extracts, and the loss of radioactivity over time was assessed by SDS-PAGE and autoradiography (Salic et al. 2000
). The legend to contains the values of rate constants calculated from the input quantities using the described system of equations. The values of all variables in the reference state are listed in the first column of . These values represent the steady state solutions of system equations using the data in as input quantities with the value of Wnt set at Wnt
Steady-State Concentrations of Pathway Compounds in the Reference State and in the Standard Stimulated State
Using the reference state as a starting point, we consider other stationary states that are attained when the pathway is permanently stimulated. To describe the strength of Wnt stimulation, we introduce a dimensionless quantity W = Wnt/Wnt0 that represents the ratio of the Wnt concentration with respect to its concentration Wnt0 in a “standard” stimulated (signaling) state. W = 0 and W = 1 characterize the reference state and a standard stimulated state, respectively, with the hyperstimulated state defined as W > 1. In order to calculate concentrations in the standard stimulated state, additional input quantities are required. These include the ratio of the active and inactive forms of Dsh (Dsha/Dshi), the relation between non-Dsh-mediated and Dsh-mediated release of GSK3β from the destruction complex (the flux ratio ν−6/ν3), and the characteristic time for the Dsh activation/inactivation cycle (τDsh.act). These values are not at present measurable. The values for these input quantities are listed in the legend of . In a later section, we analyze the effects of changes in these additional input quantities.
By setting W
= 1 and fixing all other parameters, we arrive at steady-state solutions of the systems equations (see Dataset S1
, Equations. [A-1]–[A-15]), which yield the numerical variables for the standard stimulated state (listed in the second column of ). A comparison of this state with the reference state shows that the concentration of free nonphosphorylated β-catenin increases by a factor of approximately 6, from 25 to 153 nM. Upon Wnt stimulation, the free phosphorylated β-catenin concentration decreases by 8%, from 1 nM to 0.92 nM. The increase in β-catenin levels reflects the decrease in its degradation caused by the reduction in the ability of GSK3β to phosphorylate it. The concentration of the β-catenin/TCF complex increases by a factor of 1.8. The large increase in β-catenin concentration shifts the binding equilibrium between APC and β-catenin and the concentration of free APC falls slightly. Total axin concentration decreased by a factor of 2.7 upon Wnt stimulation since addition of Dsh decreases the concentration of the various axin containing complexes. Remarkably, the steady-state concentration of free axin is constant during the transition from W
= 0 to W
= 1. This is due to the fact that under steady-state conditions, the rate of axin synthesis equals its degradation; the rate of synthesis (ν14
) is a fixed value and the rate of degradation depends solely on the concentration of free axin (and independent of other parameters such as binding constants and strength of the Wnt signal).
As expected, simulations of increasing Wnt activation (0 ≤ W
≤ 1.4) on the steady-state concentrations of β-catenin and axin reveal a nearly hyperbolic saturation of increasing concentrations of nonphosphorylated and total β-catenin with increasing strength of Wnt stimulation. Furthermore, Wnt stimulation affects the steady-state concentrations of axin and β-catenin in an opposite direction (see Figure S1
β-Catenin Degradation: Comparison of Theory and Experiment
To test whether the mathematical model represented the Wnt pathway under a variety of conditions, we ran through a series of simulations, all of which used the same set of parameters. From these we calculated simulated timecourses for β-catenin degradation under a range of different conditions (increased axin concentration, increased Dsha
concentration, inhibition of GSK3β, increased TCF concentration) (A). We then tested the results using the previously described biochemical system (Salic et al. 2000
; Lee et al. 2001
), adding purified proteins or compounds at t =
0 (B). Simulations and experimental results are each shown as plots of total β-catenin concentration versus time. The agreement between theory and experiment is excellent.
Kinetics of β-Catenin Degradation: Simulation and Experimental Results
The straight line for t < 0 represents the reference state. The simulated reference state curve (A, curve a) for β-catenin degradation is calculated for t > 0, at which there is an absence of protein synthesis for axin (ν14 = 0) and β-catenin (ν12 = 0). This reference curve is in close agreement with our experimental data (B, curve a′) with identical half-lives for β-catenin degradation (theoretical value of t½ = 60.2 min versus experimental value of t½ = 60 min). We examined a new state, where we have increased the amount of endogenous axin (0.02 nM) by 0.2 nM. As shown in A, curve b, the additional axin markedly accelerated β-catenin degradation (t½ = 11.8 min) in agreement with the experimentally obtained values (B, curve b′; t½ = 12 min). Theoretically, the effect of axin on β-catenin degradation is primarily due to the large concentration difference between the two scaffold proteins, APC and axin. Owing to the high concentration of APC, an increase in axin concentration results in a sharp increase in the concentration of the APC/axin complex, thereby accelerating β-catenin binding to the destruction complex.
Curve d in shows the effect of inhibiting GSK3β on β-catenin degradation. This effect is produced in the simulation by inhibiting GSK3β activity (steps 4 and 9). Only a small fraction of β-catenin (phosphorylated β-catenin) is available for degradation after complete inhibition of β-catenin phosphorylation (step 9), so inhibition is rapid. This is in complete agreement with our experimental data in which degradation is essentially blocked after inhibiting GSK3β activity by lithium (B, curve d′). Curve e in A predicts that β-catenin degradation is strongly inhibited after the addition of 1 μM TCF. Previously we have shown that β-catenin is sequestered by TCF, thereby resulting in a significant decrease in free β-catenin (Lee et al. 2001
). The addition of TCF would be expected to decrease the rate of β-catenin phosphorylation (step 9) and subsequently β-catenin degradation. This is also seen experimentally (B, curve e′).
The immediate inhibition by LiCl is in contrast with the action of Dsh that inhibits only after a significant delay. We were intrigued by the theoretical biphasic degradation curves of β-catenin in the presence of Dsha, as well as the experimental support for it (A and 2B, curves c and c′). In both cases, there is an initial rapid decrease in β-catenin in the first 30 min to 1 h, followed by a much slower decrease. Such a feature should allow us to distinguish mechanistic details of complex formation. Experimentally, the biphasic nature of Dsh activity is not due to a delay in Dsh activation upon its addition to the Xenopus extracts since we see the same effect with Dsh protein that has been “activated” with extracts prior to its use in our degradation assay. As shown in , the characteristic time τK.P of phosphorylation and dephosphorylation of APC and axin in the destruction complex is relatively slow (2.5 min), and it therefore takes 5 min for 75% of the complex to be dephosphorylated. If Dsha acted only on the dephosphorylated complex (through step 3) to remove GSK3β and thus block phosphorylation of the complex, then we would predict the biphasic kinetics shown in A, curve c. These data suggest that Dsh inhibits the phosphorylation of the scaffold complex by GSK3β, but does not inhibit the phosphorylation of β-catenin. When Dsh binds, the complex can go around many times binding and phosphorylating β-catenin before it dissociates and is inhibited by Dsh. One hour after the addition of Dsh, β-catenin degradation is significantly inhibited due to the removal of a significant pool of GSK3β from the degradation complex over time (through the action of Dsh). As a result, the scaffold protein axin is dephosphorylated by the phosphatase (step 5) that remains bound to the degradation complex. Dephosphorylated axin is rapidly ubiquitinated and degraded when the β-catenin degradation normally stops. The small decrease in β-catenin levels in , curve c, after a 1 h incubation with Dsh, is due to degradation of β-catenin via nonWnt pathway mechanisms (see ) that we have incorporated into our model.
To test this prediction beyond consistency with experimental data, we performed an experiment in which Dsh was either preincubated with extract before or added at the same time as radiolabeled β-catenin (). If β-catenin and Dsh are added at the same time, there is an initial rapid loss of β-catenin (, curve b) followed by pronounced inhibition of degradation after 1 h. This initial rapid loss is consistent with Dsh acting on a subpopulation of degradation complexes (presumably the unphosphorylated forms). Strikingly, preincubation with Dsh prior to the addition of radiolabeled β-catenin (, curve a) results in immediate action of Dsh. We interpret this result to simply reflect the fact that over time in the preincubated extract Dsh can remove GSK3β from the degradation complexes, thereby enhancing the activity of the phosphatase and, as a result, promoting the degradation of axin and inhibition of β-catenin degradation. The small decrease in β-catenin levels at t > 2 h in both curves a and b again suggests the existence of a slow degradation process mediated by non-Wnt pathway mechanisms.
Preincubation of Dsh in Xenopus Egg Extracts Abolishes the Lag in Dsh Activity
Clues to Axin Activity from Its Very Low Cellular Concentration
In establishing quantities for our model in , we found that the axin concentration (20 pM) is much lower than the concentration of the other major components (β-catenin, 35 nM; APC, 100 nM; Dsh, 100 nM; and GSK3β, 50 nM). This unusual finding suggests that the function of the Wnt signaling system may actually depend on a low axin concentration. Our theoretical predictions for the effects of axin, GSK3β, and Dsh on the half-lives of β-catenin are shown in A and 4B, respectively. At zero concentration of Dsh, doubling the concentration of axin (from the reference state, indicated as 0, to a state where the concentration has been increased by 0.02 nM) causes a 50% drop in the half-life of β-catenin. By contrast, a doubling of the GSK3β concentration only decreases the half-life of β-catenin by 10%. The small effect of GSK3β is predicted to be due to the fact that only a limited amount of axin can be recruited to the degradation complex through binding to additional GSK3β. On the other hand, increased axin concentrations are immediately translated into an increased concentration of the destruction complex, because the concentrations of APC and GSK3β are high. Changing the concentration of either GSK3β or of axin should also change the amount of Dsha required to inhibit β-catenin degradation, but the pathway is much more sensitive to axin concentration than it is to GSK3β concentration. In the presence of high concentrations of axin, the effect of Dsha should be blocked; high concentrations of axin will lead to high concentrations of the phosphorylated destruction complex no matter what level of Dsha activity is present. High levels of the destruction complex will require even higher levels of Dsh to overcome the inhibition. The interaction between Dsha and GSK3β is similar in principle: Dsh-mediated release of GSK3β (step 3) from the degradation complex can simply be reversed by sufficiently high concentrations of GSK3β (step 6). In this case, however, the effect is small. Thus, axin blocks the action of Dsh so effectively that it renders the Dsh pathway inoperable.
The Effect of Dsh versus Axin or GSK3β on the Half-Life of β-Catenin in Xenopus Extracts
In C and 4D, we studied experimentally the dose-dependent effects of Dsh, GSK3β, and axin on β-catenin degradation. These curves represent β-catenin half-lives for various concentrations of axin (C) and GSK3β (D) with varying concentrations of Dsh. The results are qualitatively similar to those predicted by the model. As expected, β-catenin degradation is inhibited by increasing Dsh concentration and stimulated by increasing the concentration of either axin or GSK3β. There are, however, two pronounced differences in the effects of axin and GSK3β on Dsh inhibition. Whereas axin activates β-catenin degradation over a wide range of Dsh concentrations (C), the effect of GSK3β becomes significant only at high concentrations of Dsh (D). Furthermore, the inhibitory effect of Dsh can be almost completely blocked by high concentrations of axin (10 nM). In contrast, GSK3β (1 μM) can only partially inhibit the strong inhibitory effect of Dsh on β-catenin degradation.
Our experimental results, however, show a smaller effect on the half-life of β-catenin degradation at high concentrations of Dsh as GSK3β levels are increased. Also, the concentrations of added axin in the theoretical curve and the experimental curves are very different. The quantitative difference between the model and experimental may simply reflect the fact that the specific activity of our GSK3β and axin preparations (purified from Sf9 cells and bacteria, respectively) may be low and that a significant fraction of the recombinant proteins may not be active. Alternatively, the low activity of GSK3β may point to an unidentified inhibitory activity present in our Xenopus egg extracts.
Effects of Dynamic Changes in Protein Concentrations
The dependence of flux on the concentration of a pathway component is a measure of how much the flux is sensitively controlled by that component. In metabolic control theory, the normalized concentration-dependent parameters of the total flux known as control coefficients have been very useful in defining the characteristics of pathways (Heinrich and Rapoport 1974
; Fell 1997
). Similarly, in the analysis of bacterial chemotaxis, the response of a behavioral parameter as a function of changes in specific kinetic rates has been termed robustness (Alon et al. 1999
). Such terms are rarely measured in signal transduction.
To determine the effects of changes in the levels of Wnt pathway components, we analyzed how the flux (β-catenin degradation) changes with changes in the concentrations of APC0
, and TCF0
(see Figure S2
We chose to focus on the effects of changes in the concentrations of pathway components in the reference state, because similar effects were also seen for the stimulated state. Recently, we investigated a new and important property of the Wnt pathway, namely that the degradation of axin (reaction 15) is dependent on APC (unpublished data). The degradation rate of axin is mathematically expressed in the following manner:
represents a half-saturation constant for the activating effect of APC.
The theoretical effect of APC on the concentrations of both β-catenin and axin is shown in , where we considered independently the effect of APC-mediated degradation of axin (“with regulatory loop” where Equation  is applied) or the absence of such an effect (where the linear rate equation ν15 = k15 axin is applied). With APC-mediated axin degradation, β-catenin degradation is affected very little by changes in the concentration of APC (25% decrease with a 2-fold increase in APC concentration). This resistance to changes of β-catenin levels upon changes in APC concentration is due to the APC-dependence of axin degradation (see and Equation ). Decreasing the concentration of APC inhibits the degradation of axin, thereby promoting the formation of the degradation complex. As shown in , in the absence of the regulatory loop, axin degradation is APC independent, homeostasis is lost, and β-catenin levels are greatly upregulated with decreasing APC concentrations. A comparison of the curves that represent the dependence and independence of axin degradation on APC (dashed lines in ) indicates that the regulatory loop acts in such a way that the normally inhibitory effect on β-catenin degradation as a result of lowering the concentration of APC is counteracted by an increase in axin levels.
Effect of the Regulatory Loop for Axin Degradation
We have also simulated the effects of changes in the rate of β-catenin (ν12
) and axin (ν14
) synthesis on both β-catenin and axin levels (see Figure S3
). Interestingly, changing the level of axin or β-catenin affects the concentration of the other component in different ways. An increase in the synthesis of axin results in a decrease in β-catenin, whereas increasing β-catenin synthesis leads to an increase in axin levels. This latter effect contrasts with effects observed upon changes of other parameters (see Figure S2
) that affect the concentrations of axin and β-catenin in opposite directions.
Transient Stimulation of the Pathway
Wnt stimulation in vivo is transient, likely due to receptor inactivation/internalization and/or other downregulatory processes.
We model transient Wnt stimulation by an exponential decay:
where the reciprocal of λ
represents the characteristic lifetime τW
of receptor stimulation and t0
denotes the onset of signaling. The concentration changes of all other pathway compounds resulting from Wnt stimulation can be calculated by numerical solution of the system equations (see Dataset S1
), with initial values of the variables corresponding to the reference state.
Regulating axin turnover is important for Wnt signaling. Wnt-stimulated axin turnover has been reported in cultured mammalian cells (Yamamoto et al. 1999
) and in Drosophila
(Tolwinski et al. 2003
). In a future paper we will show that axin turnover is affected inversely to β-catenin turnover by phosphorylation by GSK3β. Here we show theoretically that this regulated axin turnover sharply affects the dynamics of the response. shows the time-dependent behavior of the total concentration of β-catenin and the total concentration of axin upon transient Wnt stimulation. The concentration of β-catenin increases transiently and then returns to its initial value. In contrast, the concentration of axin is temporarily downregulated. Further analysis of reveals that the amplitude of the β-catenin signal upon transient stimulation is significantly lower than the steady-state concentration upon permanent stimulation (W =
1; see Figure S1
). The curves a and a′ in are calculated for the reference values of the rate of axin synthesis and of the rate constant of axin degradation, whereas the curves b and b′ and the curves c and c′ are obtained for the case where both parameters are increased by a factor of 5 and decreased by a factor of 5, respectively. Under these conditions, both the degradation rate and the synthesis rate are altered by the same factor, thus maintaining essentially identical steady-state concentrations of axin. As a result, the steady-state concentrations of axin are the same in the unstimulated condition (W =
0) and after diminution of the Wnt signal; however, during active signaling, the differences in the dynamic nature of signal output at differing rates of axin turnover are dramatically revealed.
Timecourse of β-Catenin and Axin Concentrations Following a Transient Wnt Stimulation
Interestingly, an increase in the turnover rate of axin leads to higher amplitudes and shorter durations of the β-catenin signal. This can be explained by the faster degradation of axin after its Dsh-mediated release from the destruction complex.Thus, β-catenin degradation is effectively inhibited for a certain time period due to a reduced availability of the scaffold axin.
Since the steady-state concentration of free axin remains unchanged (rate of axin synthesis equals the rate of its degradation) during the transition from W = 0 to W = 1, a fast axin turnover favors rapid replenishment of the axin pool after the decline of the Wnt stimulus and, in this way, fast recovery of the destruction complex. This explains why the β-catenin signal is not only amplified, but becomes more spike-like. Increasing the turnover rate of axin affects the response of axin to temporary Wnt stimulation in a similar way as the response of β-catenin; i.e., the signal is amplified and sharpened (). Closer inspection of reveals that the axin response precedes the β-catenin response. For example, in the reference case, the β-catenin concentration reaches its maximum at about 260 min (curve a), whereas the minimum of the axin concentration is reached at 130 min (curve a′). This effect can be understood by observing that it is the lowering of the axin concentration that decreases the concentration of the destruction core complexes; in turn, this stabilizes β-catenin.
Mechanistic Differences between APC and Axin as Scaffolds
As the axin concentration is several orders of magnitude lower than that of the other components in the degradation pathway (see ), we decided to test the effect of increasing axin levels (up to, equal to, and greater than the concentrations of other components in the pathway). To do this, we had to extend the model to include additional reactions, marked in blue in ; these had previously been neglected due to the very low axin concentrations. High axin concentrations affect most prominently the formation of the β-catenin/axin complex. Assuming a realistic value for the β-catenin–axin dissociation constant (K18 = 1 nM), a moderate increase in axin concentration should theoretically accelerate β-catenin degradation, whereas a much higher concentration should result in inhibition of β-catenin degradation, due to the formation of partial complexes on axin. A more extensive analysis of β-catenin half-lives over a range of axin concentrations shows such a biphasic curve (A, curve b). These effects can also be seen experimentally in extracts (B), where 10 nM axin accelerates and 300 nM axin inhibits β-catenin degradation. The t½ decrease for low amounts of added axin can be easily explained by the fact that greater amounts of APC and GSK3β can be recruited to form the destruction complex. As a result, the t½ decreases from 60 min to t½ = 3 − 4 min. The inhibitory effect of axin becomes apparent only for axin concentrations approaching that of the other components. As shown in A, the effect of axin binding only to GSK3β (K19 = 1 nM, K18 → ∞) only becomes inhibitory at higher than micromolar concentration (curve c), whereas the combined effect of binding to both β-catenin and GSK3β (K18 = 1 nM, K19 = 1 nM) shows inhibition at less than 500 nM (curve d). If, however, we model an ordered process of binding to axin, then abortive inhibitory complexes cannot form. We show this in A. Here there is no separate binding of axin to β-catenin or GSK3β. In this case, there is no increase in the t½ at high axin concentrations (A, curve a).
Effects of Increasing Axin Concentration on β-Catenin Degradation
We also examined theoretically the effects of increasing APC concentration on the half-life of β-catenin, as shown in . The black curve corresponds to a nonordered mechanism, such as that found in axin, in which the β-catenin–APC dissociation constant (reaction 17) is low. The inhibitory effect of APC at high concentrations is due to its β-catenin buffering activity. The green curve corresponds to an ordered mechanism and reflects a high β-catenin–APC dissociation constant (high K17
). In this case, increasing concentrations of APC greater than the reference concentrations does not lead to inhibition of β-catenin degradation even at very high concentrations of APC. In cultured cells, overexpression of APC stimulates β-catenin degradation (Munemitsu et al. 1995
; Papkoff et al. 1996
). Unfortunately, we are presently unable to express full-length APC in Xenopus
egg extracts to measure the effects of high levels in the extract system.
Effects of APC Concentrations on β-Catenin Degradation
β-Catenin can also be degraded by nonaxin-dependent mechanisms, which include Siah-1 and presenilin-mediated degradation. Though they are expected to contribute very little to the total flux through the pathway, the nonaxin-dependent processes may have very important influences under certain conditions. In our Xenopus system, these alternative pathways do not contribute greatly to the half-life of β-catenin. Experimentally, we have measured only a 1.5% contribution to total β-catenin degradation such that the half-life of β-catenin is 45 h when the axin-dependent processes are inhibited. If in some situations the nonaxin-dependent degradation contributed 10% to the flux, the half-life would be 6.3 h (k13 = 1.83 · 10−3 min−1). The alternative pathways have very little effect on the half-life of β-catenin at normal and supranormal concentrations of APC. However, the effect of these alternative pathways becomes much more prominent when the APC concentration is lowered, a situation that may be significant under pathological conditions. As seen in A, when APC levels are at 50% of their normal concentration, there are dramatic differences in β-catenin concentration, depending on whether the alternative degradation pathway contributes to 1.5% or 10% of the total β-catenin degradation activity. The importance of the regulatory loop involving APC-mediated axin degradation is shown in B. In the absence of the regulatory loop, a significant inhibition of APC levels would strongly inhibit axin degradation, leading to a large increase in β-catenin levels. The control of β-catenin would be very brittle in this circumstance. However, by making axin degradation dependent on APC, a loss of APC would not stabilize axin levels, and the high axin levels would support continued degradation of β-catenin. This is the situation labeled “with regulatory loop” shown in B. The control of axin degradation could be a decisive factor in the response of the system to genetic or environmental effects on APC.
Effects of the Alternative β-Catenin Degradation Pathway and of Axin Degradation at Low Concentrations of APC
Control, Modular Composition, and Robustness of the Wnt Pathway
The model contains many parameters that affect the system behavior in different ways and to various extents. We can systematically investigate these parameters and look for those whose perturbation the system is most sensitive or most robust against. We focus on the concentrations of β-catenin and axin and calculate the responses in the total concentrations of these two compounds upon changes in the rates of the individual processes. For quantifying the effects of the rate constants k+i and k−i, we use control coefficients for the total concentration of β-catenin
and corresponding definitions for the control coefficients Caxin±i
for the total axin concentration. These coefficients, originally proposed for quantifying control in metabolic networks (for reviews, see Heinrich and Schuster 1996
; Fell 1997
), describe the relative changes of the concentrations of the given compounds to relative changes of the rate constants. The control coefficients for the reference state are listed in . It should be remembered that the following discussion refers to small perturbations of the reference state.
Control Coefficients for the Total Concentrations of β-Catenin and Axin and Parameters Quantifying the Sensitivity and the Robustness of the Wnt/β-Catenin Pathway
For the reference state, there are six steps exerting strong negative control on the total β-catenin concentration (Cβcati
−1). This group includes the reactions participating in assembling the destruction complex APC*/axin*/GSK3β. The corresponding parameters involve the rate constants k7
for the binding of axin to APC, k6
for the association of GSK3β to the APC/axin complex, and k4
for the phosphosphorylation of axin and APC in the destruction complex. Similar strong negative control is exerted by β-catenin binding to the phosphorylated destruction complex (rate constant: k8
), the phosphorylation of β-catenin in the destruction complex (rate constant: k9
), and the synthesis of axin (ν14
Six other reactions exert strong positive control in the reference state on the total concentration of β-catenin (concentration (Cβcati
1). To this group belong the reactions participating in the disassembly of the destruction complex APC*/axin*/GSK3β, which are described by the rate constants k−7
for the dissociation of the APC/axin complex, k−6
for the dissociation of GSK3β from the destruction complex, and k5
for the dephosphorylation of the APC and axin in the destruction complex. Other steps with a high positive control are the dissociation of β-catenin from the destruction complex (rate constant: k−8
), axin degradation (rate constant: k15
), and β-catenin synthesis (ν12
There are many reactions exerting almost no control on β-catenin levels in the reference state. This group includes binding of β-catenin to TCF and APC (k16 and k17), and the corresponding dissociation processes (k−16 and k−17; again only valid for small perturbations). Interestingly, the effects of the two β-catenin degradation processes (rate constants: k11 and k13) are also small. Calculation of control coefficients for the standard stimulated state reveals that some steps that exert no control in the reference state become important. These are the activation and inactivation of Dsh (rate constants: k1 and k2) and, more pronounced, the Dsh-mediated release of GSK3β from the destruction complex (k3). For all other processes, the signs of the control coefficients for β-catenin and axin do not change at the transition from the reference state to the standard stimulated state. The effects of parameter changes on axin are generally opposite to those on β-catenin; i.e., processes with a positive control coefficient for β-catenin have negative control coefficients for axin and vice versa. A significant exception is the synthesis of β-catenin, which exerts a positive control not only on β-catenin but also on axin, as expected from the results obtained in the last section.
Closer inspection of reveals that the values of the control coefficients for the rate constants sum up to zero. This fact is known as the summation theorem for concentration control (Heinrich and Rapoport 1974
) and is valid for all reaction networks at steady state. This result finds its explanation in the invariance of the steady-state concentrations against simultaneous change of all rate constants by the same factor. Interestingly, in the present case there are subgroups of processes whose control coefficients separately sum up to zero, indicating a modular structure of the pathway. In , the control coefficients of the different modules are separated by horizontal lines. The main four subgroups are the Dsh module (not shown in ), the kinase/phosphatase module, the β-catenin module, and the axin module. A subgroup is defined by a set of reactions where the control coefficients of the binding reactions are opposite to those of the corresponding dissociation reactions (C+i
= 6, 7, 8, 16, 17).
For those more familiar with genetic manipulation, it is more common to vary the concentrations of individual components rather than vary the rate constant of a reaction. shows the control coefficients for β-catenin and axin calculated for changes in the total concentrations of pathway components instead of the rate constants. Using the values of , the potential tumor-supressing effects (of APC, GSK3β, and axin) and potential oncogenic effects (of PP2A, TCF, Dsh, β-catenin) can be explained and quantified. It may be worth mentioning that there is no summation theorem for the control coefficients when calculated by changing total concentrations instead of rate constants. For practical reasons, it may be easier to discuss the coefficients with respect to concentration changes (); for theoretical reasons, changing rate constants are simpler to handle. We think that eventually it will also be clearest to speak about oncogenic reactions instead of oncogenic genes, especially if we are thinking of oncogenesis in response to pharmacologic or environmental perturbations. Genetic defects then can be considered in terms of changes in activity, transcription, translation, or proteolysis.
Concentration Control Coefficients for the Total Concentrations of β-Catenin and Axin Relative to Changes in the Concentrations of Pathway Components
Clearly, the robustness of a variable towards parameter changes is higher the lower the corresponding concentration control coefficient. To arrive at an estimation of the overall effects of parameter perturbations on the system as a whole, we consider first the standard deviation σ
of the control coefficients from their mean value. According to the summation theorem, the mean value of all control coefficients for a given variable is zero. Thus, we get for the standard deviation for the control coefficients of β-catenin:
where the summation is performed over all reactions, including forward and backward steps of fast equilibria. High values of σ
indicate that the given variable is on average very sensitive towards changes of rate constants. We propose to introduce a measure for the robustness ρ
of a variable towards changes of all parameters in the following way:
As σ may vary between zero and infinity, the range of ρ is confined to the interval 1≥ ρ ≥ 0. High values of ρ resulting from low σ values for the control coefficients indicate that the variable is robust against parameter perturbations. The standard deviations σ of the control coefficients and the ρ values for β-catenin and axin are presented in the last two rows of . Because many control coefficients are close to zero and the absolute values of the others hardly exceed unity, the σ values for β-catenin as well as for axin are rather small. Since all values for σ are lower than unity, a 1% change in a rate constant leads, on average, to a response of <1% in the overall level of β-catenin. The total concentration of axin is more robust against parameter perturbations than the total concentration of β-catenin, particularly in the standard stimulated state. A transition from the reference to the standard stimulated state results in a lower robustness for β-catenin and a higher robustness for axin.