Effects of sequestration on signal transduction cascades Nils Bluthgen1,* Frank J Bruggeman2, Stefan Legewie1, Hanspeter Herzel1, Hans V Westerhoff2,3 ă and Boris N Kholodenko4 Institute for Theoretical Biology, Humboldt University Berlin, Germany Department of Molecular Cell Physiology, Institute of Molecular Cell Biology, Faculty of Earth and Life Sciences, Vrije Universiteit, Amsterdam, the Netherlands Manchester Centre for Integrative Systems Biology, Manchester Interdisciplinary Biocentre, School of Chemistry, University of Manchester, UK Department of Pathology, Anatomy and Cell Biology, Thomas Jefferson University, Philadelphia, USA Keywords MAPK; phosphorylation; sequestration; signal transduction; zero-order ultrasensitivity Correspondence N Bluthgen, Institute for Theoretical ă Biology, Humboldt University Berlin, Invalidenstr 43, 10115 Berlin, Germany Fax: +49 30 838 56943 Tel: +49 30 838 56971 E-mail: nils@itb.biologie.hu-berlin.de *Present address Molecular Neurobiology, Institute of Biology, Free University of Berlin, Germany Note Nils Bluthgen and Frank J Bruggerman ¨ contributed equally to this study (Received 21 November 2005, accepted 15 December 2005) The building blocks of most signal transduction pathways are pairs of enzymes, such as kinases and phosphatases, that control the activity of protein targets by covalent modification It has previously been shown [Goldbeter A & Koshland DE (1981) Proc Natl Acad Sci USA 78, 6840–6844] that these systems can be highly sensitive to changes in stimuli if their catalysing enzymes are saturated with their target protein substrates This mechanism, termed zero-order ultrasensitivity, may set thresholds that filter out subthreshold stimuli Experimental data on protein abundance suggest that the enzymes and their target proteins are present in comparable concentrations Under these conditions a large fraction of the target protein may be sequestrated by the enzymes This causes a reduction in ultrasensitivity so that the proposed mechanism is unlikely to account for ultrasensitivity under the conditions present in most in vivo signalling cascades Furthermore, we show that sequestration changes the dynamics of a covalent modification cycle and may account for signal termination and a signsensitive delay Finally, we analyse the effect of sequestration on the dynamics of a complex signal transduction cascade: the mitogen-activated protein kinase (MAPK) cascade with negative feedback We show that sequestration limits ultrasensitivity in this cascade and may thereby abolish the potential for oscillations induced by negative feedback doi:10.1111/j.1742-4658.2006.05105.x In most biological organisms intracellular signal processing is carried out by networks composed of enzymes that activate and inactivate each other by covalent modification Signals received at the cell membrane ripple through signalling networks via covalent modification events to reach various locations in the cell and ultimately cause cellular responses The biochemical building blocks of these networks are frequently enzyme pairs, such as a kinase and a phosphatase, that form covalent modification cycles in which the target enzyme is covalently modified at single or multiple sites in a reversible manner In some experiments, the stimulus–response curves display strong sigmoidal dependencies in vivo, for example, in the activation of the mitogen-activated protein kinase (MAPK) cascade [2] and Sic1 [3], and in vitro, for example, in the phosphorylation of isocitrate dehydrogenase [4], muscle glycolysis [5] and in postsynaptic calcium signalling [6] Sigmoidal stimulus–response curves imply that the responses are highly sensitive to changes in signals around the threshold level Thus it is more sensitive than a typical Michaelis–Menten-like response, a property that has been termed ultrasensitivity [1] Abbreviations JAK, janus kinase; MAPK, mitogen-activated protein kinase; MAPKK, mitogen-activated protein kinase kinase; MCA, metabolic control analysis FEBS Journal 273 (2006) 895–906 ª 2006 The Authors Journal compilation ª 2006 FEBS 895 Effects of sequestration N Bluthgen et al ă Sigmoid responses can be used to generate binary-like decisions [7] and to filter out noise or delay responses [8] Moreover, ultrasensitive signal transduction cascades can display oscillations in combination with a negative feedback loop [9] and bistability (hysteresis) in combination with positive feedback [10,11] Surprisingly, ultrasensitivity coupled with negative feedback also yields highly linear responses and signal fidelity in the presence of high load [12] Several mechanisms account for ultrasensitive stimulus–response curves, including cooperativity, multisite phosphorylation, feed-forward loops and enzymes operating under saturation The latter mechanism has been termed zero-order ultrasensitivity because a necessary condition is that the modifying and de-modifying enzyme of a covalent modification cycle display zero-order kinetics This mechanism was explored for the steady-states of cycles composed of enzymes with irreversible product-insensitive kinetics in pioneering work by Goldbeter & Koshland [1] Zero-order sensitivity is appealing because of its simplicity: all it needs is one modification site on a protein that acts as a substrate (e.g a phosphorylation site) and, for example, a kinase and a phosphatase in which at least one of the enzymes has a KM value for their substrate that is low compared with the total concentration of the protein substrate This mechanism might provide cells with simple ultrasensitive signalling units that can be interconnected to form networks that can display a great variety of responses [13] However, cells also use more complicated mechanisms that activate proteins by multiple modification events to bring about ultrasensitivity Examples of such protein targets are Sic1 which has at least six phosphorylation sites [3] Nuclear factor of activated T-cells (NFAT) has even more phosphorylation sites [14], and the MAPK cascades containing MAPK kinase (MAPKK) and MAPK both become fully activated by double phosphorylation It remains a puzzle, why other, more complicated means like multisite phosphorylation need to be applied to get high sensitivity when there is a simple mechanism like zero-order ultrasensitivity Goldbeter & Koshland discussed briefly that product sensitivity and a large amount of enzyme–substrate complex compared with the total concentration of the interconvertible enzyme may reduce the sensitivity of the cycle They did not analyse any of the general consequences of sequestration, however, and the severe consequences of sequestration for ultrasensitivity therefore remain unclear The effect of product sensitivity has been quantified in more detail by Ortega et al [15], who showed that ultrasensitivity disappears if the enzymes are product sensitive Data about protein 896 Table Concentrations of members of the MAPK cascade (MAPKKK, MAPKK, MAPK) in different organisms and cell types as found in the literature In many of these, the concentrations are of the same order of magnitude RU, relative units Cell type Budding yeast Chinese hamster ovary cells Xenopus oocytes HeLa cells Rat NIH 3T3 208F COS-1 MAPKKK 1 1 RU RU RU RU MAPK Ref < 35 nM 1300 nM nM MAPKK 100 nM 2800 nM [7] [7] 1200 nM 30 lM 1.6 RU 1.4 RU 2.9 RU 0.7 RU 330 nM 30 lM 2.4 RU 3.5 RU 5.9 RU RU [7] [40] [41] [41] [41] [41] abundance in signal transduction cascades are now in hand, showing that members of the cascades are present in concentrations of the same order of magnitude [16] (see Table for examples) Therefore, we decided to investigate the effect of high enzyme concentration on the sensitivity of signal transduction cascades in more detail Without loss of generality we assume that the modification is phosphorylation and the enzymes are kinases and phosphatases First, we investigate the amount of sequestered substrate in a simple covalent modification cycle (Fig 1) We then show that sequestration reduces zero-order ultrasensitivity dramatically Subsequently, we illustrate the consequences of sequestration on zero-order ultrasensitivity by numerical simulations and confirm the predictions We show that sequestration also has dramatic effects on signalling dynamics Sequestration can account for the transient transduction of a permanent signal Multisite phosphorylation and kinase sequestration can work as a sign-sensitive delay element [17], in which the rise in the signal is delayed but the dropping signal is transduced immediately Finally, we analyse the effect of sequestration on a complex signal transduction cascade, the MAPK cascade Computational studies by Kholodenko [9] have shown that oscillation can arise in this system from a combination of ultrasensitivity and negative feedback We show that sequestration abolishes those oscillations by reducing zero-order ultrasensitivity Results Sequestration in covalent modification cycles Unlike metabolic systems, the modification cycles involved in signal transduction cascades often exhibit comparable amounts of protein substrates and enzymes [16] For example, the individual concentrations of FEBS Journal 273 (2006) 895–906 ª 2006 The Authors Journal compilation ª 2006 FEBS N Bluthgen et al ă Effects of sequestration the MichaelisMenten constant of the kinase The concentration of the complex [TK] approaches the total concentration of the kinase as [T] > KM The phosphatase–substrate concentration can be calculated accordingly To illustrate this effect for a covalent modification cycle, we investigate a special case, i.e when both kinase and phosphatase have the same kinetic constants and the same concentrations Consequently, the two substrates are in equal steady-state concentrations ([T] ¼ [T*]) and the two complex concentrations are equal ([TK] ¼ [T*P]) Therefore, the total target concentration can be expressed as: TT ¼ 2[T] + 2[TK] After substitution of the resulting expression for [TK] into the Michaelis–Menten formula, we obtain: K K 1a 1b T T T T * * 2b 2a P P Fig Schematic representation of the simplest covalent modification cycle The target protein T can be covalently modified The unmodified protein T binds to the kinase K in the first reaction (1a) to form the complex TK The second reaction (1b) is the catalytic modification step yielding K and the covalently modified target protein T* In the third reaction (2a), the phosphatase P binds T* to form the complex T*P In the fourth reaction (2b) the cycle is closed by the recycling of T via catalytic demodification and the release of P Reactions 1b and 2b are assumed to be irreversible for simplicity the three kinases of the well-characterized MAPK cascade are similar in a variety of cell types and organisms (Table 1) Each of these kinases modifies its target protein and is itself a target for the upstream kinase Because the concentration of kinases and their target proteins are comparable, the kinase can sequester a significant amount of target by binding to it, provided that the kinase shows high affinity towards the substrate This sequestered fraction of the target is no longer accessible to other kinases and phosphatases Available data about phosphatase concentrations suggest that they are also likely to be of the same order of magnitude as or even exceed their substrate concentrations [18,19] The concentration of the kinase–substrate complex [TK] in the steady-state can be calculated using the MichaelisMenten formula: ẵTK ẳ ẵTẵKT ẵT ỵ KM ð2Þ From this, the amount of free substrate in the cycle, i.e [T] + [T*] ¼ 2[T] can be calculated from the total concentrations of kinase and target Importantly, the concentrations of the free substrate forms [T] and [T*] decrease below KM if [KT] > [TT] ) 2KM (see Supplementary material for mathematical details) If the catalytic activity of the phosphatase exceeds the activity of the kinase, the free substrates can be higher In this case, [T] and [T*] will still fall below KM if the kinase and phosphatase concentrations together exceed twice the target concentration, i.e [KT] + [PT] > 2[TT] ) 4KM Thus, in a signalling cycle, sequestration reduces the free target concentrations such that the concentration of the free target is below the KM value of the enzymes, provided that the enzymes are available in a concentration as high as their total protein substrate concentration The effect of sequestration on zero-order ultrasensitivity The sensitivity of simple modification cycles was explored in pioneering work by Goldbeter & Koshland [1] using methods from nonlinear dynamics Later, it was formulated in terms of metabolic control analysis (MCA) by Small & Fell [20] Small & Fell expressed the response of the active fraction to a change of the kinase concentration as a function of the concentrations of the two forms ([T] and [T*]) and the elasticities of the enzymes by the following simple relation: RT T ẳ K 1ị where [T] and [KT] are the free target concentration and total kinase concentration, respectively, and KM is ẵKT ẵT ẳ TT 2ẵT KM ỵ ẵT ẵT et2 ẵT ỵ et1 ẵT T T 3ị As discussed in the Methods, this response coefficient expresses the fractional change of the active form FEBS Journal 273 (2006) 895–906 ª 2006 The Authors Journal compilation ª 2006 FEBS 897 Effects of sequestration N Bluthgen et al ă [T*] upon a fractional change of the kinase concentration If the enzymes are unsaturated, their elasticities are em2 % and em1 % 1, and the response Tà T à coefficient is RTT < 1, corresponding to a sublinear K response In this case, no ultrasensitivity is observed In contrast, saturation of the enzymes leads to elastià cities closer to 0, hence RTT can exceed and give K rise to an ultrasensitive response In the derivation of Eqn (3), Small & Fell [20] assumed that the concentration of the substrate bound to the enzyme is negligible But as discussed above, this assumption does not hold where the concentrations of enzymes and substrate are similar, as observed in signal transduction cascades if the enzymes are saturated Therefore, the assumptions made to derive Eqn (3) may not necessarily hold If the effect of sequestration is taken into account the response coefficient modifies to: à RT T ẳ K ẵT et2 ẵT ỵ et1 ẵT ỵ et2 et1 ẵTK ỵ ẵT P T T T T ð4Þ A detailed mathematical derivation of Eqn (4) can be found in the Supplementary material Comparison of Eqn (4) with Eqn (3) reveals the effect of sequestration on zero-order ultrasensitivity as an additional term in the denominator which increases with the extent of sequestration, i.e ([TK] + [T*P]) Therefore, at constant elasticities, sensitivity should decrease with sequestration Another effect is hidden in the equations: an increase in sequestration also increases the elasticities et2à and et1 , because the available substrate T T decreases This eventually causes an additional à decrease in the sensitivity RT T K To elucidate this further, we examined the special case when both kinase and phosphatase have the same kinetic constants In this case, we expect, on the basis of symmetry, that the highest response coefficient occurs when there are equal amounts of phosphorylated and unphosphorylated target We can then express all concentrations and elasticities in terms of [T], the Michaelis–Menten constant, KM, of kinase and phosphatase In this case Eqn (4) reads: ẵT ỵ KM  RT T ẳ  K K ẵKT ỵ K MỵẵTị2 ð5Þ M à RT T increases with [T] and decreases with [KT] This K shows that the response coefficient gets smaller as the amount of free substrate [T] decreases due to sequestration As discussed previously, similar concentrations of the enzymes and target imply that the free target falls below the KM value The response is then 898 à 2KM sublinear, i.e RT T < 1, because ẵTỵKM % Also, if K KM is very small, most of the substrate is sequestered, leading to essentially zero concentrations of T and T* Goldbeter & Koshland [1] discussed the possibility that ultrasensitivity might be preserved if the phosphatase–target complex T*P is assumed to be active However, as calculated in the Supplementary material, the T combined response of T and T*P, RKTỵT P is always à < RTT Thus, the attenuation of sensitivity by sequesK tration cannot be restored by an active phosphatase– target complex The consequences of sequestration for ultrasensitivity: numerical investigations To further investigate the consequences of sequestration on ultrasensitivity, the steady-state of the cycle depicted in Fig was calculated numerically The KM value was chosen to be much smaller than the total concentration (KM ¼ 0.02[TT]) for both the kinase and the phosphatase The phosphatase concentration [PT] was increased from to 2[TT], to vary the amount of sequestration Figure 2B shows that this increase is accompanied by an increase in the sequestered fraction ([TK] + [T*P]) ⁄ [TT] The response of the cycle [T*] to the input [KT] decreases if the total levels of the phosphatase approach half of the total target concentration [TT] (Fig 2A) Taken together these two plots illustrate our argument: when the kinase and phosphatase concentrations become comparable with the total concentration of the target protein, the sequestered fraction increases, which causes the sensitivity to decrease In Fig 2C the activated fraction of the target T* is plotted, illustrating that the fraction of activated target decreases dramatically as the phosphatase concentration exceeds [TT] ⁄ These results are in good agreement with the estimates made above This suggests that in vivo, where in many cases the concentrations of the kinase, the phosphatase and the target protein are comparable, the sensitivity of covalent-modification cycles is likely to be achieved by mechanisms other than zero-order ultrasensitivity Simulations for different catalytic activities of kinases and phosphatases are shown in Fig 2D–I If the phosphatase is catalytically 10-fold more active than the kinase, the region of enhanced sensitivity is broadened slightly (Fig 2D–F) In contrast, if the kinase is more active than the phosphatase, the region if ultrasensitivity is drastically reduced (Fig 2G–I) It seems that in case of mammalian MAPK cascades high concentrations of the phos- FEBS Journal 273 (2006) 895–906 ª 2006 The Authors Journal compilation ê 2006 FEBS N Bluthgen et al ă Effects of sequestration Response Coefficient K1b,f=0.1 2 [KT] 10 [PT] 1 [PT] < 1.25 3 0.9 < 0.9 < 0.5 < 0.15 < 0.1 < 0.05 < 0.02 < 0.01 Fig Steady-state signalling characteristics of a covalent-modification cycle for equal catalytic activity of kinase and phosphatase (A–C), for 10-fold higher catalytic activity of the kinase (D–F), for 10-fold reduced catalytic activity of the kinase (G–I) (A) Contour plot of the response T coefficient RKTà as function of the total concentration of the phosphatase and the kinase (normalized to the phosphatase concentration) (B) Sequestered fraction of the target protein (C) Fraction of the activated target protein Parameter values: TT ¼ 1, k1a,f ¼ 10, k1a,r ¼ 0.1, k1b,r ¼ 0, k2a,f ¼ 10, k2b,f ¼ 0.1 and k2b,r ¼ varied to simulate different catalytic activity of the kinase: (A–C) k1b,f ¼ 0.1, (D–F) k1b,f ¼ 1, (G–I) k1b,f ¼ 0.01 KT and PT refer to the total kinase and phosphatase concentration, respectively phatases yield high sequestration which not allow for zero-order ultrasensitivity The consequences of sequestration for signalling dynamics Receptor desensitization is a relatively slow process and downstream signal transduction cascades are often in a quasi-steady-state with the receptor activity However, some downstream parameters adapt very quickly (e.g insulin receptor substrate phosphorylation after insulin and Erk after epidermal growth factor), suggesting that downstream pathways are capable of adaptation Figure 3A shows the dynamics of the covalent modification cycle for a fast kinase with low affinity and a slow phosphatase with high affinity If a permanent stimulus is given, the target displays only transient activation Thus a covalent modification cycle is capable of terminating prolonged signals The fast kinase phosphorylates the available target, but the FEBS Journal 273 (2006) 895–906 ª 2006 The Authors Journal compilation ª 2006 FEBS 899 Effects of sequestration N Bluthgen et al ă threshold time Also, deactivation must be fast in comparison with activation as removal of the signal has to be translated into an immediate response Such properties have been described for coherent feed-forward loops, which display sign-sensitive delay [17] Figure 3B shows that competition for the enzyme by two phosphorylation sites may also account for such a sign-sensitive delay and dramatically improves duration decoding The solid line shows the dynamics of double-phosphorylation in which both phosphorylation sites compete for the kinase, the dotted line shows the dynamics of the corresponding system in case there is no competition (details in the Supplementary material) If the stimulus increases it must be of a certain length to be transduced if the sites compete for the kinase However, if the stimulus falls, the change is transduced immediately Thus, sequestration and multisite phosphorylation might be a mechanism for sign-sensitive delays, similar to coherent feed-forward loops in transcriptional networks [17] Changes in the steady-state stimulus–response curve might also have a large impact on the dynamics because the onset of oscillations in a signal transduction cascade harbouring a negative feedback is determined by the sensitivity of the stimulus–response curve in the steady-state We investigated the effects of sequestration in a complex signal transduction cascade with negative feedback as described below A 50 [T*] x0.1 25 B 20 time 40 100 80 [T**] 60 40 20 0 50 55 time Fig (A) The dynamics of free phosphorylated target protein in case of more active kinase than phosphatases k1a,f ¼ 0.005, k1a,r ¼ 0.4, k1b,f ¼ 0.1, k2a,f ¼ 0.0005, k2a,r ¼ 0.004, k2b,f ¼ 0.001 TT ¼ 100 KT ¼ 300 PT ¼ (300) At zero time-point, the system is at steady-state for zero stimulus (initial conditions: [T](0) ¼ TT, [T*](0) ¼ [TK](0) ¼ [T*P](0) ¼ 0) (B) The dynamics of double-phosphorylation in case the kinase shows higher affinity towards the unphosphorylated target Solid line: the case of kinase sequestration, dotted line: no kinase sequestration Grey lines indicate the stimulus (i.e kinase concentration), scaled by 0.1 in (A) phosphorylated target is subsequently sequestered by the low-activity high-affinity phosphatase At steadystates most of the target substrate is sequestered by the phosphatase Thus substrate sequestration by a phosphatase might be a means to achieve signal termination and desensitization downstream of receptors without involving a negative feedback loop For many signals, their duration determines the biological response [21] We have pointed out that sequestration might cause short, transient signals However, interpretation of the signal by the signal transduction network requires circuits that respond only to prolonged activation As pointed out by Deshaies & Ferrell [22], such signal duration decoding requires a 900 The effect of sequestration in MAPK signal transduction cascade The MAPK cascade consists of three kinases that activate their downstream kinases by phosphorylation (Fig 4) It has the potential to be ultrasensitive because of the combination of multisite phosphorylation, zero-order kinetics [23] and cascade amplification effects [24] According to Kholodenko [9] a negative feedback that is wrapped around this ultrasensitive cascade can bring about sustained oscillations over a wide range of stimuli if sequestration is neglected (Fig 6A) As the kinases are present at similar concentrations, we investigated whether sequestration affects ultrasensitivity and oscillatory behaviour We modelled the cascade such that sequestration was taken into account (similar to Huang & Ferrell [23], with parameters adopted to reflect the catalytic and Michaelis–Menten constants from Kholodenko [9], see Supplementary material for details) We chose concentrations of the phosphatases for MAPK and MAPKK that were as high as that of their substrate (300 nm) First, we analysed the cascade without feedback Figure compares FEBS Journal 273 (2006) 895–906 ª 2006 The Authors Journal compilation ª 2006 FEBS N Bluthgen et al ă Effects of sequestration MAPKKKK P MAPKKK MAPKKK MAPKK MAPKK P P MAPKK P P MAPK MAPK P MAPK P Fig Sketch of the MAPK cascade A MAPKKKK stimulates the phosphorylation of MAPKKK, which after phosphorylation phosphorylates MAPKK at two sites The double-phosphorylated MAPKK phosphorylates MAPK also at two sites The double-phosphorylated MAPK in turn inhibits the activity of MAPKKKK a model neglecting sequestration (Fig 5A–C) and one including the effect of sequestration (Fig 5D,E) Whereas the response of the first molecule (MAPKKK) is relatively unchanged because its kinase and phosphatase are present only at low concentrations, the response of the second and third molecules (MAPKK and MAPK) is changed dramatically There are two main effects of sequestration visible in the response of MAPKK and MAPK: the ultrasensitivity of the stimulus response curves is reduced and the amount of maximally activated kinases in this cascade is decreased If we add a negative feedback loop to this model, similar to the model by Kholodenko [9], no oscillations arise (Fig 6B) The effect of lower activation of MAPK can be compensated for by a stronger feedback (lower values of kloop, see Supplementary material) However, lowering of kloop does not restore oscillations (Fig 6B) This leads us to conclude that the reduction in ultrasensitivity due to sequestration is responsible for the diminishing of oscillations We observed in the analysis of simple, isolated covalent modification cycles that an increase in the total target concentration will limit the sequestered fraction of the target and restore ultrasensitivity However, in cascades such as the MAPK cascade the kinases are both enzymes for the modification of the downstream kinase and substrate for the upstream kinase Hence, the complex of, for example, MAPK and MAPKK reduces the free concentration of both MAPK and MAPKK Therefore, an increase of the MAPK concentration in this cascade gives rise to more sequestration of MAPKK by MAPK Consequently, it is not surprising that we found that an increase in MAPK of one order of magnitude cannot restore the oscillations In addition, we investigated the effects of sequestration by phosphatases We found that oscillations can be restored if the phosphatase concentrations of MAPK- and MAPKK-phosphatase are lowered to one fifth of the kinase concentrations (while increasing their catalytic activity by factor five to keep the Vmax value constant) However, in contrast to the model that neglects sequestration, the stimulus needs to be rather low (Fig 6C) In this case, sequestration due to the phosphatase is reduced and the upstream kinases of MAPK and MAPKK are only slightly activated and can sequester only limited fractions of MAPK and MAPKK Discussion The function of the signal transduction network is to sense changes in the environment of the organism in the form of signals of physicochemical origin, e.g concentrations of molecules or mechanical stress, and to integrate these with the current cellular status to ‘compute’ an adaptive response [12] Such adaptive responses involve covalent modification of enzymes, changes in gene expression, and cell-fate decisions that occur on different time scales Many signal transduction networks have common building blocks: enzyme couples that activate and inactive their protein targets via covalent modification It is reasonable to expect that network responses can be highly sensitive to changes in the signals Ultrasensitivity can be used generate thresholds, oscillations and linear responses [12] Therefore, it may not be surprising that ultrasensitivity has been documented experimentally for some signalling systems [10] Theoretical studies by Goldbeter & Koshland [1] unveiled a potential mechanism responsible for ultrasensitivity: the kinase and phosphatase have to be saturated with their target protein This case has been referred to as zero-order ultrasensitivity Since then, many groups have analysed zero-order ultrasensitivity [15,25,26,27] Although the effect of complex formation in a substrate cycle has been addressed previously [28], the impact of sequestration on zero-order ultrasensitivity has not FEBS Journal 273 (2006) 895–906 ª 2006 The Authors Journal compilation ª 2006 FEBS 901 Effects of sequestration N Bluthgen et al ă sequestration A D B E C F 50 MAPK-PP 300 200 100 300 MAPK-PP activated kinase (nM) MAPKKK-P no sequestration 100 200 100 0 0.05 0.1 0.05 stimulus (MAPKKKK in nM) Experimental data (Table 1) indicate that the concentrations of enzymes and target proteins in signal transduction cascades are similar When the affinity of enzymes for their target is sufficiently high, it implies that a high fraction of the target concentration is bound to the enzymes, and thereby sequestered This, in turn, decreases the amount of target accessible to the enzymes, and reduces ultrasensitivity Moreover, the amount of activated target decreases dramatically Consequently, the concentrations of the complexes can no longer be neglected in the analysis of ultrasensitivity, as long as the concentrations of players in the signal transduction cascades are comparable We investigated the consequences of sequestration on zero-order ultrasensitivity using the analytical approach of MCA and numerical simulations In terms of MCA, ultrasensitivity is equivalent to a response coefficient higher than [15] We derived an analytical expression for the response coefficient (Eqn 4) that accounts for the effect of sequestration Comparison with a response coefficient that neglects sequestration (Eqn 3) suggests that sequestration may significantly reduce and even eliminate ultrasensitivity Eqn (5) corroborates this for a simple example in which the kinetic parameters of both enzymes are equal It shows 902 0.1 Fig Stimulus–response curves for the three layers of the MAPK cascade in the model considered by Kholodenko [9] (A–C), which neglects sequestration and the corresponding model that takes the effects of sequestration into account (D–F) that the response coefficient decreases below 0.5: hence, ultrasensitivity is absent The results of numerical simulations illustrated that if the total concentrations of both enzymes are increased simultaneously, ultrasensitivity decreases and ultimately vanishes when these concentrations exceed 70% of the total target concentration This correlated with high sequestration of the target protein by the enzymes, which illustrates that sequestration reduces ultrasensitivity Another problem of zero-order ultrasensitivity arises due to the sequestration of the enzyme by the substrate: The saturated enzyme may then not be available for other reactions This is of special importance if the enzyme itself is the substrate of a modification cycle like MAPKK, which is itself controlled by phosphorylation and is the enzyme that phosphorylates MAPK Here sequestration reduces the zero-order ultrasensitivity in both cycles: the cycle in which the enzyme drives the modification and that in which the enzyme is subject to modification In such signalling cascades sequestration can be significant even if the kinase concentrations increase along the cascades due to the sequestration of the enzymes The extent of ultrasensitivity that can be generated by signal transduction cascades is thereby limited by sequestration This effect FEBS Journal 273 (2006) 895–906 ª 2006 The Authors Journal compilation ª 2006 FEBS N Bluthgen et al ă Effects of sequestration activated MAPK (nM) A B 50 300 200 25 100 0 MAPKKKK (nM) 0 MAPKKKK (nM) C no sustained oscillations Phophatase concentrations Fig Bifurcation diagrams for the models that neglect (A) and include (B,C) the effects of sequestration Solid lines show stable steady-states, dotted lines indicate unstable steady-states The dashed lines mark the amplitude of the oscillations observed in the model that neglects sequestration The four lines in (B) show situations for different feedback parameters (from top to bottom: kloop ¼ 9, 1, 0.1, 0.01 nM) (C) Two-dimensional bifurcation diagram for the model that includes the effect of sequestration Concentrations of the MAPK- and MAPKK-phosphatases (vertical axis) and the stimulus (horizontal axis) are changed The dashed area shows the region where sustained oscillations occur Insets show qualitatively the dynamics in the corresponding areas 60 30 sustained oscillations 0 might be responsible for the fact that sustained oscillations have not yet been documented in the MAPK cascade as opposed to the NF-jB cascade [29] Because each enzyme usually targets more than one reaction, as, for example, most phosphatases, modification cycles compete for the enzymes After a pathway is activated it recruits its phosphatases, which are no longer accessible to others We show that sequestration of the kinase in a double-phosphorylation cycle may account for a sign-sensitive delay element, such that the activation of a target enzyme upon a signal is delayed, but it is in-activated immediately after removal of the signal Such a delay element provides cells with units that neglect short fluctuations in signals, but transduce long signals In addition, sequestration might mediate cross-talk between pathways if an enzyme is shared This has been observed in the JAK ⁄ STAT pathway, in which the receptors share the janus kinase (JAK) and multiple receptors compete for it Upregulation of one 0.01 0.02 MAPKKKK (nM) 0.03 0.04 receptor downregulates the response of the other by sequestration of JAK [30] Our results suggest that to generate ultrasensitivity, cells need to exploit mechanisms that not require enzyme saturation Such mechanisms include multisite phosphorylation, which generates ultrasensitivity without the need for sequestration Moreover, not only ultrasensitivity, but also bistability and hysteresis arise from multisite covalent modification in signalling cascades [31] Ultrasensitivity and bistability induced by multisite phosphorylation may be a widespread mechanism for the control of protein activity in signalling networks, whereas zero-order ultrasensitivity is unlikely to be the major means of generating switch-like behaviour in such systems One thing is clear, the covalent cycle is extremely versatile for eliciting different kinds of behaviour [12,32] This great versatility may partly explain why signalling pathways, in both prokaryotic and eukaryotic systems, employ this motif in so many instances FEBS Journal 273 (2006) 895–906 ª 2006 The Authors Journal compilation ª 2006 FEBS 903 Effects of sequestration N Bluthgen et al ă Unfortunately, the lack of any clear guidance from experimental data means we are unable to determine exactly the true functional role played by these motifs Although many signalling networks have been mapped in great detail we still have very little understanding of their actual dynamical behaviour Until experimentalists embrace a systems approach we will remain in the dark regarding this question Methods The model files used to perform numerical simulations are available from the authors upon request Metabolic control analysis To analyse ultrasensitivity, we adopt some methods and terms from MCA [33–35], for application to conserved moieties, see Hofmeyr et al [27] MCA has been successfully applied to intracellular signal transduction systems in the past [36–38] MCA links ‘global’ control properties of a network to ‘local’ properties (e.g mechanistic details of enzyme-catalysed reactions) The local properties are called v @v elasticity coefficients and are defined by exji ẳ xij @ẵxji Elastiv cities evaluate the relative change in a reaction rate as a result of an infinitesimal relative change in one of its substrate, product, or effector concentrations (e.g [xi]) The elasticities of an enzyme ej following irreversible Michaelis– Menten kinetics with the Michaelis–Menten constant KM v are eejj ¼ with respect to the enzyme concentration and vj eS ẳ ẵSKMKM for the substrate S ỵ Global properties are called response coefcients and describe the response of the entire system to small perturbap tions in parameters, RSij ẳ ẵSij dẵSij Here, RSij accounts for a p p dp relative change in steady-state metabolite concentration [Si] upon infinitesimal relative change in the value of the parameter pj Model of a simple interconversion cycle In the first part of this paper we analyse a simple covalent modification cycle that consists of two enzymes K and P that phosphorylate and dephosphorylate a target protein T, respectively (Fig 1) T can be in a modified and unmodified form, denoted by T and T, respectively To investigate the effect of sequestration, we model the reactions catalysed by the two enzymes K and P We explicitly take the enzyme– target complex into account In the case of reversibility and product sensitivity, this system has been shown not to be ultrasensitive, and therefore such effects are not considered here [15] However, Ortega et al [15] did not consider sequestration The total concentrations of the three enzymes involved are denoted by [TT], [KT] and [PT] The enzyme–substrate complexes are called TK and T*P We 904 describe the dynamics of this kinetic scheme depicted in Fig by a system of three ordinary differential equations using mass-action kinetics Models of the MAPK cascade We shall also analyse the effect of sequestration in a more complicated system, the MAPK cascade We construct two models: One according to Kholodenko [9], which neglects sequestration, and a second one similar to Huang & Ferrell [23], which takes enzyme–substrate complexes into account In the second model, the parameters are adopted such that they reflect the catalytic constants and KM values of the model by Kholodenko [9] The details of the kinetic model can be found in the appendix The numerical analysis of the equations was carried out using mathematica and xpp-auto [39] Acknowledgements We would also like to thank Herbert M Sauro for critically reading the manuscript and for assisting NB in the development of some of the theory outlined here during NB’s stay at Sauro’s Laboratory in Los Angeles NB acknowledges support from DFG (SFB 618) FB was supported by the European Union through the Network of Excellence BioSim, Contract No LSHBCT-2004-005137 Cooperation between NB and FB was supported by the DFG Graduate Program GK268 ‘Dynamics and Evolution of Cellular and Macromolecular Processes’ BNK 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Discussion The function of the signal transduction network is to sense changes in the environment of the organism in the form of signals of physicochemical origin, e.g concentrations of molecules or