Bell-shaped and ultrasensitive dose-response in phosphorylation-dephosphorylation cycles: the role of kinase-phosphatase complex formation
© Szomolay and Shahrezaei; licensee BioMed Central Ltd. 2012
Received: 30 August 2011
Accepted: 24 April 2012
Published: 24 April 2012
Phosphorylation-dephosphorylation cycles (PDCs) mediated by kinases and phosphatases are common in cellular signalling. Kinetic modelling of PDCs has shown that these systems can exhibit a variety of input-output (dose-response) behaviors including graded response, ultrasensitivity and bistability. In addition to proteins, there are a class of lipids known as phosphoinositides (PIs) that can be phosphorylated. Experimental studies have revealed the formation of an antagonistic kinase-phosphatase complex in regulation of phosphorylation of PIs. However, the functional significance of this type of complex formation is not clear.
We first revisit the basic PDC and show that partial asymptotic phosphorylation of substrate limits ultrasensitivity. Also, substrate levels are changed one can obtain non-monotonic bell-shaped dose-response curves over a narrow range of parameters. Then we extend the PDC to include kinase-phosphatase complex formation. We report the possibility of robust bell-shaped dose-response for a specific class of the model with complex formation. Also, we show that complex formation can produce ultrasensitivity outside the Goldbeter-Koshland zero-order ultrasensitivity regime through a mechanism similar to competitive inhibition between an enzyme and its inhibitor.
We conclude that the novel PDC module studied here exhibits new dose-response behaviour. In particular, we show that the bell-shaped response could result in transient phosphorylation of substrate. We discuss the relevance of this result in the context of experimental observations on PI regulation in endosomal trafficking.
Biochemical networks have a modular structure . The functional modules have different dynamical and input-output properties [2, 3]. For example positive feedback loops can produce bistability while negative feedback loops filter noise. An important biochemical module in cellular signalling is a phosphorylation-dephosphorylation cycle (PDC). Phosphorylation is a common post-translational covalent modification of proteins and lipids, that is mediated by kinases and needs ATP to proceed. However, dephosphorylation is mediated by phosphatases and does not need ATP to proceed. Phosphorylation can affect binding properties, localization and activity of proteins and receptors .
Systems with phosphorylation-dephosphorylation cycles can exhibit a variety of input-output or dose-response behaviors . The level of phosphorylated substrate at steady-state is controlled by the kinase-phosphatase balance (KPB), i.e., the ratio of total active kinase to active phosphatase concentration. If the enzymes are far from saturation, the phosphorylated level of substrate is a graded function of the KPB. However, if the enzymes are saturated and are operate in zero-order regime, an ultrasensitive switch-like response is achieved where a small change in the KPB can produce a large change in the level of phosphorylated substrate . Small modifications to the structure of these biochemical modules, such as the introduction of cooperativity or product inhibition can significantly affect their dynamical properties [7, 8]. Cascades of phosphorylation-dephosphorylation cycles such as MAP kinase cascades also produce ultrasensitivity and amplification . Multisite protein phosphorylation with distributive mechanism can produce robust ultrasensitivity -. Systems with multiple phosphorylation sites can exhibit additional dynamical properties including multistability [12, 13]. Multisite phosphorylation can also result in robust ultrasensitivity outside zero-order regime through local saturation [14, 15]. In addition, the order and the distributivity of phosphorylation affects the response properties of the system .
Another example of an antagonistic kinase-phosphatase pair in yeast and mammals is the Fab1 PI(5)P 5-kinase (also called PIP5K3 or ’PIKfyve’) and the Fig4 PI(3,5)P25-phosphatase (also called Sac3) -. Moreover, the existence of two new kinase-phosphatase complexes which play a role in the regulation of PI(5)P has also been hypothesized . In particular, it has been proposed that a PIK3C (class-I PI 3-kinase) and myotubularin complex regulates the interconversion between PI(3,5)P2 and PI(5)P . Less is known about the PIP5K2 4-kinases and PI(4,5)P2 4-phosphatases which regulate the interconversion between PI(4,5)P2 and PI(5)P, however, the existence of a corresponding kinase-phosphatase complex has also been speculated . The identified and hypothesized kinase-phosphatase pairs are summarized in Figure 1. In summary, there is a growing evidence that a number of kinases and phosphatases can form a complex, but the exact role of such a kinase-phosphatase duo remains to be investigated . Moreover, there is some evidence that this kind of complex formation is evolutionarily conserved which suggests functional significance . Appropriate mathematical modelling may shed light on the functional roles of these complexes.
To investigate the properties of PDC with complex formation, we need to assume enzymatic activity of the kinase-phosphatase complex. Thus, we study four different variants of the extended PDC module, where the complex exhibits kinase and/or phosphatase activity (Figure 2b-e). We observe that the module can produce an ultrasensitive response outside the zero-order regime. In addition, we observe robust bell-shaped non-monotonic dose-responses in a variant of the module where the complex has phosphatase activity. Finally, we discuss our results in the context of PI regulation and endosome trafficking.
Results and discussion
Basic model of phosphorylation-dephosphorylation cycle
where the Michaelis constants (i = 1,2) are in μ M.
The system (3)-(8) has been extensively studied over the past 30 years since the key study of zero-order ultra-sensitivity by Goldbeter and Koshland . Most studies look at the steady-state dose-response of the average phosphorylated substrate versus the kinase-phosphatase balance (K t /P t ). As a measure of phosphorylated substrate one can either look at the fraction of total phosphorylated substrate (R = ([S p ] + [S p P])/S t ) or at the fraction of free phosphorylated substrate (R ′ = [S p ]/S t ). The relevant functional form could be either free or total phosphorylated substrate depending on the specificity of substrate binding sites for phosphatases and recognized targets. In this study, we follow other works that use the fraction of total phosphorylated substrate (R) to study the dose response in the PDCs [7, 27]. An important approximation used to analyse these systems is the quasi steady-state assumption (QSSA)  which assumes that the enzyme-substrate complex concentration remains approximately constant over time (apart from a fast initial transient). Under QSSA, total (R) and free (R ′ ) follow the same trends.
Most studies assume that the kinase-phosphatase balance is changed by varying the level of kinase (K t ), while keeping the phosphatase levels (P t ) constant. This is probably the most relevant regulation that happens in cellular systems. For example, in a MAPK cascade the activity of upstream kinase is increased while effectively the level of phosphatase remains constant. However, this way of varying KPB is asymmetric with respect to R. This is why our analysis shows that for low KPB the fraction R is always close to zero (no phosphorylation of substrate), however for large KPB the fraction R can take any value smaller than one (partial asymptotic phosphorylation of substrate). There are two other ways that KPB can be changed. Another asymmetric method is to keep kinase levels constant and change the level of phosphatase. An example of this kind of regulation has been recently reported in the regulation of yeast mating pathway . A symmetric way of changing KPB is to keep the total level of kinase and phosphatase constant and change the ratio only. This is probably less relevant to cellular systems. However, this way of changing KPB will always result in full asymptotic phosphorylation of substrate at large KPB. Finally, it should be noted that while over some signalling time scales KPB might increase monotonically, over longer time scales there are typically negative feedbacks in place that would decrease KPB, e.g., by increasing the levels of phosphatases [29, 30]. In this study, we focus on short-term monotonic changes of KPB ignoring the slow negative-feedback processes.
Partial asymptotic phosphorylation of substrate and Hill numbers in the basic model
Goldbeter and Koshland have introduced analytical formulas for the response coefficient (which is defined as the ratio of the kinase concentration required to give 90% response relative to the concentration required to give 10% response; assuming constant phosphatase concentration), both using QSSA and the exact steady-state solution. These results, however, do not take into account the limit of R which in general can be smaller than one and based on which the 10% and 90% value should be calculated. Here, we have used results for R obtained analytically to introduce a new response coefficient formula and hence, to find the Hill-number (see methods). To illustrate our analysis, we have looked at the dependence of the analytically calculated n H with respect to relevant parameter ratios S t /P t ,α,λ1/λ2 (Figure 3(b)). Comparison with numerically obtained values of and n H (obtained by fitting a Hill function; not shown) confirms our analytical results. As expected, n H is very large at high substrate to phosphatase concentration ratio (zero-order regime; Figure 3(b)). Interestingly, large changes in n H are also obtained when α and λ1/λ2 are varied. Comparing results in Figure 3(a) and Figure 3(b) suggests that high Hill numbers are only achieved when is close to zero. However, modest ultrasensitivity is present even at very low asymptotic phosphorylation levels of substrate. These results are also consistent with another work which suggests that substrate sequestration can significantly reduce ultrasensitivity .
Bell-shaped dose-response in the basic model
Most biochemical signalling that involves phosphorylation is mediated by changes of KPB by either regulating the level of active kinase or active phosphatase as explained above. However, one can imagine cases where the substrate level can change, while the KPB is relatively constant. Therefore, it is interesting to ask what kind of dose-response one can expect when the substrate levels are changed.
Model with kinase-phosphatase complex formation
In (10)-(12), the parameters κ1,λ3,λ4 and κ−1,λ−3,λ−4 correspond to the association/dissociation rates, respectively, and k3,k4 represent the catalytic reactions. Here we define a parameter ω = κ−1/κ1, similar to the concept of a ’dissociation constant’, which expresses the affinity of binding between the phosphatase P and kinase K.
We note that the the extended model (13)-(21) is too complicated to find the corresponding steady-states analytically. In the following we have used numerical integration of the kinetic equations above and variants of the QSSA to study the dose-response in this system. The ration of phosphorylated substrate in the extended model is now defined as R=([S p ] + [S p P] + [S p PK])/S t . Our goal is to determine the dose-response for R as a function of KPB in the extended model. We have mainly focused on the kinase active case and the phosphatase active case which are the two extreme scenarios (Figure 2(c)-(d)).
The mechanism of ultrasensitivity for the kinase active case is novel. In this case, when the level of K is increased, it binds to P and forms KP that acts as a kinase. Therefore, increasing levels of K has the compound effect of increasing the kinase activity and at the same time reducing the phosphatase activity, resulting in a nonlinear increase in R and an ultrasensitive response. As shown in the Figure 5, both ultrasensitivity mechanisms are independent of the enzyme saturation and can produce sharp responses in the parameter regime where the basic model cannot.
Bell-shaped dose-response curves
The active phosphatase case, when the kinase is ’off’ and the phosphatase is ’on’ in the protein complex is interesting since it can exhibit a non-monotonic response. Suppose that the affinity between P and K is low and the PK complex is a stronger phosphatase than P alone (with a lower ). In this case, when the level of K is increased initially, the KP production is small. However, the kinase activity soon takes over the phosphatase activity and R is increased close to . As K is increased, the KP production becomes more dominant, and, since KP is a stronger phosphatase, the is dropped to a lower level, hence, producing a bell-shaped response. As K t → ∞, the limit of the response curve has been derived earlier for the basic model. This measure can give an estimate of the peak level of R. However, this -formula could be used for the active P case as well. Indeed, at high K t concentrations the free phosphatase is close to zero and we effectively reduce to the basic model with only K and PK around. Hence, replacing the parameters λ1,λ−1,k1 by λ3,λ−3,k3 in (27) gives the desired result for the asymptotic level of R.
The steady-state solutions of the extended model can only be defined implicitly. Hence, we can use the total quasi-steady-state-approximation (tQSSA), which is more generally valid than the QSSA [33, 34]. The tQSSA method has been successfully applied to a model with a pair of enzymes and substrates (analogous to the basic PDC)  and to coupled PDCs . The details of the tQSSA method are explained in the Methods section. The results of tQSSA calculation is consistent with the numerical results in Figures 6(a)-(b). The fact that tQSSA but not QSSA can explain the bell-shaped response suggests that the dynamic of enzyme-substrate complex contributes to the presence of the non-monotonic response.
Possible role for complex formation in phosphoinositides regulation
We propose that complex formation can contribute to the transient regulation of PIs observed experimentally in endosome trafficking . Suppose the kinase is being recruited to the vesicle with a constant rate (slower than the time scale for the transient behaviour in Figure 8) and there is a fixed level of phosphatase on the vesicle. This produces a gradual change in KPB. If there is complex formation and the complex has phosphatase activity, in the parameter regime where we have bell-shaped dose-response as discussed above, the level of phosphorylated PI goes through a maximum as KPB is changed over time. This can produce a fast and accurate transient change in the phosphorylation of PI.
Botelho  has speculated on the possible role of enzymatic complex formation in transient regulation of PIs but he has proposed an alternative mechanism to achieve a transient behaviour. This alternative explanation assumes that there is a signaling trigger that activates the phosphatase relative to the kinase in the enzymatic complex leading to the down-regulation of phosphorylated PI on the membrane. Therefore enzymatic activity of the complex switches from kinase to phosphatase upon signalling. The signalling trigger might be the recruitment of other vesicles or GTP hydrolysis by GTPases . The Fab1-Figure 4 and Vps34-MTM complexes in PI regulation could follow this concept (Figure 1).
In this paper, we have studied the steady-state dose-response in basic PDC motif and a modified PDC motif using analytical and numerical methods. We first showed that there are parameter regimes where even at large KPB the phosphorylated ratio of substrate remains small. This is not expected from the QSSA analysis of the basic PDC, and it is a consequence of efficient phosphatase binding to a substrate. This results in modified estimates of the Hill number in zero order ultrasensitivity. In addition, we observe that partial asymptotic phosphorylation of substrate limits the ultrasensitive behavior. Over a narrow parameter regime, one can obtain a bell-shaped dose-response when the substrate level is changed for a fix KPB level. We then studied a modified PDC motif where the antagonistic kinase and phosphatase can form a complex. Depending on the activity of the kinase and phosphatase within the complex, the modified system has four different cases. We investigated the possible dose-response curves in all scenarios. We observed a novel form of ultrasensitivity arising from complex-formation that can function outside the zero-order regime. Also, we report a non-monotonic bell shaped dose-response for the phosphatase active-kinase inactive case that can function over a wide-range of parameters.
Complex formation between antagonistic enzymes have been reported in the PI regulatory network (Figure 1). Motivated by the observed bell shaped dose-response, we propose the complex formation between the kinase and phosphatase may contribute to the observed transient regulation of PIs during endosome maturation . It has been recently proposed that the complex formation observed in several kinase-phosphatase pairs in the PI regulation may enhance the temporal regulation of the PIs . Here we have shown that if the complex has enhanced phosphatase activity and negligible kinase activity, then the dose-response will be bell-shaped. This mechanism produces a transient increase in the phosphorylation level of the PIs as enzymes are recruited to the endosomes and PKB is changed over time. There is no need for additional regulators or triggering mechanisms, but only monotonic change of the PKB on the endosome by enzymatic recruitment. Experimental measurement of enzymatic activity within the complexes are still missing. Our model predicts that an enhanced phosphatase activity within the complex gives rise to bell-shaped dose-response. Also, we predict that alteration of the total level of enzymes or the affinity of enzymes in the complex will have a significant effect on the height of the bell-shaped response.
Our study illustrates that simple alterations of the regulatory motifs in biochemical networks can have a significant consequence on the response and functionality of these systems. Signalling systems are typically composed of several interlinked regulatory motifs. To ultimately understand the function of the whole system one needs to put the motifs together and investigates the coordinated dynamic behaviour of all the components as they interact with one another. In the context of the PI regulation, as shown in Figure 1, there are several PDC motifs with enzymatic complex formation. As shown here, these motifs can produce ultrasensitive or non-monotonic dose-response depending on their biochemical parameters. The challenge that remains is to quantify these parameters and explain the overall function by investigating the interplay between the dynamic of individual motifs.
Limit of the average fraction of phosphorylated substrate for K t → ∞ and S t → ∞
where is given by (25). It is shown in the Methods section (Corollary 1) that R ≤ 1 as expected.
and this concludes the proof. □
Corollary 2. If , then and R → 1 as S t → ∞. If , then [SK] → K t and R → 0 as S t → ∞.
Thus, if and otherwise. Obviously, if , then . To see that if , then , we show that [Sp]→S t as S t →∞. Indeed, note that as S t →∞. Hence, [S p ]=S t −[SK]−[S p P]−[S]→S t as S t →∞ and so when . □
The following Corollary is introduced without its proof, although, the arguments would be similar to Corollary 2.
as S t → ∞ and as S t → ∞
ii) as α → ∞ and as α → ∞
iii) If , then as λ1 → ∞ and as λ1 → ∞. If , then as λ1 → ∞ and as λ1 → ∞.
Given , we have and . Then we can obtain S K90 and S K10 from (30) and finally, and from (31). This gives the Hill number n H .
Steady-state approximation using tQSSA
The tQSSA replaces the free substrate S p as the slow variable by the total intact substrate concentration S p + S p P, while retaining the quasi-steady-state assumption for the intermediate complexes. Since the extended model has 3 products (S p ,S and PK), we introduce 2 slow variables (total S p and total PK concentrations).
The simulations of the tQSSA approach are shown in Figure 5.
The research leading to these results has received funding from the European Community’s Seventh Framework Programme FP7/2007-2013 under grant agreement no. HEALTH-F4-2008-223451. The authors would like to thank Dr. Brian Robertson for coordinating this project. The authors also would like to thank Dr. Omer Dushek for his helpful comments on this work.
- Hartwell LH, Hopfield JJ, Leibler S, Murray AW: From molecular to modular cell biology. Nature 1999, 402: C47-C52. 10.1038/35011540View ArticleGoogle Scholar
- Tyson JJ, Chen KC, Novak B: Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr Opin Cell Biol 2003, 15: 221-231. 10.1016/S0955-0674(03)00017-6View ArticleGoogle Scholar
- Alon U: An Introduction to Systems Biology: Design Principles of Biological Circuits. London: CRC Press; 2006.Google Scholar
- Salazar C, Hofer T: Multisite protein phosphorylation–from molecular mechanisms to kinetic models. FEBS J 2009, 276: 3177-3198. 10.1111/j.1742-4658.2009.07027.xView ArticleGoogle Scholar
- Ferrel JE: Tipping the switch fantastic: how a protein kinase cascade can covert graded inputs into switch-like outputs. Trends in Bioch Sci 1996, 21: 460-466. 10.1016/S0968-0004(96)20026-XView ArticleGoogle Scholar
- Goldbeter A, Koshland DE: An amplified sensitivity arising from covanlent modification in biological systems. PNAS 1981, 78: 6840-6844. 10.1073/pnas.78.11.6840View ArticleGoogle Scholar
- Salazar C, Hofer T: Kinetic models of phosphorylation cycles: a systematic approach using the rapid-equilibrium approximation for protein-protein interactions. Biosystems 2006, 83: 195-206. 10.1016/j.biosystems.2005.05.015View ArticleGoogle Scholar
- Xing J, Chen J: The Goldbeter-Koshland switch in the first-order region and its response to dynamic disorder. PLoS ONE 2008, 3: e2140. 10.1371/journal.pone.0002140View ArticleGoogle Scholar
- Ferrell JEJr, Bhatt RR: Mechanistic studies of the dual phosphorylation of mitogen-activated protein kinase. J Biol Chem 1997, 272: 19008-19016. 10.1074/jbc.272.30.19008View ArticleGoogle Scholar
- Deshaies RJ, Ferrell Jr JE: Multisite phosphorylation and the countdown to S phase. Cell 2001, 107: 819-822. 10.1016/S0092-8674(01)00620-1View ArticleGoogle Scholar
- Bluthgen N, Herzel H: How robust are switches in intracellular signaling cascades? J Theor Biol 2003, 225: 293-300. 10.1016/S0022-5193(03)00247-9View ArticleGoogle Scholar
- Markevich NI, Hoek JB, Kholodenko BN: Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades. J Cell Biol 2004, 164: 353-359. 10.1083/jcb.200308060View ArticleGoogle Scholar
- Thomson M, Gunawardena J: Unlimited multistability in multisite phosphorylation systems. Nature 2009, 460: 274-277. 10.1038/nature08102View ArticleGoogle Scholar
- Malleshaiah MK, Shahrezaei V, Swain PS, Michnick SW: The scaffold protein Ste5 directly controls a switch-like mating decision in yeast. Nature 2010, 465: 101-105. 10.1038/nature08946View ArticleGoogle Scholar
- Dushek O, van der Merwe PA, Shahrezaei V: Ultrasensitivity in multisite phosphorylation of membrane-anchored proteins. Biophys J 2011, 100: 1189-1197. 10.1016/j.bpj.2011.01.060View ArticleGoogle Scholar
- Behnia R, Munro S: Organelle identity and the signposts for membrane traffic. Nature 2005, 438: 597-604. 10.1038/nature04397View ArticleGoogle Scholar
- Matteis MAD, Godi A: PI-loting membrane traffic. Nat Cell Biol 2004, 6: 487-492. 10.1038/ncb0604-487View ArticleGoogle Scholar
- Robinson FL, Dixon JL: Myotubularin phosphatases: policing 3-phosphoinositides. TRENDS Cell Biol 2006, 16: 403-412. 10.1016/j.tcb.2006.06.001View ArticleGoogle Scholar
- Backer JM: The regulation and function of class III PI3Ks: novel roles for Vps34. Biochem J 2008, 410: 1-17. 10.1042/BJ20071427View ArticleGoogle Scholar
- Cao C, Baker JM, Wandinger-Ness A, Stein MP: Myotubularin lipid phosphatase binds the hVPS15/hVPS34 lipid kinase complex on endosomes. Traffic 2007, 8: 1052-1067. 10.1111/j.1600-0854.2007.00586.xView ArticleGoogle Scholar
- Cao C, Baker JM, Laporte J, Bedrick EJ, Wandinger-Ness A: Sequential actions of myotubularin lipid phosphatases regulate endosomal PI(3) P and growth factor receptor trafficking. Mol Biol Cell 2008, 19: 3334-3346. 10.1091/mbc.E08-04-0367View ArticleGoogle Scholar
- Lecompte O, Poch O, Laporte J: PtdIns5P regulation through evolution: roles in membrane trafficking. Trends in Bioch Sciences 2008, 33: 453-480. 10.1016/j.tibs.2008.07.002View ArticleGoogle Scholar
- Sbrissa D, Ikonomov O, Fu Z, Ijuin T, Gruenberg J, Takenawa T, Shisheva A: Core protein machinery for mammalian phosphatidylinositol 3,5-bisphosphate synthesis and turnover that regulates the progression of endosomal transport complex. J Biol Chem 2008, 282: 23878-23891.View ArticleGoogle Scholar
- Botelho RJ, J A Efe DT: Assembly of a Fab1 phosphoinositide kinase signaling complex requires the Fig4 phosphoinositide phosphatase. Mol Biol Cell 2008, 19: 4273-4286. 10.1091/mbc.E08-04-0405View ArticleGoogle Scholar
- Ho CY, Alghamdi TA, Botelho RJ: Phosphatidylinositol-3,5-Bisphosphate: No longer the poor PIP2. Traffic 2012, 13: 1-8. 10.1111/j.1600-0854.2011.01246.xView ArticleGoogle Scholar
- Mruk DD, Cheng CY: The myotubularin family of lipid phosphatases in disease and in spermatogenesis. Biochem J 2011, 433: 253-262. 10.1042/BJ20101267View ArticleGoogle Scholar
- Gomez-Uribe C, Verghese GC, Mirny LA: Operating regimes of signaling cycles: statics, dynamics, and noise filtering. PLoS Comput Biol 2007, 3: 2487-2497.Google Scholar
- Segel LA, Slemrod M: The quasi-steady-state assumption: a case study in perturbation. SIAM Review 1989, 31: 446-477. 10.1137/1031091View ArticleGoogle Scholar
- Amit I, Citri A, Shay T, Lu Y, Katz M, Zhang F, Tarcic G, Siwak D, Lahad J, amd N Amariglio JJH, Vaisman N, Segal E, Rechavi G, Alon U, Mills GB, Domany E, Yarden Y: A module of negative feedback regulators defines growth factor signaling. Nat Genet 2007, 39: 503-512. 10.1038/ng1987View ArticleGoogle Scholar
- Legewie S, Herzel H, Westerhoff HV, Bluthgen N: Recurrent design patterns in the feedback regulation of the mammalian signalling network. Mol Syst Biol 2008, 4: 190.View ArticleGoogle Scholar
- Bluthgen N, Bruggeman FJ, Legewie S, Herzel H, Westerhoff HV, Kholodenko BN: Effects of sequestration on signal transduction cascades. FEBS J 2006, 273: 895-906. 10.1111/j.1742-4658.2006.05105.xView ArticleGoogle Scholar
- Legewie S, Sers C, Herzel H: Kinetic mechanisms for overexpression insensitivity and oncogene cooperation. FEBS Lett 2006, 273: 895-906.Google Scholar
- Tzafiri AR, Edelman E: Quasi-steady-state at enzyme and substrate concentrations in the excess of the Michaelis-Menten constant. J Theor Biol 2007, 245: 737-748. 10.1016/j.jtbi.2006.12.005View ArticleGoogle Scholar
- Tzafiri AR: Michaelis-Menten kinetics at high enzyme concentrations. Bull of Math Biol 2003, 65: 1111-1129. 10.1016/S0092-8240(03)00059-4View ArticleGoogle Scholar
- A Ciliberto Capuani, Tyson JJ: Modeling networks of coupled enzymatic reactions using the total Quasi-Steady state approximation. PLOS Comput Biol 2007, 3: e45. 10.1371/journal.pcbi.0030045View ArticleGoogle Scholar
- Yeung T, Grinstein S: Lipid signaling and the modulation of surface charge during phagocytosis. Immun Rev 2007, 219: 17-36. 10.1111/j.1600-065X.2007.00546.xView ArticleGoogle Scholar
- Botelho RJ: Changing phosphoinositides “on the fly”: how trafficking vesicles avoid an identify crisis? BioEsseays 2009, 31: 1127-1136. 10.1002/bies.200900060View ArticleGoogle Scholar
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.