 Research article
 Open Access
 Published:
Bellshaped and ultrasensitive doseresponse in phosphorylationdephosphorylation cycles: the role of kinasephosphatase complex formation
BMC Systems Biologyvolume 6, Article number: 26 (2012)
Abstract
Background
Phosphorylationdephosphorylation 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 inputoutput (doseresponse) 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 kinasephosphatase complex in regulation of phosphorylation of PIs. However, the functional significance of this type of complex formation is not clear.
Results
We first revisit the basic PDC and show that partial asymptotic phosphorylation of substrate limits ultrasensitivity. Also, substrate levels are changed one can obtain nonmonotonic bellshaped doseresponse curves over a narrow range of parameters. Then we extend the PDC to include kinasephosphatase complex formation. We report the possibility of robust bellshaped doseresponse for a specific class of the model with complex formation. Also, we show that complex formation can produce ultrasensitivity outside the GoldbeterKoshland zeroorder ultrasensitivity regime through a mechanism similar to competitive inhibition between an enzyme and its inhibitor.
Conclusions
We conclude that the novel PDC module studied here exhibits new doseresponse behaviour. In particular, we show that the bellshaped 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.
Background
Biochemical networks have a modular structure [1]. The functional modules have different dynamical and inputoutput 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 phosphorylationdephosphorylation cycle (PDC). Phosphorylation is a common posttranslational 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 [4].
Systems with phosphorylationdephosphorylation cycles can exhibit a variety of inputoutput or doseresponse behaviors [5]. The level of phosphorylated substrate at steadystate is controlled by the kinasephosphatase 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 zeroorder regime, an ultrasensitive switchlike response is achieved where a small change in the KPB can produce a large change in the level of phosphorylated substrate [6]. 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 phosphorylationdephosphorylation cycles such as MAP kinase cascades also produce ultrasensitivity and amplification [5]. Multisite protein phosphorylation with distributive mechanism can produce robust ultrasensitivity [9][11]. Systems with multiple phosphorylation sites can exhibit additional dynamical properties including multistability [12, 13]. Multisite phosphorylation can also result in robust ultrasensitivity outside zeroorder regime through local saturation [14, 15]. In addition, the order and the distributivity of phosphorylation affects the response properties of the system [4].
During endosome trafficking tagging molecules that specify the identity and the fate of different vesicles need to be tightly regulated in time. Two classes of molecules that have a fundamental role as molecular tags in membrane traffic are small GTPases and phosphopinositides (PIs) [16]. PIs are a type of cellular phospholipids that can undergo phosphorylationdephosphorylation cycles. Although PIs are present at relatively low abundance, they have important roles in cellular regulation and membrane trafficking [17, 18]. In particular, PIs can get phosphorylated at positions 3, 4 and 5 of their inositol headgroup, through organelle specific kinases and phosphatases [17, 18]. Interestingly, several antagonistic kinase and phosphatase pairs regulating phosphorylation in PIs form complexes (Figure 1). Since these kinsasephosphatase complex pairs have opposing effects, the formation of this kind of complexes between them is puzzling and their functional significance is not clear. For example, the Vps34 kinase generates PI(3)P on endosomes by phosphorylating the third position of the PI inositol headgroup and is regulated by Vps15 protein kinase [19]. Cao et al. [20, 21] demonstrated that Vps34Vps15 subcomplex could also interact in vivo and in vitro with the direct antagonist MTM1 and MTMR2 phosphatases, where Vps15 mediates the interaction between the Vps34 and the myotubularin isoforms.
Another example of an antagonistic kinasephosphatase pair in yeast and mammals is the Fab1 PI(5)P 5kinase (also called PIP5K3 or ’PIKfyve’) and the Fig4 PI(3,5)P_{2}5phosphatase (also called Sac3) [23][25]. Moreover, the existence of two new kinasephosphatase complexes which play a role in the regulation of PI(5)P has also been hypothesized [22]. In particular, it has been proposed that a PIK3C (classI PI 3kinase) and myotubularin complex regulates the interconversion between PI(3,5)P_{2} and PI(5)P [22]. Less is known about the PIP5K2 4kinases and PI(4,5)P_{2} 4phosphatases which regulate the interconversion between PI(4,5)P_{2} and PI(5)P, however, the existence of a corresponding kinasephosphatase complex has also been speculated [22]. The identified and hypothesized kinasephosphatase 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 kinasephosphatase duo remains to be investigated [26]. Moreover, there is some evidence that this kind of complex formation is evolutionarily conserved which suggests functional significance [22]. Appropriate mathematical modelling may shed light on the functional roles of these complexes.
In this paper, we use mathematical modelling to investigate the doseresponse in an extended phosphorylationdephosphorylation cycle that includes complex formation between the kinase and phosphatase. In particular, we ask under what conditions the PDCs can exhibit a nonmonotonic doseresponse. To this end, we first revisit the basic PDC (Figure 2a) and show that depending on the parameters of the system it is possible to only achieve partial phosphorylation of the substrate even for large KPB. We then use this results to refine estimates of zeroorder ultrasensitivity. We further show that at a fix kinasephosphatase balance, if we change the substrate levels, it is possible to obtain nonmonotonic bellshaped doseresponse. However, this behavior is only obtained over a narrow range of parameters.
To investigate the properties of PDC with complex formation, we need to assume enzymatic activity of the kinasephosphatase complex. Thus, we study four different variants of the extended PDC module, where the complex exhibits kinase and/or phosphatase activity (Figure 2be). We observe that the module can produce an ultrasensitive response outside the zeroorder regime. In addition, we observe robust bellshaped nonmonotonic doseresponses 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 phosphorylationdephosphorylation cycle
We shall consider a system in which a substrate can exist in the unphosphorylated form S and phosphorylated form S_{ p }. The substrate could represent a protein or a PI. A kinase K and a phosphatase P mediates phosphorylation and dephosphorylation (Figure 2a). S and S_{ p }are controlled by the following enzymatic reactions:
where, the rate constants are denoted λ_{1},λ_{2} for binding an enzyme and substrate, λ_{−1},λ_{−2} for dissociation of the complex and k_{1},k_{2} for the catalytic reaction. The association rates (λ_{ i }) have units μ M^{−1}s^{−1}, whereas dissociation (λ_{−i}) and catalytic (k_{ i }) rates have units s^{−1}. The kinetic equations governing the time evolution are
This system is complemented by conservation equations
where the total enzyme K_{ t },P_{ t }and substrate S_{ t }concentrations are in μ M and are often assumed to be constants. At equilibrium,
where the Michaelis constants ${M}_{i}=\frac{{\lambda}_{i}+{k}_{i}}{{\lambda}_{i}}$(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 zeroorder ultrasensitivity by Goldbeter and Koshland [6]. Most studies look at the steadystate doseresponse of the average phosphorylated substrate versus the kinasephosphatase 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 steadystate assumption (QSSA) [28] which assumes that the enzymesubstrate 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 kinasephosphatase 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 [14]. 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 shortterm monotonic changes of KPB ignoring the slow negativefeedback processes.
Partial asymptotic phosphorylation of substrate and Hill numbers in the basic model
Under QSSA, the asymptotic limit of R for large kinasephosphatase balance (e.g., as K_{ t }→ ∞), denoted by $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$ is always one, meaning that the substrate gets fully phosphorylated. However, this is not the case in general when QSSA is not valid, as shown using the exact steadystate solution of (3)(8) (see Methods). In fact, the asymptotic phosphorylation can be very small for some parameter values as seen using the analytically calculated $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$ dependence on the relevant parameter ratios S_{ t }/P_{ t },α,λ_{1}/λ_{2} in Figure 3(a). Intuitively, the substrate is only partially asymptotically phosphorylated when the phosphatase can sequester a significant fraction of substrate by either saturating the substrate (small $\frac{{S}_{t}}{{P}_{t}}$), having higher catalytic rate (small $\alpha =\frac{{k}_{2}}{{k}_{1}}$), or having a higher affinity for substrate than kinase (large $\frac{{\lambda}_{1}}{{\lambda}_{2}}$) (Figure 3(a)) [31]. Similar results have been previously obtained for other protein modification cycles [32]. The observed limits in Figure 3(a) are $\left(\right)close="">{R}_{\infty}^{{K}_{t}}\to 1$ for S_{ t }/P_{ t }→ ∞ and $\left(\right)close="">{R}_{\infty}^{{K}_{t}}\to \alpha /(\alpha +1)$ for λ_{1}/λ_{2}→∞; these results are obtained in the Methods section.
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 steadystate 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 Hillnumber (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,λ_{1}/λ_{2} (Figure 3(b)). Comparison with numerically obtained values of $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$ 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 (zeroorder regime; Figure 3(b)). Interestingly, large changes in n_{ H }are also obtained when α and λ_{1}/λ_{2} are varied. Comparing results in Figure 3(a) and Figure 3(b) suggests that high Hill numbers are only achieved when $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$ 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 [31].
Bellshaped doseresponse 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 doseresponse one can expect when the substrate levels are changed.
To address this question we introduce asymptotic phosphorylation of substrate for large substrate levels (as K_{ t }→ ∞) denoted by $\left(\right)close="">{R}_{\infty}^{{S}_{t}}$. As shown in the Methods, when the maximum speed of kinase activity is greater than the maximum speed of phosphatase activity ($\frac{{P}_{t}}{\alpha}<{K}_{t}$), the asymptotic phosphorylation is $\left(\right)close="">{R}_{\infty}^{{S}_{t}}=1$, since kinase activity wins at large S_{ t }. For this case we observe a monotonic increasing doseresponse with mild ultrasensitivity with Hill coefficients about 2 (Figure 4(a)). More interesting behaviour is achieved for the case when the maximum speed of kinase activity is less than the maximum speed of phosphatase activity ($\frac{{P}_{t}}{\alpha}>{K}_{t}$) and the asymptotic phosphorylation is $\left(\right)close="">{R}_{\infty}^{{S}_{t}}=0$. In this case, there are specific parameters, where a bellshaped nonmonotonic doseresponse can be observed (Figure 4(a)). As S_{ t }is increased from zero, R is initially a constant that is set by the KPB. For large S_{ t }, it eventually approaches its asymptote of zero. For most parameters, the transition from constant R to zero is monotonic (Figure 4(a)). However, as shown in Figure 4(b), over a narrow region of parameters above the line P_{ t }/K_{ t }= α, it can exhibit a significant nonmonotonic bellshaped response. The bellshaped response is produced as the transition from $\left(\right)close="">{R}_{\infty}^{{S}_{t}}=0$ to $\left(\right)close="">{R}_{\infty}^{{S}_{t}}=1$ is approached at P_{ t }/K_{ t }= α (See Eq. 26 in the Methods). Since, this behaviour is only achieved with fine tuning of the parameters and therefore is not expected to be of biological relevance.
Model with kinasephosphatase complex formation
In this section we consider an extension to the basic model of phosphorylationdephosphorylation cycle (Figure 2). We assume that the phosphatase and kinase can form a single protein complex PK given through a reversible reaction (10):
Depending on which enzyme is ’on’ or ’off’ within the kinasephosphatase complex, the substrates S/S_{ p }could be further phosphorylated/dephosphorylated by PK as shown in (11) and (12):
In (10)(12), the parameters κ_{1},λ_{3},λ_{4} and κ_{−1},λ_{−3},λ_{−4} correspond to the association/dissociation rates, respectively, and k_{3},k_{4} represent the catalytic reactions. Here we define a parameter ω = κ_{−1}/κ_{1}, similar to the concept of a ’dissociation constant’, which expresses the affinity of binding between the phosphatase P and kinase K.
There are four possibilities as shown in Figure 2; both enzymes in the complex could be enzymatically inactive (Figure 2(b)), only phosphatase active (Figure 2(c)), only kinase active (Figure 2(d)) or both kinase and phosphatase active (Figure 2(e)). This system is described by the following kinetic equations
coupled with conservation equations
We note that the the extended model (13)(21) is too complicated to find the corresponding steadystates analytically. In the following we have used numerical integration of the kinetic equations above and variants of the QSSA to study the doseresponse 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 doseresponse 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)).
Ultrasensitivity
We observe ultrasensitivity for both the kinase activephosphatase inactive and the phosphatase activekinase inactive cases outside the zeroorder regime, where S_{ t }< P_{ t }(Figure 5). For the phosphatase active case, the mechanism is similar to the previously described reversible competitive inhibition, where the substrate and the inhibitor cannot bind the enzyme at the same time [5]. In this case, when the level of K is increased, initially it binds to P and forms KP that acts as a phosphatase and R remains close to zero. However, at some level of K, most of the free P will be in complex with K. Above this level a small increase in K changes the phosphorylation level of substrate (R) nonlinearly, producing ultrasensitivity.
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.
Bellshaped doseresponse 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 nonmonotonic 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 $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$). 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 $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$. As K is increased, the KP production becomes more dominant, and, since KP is a stronger phosphatase, the $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$ is dropped to a lower level, hence, producing a bellshaped 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 $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$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 λ_{1},λ_{−1},k_{1} by λ_{3},λ_{−3},k_{3} in (27) gives the desired result for the asymptotic level of R.
Bellshaped responses could be obtained both at high enzyme saturation (S_{ t }≫ P_{ t }), and at low enzyme saturation (S_{ t }≪ P_{ t }). Figures 6(a)(b) compare the response curve for the active P with the response curve for the corresponding basic model for both scenarios. The only difference is that for the low enzyme saturation we have an ultrasensitive rise to the peak of the bellshaped response, whereas for the high enzyme saturation there is a linear rise to the peak.
The steadystate solutions of the extended model can only be defined implicitly. Hence, we can use the total quasisteadystateapproximation (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) [27] and to coupled PDCs [35]. 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 bellshaped response suggests that the dynamic of enzymesubstrate complex contributes to the presence of the nonmonotonic response.
We have investigated the robustness of the bellshaped response by varying some of the key parameters. We have characterised the bellshape response by looking at the ratio of the maximum response to asymptotic phosphorylation as a function of dissociation constant of the phosphatasekinase complex $\omega =\frac{{\kappa}_{1}}{{\kappa}_{1}}$, the MichaelisMenten constant ${M}_{3}=\frac{{\lambda}_{3}+{k}_{3}}{{\lambda}_{3}}$ of the complex and the total phosphatase level P_{ t }(Figure 7). It is possible to obtain significant bellshaped responses over a large range of parameters. The response has the highest relative peak for large values of ω and P_{ t }(Figure 7(a)), however for each parameter set there is an optimal value of M_{3}(Figure 7(b)). Also, we have compared the transient time needed to approach steadystate in the extended model with active P (Figure 8). We observe that the transient time for the extended model follows closely the transient time for the basic model and it goes through a maximum at the threshold of ultrasensitivity. This is because the enzymatic complex formation is fast compared to phosphorylation and dephosphorylation.
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 [36]. 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 bellshaped doseresponse 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 [37] 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 downregulation 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 [37]. The Fab1Figure 4 and Vps34MTM complexes in PI regulation could follow this concept (Figure 1).
In Figure 9 we have compared the time evolution of phosphorylated substrate (PI) following the two alternative mechanisms mentioned above. In the first scenario, we assume a hypothetical recruitment of kinases over time to the membrane, fixed levels of phosphatase on the membrane and the active phosphatase activity in the PK complex. In the second scenario following Botelho [37], we assume a fixed level of kinase and phosphatase and we assume a switch in the enzymatic activity of the PK complex from kinase to phosphatase at t = 300 s. Both scenarios can result in fast transient phosphorylated substrate which peaks at 300 seconds, while our proposed mechanism results in faster decay as K_{ t }increases over time. While both phosphorylation profiles in Figure 9 are in qualitative agreement with the observed profiles in the process of endocytosis [36], however, the dynamic of available kinase is quite different in the two models and can be used to experimentally test our proposed hypothesis. Note that in our proposed mechanism, the asymptotic phosphorylation level of the substrate does not get to zero, however, by increasing the phosphatase activity of the complex, one can make this asymptotic phosphorylation arbitrarily small. Also, a counter intuitive aspect of our proposed model is that if the kinase levels are downregulated, then the phosphorylation of the substrate could even increase again. So, one requires no downregulation of the upstream kinase over the relevant timescales of the transient PI profiles. Ultimately, the response of the PDC module should be considered in the context of the larger PI regulation network (Figure 1) to be able to obtain a full understanding of the role of enzymatic complex formation in the endosome trafficking.
Conclusions
In this paper, we have studied the steadystate doseresponse 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 bellshaped doseresponse 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 doseresponse curves in all scenarios. We observed a novel form of ultrasensitivity arising from complexformation that can function outside the zeroorder regime. Also, we report a nonmonotonic bell shaped doseresponse for the phosphatase activekinase inactive case that can function over a widerange of parameters.
Complex formation between antagonistic enzymes have been reported in the PI regulatory network (Figure 1). Motivated by the observed bell shaped doseresponse, we propose the complex formation between the kinase and phosphatase may contribute to the observed transient regulation of PIs during endosome maturation [36]. It has been recently proposed that the complex formation observed in several kinasephosphatase pairs in the PI regulation may enhance the temporal regulation of the PIs [37]. Here we have shown that if the complex has enhanced phosphatase activity and negligible kinase activity, then the doseresponse will be bellshaped. 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 bellshaped doseresponse. 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 bellshaped 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 nonmonotonic doseresponse 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.
Methods
Limit of the average fraction of phosphorylated substrate for K_{ t }→ ∞ and S_{ t }→ ∞
We derive limiting values for the average fraction of phosphorylated substrate R = ([S_{ p }] + [S_{ p }P])/S_{ t }at steadystate as K_{ t }→ ∞ and S_{ t }→ ∞, respectively. First, we will implicitly define the steadystate [SK] as a solution of the cubic equation
The coefficients Q_{ i }(i = 1,2,3) are of the form Q_{1} = −K_{ t }+ v_{1}, Q_{2} =u_{1}K_{ t }+ v_{2}, Q_{3} =u_{2}K_{ t }with
where $\alpha =\frac{{k}_{2}}{{k}_{1}}$. It follows from (3) and (9) that α satisfies
and hence,
for P_{ t }> 0. The asymptotic expansion of [SK] with respect to powers of $\frac{1}{{K}_{t}}$ in (22) implies that the asymptotic limit of [SK], i.e., $\left(\right)close="">S{K}_{\infty}^{{K}_{t}}={\text{lim}}_{{K}_{t}\to \infty}[\mathit{\text{SK}}]$ satisfies the quadratic equation
By condition (24) we obtain that
Note that ${u}_{1}=\frac{{P}_{t}}{\alpha}+\frac{{M}_{1}+{S}_{t}}{\alpha +1}$ and ${u}_{2}=\frac{{P}_{t}{S}_{t}}{\alpha (\alpha +1)}$ do not depend on M_{2} and hence, $\left(\right)close="">S{K}_{\infty}^{{K}_{t}}$ does not change when λ_{2} or λ_{−2} is varied. Intuitively, this is because for high kinase levels the substrate will be saturated and the $\left(\right)close="">S{K}_{\infty}^{{K}_{t}}$ will not be influenced by the dissociation constant of kinasesubstrate binding. By (9) and (23) the phosphorylated substrate fraction R could be rewritten as
For K_{ t }→ ∞, the limiting value of R denoted by $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$ becomes
where $\left(\right)close="">S{K}_{\infty}^{{K}_{t}}$ is given by (25). It is shown in the Methods section (Corollary 1) that R ≤ 1 as expected.
For S_{ t }→ ∞, we also include a statement about the limiting value of R denoted by $\left(\right)close="">{R}_{\infty}^{{S}_{t}}$, which will be useful later on (for proof see Corollary 2 in the Methods section). If $\frac{{P}_{t}}{\alpha}<{K}_{t}$, then $\left[\mathit{\text{SK}}\right]\to \frac{{P}_{t}}{\alpha}$ and R → 1 as S_{ t }→ ∞, otherwise, [SK] → K_{ t }and R → 0 as S_{ t }→ ∞. That is,
Corollary 1–3
Corollary 1. The ratio R satisfies
Proof. It is sufficient to show that $\left(\right)close="">{R}_{\infty}^{{K}_{t}}\le 1$ since $\frac{\mathrm{dR}}{d{K}_{t}}=\frac{\mathrm{dR}}{d\left[\mathit{\text{SK}}\right]}\frac{d\left[\mathit{\text{SK}}\right]}{d{K}_{t}}>0$ and hence, R is monotone increasing with respect to K_{ t }and $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$ is its maximum. $\left(\right)close="">{R}_{\infty}^{{K}_{t}}\le 1$ is equivalent to the condition
To show that (28) holds we use that $\left(\right)close="">{\left(S{K}_{\infty}^{{K}_{t}}\right)}^{2}{u}_{1}S{K}_{\infty}^{{K}_{t}}{u}_{2}=0$ and that $\left(\right)close="">{P}_{t}\mathrm{\alpha S}{K}_{\infty}^{{K}_{t}}$ (this follows from (24)). Multiply (28) by $\frac{1}{\alpha (\alpha +1)}$ to get
Note that the lefthand side of (29) is equivalent to
and this concludes the proof. □
Corollary 2. If $\frac{{P}_{t}}{\alpha}<{K}_{t}$, then $\left[\mathit{\text{SK}}\right]\to \frac{{P}_{t}}{\alpha}$ and R → 1 as S_{ t } → ∞. If $\frac{{P}_{t}}{\alpha}>{K}_{t}$, then [SK] → K_{ t }and R → 0 as S_{ t }→ ∞.
Proof. Denote $\left(\right)close="">S{K}_{\infty}^{{S}_{t}}=\underset{{S}_{t}\to \infty}{\text{lim}}\left[\mathit{\text{SK}}\right]$. It follows from (22) that $\left(\right)close="">S{K}_{\infty}^{{S}_{t}}$ satisfies the quadratic equation
and then
Thus, $S{K}_{\infty}^{{S}_{t}}=\frac{{P}_{t}}{\alpha}$ if $\frac{{P}_{t}}{\alpha}<{K}_{t}$ and $\left(\right)close="">S{K}_{\infty}^{{S}_{t}}={K}_{t}$ otherwise. Obviously, if $\left(\right)close="">S{K}_{\infty}^{{S}_{t}}={K}_{t}$, then $\left(\right)close="">{R}_{\infty}^{{S}_{t}}=0$. To see that if $S{K}_{\infty}^{{S}_{t}}=\frac{{P}_{t}}{\alpha}$, then $\left(\right)close="">{R}_{\infty}^{{S}_{t}}=1$, we show that [Sp]→S_{ t }as S_{ t }→∞. Indeed, note that $\left[S\right]=\frac{{M}_{2}\left[\mathit{\text{SK}}\right]}{{K}_{t}\left[\mathit{\text{SK}}\right]}\to \frac{{M}_{2}{P}_{t}}{\alpha {K}_{t}{P}_{t}}$ as S_{ t }→∞. Hence, [S_{ p }]=S_{ t }−[SK]−[S_{ p }P]−[S]→S_{ t }as S_{ t }→∞ and so $\left(\right)close="">{R}_{\infty}^{{S}_{t}}=1$ when $\frac{{P}_{t}}{\alpha}<{K}_{t}$. □
The following Corollary is introduced without its proof, although, the arguments would be similar to Corollary 2.
Corollary 3. It holds that:

i)
$S{K}_{\infty}^{{K}_{t}}\to \frac{{P}_{t}}{\alpha}$ as S_{ t }→ ∞ and $\left(\right)close="">{R}_{\infty}^{{K}_{t}}\to 1$ as S_{ t }→ ∞
ii) $\left(\right)close="">S{K}_{\infty}^{{K}_{t}}\to 0$ as α → ∞ and $\left(\right)close="">{R}_{\infty}^{{K}_{t}}\to 1$ as α → ∞
iii) If $\frac{{P}_{t}}{\alpha}<\frac{{S}_{t}}{\alpha +1}$, then $S{K}_{\infty}^{{K}_{t}}\to \frac{{P}_{t}}{\alpha}$ as λ_{1} → ∞ and $\left(\right)close="">{R}_{\infty}^{{K}_{t}}\to 1$ as λ_{1} → ∞. If $\frac{{P}_{t}}{\alpha}>\frac{{S}_{t}}{\alpha +1}$, then $S{K}_{\infty}^{{K}_{t}}\to \frac{{S}_{t}}{\alpha +1}$ as λ_{1} → ∞ and ${R}_{\infty}^{{K}_{t}}\to \frac{\alpha}{\alpha +1}$ as λ_{1} → ∞.
Hill numbers
The Hill number is defined as
where $\left(\right)close="">{K}_{t}^{90}/{K}_{t}^{10}$ is the response coefficient [6]. Solving for SK from (26) and by condition (24) we obtain
We also find K_{ t }from (22) to be
Given $\left(\right)close="">{R}_{\infty}^{{K}_{t}}$, we have $\left(\right)close="">{R}^{90}=0.9{R}_{\infty}^{{K}_{t}}$ and $\left(\right)close="">{R}^{10}=0.1{R}_{\infty}^{{K}_{t}}$. Then we can obtain S K^{90} and S K^{10} from (30) and finally, $\left(\right)close="">{K}_{t}^{90}$ and $\left(\right)close="">{K}_{t}^{10}$ from (31). This gives the Hill number n_{ H }.
Steadystate 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 quasisteadystate 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).
Denote C_{1} = [S_{ p }P],C_{2} = [S_{ p }PK],C_{3} = [SK],C_{4} = [PK] and let X = C_{2} + C_{4} to be the total PK concentration and A = [S_{ p }] + C_{1} to be the total S_{ p }concentration, respectively. Then it holds that
The variables A and X satisfy the following equations
We hypothesize that the intermediate complexes C_{1},C_{2},C_{3} have faster dynamics then the active protein A. At steadystate, the intermediate complexes satisfy
The simulations of the tQSSA approach are shown in Figure 5.
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Acknowledgements
The research leading to these results has received funding from the European Community’s Seventh Framework Programme FP7/20072013 under grant agreement no. HEALTHF42008223451. 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.
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VS conceived the study and BSz performed the research. BSz and VS analyzed results and wrote the paper. Both authors read and approved the final manuscript.
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Keywords
 Cellular signalling
 Zeroorder ultrasensitivity
 Phosphoinositide regulation
 Endosomal trafficking