( ) ( ), (S3) ( ). (S4)

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1 Ultrasesitivity i phosphorylatio-dephosphorylatio cycles with little substrate: Supportig Iformatio Bruo M.C. Martis, eter S. Swai 1. Derivatio of the equatios associated with the mai model From the differetial equatios that describe the dyamics of the system (Eqs. (1 4) i the mai text), the cocetratios of the chemical species at steady state are straightforward to obtai: the left had side of of Eqs. (1 4) becomes zero ad we get Eqs. (7 10), reproduced below.. S j = 1 b + j k + k ' ( ). S j = 1 b + j ( )( k + k ' ) ( f [ ] S j ( )k '. S j+1 ), (S1) + j +1 f [ ] S j ( )k ' (. S j 1 ), (S) + j +1 ( ), (S3) S j = 1, j + f [ ] f L, j S j + b. S j + ( j +1)k. S j+1 S j = 1 f L, j + f ( ). (S4) [ ] b L, j S j + b. S j + ( j +1)k. S j 1 where the rates are as described i the mai text. We wish to compute the Hill umber of the dose-respose curve, which ca be defied as proportioal to the local sesitivity at the level of iput that geerates a half respose [1] (Eq. (1)): h = d log g d log [ ]= h. (S5) This is the mathematical defiitio of the Hill umber ad ca be obtaied from a Hill equatio. rovided that the type of curve we are aalysig is of a similar shape to a sigmoidal Hill curve or a hyperbolic Michaelis-Mete curve, it is similar to other 1

2 proxy measures used to estimate ultrasesitivity [], ad it is particularly useful for its aalytical tractability. We keep the kiase levels fixed ad vary the phosphatase levels, hece the cocetratio of the phosphatase ormalised to the costat levels of the kiase is the iput, ad Eq. (S5) is evaluated at the level [ ] h that geerates a half maximal respose. The respose is give by the fractio of active states of the substrate g = j=0 S j +. S j=0 j S j + S j +. S j +. S j, (S6) hece we must solve the system of equatios S1 S4. We solve the system i a algebra package, such as Mathematica (Wolfram Research, Illiois), costraied to. S 1 = 0 i Eqs. (S1, S3) ad [. S +1 ] = 0 i Eqs. (S, S4) (because both. S 1 ad. S +1 do ot correspod to ay real states). Oe ca, for example, use Eqs. (S1 S4) to first determie. S 0, [. S 0 ], [. S 1 ] ad [ S 0 ] as a fuctio of S 0 ; the to use these to determie. S 1, [. S 1 ], [. S ] ad [ S 1 ] as a fuctio of S 0 ; ad cotiue util the!-th terms. If oe does the above calculatios for = 1, =, = 3, etc., ad cosiders a idealised symmetric system whe the parameters that gover the activity of the kiase are idetical to the equivalet parameters for the phosphatase ( f = f = f, b = b = b, k = k =, k ' = k ' = k ' cat ; ad so [ ] h = T ), as well as idetical forward ad backward allosteric rates (, j = f L, j = ), oe obtais: h =1 = b + B, h = = 3 3b + B, h =3 = 4 4b + B, (S7) (S8) (S9)

3 from whece we coclude:! h = ( +1)b L ( +1)b + B, (S10) where A = λ f ( ( k ' cat + b) + λ f ( + k ' cat )), B = b + λ f (( + k ' cat )( + λ f ) + k ' cat ) + b λ f + + k ' cat ( ( ) + k ' cat ). Rearragig Eq. (S10), oe obtais Eq. (15) i the mai text. Eq. (S10) has a upper boud of +1 because, uder the coditios described i the mai text, the first term i the deomiator vaishes whe b is small. Eq. (17) i the mai text is derived as above for = 1, but with oidetical allosteric rates ad disregardig the processive catalytic rates ( k ' k = k ' = 0 ).. The geeral case whe k ' k = k ' = 0 If we are simply iterested i the case where oly distributive reactios occur (diagoal arrows i Fig. ), the the aalytical calculatios are more ameable ad we ca write more elegat solutios for the steady state levels of the states of the system. From Eqs. (1 4) i the mai text, we obtai:. S j = 1 M, j. S j = 1 M, j [ ] S j, [ ] S j, (S11) (S1) S j = S 0 b M, x ( j + x L, x + x.k [ ] j )k M, x M, x 1 x.k f L, x + ( x)k, (S13) x=1 M, x j 3

4 S j = S j f L, j + ( j)k M, j, j + j.k M, j [ ], (S14) [ ] which have bee rewritte as a fuctio of the classical Michaelis-Mete costats M, j = b + j.k f ad M, j = b + ( j)k. f Eqs. (S11) ad (S1) are obtaied from Eqs. (1) ad (). The geeral expressios for the cocetratios of the substrate states S j ad S j are obtaied from the respective ordiary differetial equatios at steady state: ( ) S j d S j dt = 0 = f L, j S j + b. S j + ( j +1)k. S j+1, j + f [ ] d dt S j = 0 =, j S j + b. S j + j +1 ( )k. S j 1, (S15) f ( L, j + f [ ]) S j. (S16) Startig with j = 0 (deotig uphosphorylated substrate), we use Eqs. (S16) ad (S) to fid [ S 0 ] as a fuctio of S 0 : [ S 0 ] = 1,0 f L,0 + k M,0 [ ] S 0, (S17) ad the replace Eqs. (S1) ad (S17) ito Eq. (S15) obtai [ S 1 ] as a fuctio of S 0 : k M,1 S 1 = b M,1 k + L,1 [ ] M,0 k f L,1 + ( 1)k M,1 [ ] [ ]. (S18) S 0 Repeatig the steps above for j = 1, we obtai:! 4

5 [ S 1 ] = M,1 M,0 = M,1 S k k M, S 0 M,0 M,1, (S19) ( 1)k k,1 + k M,1 f L,1 + ( 1)k M,1 [ ], + k M, f L, + ( )k M, [ ] [ ] S 0. (S0) By repeatig this util we reach the -th terms, Eqs. (S13) ad (S14) ca be obtaied recursively, ad used directly to calculate the Hill umbers for purely distributive systems. For the alterative model of Fig. 5A, Eq. (0) i the mai text is obtaied recursively from the ordiary differetial equatio for S j described above. at steady state i the same maer as 3. Compariso to a Mood-Wyma-Chageux model The Mood-Wyma-Chageux (MWC) model of allostery [3] is a well kow mechaism that ca geerate ultrasesitivity i respose to a iput. Allosteric proteis are assumed to trasitio betwee two coformatioal states, active (typically called the R, or relaxed, state) ad iactive (typically called the T, or tese, state). The trasitios are cocerted, i.e., if the protei is multimeric, all of its subuits chage their coformatio simultaeously. I the absece of ay iput there is a equilibrium bias that favours oe of the coformatios, e.g., the iactive. Additio of the iput reveals a couterbalacig bias: the molecules of iput have greater affiity for the active form. Cosequetly, the presece of iput stabilises the active coformatio ad switches the system o. The dose-respose curve of the MWC system ca be sigmoidal with the upper bouds of the Hill umber beig equal to the umber of subuits (or the umber of bidig sites for the iput molecules). 5

6 Oe importat differece betwee the MWC system ad phosphorylatio cycles is that the former does ot eed to sped eergy all reactios occur i closed thermodyamic cycles ad reach chemical equilibrium. We therefore asked what are the cosequeces of that observatio for the dose-respose of a MWC system whose allosteric iteractios with the molecules of iput are govered by ezymes. Let us cosider the followig modificatio of the origial MWC model: a ezyme E bids to the substrate i the R state, formig the complexes E. R j. Molecules of iput S are the free to bid the substrate, formig the complexes E. R j+1 ad, if the ezyme ubids, R j+1. There is a ezyme F, which performs a aalogous activity whe the substrate is i the T state. Whe the system is at chemical equilibrium, calculatig the levels of the states of the system is straightforward: j R j = S R j 0 R E. R j = E, (S1) [ S] j E j j [ R 0 ], (S) R T j = L S j [ R 0 ], (S3) T F. T j = L F [ S] j F j [ R 0 ], (S4) R where R is the dissociatio costat of the molecules of iput i the R state, T is the dissociatio costat of the molecules of iput i the T state, E is the dissociatio costat of the ezyme E, F is the dissociatio costat of the ezyme F, ad L is the allosteric equilibrium costat betwee the R ad T states. The dose-respose fuctio is give by the proportio of active states f = j=1 j=1 R j + E. R j R j + E. R j + T j + F. T j. (S5) 6

7 Assumig the cocetratios of ezymes ad iput are much greater tha the cocetratio of substrate ad takig E = F =, we replace Eqs. (S1 S4) ito Eq. (S5) to obtai: f = [ 1+ S ] R 1+ S R + L 1+ c S R + F + E, (S6) where c = R F. Eq. (S6) is ultrasesitive for sigals that affect the ligad S (with Hill umbers of up to ), but ot for chages i the ezyme cocetratios (if S is fixed), presumably because the actio of the ezymes ca be easily reverted i eergy-free thermodyamic cycles. A phosphorylatio-dephosphorylatio cycle, as we have studied i the mai text, is ot so costraied ad ca geerate ultrasesitive behaviour as the ratio of ezyme cocetratio chages. Refereces 1. Beard DA, Qia H (008) Chemical biophysics: quatitative aalysis of cellular systems. Cambridge: Cambridge Uiversity ress. pp Goldbeter A, oshlad DE (1981) A amplified sesitivity arisig from covalet modificatio i biological systems. roc Natl Acad Sci U S A 78: Mood J, Wyma J, Chageux J (1965) O the ature of allosteric trasitios: a plausible model. J Mol Biol 1: