Supporting Information

Transcription

1 Supporting Information Olsman and Goentoro /pnas SI Materials Analysis of the Sensitivity and Error Funtions. We now define the sensitivity funtion Sð, «0 Þ, whih summarizes the steepness of the slope of the ativity urve is as a funtion of and «0, Sð, «0 Þ e «0 1 + e «0 2. [S1] Looking at the dynamis of ativity with respet to ligand hanging in time, we get the equation da dt a d dt Sð, «0Þ K A d Sð, «0 Þ d ln. [S2] dt dt limits give the ligand onentration range over whih an MWC protein behaves as a logarithmi sensor, «0 lnðτþ < ln < «0 + lnðτþ. [S7] This range is shown in Fig. 2 B and C, where the sensitivity regimes for different values of «0 are shaded in gray. We see that the range of ligand over whih the MWC systems funtions as a logarithmi sensor is set by a threshold for sensitivity S min. The error between the MWC ativity urve and the idealized sensor is parametrized by τ. The range over whih the MWC protein behaves as a logarithmi sensor depends on how muh error the system an tolerate. To derive the error, we first write the formal expression for an ideal logarithmi sensor, Here,weseethefirstrequirementsforaproteintogiverisetologarithmi sensing: the rate of hange of ativity is naturally a funtion of the logarithm of the ligand onentration.eq.5 is ompliated by the sensitivity funtion Sð, «0 Þ,whihvarieswith and is therefore not a simple proportional fator. An ideal logarithmi sensor requires that the ativity funtion depends stritly on ln, as illustrated by the blue dashed line in Fig. 2B. To measure how well an MWC protein an at as a logarithmi sensor, let us quantify the extent to whih Sð, «0 Þ varies as a funtion of. First, we note that an ideal logarithmi sensor oinides exatly with an MWC protein at the midpoint of the ativity urve (a 12, at the infletion point of a). This point also orresponds to the maximum of the sensitivity funtion S max 4 in Eq. 4 (i.e., the peak in Fig. 2C). Any variation in ligand whih pushes ativity away from the midpoint will lower the sensitivity and will do so in a nonlinear way. Our first task here is to define a regime of the sensitivity urve (the gray region in Fig. 2 B D) where the MWC protein an approximate a logarithmi sensor, and ompute the orresponding error. To parametrize variation from S max, we define the effetive ligand onentration to be Lð, «0 Þ e «0 ð Þ. [S3] Deriving Eq. 4 in terms of L, we obtain a natural representation of the sensitivity funtion, SðLÞ da dl L ð1 + LÞ 2. [S4] In this representation, the sensitivity is now maximized at L 1. ext, we derive a lower limit on the sensitivity funtion. Let us define the parameter τ, suh that for distane τ > 1 from the midpoint of the ativity urve, we have a minimum sensitivity, τ S min ðτþ ð1 + τþ 2. [S5] With these lower and upper limits on sensitivity, we now define the regime in the response urve over whih the MWC protein approximates a logarithmi sensor as S min < SðLÞ < S max. [S6] Using Eq. S3, we an derive a orresponding lower limit on for the range of L over whih the bound holds, 1 τ < L < τ. These a p ðlþ 1 4 ln L [S8] We will now use this expression to define an error funtion r 1 a a p to quantify the deviation of the atual ativity funtion a from the idealized one a p. Combining Eq. 3 and Eq. S8, wehave rðlþ 1 aðlþ a p ðlþ 1 2L. [S9] ð1 + LÞ ln L At the midpoint of ativity (L 1) the error funtion is minimized at rð1þ 0, beause this is the point where MWC ativity oinides exatly with the ideal logarithmi sensor. We observe that the error rðlþ inreases as L moves away from 1. Consequently, the error at the threshold τ, rðτþ, orresponds to the worst ase error in the sensitive regime. For example in Fig. 2E, where we set τ 6, we have rðτþ 0.1, so the MWC response differs by at most 10% from the ideal logarithmi response in the sensitive regime. The threshold τ serves as a way to analyze how muh the response of an MWC protein differs from an ideal logarithmi sensor as we expand the range of ligand onentration over whih it is used. The maximum error rðτþ inreases at an asymptoti rate of lim rðτþ 1 4 τ1 ln τ. This limit shows that the error inreases slowly with τ and that the MWC protein an approximate well an ideal logarithmi sensor over a wide range of the ativity urve. With the error funtion, we an now define the logarithmi regime of an MWC protein, as the ratio of the maximum and minimum ligand onentrations in the sensitive regime S ðτþ max e «0 +lnðτþ min e «0 lnðτþ e 2 lnðτþ τ 2. [S10] If we tolerate, for example, 10% error from the ideal logarithmi sensor (orresponding to τ 6), then an MWC protein with ooperativity 4 (as is the p ase, for example, with hemoglobin and PFK1), we have S ðτþ ffiffi τ 2.45, so the protein an at as a logarithmi sensor over a 2.45 range of fold hange in signal. If, for example, the protein of interest were a monomer (i.e., 1) that laks ooperativity, we would have S ðτþ τ 2 36, so the Olsman and Goentoro 1of6

2 protein an at as a logarithmi sensor over a 36-fold range of signal. We see from these results that an MWC protein an approximate an ideal logarithmi sensor over a substantial range of ligand onentration. Reduing ooperativity effetively inreases the regime over whih an MWC protein responds logarithmially to ligand. Eq. S10 tells us that there is an intrinsi trade-off between sensitivity and signaling range. Beause orresponds to ooperativity and s orresponds to the width of the sensitivity regime, we see diretly that inreasing for a given τ narrows the range over whih the sensor an funtion. Effets of the Allosteri Constant on the Sensitivity Funtion. We first analyze the effets of «0 on Sð, «0 Þ when is in the range K I and then analyze the general ase where ould be near saturation. In the former ase, as we derive in the main text, Sð, «0 Þ e «0 1 + e «0 2. Fig. S1 shows that Sð, «0 Þ shifts logarithmially as «0 is varied, in the same way as the ativity urve að, «0 Þ does. ext, we analyze the general ase for all values of ligand onentration. The general sensitivity funtion Sð, «0 Þ is defined in terms of the expression, a t Sð, «0Þ d log. [S11] dt For an MWC protein, we have from Eq. 2 that a t 1 A K 1 I K að1 aþ ð1 + Þð1 + K I Þ K 1 A K 1 I að1 aþ ð1 + Þð1 + K I Þ K 1 A K 1 I Sð, «0 Þ að1 aþ ð1 + Þð1 + K I Þ d dt d log dt [S12] [S13] ext, assuming that K I, we an rewrite the sensitivity funtion as Sð, «0 Þ að1 aþ ð1 + Þð1 + K I Þ. [S14] We see that, in the limit K I (i.e., 1, K I 1), Eq. S14 redues to the sensitivity funtion we derive in the main text, Sð, «0 Þ að1 aþ e «0 1 + e «0 2. Logarithmi tuning fails for very high or low values of «0,when is near saturation. To see why logarithmi tuning fails, we derive the limit of Sð, «0 Þ as, orresponding to the ligand onentration being near the lower saturation limit. We an make the simplifiation K I 1, whih yields S lower ð, «0 Þ að1 aþ. [S15] 1 + Fig. S2B shows the full sensitivity funtion (Eq. S14) as a solid blak line and the approximation in Eq. S15 as a blue dotted line. We see that, as approahes 1, Sð, «0 Þ is saled down by a K fator of A 1 + (shown as a dotted blak line). This effet will beome notieable when «0 is small enough to push the enter of the sensitivity funtion lose to. At the upper limit, as K I, whih give the simplifiation 1, we an derive 1 S upper ð, «0 Þ að1 aþ. [S16] 1 + K I Fig. S2C shows the Eq. S16 in red. As approahes K I, Sð, «0 Þ 1 sales down by a fator of 1 + K I. This effet will beome notieable when «0 is large enough to push the enter of the sensitivity funtion lose to K I. This analysis shows how and «0 ombine to determine the shape of the full sensitivity funtion and how logarithmi sensing breaks down as ligand onentration nears saturation. Allosteri Ativators and Inhibitors in the MWC Model. In their original model (21), Monod et al. did not express their allosteri onstant in the general form e «0 but rather proposed a more detailed model where the binding of allosteri ativators and inhibitors are expliitly aounted for, in muh the same way as the primary ligand. In terms of our notation, their model an be expressed in the form að, a, i Þ ni 1 + i K i where L e «0 1 + a Ka 1 +, [S17] L 1 + K I na. This version of the model assumes that the ativator and inhibitor have n a and n i binding sites with dissoiation onstants K a and K i, respetively. Rewriting the expression for L, we get L exp «0 + n i ln 1 + i n a ln 1 + a. [S18] K i K a If the allosteri effetors are far from saturation, then from the Taylor expansion of lnð1 + xþ, we have i a L exp «0 + n i n a. [S19] K i K a This approximation gives a mehanism for the linear dependene of free energy on the onentrations of allosteri regulators. Shimizu et al. found just suh a dependene in experiments on reeptor methylation in the baterial hemotaxis pathway (28). Logarithmi Tuning in the KF Model. Shortly after Monod, Wyman, and Changeux published their MWC model of allostery via onformational seletion, Koshland et al. put forth what is now alled the indued fit or KF model of allostery to explain hemoglobin binding kinetis (24). This model proposes that instead of undergoing spontaneous onformational hange, individual binding events in one subunit ould diretly hange the binding kinetis of another. This model has the advantage that it an both enapsulate positive ooperativity (like the MWC model) and negative ooperativity, where a given binding even ould potentially inhibit the next. In the years after both models of hemoglobin were published, strutural work by Perutz gave evidene that the MWC model was indeed more aurate. In referene to the work of Monod Olsman and Goentoro 2of6

3 et al., Perutz wrote, These words ring prophetially if we look at the mehanism in terms of quaternary struture (51). Be that as it may, the onept of indued fit proved useful for desribing other lasses of allosteri systems, in partiular, those in whih negative ooperativity plays an important role (52). Here, we show under what onditions the KF model an be logarithmially tuned. The KF model differs from other models of binding typially disussed beause the speifi geometry of the protein plays an important role, we will use as an example the tetrahedral geometry disussed in the original paper by Koshland et al. (24), whih results in the saturation funtion Yð, K BB Þ KAB 3 +3KAB 4 K 2 BB + 3K 3 AB KBB K 6 BB 1+4KAB 3 +6KAB 4 K 2 BB +4K 3 AB KBB 3 3 4, +K 6 BB [S20] where is the ligand dissoiation onstant, and the subunit onformations are denoted A and B. By onvention, A will be the low affinity inative state and B will be the high-affinity ative state. The interation strengths B and K BB represent the relative strengths of interations between the A and B onformations and the B onformation with itself, respetively. Here, we allow allosteri effets to enter through K BB. The motivation for this assumption is the underlying model that the allosteri effetors alter the stability of the bonds between the ative onformation. The authors use A 1 as a referene interation strength against whih to measure the other two, so it does not have to be expliitly aounted for in Eq. S20. In this model, high ooperativity omes from high stability of the ative state B (i.e., K BB 1 and B A 1). Under these onditions, the intermediate terms in the KF model drop out the saturation funtion and we have the simplified expression K 6 4 BB Y ð, K BB Þ 1 + KBB 6 4 e « e «0 4, [S21] where «0 6lnðK BB Þ. Here, we see that, in the limits of strong ooperativity, the KF model satisfies the logarithmi tuning requirement in relationship 9. This observation is onsistent with the data originally fitted by Koshland et al. (24), where they use A B 1 and, for the tetrahedral ase, find K BB ½1.8, 6.8Š. Even for the lower end of this range, we have KBB 6 34, whih is muh greater than the next largest oeffiient in Eq. S20, KAB 3 K3 BB 5.8. Detailed Analysis of the GPCR Model. Here, we present a more detailed derivation of the ativation funtion derived in Eq. 8.We begin again from the system of differential equations for GPCR ativation: _R ð1 RÞ k 2 R _T GDP α GDP T GDP R _T GTP T GDP R T GTP _α GTP T GTP α GTP _α GDP α GTP α GDP. From this system of equations, we will solve for ^α GTP, the relative level α GTP ompared with the total level of G protein T tot T GDP + T GTP + α GDP + α GTP. Just by setting derivatives equal to zero, we get α GTP ^α GTP α GTP T tot T GDP + T GTP + α GDP + α GTP T GTP T GDP + T GTP + α GDP + T GTP T GDP R T GDP + k4 T GDP R + α GDP + k4 T GDP R R 1 + k4 R + k4 R + k4 R. + R k6 + 1 R We then use the fat steady-state relationship to find ^α GTP + k2 + R k 2 + k 2 + k6 k 2 k k6 + 1, 1 k 2 just as in Eq. 8. Beause k2 is effetively a for the reeptors, it is taken as a fixed quantity. On the other hand, β-arrestin signaling alters the rate ( ) at whih T GDP binds to ative reeptors. To this end, we will rewrite Eq. 8 to see whether it an be made to look like the form desribed in relationship 9, withthe definition k2, ^α GTP 1 k k4. [S22] Here, we see that, if we allow k4 to play the role that e «0 plays in the MWC model, with variations in β-arrestin signaling effetively shifting the free energy «, then the GPCR model almost fits the logarithmi tuning requirement in relationship 9. The onfounding element is the fator of 1 that depends on and thus ould potentially ompliate things. Rearranging this term, we get k k3 1k4 + 1k5. From this equation, we see that there dependene on will vanish so long as either or, and onsequently under these onditions the system will behave as a logarithmi sensor. In terms of the biohemistry of the GPCR pathway, this means that β-arrestin binding is far from saturation so that T GDP is always able to find ative reeptors, be it at an attenuated rate. Olsman and Goentoro 3of6

4 Fold-Change Detetion Arises from Logarithmi Sensing and egative Feedbak. We present here simulations showing fold-hange detetion arising from a iruit ontaining allosteri regulation and negative feedbak. We use as speifi examples the Tar/Tsr reeptor system (disussed in Results) and the GPCR system. Tar/Tsr Reeptor and egative Feedbak. We model the allosteri regulation of the reeptor using the MWC model and the negative feedbak as desribed in Shimizu et al. (28) and Pontius et al. (38). The negative feedbak via methylation ats on a slower time sale than reeptor ativation, suh that að, «0 Þ instantaneously responds to hanges in ligand and allosteri effetor onentrations. Furthermore, «0 and a are related by a linear feedbak oupling, suh that að, «0 Þ e «0 1 + K I _«0 mða a 0 Þ, [S23] where a 0 is the basal ativation level to whih the system adapts and m is a onstant orresponding to the rate of adaptation. When _«0 0, we have a a 0, and the system will show preise adaptation, as expeted. Fig. S4 shows the hange in Tar/Tsr reeptor ativity (blue) in response to sequential threefold step inreases in ligand onentration (orange). The system gives idential responses for all three steps, performing fold-hange detetion. GPCR and egative Feedbak. The dynamis of the GPCR system are desribed in the main text (Eq. 8). As desribed in the main text, allosteri regulation is implemented through, whih haraterizes the rate of reeptor phosphorylation and β-arrestin binding. Although we ould express feedbak in terms of diretly, we may run into problems beause is a reation rate and therefore must be nonnegative. To avoid running into negative values, we rewrite βe «0, for some onstant β. The differential equations desribing the GPCR system are now _R ð1 RÞ k 2 R _T GDP α GDP βe «0 T GDP R _T GTP βe «0 T GDP R T GTP _α GTP T GTP α GTP _α GDP α GTP α GDP _«0 mða 0 ^α GTP Þ, [S24] where ^α GTP αgtp T tot. Fig. S5 shows the response of the GPCR system to sequential threefold step inrease in signal. We see again that a logarithmi sensor oupled with negative feedbak yields fold-hange detetion. Fig. S1. Effets of «0 on the sensitivity funtion. (A) Ativation urves for the MWC model in Eq. 1. The parameters used here are 10 2, K I 10 2, 4, and «0 ½15,25Š. (B) Sensitivity funtions orresponding to the MWC ativation urves in A. Olsman and Goentoro 4of6

5 Fig. S2. Saturation effets in the sensitivity funtion. (A) Ativation urves for the MWC model in Eq. 1, aross a full range of ligand onentration,. The parameters used here are 10 2, K I 10 2, 4, and «0 ½0,60Š. (B) Sð, «0 Þ as nears lower saturation. The solid blak urves are Sð, «0 Þ for the same parameters as in A, the dashed blue line is S lower ð, «0 Þ from Eq. S15, and the dotted blak line is the saling funtion KA (C) Sð, «0Þ as nears upper saturation. The solid blak urves are plots of Sð, «0 Þ for the same parameters as in A, the dashed red line is from Eq. S16, and the dotted blak line is the saling KA funtion KI Fig. S3. Logarithmi tuning in the KF model. Here, we show the apaity of the KF model to be logarithmially tuned. This plot uses 10 2, K ab 1, and K bb ½10 0,10 2 Š. For these parameters, we observe approximately three orders-of-magnitude in logarithmi shifting before the response urve begins to hange shape. Muh like the MWC and GPCR models, the KF model an potentially at as a logarithmi sensor over a broad range of signal. Olsman and Goentoro 5of6

6 Fig. S4. Fold-Change detetion with an MWC Tar/Tsr Sensor. In this simulation, 10 2, K I 10 2, 4, m 10, and a The blue line indiates the ativity of the Tar/Tsr reeptor. The orange line indiates ligand onentration, varied by threefold at eah step inrease. Fig. S5. Fold-Change detetion with a GPCR sensor. In this simulation, 0.01, k 2 15, 10, 20, 0.05, β 1, m 0.15, and a The blue line indiates the ativity of the GPCR system. The orange line indiates ligand onentration, varied by threefold at eah step inrease. Olsman and Goentoro 6of6