Báo cáo khoa học: Quantitative analysis of ultrasensitive responses pot

Quantitative analysis of ultrasensitive responses Stefan Legewie, Nils Blu ¨ thgen and Hanspeter Herzel Institute for Theoretical Biology, Humboldt University Berlin, Germany In cellular signal transduction, a stimulus (e.g. an extracellular hormone) brings about intracellular responses such as transcription. These responses may depend on the extracellular hormone concentration in a gradual or an ultrasensitive (i.e. all-or-none) manner. In gradual systems, a large relative increase in the sti- mulus is required to accomplish large relative changes in the response, while a small relative alteration in the stimulus is sufficient in ultrasensitive systems. Ultra- sensitive responses are common in cellular information transfer [1–5] as this allows cells to reject background noise, while amplifying strong inputs [6,7]. In addition, ultrasensitivity embedded in a negative-feedback loop may result in oscillations [8], while bistability can be observed in combination with positive feedback [9,10]. Surprisingly, ultrasensitive signalling cascades equipped with negative feedback may also exhibit an extended linear response [11]. Finally, spatial gradients known to be important in development can be converted to sharp boundaries if they elicit ultrasensitive responses [5]. Previous theoretical work has demonstrated that ultrasensitivity in the fundamental unit of signal trans- duction, the phosphorylation–dephosphorylation cycle, can arise if the catalyzing enzymes operate near satura- tion [12] and ⁄ or if an external stimulus acts on both the phosphorylating kinase and the dephosphorylating phosphatase in opposite directions [13,14]. In addition, multisite phosphorylation [1], stoichiometric inhibition [15], regulated protein translocation [16] and cascade amplification effects [17] have been shown to contri- bute to ultrasensitive behaviour in more complex sys- tems. Biochemical responses are usually analyzed by fitting the Hill equation, and the estimated Hill coefficient is taken as a measure of sensitivity. However, this approach is not appropriate if the response under con- sideration significantly deviates from the best-fit Hill equation. In addition, Hill coefficients greater than unity do not necessarily imply ultrasensitive behaviour if basal activation is significant. In order to circumvent these problems, we present a general framework for the quantitative analysis of sensitivity, the relative amplification approach, which is based on the response coefficient defined in metabolic control analysis [18]. The relative amplification approach allows quantifica- tion of sensitivity, at both local and global levels. In addition, our approach also applies for monotonically decreasing, bell-shaped or nonsaturated responses. Keywords basal activation; Hill coefficient; metabolic control analysis; response coefficient; ultrasensitivity Correspondence S. Legewie, Institute for Theoretical Biology, Humboldt University Berlin, Invalidenstrasse 43, Berlin, D-10115, Germany Fax: +49 30 20938801 Tel: +49 30 20938496 E-mail: s.legewie@biologie.hu-berlin.de (Received 23 March 2005, revised 7 June 2005, accepted 14 June 2005) doi:10.1111/j.1742-4658.2005.04818.x Ultrasensitive responses are common in cellular information transfer because they allow cells to decode extracellular stimuli in an all-or-none manner. Biochemical responses are usually analyzed by fitting the Hill equation, and the estimated Hill coefficient is taken as a measure of sensi- tivity. However, this approach is not appropriate if the response under con- sideration significantly deviates from the best-fit Hill equation. In addition, Hill coefficients greater than unity do not necessarily imply ultrasensitive behaviour if basal activation is significant. In order to circumvent these problems we propose a general method for the quantitative analysis of sensitivity, the relative amplification plot, which is based on the response coefficient defined in metabolic control analysis. To quantify sensitivity globally (i.e. over the whole stimulus range) we introduce the integral-based relative amplification coefficient. Our relative amplification approach can easily be extended to monotonically decreasing, bell-shaped or non- saturated responses. FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS 4071 The response coefficient In ultrasensitive responses, a small relative increase in the stimulus results in a large relative change in the response. This phrase is reflected in the definition of the response coefficient used in metabolic control ana- lysis [19,20]: R X S ¼ S X Á dX dS ¼ d ln X d ln S ; ð1Þ where S is the stimulus and X is the response. The response coefficient equals the relative alteration in the response divided by the relative change in the stimulus. Response coefficients greater than unity refer to rela- tive amplification. This means that a relative change in the stimulus is increased by the factor R X S ; in other words the relative alteration in the response is R X S -times greater. Thus, highly ultrasensitive responses are char- acterized by large response coefficients [17]. Limitations of the Hill approach Sensitivity in biochemical stimulus–response relation- ships is usually analyzed by fitting the Hill equation [21]: X ¼ X basal þ X max À X basal ðÞÁ S n H S n H 50 þ S n H : ð2Þ Here, X basal and X max are the basal and the maximal responses, respectively, while S 50 refers to the stimulus required to reach half-maximal activation. Depending on the Hill coefficient, n H , the system is referred to as ultrasensitive (n H > 1), hyperbolic (n H ¼ 1) or subsen- sitive (n H < 1). By using this approach, the term ultrasensitivity is used synonymously with the phrase ‘more sensitive than the Michaelis–Menten equation’ (Eqn 2 with n H ¼ 1). In addition, sensitivity is ana- lyzed globally (i.e. over the full range of stimulus levels). However, fitting the Hill equation is inappropriate if the shape of the stimulus–response under conside- ration significantly deviates from that of the Hill equation. As an example consider the scheme in Fig. 1A, which depicts a common motif in signal transduction, the positive feedback (Appendix I). The corresponding response (Fig. 1B ¾) appears to be highly ultrasensitive upon weak stimulation, but sub- sensitive for stronger stimuli. The Hill equation is usually assumed to fit biochemical responses well in the range between 10 and 90% activation. However, the best-fit Hill equation (Fig. 1B - - -) using this range appears to be hyperbolic and thus does not reflect the behaviour of the positive feedback model. Accordingly, the Hill coefficient of the best-fit Hill equation (n H ¼ 1.19) suggests that the positive feed- back scheme exhibits only very weak ultrasensitivity. Similar conclusions also hold for an alternative defini- tion of the Hill coefficient proposed by Goldbeter & Koshland [12] for the analysis of responses, whose shape differs from the Hill equation: Fig. 1. Limitations of the Hill approach: deviation from the Hill equa- tion. (A) Schematic representation of the positive feedback model (Appendix I). (B) The response of the positive feedback model (solid line) significantly deviates from the best-fit Hill equation (dashed line) with n H ¼ 1.19, so that fitting to the Hill equation is inappropri- ate for the quantification of ultrasensitivity. For comparison, the Michaelis–Menten equation is also shown (dotted line). Parameters assumed in the positive feedback model (see Eqn A2 in Appendix I): k K ¼ 1, k A ¼ 1.3, m ¼ 3, X 50 ¼ 0.5, X tot ¼ 1andk P ¼ 1. The Hill equation was fitted by using the least-squares method with the sti- mulus values S i ¼ 1.0233 I Æ 10 )2 and I 2 [100, 101,. , 300], which cover 10–90% of the maximal response (i.e. 0.1 < X < 0.9) in the positive feedback model. Quantification of ultrasensitivity S. Legewie et al. 4072 FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS n H ¼ log 81ðÞ log S 90  S 10 ÀÁ : ð3Þ Here, S 10 and S 90 equal the stimulus levels required to achieve 10% and 90% activation, respectively. For the response of the positive feedback model depicted in Fig. 1B, one obtains n H ¼ 1.01. Thus, both the Hill coefficients obtained by fitting or by using Eqn (3) suggest that the positive feed- back model is not ultrasensitive. In addition, none of the two approaches allows quantitative local analysis for which stimuli ultrasensitivity is especially pro- nounced. Based on the preceding discussion, one may conclude that the Hill approach is inappropriate for the quantitative analysis of sensitivity if a response consists of two parts that differ in their steepness relative to the Michaelis–Menten equation and thus cannot be described simultaneously by a single Hill coefficient. As further outlined in the Dis- cussion, such ‘discontinous’ behaviour has indeed been shown experimentally for a variety of biochemi- cal responses. Experimentally measured biochemical responses often exhibit basal activation [4]. Figure 2 shows the Hill equation with (¾) or without (- - -) basal activa- tion. The Hill coefficient (n H ¼ 4), the maximal activa- tion level (X max ¼ 1) and the half-maximal stimulus (S 50 ¼ 1) were assumed to be equal in both plots, which allows direct comparison of their sensitivities. Importantly, a twofold increase in the stimulus level, from S ¼ 0.5 to S ¼ 1, results in an approximately fivefold increased response without basal activation, but in only a 1.5-fold increased response with basal activation. Thus, the biologically relevant sensitivity (i.e. the response coefficient as defined in Eqn 1), strongly decreases with increasing basal activation. As similar conclusions hold over the whole stimulus range, one can conclude that Hill coefficients significantly greater than unity do not necessarily imply ultrasensi- tive responses if basal activation is significant. Relative amplification approach Relative amplification plot Owing to the conclusions made in the section above, it seems more reasonable to analyze the sensitivity of biochemical responses by means of the response coeffi- cient defined in Eqn (1) rather than by fitting the Hill equation. The response coefficient of the Hill equation devoid of basal activation (Eqn 2 with X basal ¼ 0) is a func- tion of the stimulus S, the Hill coefficient, n H , and the half-maximal stimulus, S 50 (data not shown). Thus, a plot of the response coefficient vs. the stimulus not only depends on the sensitivity of the response (i.e. the Hill coefficient, n H ), but also on the half-maximal stimulus, S 50 , and is therefore inappropriate for the quantitative analysis of sensitivity. To circumvent this problem, we propose to plot the response coefficient, defined in Eqn (1), against the activated fraction, f, which is given by: f ¼ X À X basal X max À X basal : ð4Þ Expressing the response coefficient of the Hill equation devoid of basal activation as a function of the activa- ted fraction f ¼ S n H  S n H þ S n H 50 ðÞ(Eqn 2 and Eqn 4) yields [22]: R X S ¼ n H Á 1 À fðÞ: ð5Þ This is a linear relationship, which solely depends on the Hill coefficient, n H , and thus allows the quantifi- cation of sensitivity. Also, more in general, a plot of the response coefficient against the activated fraction, f, which will be referred to as ‘relative amplification plot’ in the following, solely depends on the sensiti- vity of the response considered: the activated fraction defined in Eqn (4) refers to a per cent response, so that the relative amplification plot does not depend on the threshold (i.e. the half-maximal stimulus), as, at the threshold, f ¼ 0.5 holds for all responses. Like- wise, the relative amplification plot is independent of Fig. 2. Limitations of the Hill approach: basal activation. Stimulus– response of the Hill equation with (¾; X basal ¼ 0.5) and without (- - -; X basal ¼ 0) basal activation. As n H ¼ 4, S 50 ¼ 1andX max ¼ 1 in both plots, the sensitivities can be compared directly, which reveals that basal activation decreases the sensitivity (see the main text). S. Legewie et al. Quantification of ultrasensitivity FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS 4073 the maximal response, as the maximal activated frac- tion equals unity for all responses. In other words, all responses are treated as if they had the same half- maximal stimulus and the same maximal activation level. A relative amplification plot of the Michaelis–Men- ten equation (Eqn 5 with n H ¼ 1) is shown in Fig. 3A (black dashed line). If other relative amplification plots reside below or above the linear plot of the Michaelis– Menten equation, the corresponding systems can be considered to be sub- or ultrasensitive. Here, we use the term ultrasensitivity synonymously with ‘more sen- sitive than the Michaelis–Menten equation’ to allow direct comparison with the Hill approach described above (Discussion). As an example, consider the rela- tive amplification plot of the positive feedback scheme (¾ in Fig. 3A), whose stimulus–response is depicted in Fig. 1B. The positive feedback scheme is ultrasensitive in the range of weak activation levels (f < 0.45), but subsensitive upon strong stimulation (f > 0.45). This demonstrates that the relative amplification plot can be used to quantify the sensitivity locally (i.e. for a given response), even if the shape of the response under con- sideration differs from that of the Hill equation. Relative amplification coefficient Often it is more reasonable to analyze sensitivity globally (i.e. over the whole stimulus range). As mentioned above, the Hill coefficient obtained by fit- ting Eqn (2) or by using Eqn (3) is generally used to measure sensitivity globally. Based on Eqn (5) we can define a more general metric of global sensitivity, which circumvents the problems associated with fitting the Hill equation. Consider the relative amplification plots of the Hill equation devoid of basal activation (Eqn 2 with X basal ¼ 0) shown in Fig. 3A: the area below the ultrasensitive Hill function (n H ¼ 2; grey line), divided by the area of the hyperbolic Michaelis– Menten equation (n H ¼ 1; - - -), equals two (i.e. it equals the Hill coefficient of the ultrasensitive Hill function). Thus, we can define the ‘relative amplifica- tion coefficient’ as a measure of global sensitivity: n R ¼ R f H f L R X Sðf Þ df           R f H f L R X R S R ðf Þ df           ð6Þ Here, f L and f H specify the range of activated fractions over which the sensitivity of the response X under consideration is compared to that of the reference response X R . The relative amplification coefficient n R equals the mean response coefficient of the response X divided by the mean response coefficient of the refer- ence response X R . In principle, the reference response X R can be any monotonically increasing or decreasing function (see AB Fig. 3. Relative amplification approach. (A) Relative amplification plot of the positive feedback model shown in Fig. 1B (¾). The response coefficient (Eqn 1) is plotted as a function of the activated fraction (Eqn 4). A comparison with the reference Michaelis–Menten equation (- - -) reveals that the positive feedback model is ultrasensitive for f < 0.45 and subsensitive for f > 0.45. The corresponding relative amplifica- tion coefficients n R (Eqn 6) are indicated on the top. The grey line corresponds to a Hill equation devoid of basal activation with n H ¼ 2. See Fig. 1B for parameters chosen in the feedback model. (B) Relative amplification plot of the Michaelis–Menten equation for varying basal activation levels X basal . The maximal activation level X max was kept constant and assumed to be unity. The corresponding relative amplification coefficients n R (Eqn 6) calculated over the whole stimulus range (f L ¼ 0andf H ¼ 1) are indicated in the legend. Quantification of ultrasensitivity S. Legewie et al. 4074 FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS also the Discussion). In order to obtain values for the relative amplification coefficient that are compar- able to the Hill coefficient discussed above, we will use the Michaelis–Menten equation as the reference response, so that ultrasensitivity (i.e. n R > 1) again refers to ‘more sensitive than the Michaelis–Menten equation’. As an example consider the response of the positive feedback model: the relative amplification coefficients calculated for f L ¼ 0 and f H ¼ 0.45, as well as for f L ¼ 0.45 and f H ¼ 1 reflect ultrasensitive behaviour in the range of weak activation (n R ¼ 2.54) and subsensi- tivity in the range of strong activation (n R ¼ 0.72), as indicated in Fig. 3A. Furthermore, the relative amplifi- cation coefficient calculated over the full range of acti- vated fractions (i.e. f L ¼ 0 and f H ¼ 1) classifies the response of the positive feedback scheme as ultrasensi- tive (n R ¼ 1.93). Basal activation As fitting the Hill equation is inappropriate for the quantification of sensitivity if basal activation is signifi- cant (see above), we will now analyze the Hill equation with basal activation by using the relative amplification approach. Expressing the response coefficient of the Hill equation with basal activation (Eqn 2 with X basal „ 0) as a function of the activated fraction f ¼ S n H =ðS n H þ S n H 50 Þ (Eqn 2 and Eqn 4) yields: R X S ¼ n H Á 1 À fðÞÁf Á X max X basal À 1  1 þ f Á X max X basal À 1  ð7Þ Importantly, the response coefficient solely depends on the ratio of maximal and basal activation, so that the impact of basal activation on the sensitivity can easily be analyzed. As an example, the relative amplification plots of the Michaelis–Menten equation with X max ¼ 1 are shown in Fig. 3B for varying basal activation lev- els, X basal . Sensitivity strongly decreases with increasing basal activation levels and this effect is especially pro- nounced for weak responses. This is a result of the fact that upon weak stimulation (i.e. for f fi 0) the Hill equation with basal activation (Eqn 2 with X basal „ 0) is approximately given by X % X basal , while X % X max Á S  S 50 ÀÁ n H for the Hill equation devoid of basal activation (Eqn 2 with X basal ¼ 0). Even if X max ⁄ X basal ¼ 10, the relative amplification coefficient calculated over the full stimulus range (i.e. for f L ¼ 0 and f H ¼ 1) is reduced by one-third (see the legend to Fig. 3B) when compared to the Michaelis–Menten equation without basal activation (X basal ¼ 0). Thus, the impact of basal activation on sensitivity is likely to be physiologically relevant, as signalling intermediates are known to exhibit basal activation levels of 5–10% and, in some cases, even > 20% [23–27]. Similarly, even saturating concentrations of extracellular hor- mones often induce less than 10-fold transcriptional induction or repression of target genes [28], so that X max ⁄ X basal < 10. Hence, we can conclude that Hill coefficients obtained by fitting the Hill equation to responses with basal activation [4] overestimate sensi- tivity in biochemical systems. However, depending on the accuracy of the fit, the Hill equation obtained may be reanalyzed in a relative amplification plot to esti- mate biologically relevant sensitivity. Discussion Owing to the problems associated with fitting the Hill equation to biochemical responses (Figs 1B and 2), we have presented a general framework for the quantita- tive analysis of sensitivity, the relative amplification approach, which is based on the response coefficient (Eqn 1) defined in metabolic control analysis. We pro- pose to analyze the response coefficient as a function of the activated fraction f (‘relative amplification plot’), as this allows quantitative comparison of sensitivities, regardless of model structure and ⁄ or parameters. In addition, expressing analytically derived response coef- ficients as a function of an activated fraction deter- mines which parameters determine sensitivity, while those affecting the activated fraction (i.e. the response) and the sensitivity to the same extent cancel out. Thus, the relative amplification approach provides more detailed insight into mathematical models of biochemi- cal systems. To quantify sensitivity globally (i.e. over the whole stimulus range), we introduced the integral- based relative amplification coefficient (Eqn 6), which is equivalent to the mean response coefficient of the response under consideration divided by the mean response coefficient of a reference response. The relative amplification approach requires that the maximal activation level of a saturated biochemical response can be measured experimentally, which may be difficult in some cases. However, complete satura- tion was undoubtedly observed in a variety of studies [4,27,29–31] and can generally be achieved if signal transduction is studied in vitro [1,2,32] or by using pep- tide hormone stimulation in culture [23,26,28,33]. As fitting the Hill equation to data devoid of saturation serves only as a guess to what the global behaviour might be, the sensitivity of the response can only be quantified locally (e.g. by plotting the response coeffi- cient as a function of the stimulus). S. Legewie et al. Quantification of ultrasensitivity FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS 4075 In the present article we have used the term ‘ultra- sensitivity’ synonymously with the phrase ‘more sensi- tive than the Michaelis–Menten equation’ in order to directly compare the relative amplification approach with the established methodology, which is based on the Hill equation (Eqn 2). However, it may be more reasonable to define ultrasensitivity as relative amplifi- cation (R X S > 1), where a relative change in the stimu- lus elicits an R X S -times greater relative alteration in the response, so that ultrasensitivity has direct biochemical meaning. Then, the relative amplification coefficient (Eqn 6) as a measure of global ultrasensitivity should be calculated by formally setting the reference response coefficient to unity (i.e. R X R S R ¼ 1). Application of the relative amplification approach to the Hill equation with basal activation (Eqn 2) reveals that sensitivity significantly decreases with increasing basal activation (Fig. 3B). Even if the basal activation level is only 10% of the maximal response, as com- monly observed in biochemical responses [4,23–28], the relative amplification coefficient (i.e. the mean response coefficient) is reduced by one-third when compared to a system devoid of basal activation. Thus, Hill coeffi- cients obtained by fitting the Hill equation to responses with basal activation [4] overestimate sensi- tivity in biochemical systems. However, depending on the accuracy of the fit, the Hill equation obtained may be reanalyzed in a relative amplification plot to esti- mate biologically relevant sensitivity. In addition, fit- ting the Hill equation to data with basal activation may be reasonable to quantify the degree of apparent cooperativity (i.e. to obtain a hint of the biochemical mechanisms involved). By using the positive feedback model (Fig. 1A) as an example, we have shown that the relative amplifica- tion approach allows quantitative analysis of local and global sensitivities, even if the shape of the response under consideration deviates from that of the Hill equation (Fig. 3A). The Hill approach is inappropriate for the analysis of the feedback model, as the response is more sensitive than the Michaelis–Menten equation for weak stimuli, while being less sensitive for strong stimuli (Fig. 3A). Similar ‘discontinous’ behaviour was also reported for multisite phosphorylation [1,34] and stoichiometric inhibition [15]. Yet, other responses, such as insulin-induced PKBa (protein kinase B alpha) activation in hepatocytes [33], phorbol ester-induced p54JNK (c-Jun N-terminal kinase) activation [23], and anisomycin-induced JNK activation in 293 cells [26], are shallow for weak stimuli, but switch-like as the activation level is further increased. By using the Hill approach, such ‘discontinuities’ are averaged out (Fig. 1B). However, quantitative insights into the local behaviour of biochemical responses are needed because common upstream activators often induce multiple downstream pathways, each of which exhibits a dis- tinct threshold activator concentration [5,23,35]. Even if the best-fit Hill equation deviates significantly from a given ‘discontinous’ response only in the range of 0–30% and 70–100%, important biological informa- tion may be lost, as it was, for example, shown that 10% receptor occupation already drives some phero- mone responses in yeast [15]. Theoretical studies [36,37] suggest that biochemical responses can exhibit multiple thresholds (‘staircase response’), that is, the system exhibits two ranges of high sensitivity that are separated by a plateau of very low sensitivity. Indeed, such behaviour has been confirmed experimentally for allosteric enzymes [37], for Senseless-induced transcription [38], for mTOR- induced DNA synthesis [39], for insulin-induced DNA synthesis [40] and for phosphatidic acid-induced 1-phosphatidylinositol 4,5-bisphosphate production [41]. While the Hill approach obviously fails for these staircase responses, the relative amplification approach allows the quantitative analysis of sensitivity. Import- antly, the relative amplification approach also applies for monotonically decreasing responses, which occur frequently if an inhibitor diminishes signal transduc- tion. In addition, bell-shaped functions, where the response increases with increasing stimulus up to a maximum and subsequently decreases for supramaxi- mal stimuli [42,43], can also be analyzed. In this case the monotonically increasing and decreasing parts need to be quantified separately, as Eqn (6) may be unde- fined if the sign of the response coefficient changes. Finally, appropriate activated fractions can also be defined for many nonsaturated responses, which are nonlinear for weak stimuli but linear upon strong sti- mulation (S. Legewie, N. Blu ¨ thgen, R. Scha ¨ fer and H. Herzel, unpublished observations). Then, the relat- ive amplification coefficient defined in Eqn (6) gives a measure of sensitivity in the nonlinear range. We believe that future signal transduction research will focus on the processing of transient signals. There is ample evidence that signaling networks must be able to discriminate transient signals of different amplitudes and ⁄ or different durations [44–46]. Likewise, the fre- quency of Ca 2+ oscillations [47], or the number of repetitive Ca 2+ spikes [48], are known to determine biological outcomes. Obviously, signaling networks will be able to efficiently discriminate such transient inputs if they respond in an ultrasensitive manner (e.g. with respect to signal duration) [49]. Previous studies suggest that such discrimination curves often do not match the Hill equation [50,51] or that they exhibit Quantification of ultrasensitivity S. Legewie et al. 4076 FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS pronounced basal activation [52]. In addition, mono- tonically decreasing [47] and bell-shaped [53] relation- ships are frequently observed. 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Biophys Chem 106, 193–202. 52 Eshete F & Fields RD (2001) Spike frequency decoding and autonomous activation of Ca2+-calmodulin- dependent protein kinase II in dorsal root ganglion neu- rons. J Neurosci 21, 6694–6705. 53 Dupont G, Houart G & De Koninck P (2003) Sensiti- vity of CaM kinase II to the frequency of Ca2+ oscilla- tions: a simple model. Cell Calcium 34, 485–497. Appendix I Figure 1A schematically depicts a positive feedback in signal transduction: The stimulus S catalyzes the phos- phorylation of the inactive precursor X 0 , which yields the active species X (¼ response). The phosphorylated species X activates the intermediate Y in an ultrasensi- tive manner (e.g. via an ultrasensitive phosphorylation cascade) [1], and Y, in turn, catalyzes the formation of X. For example, X may be the Raf protein, which is known to induce its own enzymatic activator protein kinase C (PKC) via a Mek-Erk-PLA 2 -cascade [10]. By using the mass-conservation relationships X tot ¼ X 0 + X and Y tot ¼ Y 0 + Y, the differential equations can be written as: dX dt ¼ k K Á S þ k K1 Á YðÞÁX tot À XðÞÀk P Á X dY dt ¼ k K2 Á X m X m þ K m Á Y tot À YðÞÀk P1 Á Y ðA1Þ Here, we have assumed linear kinetics in all (de)phos- phorylation reactions despite the phosphorylation of Y 0 by X, which is modeled by using a Hill term to reflect ultrasensitivity. As we also assume that the (de)phosphorylation of Y proceeds much faster than that of X , we can approximate Y by using a quasi- equilibrium assumption (i.e. dY ⁄ dt ¼ 0), so that Eqn A1 reduces to: dX dt ¼ k K Á S þ k A Á X m X m 50 þ X m  Á X tot À XðÞÀk P Á X ðA2Þ Here, k A and X 50 are lumped constants, which can be deduced from Eqn (A1). S. Legewie et al. Quantification of ultrasensitivity FEBS Journal 272 (2005) 4071–4079 ª 2005 FEBS 4079 . Quantitative analysis of ultrasensitive responses Stefan Legewie, Nils Blu ¨ thgen and Hanspeter. approach allows quantitative analysis of local and global sensitivities, even if the shape of the response under consideration deviates from that of the Hill equation