 Research
 Open Access
 Published:
Examination of effects of GSK3β phosphorylation, βcatenin phosphorylation, and βcatenin degradation on kinetics of Wnt signaling pathway using computational method
Theoretical Biology and Medical Modelling volume 6, Article number: 13 (2009)
Abstract
Background
Recent experiments have explored effects of activities of kinases other than the wellstudied GSK3β, in wnt pathway signaling, particularly at the level of βcatenin. It has also been found that the kinase PKA attenuates βcatenin degradation. However, the effects of these kinases on the level and degradation of βcatenin and the resulting downstream transcription activity remain to be clarified. Furthermore, the effect of GSK3β phosphorylation on the βcatenin level has not been examined computationally. In the present study, the effects of phosphorylation of GSK3β and of phosphorylations and degradation of βcatenin on the kinetics of the wnt signaling pathway were examined computationally.
Methods
The wellknown computational LeeHeinrich kinetic model of the wnt pathway was modified to include these effects. The rate laws of reactions in the modified model were solved numerically to examine these effects on βcatenin level.
Results
The computations showed that the βcatenin level is almost linearly proportional to the phosphorylation activity of GSK3β. The dependence of βcatenin level on the phosphorylation and degradation of free βcatenin and downstream TCF activity can be analyzed with an approximate, simple function of kinetic parameters for added reaction steps associated with effects examined, rationalizing the experimental results.
Conclusion
The phosphorylations of βcatenin by kinases other than GSK3β involve free unphorphorylated βcatenin rather than GSK3βphosphorylated βcatenin*. In order to account for the observed enhancement of TCF activity, the βcatenin dephosphorylation step is essential, and the kinetic parameters of βcatenin phosphorylation and degradation need to meet a condition described in the main text. These findings should be useful for future experiments.
Background
The Wnt/βcatenin signaling pathway (named wnt pathway hereafter for simplicity) plays a significant role in cell proliferation, differentiation, and apoptosis. These have implications for aspects of cell development, stem cells and cancer [1]. Many characteristics of this pathway and its role in cell signaling have been revealed in experimental studies (for review, see for example [1], and references therein and the literature listed at http://www.stanford.edu/~rnusse/wntwindow.html). Briefly, wnt signaling enhances the level of the output signal protein, unphosphorylated βcatenin, which then binds with TCF to induce associated gene expression in the nucleus. At steady state (SS), the level of βcatenin is balanced by its synthesis and degradation. The socalled destruction cycle is a major mechanism of degradation, in which phosphorylation of βcatenin by GSK3b is a key step [2]. When the wnt signal acts on cell, wnt recruits several proteins to attenuate the reaction rate of this key step, slowing down the degradation. Therefore, βcatenin accumulates, enhancing the level of TCF/βcatenin complex and the resulting associated gene expression.
Recent advances have further illustrate how phosphorylation and dephosphorylation of major components in the wnt pathway affect the stability of βcatenin and its TCF transcription activity [3–7]. It has been found that LRP6 phosphorylates GSK3β and regulates βcatenin independently of the axin pathway [3]. PKA phosphorylates GSK3β and affects the βcatenin level in Saos2 cells [4]. βcatenin is also phosphorylated by AKT at Ser552, promoting TCF activity [4]. Furthermore, PKA phosphorylates Ser552 and Ser675 of βcatenin but this does not affect the βcatenin level in COS7 cells [5]. Moreover, CK1 phosphorylates not only axin and APC but also GSK3β and βcatenin [6]. A newly identified component, PP1, dephosphorylates axin [6]. In addition, phosphorylation of βcatenin at Ser675 by PKA attenuates βcatenin degradation, stabilizing βcatenin and enhancing TCF activity in the cells investigated [7]. In aspects of pathology, βcatenin has a role in carcinogenesis although the extent of its effect varies among cancers [8]. For example, the effect on colorectal cancer is more significant than on lung cancer. In a recent study [9], a derivative of celecoxib, derivatives of which have been extensively examined for anticancer treatment, was found to have potential for treating lung cancer. Proteomics examination showed that PKA activity has a significant effect on the wnt signaling pathway and in differentiating lung from normal cells.
In addition to experimental studies, computational studies have also aided understanding of the dynamical behaviors of this pathway and how it interacts with other pathways [2, 10–13]. The wnt pathway is one of the computationally beststudied signaling pathways [14–16]. The kinetics of βcatenin, axin, and other proteins have been examined in the wellestablished LeeHeinrich model [2], following the nomenclature in [12], which was built on the basis of experiments with Xenopus. Briefly, the LeeHeinrich model contains a number of key protein components in the wnt pathway shown in Figure 1[2]. (Note that a kinase, casein kinase 1 (CK1), was recently implicated in the activation of βcatenin phosphorylation by GSK3β [6], and this was not included in the LeeHeinrich model. This kinase is not directly related to the three effects examined in the present study. Therefore, it is not included here.) Binding of proteins, catalytic activities, phosphorylation reactions, etc. are described by their corresponding kinetic rate laws with the associated parameters, which were obtained from measurements or estimated [2]. Effects of wnt signaling and changes of activity of some components on the kinetics of the pathway can be investigated by solving kinetic rate law equations [2]. A number of the effects examined and the kinetics of components were in excellent agreement with experimental results [2].
In addition to the LeeHeinrich model of the wnt pathway itself, this model has been used and extended to examine the effect of Apc mutations on the wnt signaling pathway [10], crosstalk with the ERK pathway [11], and the interaction of axin2 proteins with the wnt pathway [12]. While many characteristics of this pathway have been elucidated both experimentally and computationally, much of its role in cell signaling and the means by which it interacts with other pathways remain to be explored.
In the present study, in the light of the recent experimental studies described above [3–7], I aimed to examine the effects of GSK3β phosphorylation, βcatenin phosphorylation by kinases other than GSK3β (referred to as βcatenin nonGSK3β phosphorylation hereafter), and βcatenin degradation on the kinetics of the wnt pathway using a computational method based on the LeeHeinrich model. These effects were not included explicitly in the LeeHeinrich model, and to my knowledge, they have not previously been examined computationally. The present computational study should elucidate how these effects affect the wnt pathway. The reaction steps of GSK3β phosphorylation, βcatenin nonGSK3β phosphorylation, and βcatenin degradation were added to the LeeHeinrich model (see added reaction steps in blue in Figure 1). The dependence of SS concentrations on the associated kinetic parameters of interest in these reaction steps was examined. Control coefficients of selected parameters and time courses of selected components with wnt signalling were calculated and examined as well. The method is described in Section II. Section III presents the results and discussion. Conclusions are given in the final section.
Methods
The COPASI program [17] was used to solve kinetic rate equations numerically in order to simulate the wnt pathway. I first built the LeeHeinrich model of the wnt pathway and reproduced the SS concentrations and several time course concentration curves, shown in Table 2 and Figure 6 in Ref. [2], respectively. The reproduced results are shown in Table 1 and the solid lines in Figure 3. The newlyadded and examined effects of GSK3β and βcatenin nonGSK3β phosphorylations were assumed to involve free GSK3β and βcatenin only. An experimental study [7] showed that these two proteins are less likely to be phosphorylated in the complexes than in the free forms. Additions of these two reaction steps along with their dephophorylation steps, assigned as reaction steps 18 and 19, respectively, and degradation of βcatenin, reaction 20, are shown in blue in Figure 1. The rate laws for these reactions are the kinetic equations for standard first order reversible (steps 18 and 19) and irreversible (step 20) reactions. βcatenin nonGSK3β phosphorylation and degradation were assumed to take place independently with unphosphorylated βcatenin and GSK3βphosphorylated βcatenin, as shown in the lower and upper parts of Figure 1, respectively. These processes were also examined separately. The effects of these added reactions steps on the kinetics of the pathway were examined.
The forward/backward reaction rate constants of GSK3β and βcatenin phosphorylations are denoted k_{G}/k_{G}' and k_{β}/k_{β}', respectively. The rate constant of βcatenin degradation is denoted k_{βdeg}. For simplicity, these kinetic parameters were all set equal to 1 min^{1} except in some cases (see Section III). The parameters of interest were then varied to examine their effects on the kinetics of the wnt pathway. Initial concentrations were values of SS concentrations used in the LeeHeinrich model [2] with additional components, GSK3β_{+}, βcatenin_{+}, and βcatenin*_{+}, with initial concentrations all set at zero. The superscript * denotes phosphorylation by GSK3β. Other phosphorylations are denoted by the subscript_{+}. The differential equations were solved for 20000 minutes, or as long as 40000 minutes, to ensure they reached SS.
To examine how selected parameters affect the SS concentration of βcatenin in a modified model, control coefficients were calculated as well. These were defined by [2]
where βcat is concentration of βcatenin and k is the parameter of interest. This coefficient was calculated numerically by varying the associated parameters by 1% and solving the kinetic equations over enough time to obtain SS concentrations, in order to calculate control coefficients as in [2]. Finally, the wnt signaling effect in a modified model (see below) was also examined with constant wnt signaling and transient wnt signaling separately, as in [2]. Initial concentrations in the modified model were obtained from computations for SS concentrations. Computed results are presented and discussed in the next section.
Results and discussion
Effects of GSK3β phosphorylation and βcatenin phosphorylations
Initially, I examined these two effects without including the βcatenin degradation step (step 20 in Figure 1) in computing βcatenin concentration, and assumed that βcatenin nonGSK3β phosphorylation involves βcatenin only (reaction step 19 in lower part of Figure 1). Computed SS concentrations of selected components and their fold changes are listed in Table 1, along with values in the original LeeHeinrich model [2]. The concentration of GSK3β was reduced to half its original value. The values of complexes with APC and axin were also reduced to approximately half their original values while βcatenin*/APC*/axin*/GSK3β, βcatenin*, and axin remained approximately the same. The unphosphorylated βcatenin shows a twofold change and significant enhancement in its absolute concentration because of its high concentration compared with most of the other components.
To examine how the kinetic parameters of the added reaction steps affect the kinetics of the wnt pathway, selected parameters were varied and the rate laws of the modified model were solved. The SS solutions were examined first. The effect of varying k_{G} on the total βcatenin level and free unphosphorylated βcatenin are listed in Table 2. The higher the k_{G} value, the higher the βcatenin level. This dependence qualitatively demonstrates the negative role of GSK3β in wnt/βcatenin signaling. This is because a decrease in unphosphorylated GSK3β level results in a decrease in the APC/axin/GSK3β complex. This complex is a central component of the βcatenin destruction cycle. Therefore, the level of βcatenin accumulates and increases. Experimentally, it was found that phosphorylation of GSK3β led to enhancement of the βcatenin level in HEK293TPα cells [18] but not HEK293 cells [19]. The discrepancy is due to differences in cell type, experimental conditions, etc. The present result is consistent with the former cell type. The GSK3β/GSK3β_{+} ratio is determined by the k_{G}/k_{G}' ratio. The dependence of the k_{G}/k_{G}' ratio on βcatenin level is shown in Figure 2, a plot of ratio of βcatenin level versus k_{G} that represents the strength of phosphorylation of GSK3β by other kinases. This plot shows an almost linear dependence. Because of the significant effect when k_{G}/k_{G}' = 1 (Table 2), this parameter set was used as a reference parameter set to analyze the effects of βcatenin nonGSK3β phosphorylation on the wnt pathway in the discussion below.
In addition to the effects of GSK3β phosphorylation, the effect of βcatenin nonGSK3β phosphorylation was examined. I first examined this effect in free unphosphorylated βcatenin (lower part of Figure 1). Analysis of pathway fluxes showed that no SS concentrations of components are affected by the k_{β}/k_{β}' ratio, excepting concentrations of βcatenin_{+}. This was confirmed by computations with several values of k_{β}/k_{β}'. This is because βcatenin is one of two components, in addition to axin, that have flux turnovers in the pathway. Because of this, addition of reaction step 19 to the LeeHeinrich model does not change the fluxin or fluxout at SS. This is in contrast to phosphorylation of GSK3β, when the k_{G}/k_{G}' ratio does affect the concentrations of components at SS. This is because the total amount of GSK3β is conserved and has no turnover. Therefore, reaction step 18 decreases the level of unphosphorylated GSK3β and increases the level of βcatenin. This shows that phosphorylation of GSK3β has a significant effect on the βcatenin, but βcatenin nonGSK3b phosphorylation has no significant effect on concentrations of components in the destruction cycle in this model. In addition to this calculation examining the effect of nonGSK3β phosphorylation of free unphosphorylated βcatenin, I also considered this effect as taking place separately at the βcatenin* shown in the upper part of Figure 1. Because the pattern of reaction steps is similar, similar results for the dependence of βcatenin*_{+} level on the kinetic parameter k_{β}/k_{β}' were expected and obtained in the computations.
In addition to SS concentrations, control coefficients of selected parameters were also calculated to understand how these parameters affect SS concentrations. Because parameter k_{G} in reaction step 18 has a significant effect on the level of βcatenin, I calculated the control coefficients of this parameter along with several selected parameters, k_{4}, k_{5}, and v_{12}. The results are listed in Table 3. The control coefficient of k_{G} was 0.449, close to half of the control coefficient of GSK3β, which is 0.89, in the original LeeHeinrich model. Other control coefficients do not differ significantly from the original LeeHeinrich model. These results showed that addition of reaction steps 18 and 19 in the modified model does not significantly change the control coefficients of βcatenin of rate constants k_{4} and k_{5} and the influx rate, v_{12}. It does significantly change in the case of k_{G} in reaction step 18.
Effect of wnt signal
In examining the wnt signaling effect, computations for the present modified model were carried out similarly to those in the previous work [2]. Computations with constant wnt signaling of W = 1 and with transient signaling were carried out separately. For constant signaling, the calculated SS concentrations of selected components when W = 1 are listed in right part of Table 4. Addition of reaction steps 18 and 19 does not significantly affect the concentration of Dsha. A noticeable change is βcatenin, which is again about twice the value obtained in the original LeeHeinrich model. This is similar to the case without wnt signaling (W = 0). This shows that the effect of constant wnt signaling in SS in the present modified model with (W = 1) is similar to the original LeeHeinrich model.
Because wnt signaling interacts with the destruction cycle directly through the the APC/axin/GSK3β complex and because GSK3β has a significant effect on the βcatenin level, I examined how varying k_{G} affected the time course of βcatenin and axin. As in [2], I used an exponential decay wnt signaling with a decay rate of 20 minutes. The time course of total βcatenin concentration is shown in Figure 3. The response of axin is similar to that in the original LeeHeinrich model, but the response of βcatenin lags behind slightly.
Effect of βcatenin degradation
Similar to the examinations of the effect of βcatenin nonGSK3β phosphorylation described above, degradation of βcatenin may involve free unphosphorylated βcatenin or βcatenin* as shown in Figure 1. To examine this effect, separate computations were carried out for degradation taking place in these two components. The behaviors of SS concentrations were qualitatively similar for these two cases when the associated parameters were varied, although the former case involved more reaction steps and the latter was more 'isolated' from the destruction cycle. Qualitative behaviors of SS concentrations can be understood by a simplified kinetic model shown in Figure 4. In this simplified model, the SS concentration of unphosphorylated βcatenin, and the sum of βcatenin and βcatenin_{+}, can be expressed by
and
respectively. The computations for the added reaction steps taking place at βcatenin*_{+} (upper part of Figure 1) showed that the dependence of βcatenin level on the kinetic parameter is qualitatively similar to the simple functions shown above. Similar results were obtained with computations for the added reaction steps at the βcatenin site (lower part of Figure 1). To elucidate the downstream TCF activity, only added reaction steps on free unphosphorylated βcatenin shown in the lower part of Figure 1 are considered here. This is because these reaction steps can alter the concentration of free unphosphorylated βcatenin, which in turn will interact with TCF to induce downstream transcription. To account for observed enhancement of TCF activity induced by increased kinases such as PKA, which phosphorylates βcatenin and inhibits βcatenin degradation [7], the dephosphorylation step is essential. In the absence of this reaction step, which means k_{β}' is virtually zero in the equation above in this paragraph, more phosphorylation of βcatenin by PKA will decrease the βcatenin level. This is because, without the dephosphorylation step the influx at SS needs to equal to the sum of both outfluxes shown in Figure 4. Increase of k_{β} will decrease the βcatenin. In addition, the k_{β}'/k_{βdeg} ratio needs to be much smaller than 1. The latter term in the denominator, the term k_{β}/(1+k_{β}'/k_{βdeg}) (when k_{βdeg}and k_{β} were varied due to the change of PKA activity), needs to dominate over k_{13} so that increased inhibition of degradation can enhance TCF activity. A computed result is shown in Figure 5.
To test this analysis further, computations with varied k_{β} values, which make the term k_{β}/(1+k_{β}'/k_{βdeg}) smaller or larger than k_{13}, were carried out as well. When k_{β} was equal to or 1 order smaller than k_{13}, the level of unphosphorylated βcatenin did not change significantly when the βcatenin_{+} degradation constant, k_{βdeg}, was varied (shown in the dashed line in Figure 5), consistent with the above analysis. When k_{β} was increased, the term k_{β}/(1+k_{β}'/k_{βdeg}) gradually dominates over k_{13}. The k_{β} value was increased up to 1.0. In the range of 0.5–1.0, a significant change in the unphosphorylated βcatenin level due to varying k_{βdeg}was seen, as shown in Figure 5 (thin solid line with open circles). The TCF/βcatenin level varied by about 1 order within the range examined in Figure 5 (thin solid line with solid circles). The TCF/βcatenin complex varied less than βcatenin because of the rapid equilibrium of the reaction βcatenin + TCF ↔ βcatenin/TCF with dissociation constant 30 nM; the total concentration of TCF was 15 nM [2]. The results then account for the observed, significant change in TCF activity described above. It is noted here that nonGSK3β phosphorylation of βcatenin by PKA takes place at Ser675 of βcatenin, which is distinct from the sites of phosphorylation by GSK3β [7]. Mutation of Ser675 to alanine stabilizes βcatenin, attenuates inhibition of βcatenin degradation, and enhances TCF activity [7]. On the basis of these observations, reaction step 20 was added to the LeeHeinrich model and is a degradation channel for βcatenin additional to those in the LeeHeinrich model. PKA was assumed to inhibit degradation at reaction step 20 only. The above analysis of Eqs. (1) and (2) is based on these assumptions. Furthermore, to examine whether the added reactions at βcatenin*_{+} affect the added reactions at βcatenin_{+} and the level of unphosphorylated βcatenin, computations with both added reaction parts present in the model were also carried out. Variations of both degradation rate constants in the added reaction steps, which may change the level of βcatenin significantly, showed that the computed results are similar to those obtained with each part computed separately in the model as shown in Figure 5. This is because, in the model, one key assumption is the rapid equilibration of reaction step 8, βcatenin/APC*/Axin*/GSK3β ↔ APC*/Axin*/GSK3β + βcatenin, with dissociation equilibrium constant of 120 nM [2]. Because of the low level of Axin and APC*/Axin*/GSK3β, on the 10^{4} and 10^{2} nM scales, respectively, the level of unphosphorylated βcatenin (on the tens of nM scale) is roughly 4 and 1–2 orders higher than the APC*/Axin*/GSK3β/βcatenin complex and the downstream phosphorylated βcatenin, βcatenin*, respectively. Therefore, the presence of the added reaction steps at βcatenin*_{+} has no significant effect on the SS level of components in the added reaction step at βcatenin_{+} or the unphosphorylated βcatenin. The added reaction steps at βcatenin_{+} alone can describe the parameter dependence of βcatenin level due to nonGSK3β phosphorylation, the succeeding degradation, and the resulting TCF activity. Finally, to verify the findings in the present computational study, it is suggested that the following be investigated in future experiments: (1) to seek the phosphatase of βcatenin_{+} and obtain its kinetic rate constant, k_{β}'. (2) to obtain the kinetic constants, k_{β} and k_{βdeg}, of the added reaction steps 19 and 20, respectively, associated with nonGSK3β phosphorylation of βcatenin. In the longrun, the model will be extended to include more unexamined effects caused by other components in the course of wnt signaling pathway research in order to draw a more complete picture.
Conclusion
The present computational study gave the following results: (1) Phosphorylation of GSK3β increases the level of βcatenin in an approximately linear fashion. (2) βcatenin nonGSK3β phosphorylation is more likely to take place at free unphorphorylated βcatenin rather than GSK3βphosphorylated βcatenin*, thus accounting for the observed enhancement of TCF activity. (3) To account for the effects of kinases such as PKA, which phosphorylates βcatenin and inhibits βcatenin degradation at the same time, the dephosphorylation step is essential. In addition, the kinetic parameters of the reactions taking place at the unphosphorylated βcatenin site need to meet the conditions described above in describing the simplified mathematical functions (1) and (2). Finally, the qualitative behaviors found in the present study should be useful for future experiments in charactering how the effects of GSK3β phosphorylation, βcatenin nonGSK3β phosphorylations, and βcatenin degradation, affect the kinetics of the wnt pathway, and can be verified in future experiments.
Abbreviations
 GSK3:

Glycogen synthase kinase 3
 PKA:

Protein kinase A
 TCF:

Tcell factor.
References
 1.
Fuerer C, Nusse R, ten Berge D: Wnt signalling in development and disease. Max Delbrück Center for Molecular Medicine meeting on Wnt Signaling in Development and Disease. EMBO Rep. 2008, 9: 134138. 10.1038/sj.embor.7401159.
 2.
Lee E, Salic A, Kruger R, Heinrich R, Kirschner MW: The Roles of APC and Axin Derived from Experimental and Theoretical Analysis of the Wnt Pathway. PLoS Biol. 2003, 1: 116132.
 3.
Cselenyi CS, Jernigan KK, Tahinci E, Thorne CA, Lee LA, Lee E: LRP6 transduces a canonical Wnt signal independently of Axin degradation by inhibiting GSK3's phosphorylation of betacatenin. Proc Nat Acad Sci. 2008, 105: 80328037. 10.1073/pnas.0803025105.
 4.
Suzuki A, Ozono K, Kubota T, Kondou H, Tachikawa K, Michigarni T: PTH/cAMP/PKA Signaling Facilitates Canonical Wnt Signaling Via Inactivation of Glycogen Synthase Kinase3 beta in Osteoblastic Saos2 Cells. J Cell Biochem. 2008, 104: 304317. 10.1002/jcb.21626.
 5.
Taurin S, Sandbo N, Qin YM, Browning D, Dulin NO: Phosphorylation of betacatenin by cyclic AMPdependent protein kinase. J Biol Chem. 2006, 281: 99719976. 10.1074/jbc.M508778200.
 6.
Huang H, He X: Wnt/[beta]catenin signaling: new (and old) players and new insights. Curr Opin Cell Biol. 2008, 20: 119125. 10.1016/j.ceb.2008.01.009.
 7.
Hino SI, Tanji C, Nakayama KI, Kikuchi A: Phosphorylation of {beta}Catenin by Cyclic AMPDependent Protein Kinase Stabilizes {beta}Catenin through Inhibition of Its Ubiquitination. Mol Cell Biol. 2005, 25: 90639072. 10.1128/MCB.25.20.90639072.2005.
 8.
Mazieres J, He B, You L, Xu Z, Jablons DM: Wnt signaling in lung cancer. Cancer Lett. 2005, 222: 110. 10.1016/j.canlet.2004.08.040.
 9.
Tan YH, Lee KH, Lin T, Sun YC, HsiehLi HM, Juan HF, Wang YC: Cytotoxicity and Proteomics Analyses of OSU03013 in Lung Cancer. Clin Cancer Res. 2008, 14: 18231830. 10.1158/10780432.CCR071806.
 10.
Cho KH, Baek S, Sung MH: Wnt pathway mutations selected by optimal [beta]catenin signaling for tumorigenesis. FEBS Lett. 2006, 580: 36653670. 10.1016/j.febslet.2006.05.053.
 11.
Kim D, Rath O, Kolch W, Cho KH: A hidden oncogenic positive feedback loop caused by crosstalk between Wnt and ERK Pathways. Oncogene. 2007, 26: 45714579. 10.1038/sj.onc.1210230.
 12.
Wawra C, Kuhl M, Kestler HA: Extended analyses of the Wnt/betacatenin pathway: Robustness and oscillatory behaviour. Febs Lett. 2007, 581: 40434048. 10.1016/j.febslet.2007.07.043.
 13.
Cho KH, Shin SY, Kim HW, Wolkenhauer O, McFerran B, Kolch W: Mathematical Modeling of the Influence of RKIP on the ERK Signaling Pathway. Computational Methods in Systems Biology. Edited by: Priami C. 2003, Berlin: Springer, 127141.
 14.
Kestler HA, Wawra C, Kracher B, Kuhl M: Network modeling of signal transduction: establishing the global view. Bioessays. 2008, 30: 11101125. 10.1002/bies.20834.
 15.
Kestler HA, Kuhl M: From individual Wnt pathways towards a Wnt signalling network. Philos Trans R Soc BBiol Sci. 2008, 363: 13331347. 10.1098/rstb.2007.2251.
 16.
Klipp EL, Wolfram : Mathematical modeling of intracellular signaling pathways. BMC Neurosci. 2006, 7: S1010.1186/147122027S1S10.
 17.
Hoops S, Sahle S, Gauges R, Lee C, Pahle J, Simus N, Singhal M, Xu L, Mendes P, Kummer U: COPASI–a COmplex PAthway SImulator. Bioinformatics. 2006, 22: 30673074. 10.1093/bioinformatics/btl485.
 18.
Yan WL, Tai HH: Glycogen synthase kinase3 phosphorylation, Tcell factor signaling activation, and cell morphology change following stimulation of thromboxane receptor alpha. J Pharmacol Exp Ther. 2006, 317: 267274. 10.1124/jpet.105.096826.
 19.
Fang XJ, Yu SX, Tanyi JL, Lu YL, Woodgett JR, Mills GB: Convergence of multiple signaling cascades at glycogen synthase kinase 3: Edg receptormediated phosphorylation and inactivation by lysophosphatidic acid through a protein kinase Cdependent intracellular pathway. Mol Cell Biol. 2002, 22: 20992110. 10.1128/MCB.22.7.20992110.2002.
Acknowledgements
The author thanks NSC and NTNU (Top Research Grant) for providing support. My gratitude also goes to the Academic Paper Editing Clinic, NTNU.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Sun, Y. Examination of effects of GSK3β phosphorylation, βcatenin phosphorylation, and βcatenin degradation on kinetics of Wnt signaling pathway using computational method. Theor Biol Med Model 6, 13 (2009). https://doi.org/10.1186/17424682613
Received:
Accepted:
Published:
Keywords
 Reaction Step
 Steady State Concentration
 Control Coefficient
 Present Computational Study
 Destruction Cycle