S3: Ultrasensitization is Preserved for Transient Stimuli

Transcription

1 S3: Ultrasesitizatio is Preserved for Trasiet Stimuli I the followig we show that ultrasesitizatio is preserved (albeit weaeed) upo trasiet stimulatio (e.g. due to receptor dowregulatio) as log as the stimulus duratio is sufficietly log to elicit ay sigal trasmissio. To model trasiet stimulatio, we shall assume that the phosphorylatig iase, K, is a receptor, which is subject to deactivatig iteralizatio. Sice receptor iteralizatio is usually slow whe compared to receptor-ligad associatio ad receptor (de)phosphorylatio, the time courses of receptors ca be approximated by a decayig expoetial (Heirich et al., 00). I the models, which were aalyzed i the paper, this was implemeted by assumig that the free iase is removed with a first-order rate costat it (see Fig. SA ad SB). Hece, we assumed that the substrate, S, ad factors which mediate receptor iteralizatio (or deactivatig phosphorylatio) bid competitively to the iase. Uless otherwise metioed, ultrasesitizatio was measured by plottig ormalized pea respose upo trasiet stimulatio (i.e. the maximum of the time course) as a fuctio of protei expressio (see e.g. Fig. SA). Additioally, the respose coefficiet of this pea respose (similar to Eq. i Protocol S) was plotted as a fuctio of a activatio fractio (similar to Eq. 4 i Protocol S) to allow direct compariso of ultrasesitizatio for differet iteralizatio ietics (e.g. Fig. SB; see also Legewie et al., 005) Fig. S: Ultrasesitizatio upo trasiet stimulatio ) Ultrasesitizatio due to substrate sequestratio: Reaalysis of Fig. 3 i the paper upo trasiet stimulatio reveals that the maximal ormalized respose, S / S tot, observed for strog substrate expressio levels, S tot, decreases with icreasig iteralizatio rates it (see Fig. SA). This is due to the fact that uder these coditios the stimulus duratio is too short to elicit sigificat substrate phosphorylatio. Nevertheless, ultrasesitizatio is preserved (albeit slightly weaeed) eve for such fast iteralizatio rates as ca be see from Fig. SB.

2 Fig. S: Ultrasesitizatio due to substrate sequestratio for varyig iteralizatio rates, it (Parameters chose: o, = ; off, = cat, = off,p = cat,p = ; o,p = ; K tot = 00; P tot = 0) Note: At a first glace it seems suprisig that trasiet stimuli ca elicit sigificat resposes eve for strog substrate expressio levels, where the respose time of a (de)phosphorylatio cycle is very slow due to proouced ezyme saturatio (ot show). This is due to the fact that very high substrate expressio levels, S tot, result i sequestratio of the iase i the S 0 K- complex ad thereby delay iteralizatio. Importatly, such delays are ot sigificat i the rage of ultrasesitizatio (ot show), i.e. the respose coefficiets plotted i SB were obtaied for trasiets with the half-life, τ / it. Similar coclusios also hold for Figs. S3 ad S4. Fig. S3: Ultrasesitizatio due to activity switchig for varyig iteralizatio rates, it (Parameters chose: o, = 0.; off, = cat, = off,p = 0; cat,p = ; o,p = 0; K tot = ; P tot = )

3 ) Ultrasesitizatio due to activity switchig: Reaalysis of Fig. 4 i the mauscript upo trasiet stimulatio (see Fig. S3) reveals that the maximal ormalized respose, S / S tot, agai decreases for icreasig iteralizatio rates it., sice stimulus duratio is too short (see above). As show i Fig. S3B, ultrasesitizatio due to activity switchig is preserved (albeit weaeed) as log as the stimulus duratio is sufficietly log to elicit strog sigal trasmissio. 3) Ultrasesitizatio due to syexpressio withi a iase cascade: Before aalyzig ultrasesitizatio due to syexpressio umerically, we shall discuss aalytical results obtaied by Heirich et al. (00) for a wealy iase activated cascade upo trasiet stimulatio. The correspodig model is show i Fig. SC, ad the receptor (i.e. the iput) time course was agai modelled by a decayig expoetial. For (a itegralbased defiitio of) the trasiet amplitude, S, of a wealy activated cascade with cascade stages Heirich et al. (00) derived: S = R 0 + P,l it j = P,j () To aalyze ultrasesitizatio due to syexpressio i Eq. we shall assume that a icrease i the regulator, r (see Fig. 5 i the mauscript), leads to a proportioal icrease i all iase ( K ) or all phosphatase ( P ) rate costats. Accordig to the results obtaied i Protocol S, a m-fold chage i the regulator, r, affects (o-ormalized) steady state sigal trasmissio upo wea stimulatio i a cascade with stages [m*]-fold (see Eq. 0 i Protocol S) a) The regulator, r, affects the expressio of all iases: It ca easily be see from Eq. that ultrasesitizatio due to syexpressio of all iases (where all K,i are simultaeously altered by the regulator, r) is always preserved upo trasiet stimulatio. b) The regulator, r, affects the expressio of all phosphatases: Ultrasesitizatio due to syexpressio of all phosphatases (where all P,i are simultaeously altered by the regulator, r) is also perfectly preserved as log as the coditio P,i >> it holds, sice the Eq. simplifies to: S R0 () P,l As expected this result equals that observed at steady state for the costat iput R = R 0, sice the iase cascade operates at steady state eve for trasiet stimuli if the timescale of dephosphorylatio is much faster tha that of receptor iteralizatio. By cotrast, ultrasesitizatio due to syexpressio of all phosphatases is weaeed as soo as a sigle phosphatase operates o the same timescale as receptor iteralizatio (i.e. as soo as a sigle P,i it ). I the extreme case, where all P,i << it, Eq. reduces to: 3

4 S R0 R0 P,l KP,l = r it it j= P,j j= KP,j (3) Here, we have used the relatioship P,i = r K P,i to demostrate that a m-fold chage i the regulator, r, alters the trasiet amplitude, S, oly [(-) m]-fold, so that ultrasesitizatio is slightly weaeed whe compared to the steady state result (see above). The preceedig disccusio implies that ultradesesitizatio due to syexpressio of all phosphatases is slightly weaeed upo trasiet stimulatio oly for low levels of the regulator, r (i.e. low phosphatase expressio), while ultrasesitizatio is perfectly preserved as the amout of regulator, r (i.e. phosphatase expressio), is further icreased. Reaalysis of Fig. 5B i the paper upo trasiet stimulatio (accordig to Fig. SB) reveals that strog sigal trasmissio i the cascade (i.e. T T tot ) occurs eve for very short stimuli (i.e. large it ), which is i cotrast to the observatios for a sigle phosphorylatiodephosphorylatio cycle (see Figs. SA, S ad S3). This is due to the fact that eve wea fractioal activatio levels of the upstream species, S (i.e. S << S tot ), elicited by short trasiet iputs outweigh the phosphatase activity towards T for sufficietly large regulator levels, r = S tot = T tot, so that phosphorylatio of T still occurs. Figure S4B cofirms that ultrasesitizatio due to syexpressio is preserved (albeit weaeed) upo trasiet stimulatio. Fig. S4: Ultrasesitizatio due to activity switchig for varyig iteralizatio rates, it (Parameters chose: off, = off,5 = cat, = cat,6 = off,3 = off,7 = cat,8 = ; o, = 0.0; o,5 = 0.; o,3 =.; o,7 = ; cat,4 =.; K tot = 0; P S,tot = P T,tot = ; S tot = T tot = r) 4

5 Refereces Heirich R, Neel BG, Rapoport TA (00) Mathematical models of protei iase sigal trasductio. Mol Cell 9: Legewie S, Blüthge N, Herzel H (005) Quatitative aalysis of ultrasesitive resposes. FEBS J 7: