Inferring dynamic architecture of cellular networks using time series of gene expression, protein and metabolite data. Supplementary Information

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1 Inferring ynamic architecture of cellular networks using time series of gene expression, protein an metabolite ata Euaro Sontag, Anatoly Kiyatkin an Boris N. Kholoenko *, Department of Mathematics, Rutgers Uniersity, Pisaway, NJ 0885, USA an Department of Pathology, Anatomy an Cell Biology, Thomas Jefferson Uniersity, 00 Locust Street, Philaelphia, PA 907, USA * Supplementary Information

2 Supplementary Proof This supplement iscusses the fact that, in contrast to the stationary case, for time-arying responses, the rank of the matrix R(t, P i generically equals n at any gien time, for any n inepenent perturbations selecte accoring to Eq. (that is, if f i / p j (x, p 0 for all p j P i. Moreoer, we show that, in a precise mathematical sense, this rank generically equals n een when only a single network noe is irectly affecte by n experimental interentions, each of which changes an inepenent parameter influencing that particular noe. We will pick the single network noe which is irectly affecte by n experiments to be the first one, an we will rop the subscript i in P i (since we are only intereste in i. Since we are intereste in properties of the Jacobian of f, the genericity statement is expresse in terms of linearizations of f. For simplicity, we stuy the case of time-inariant linear systems ẋ Fx (ot inies time eriatie whose matrix F has the following form: F + p F + p F n + p n F F F n F.... F n F n F nn where the F ij s are fixe coefficients an the p j s are the parameters being perturbe. (We iew such a matrix as a possible linearization of f aroun a particular state. A result may also be proe for time-epenent matrices, corresponing to linearizations along trajctories, but the present approach is sufficient in orer to show that one gets a full rank een when perturbations only irectly affect one of the ariables. We will show that for this system, an for generic alues of the F ij s, the parameters p j s, the initial conition x(0, an the time T, the sensitiity matrix R(t, P i with respect to the parameters p j at time T has full rank. Let us first efine eerything precisely. For any gien n n matrix F (F ij an n-ector P (p,..., p n, we write: F + p F + p F n + p n F F F n F(F, P..... F n F n F nn

3 Obsere that, for any n-ector x col (x,..., x n, we hae that: F(F, px p j b j (x x j 0. 0 for each j,..., n. We next introuce the sensitiity matrices with respect to the parameter ector P along solutions of ẋ F(F, Px. Pick any n- ector ξ an consier the following initial alue problem, a system of (n+n ifferential equations: ẋ F(F, Px, x(0 ξ ż F(F, Pz + b (x, z (0 0 ż F(F, Pz + b (x, z (0 0. ż n F(F, Pz n + b n (x, z n (0 0. We efine the sensitiity matrix as folows R(T, F, ξ, P (z (T,..., z n (T for any positie time T. This is the same as R(t, P i in the main text, except that we are showing the epenence on initial states an constants efining the system. Theorem. For generic alues of T, F, ξ, P, et R(T, F, ξ, P 0. The meaning of generic is: consier the set of ectors of size + n + n + n which list T an the entries of F, ξ, P; for all such ectors except for a set of measure zero, the matrix is nonsingular. Proof. Solutions of the shown initial alue problem are real-analytic functions of time an of the parameters efining the system (see for instance []. Therefore, to show that the eterminant is generically nonzero, it is enough to show that R(T, F, ξ, P 0 for just one choice of (T, F, ξ, P. We pick T, P 0 (zero ector, 0 ξ, 0. 0

4 an: F Notice that F(F, P F, because P 0. We hae that the jth coorinate of the solution of ẋ Fx with initial conition ξ is: x j (t t j, j,..., n. From this, one erifies inuction that: z j (t col (z j (t,..., z nj (t where: z ij (t (i!(j! (j + i! t i+j. We pick, in particular, t. Therefore, R(, F, 0, 0 Z, where Z is the following n n matrix: ( (i!(j!. (i + j! To show that Z is nonsingular, we first multiply the i-th row by (i! an the j-th column by (j!, so we may without loss of generality assume that Z is the n n matrix with entries: (. (i + j! This is the n-th Hankel matrix H n of the expansion of the exponential. The matrix H n has full rank, since its eterminant is nonzero: n p et H n (q (q p q (this well-known formula can be erie using the computer-algebra system CLD, eelope by Doron Zeilberger, an can be erifie by appliion of Dogson s rule for eterminants, cf. []. References [] Zeilberger, D. (00 Liebe Opa Paul, Ich Bin Auch Ein Experimental Scientist, A. Appl. Math.,, 5-5. [] E.D. Sontag (998 Mathematical Control Theory: Deterministic Finite Dimensional Systems, Secon Eition, Springer, New York.

5 Supplementary Table. Rate expressions, ifferential equations an parameter alues of the gene network moel of Fig.. Concentrations ([mrna i ], i - an Michaelis constants (K a i, K I i; K i are gien in nm. Maximal enzyme rates ( s i, i are expresse in nm/hr. synth synth synth synth egr egr egr egr + s + s Rate equation s + A mrna a n mrna K + a n ([ ] K I n ([ ] ([ mrna ] K a n ( + A ([ mrna ] K a n + ([ mrna ] K s a n ([ ] + A mrna K a n ([ ] n ([ ] + I mrna K mrna K + a n A ([ mrna ] K ( a n + [ mrna ] K [ mrna ] ( K + [ mrna ] [ mrna ] ( K + [ mrna ] [ mrna ] ( K + [ mrna ] [ mrna ] K + [ mrna ] Parameter alues s ; A ; K a.6; n ; K I 0.5; n s 0.7; A ; K a.6; n s 0.6; A 5; K a.5; n ; K I 0.7; n s 0.8; A ; K a 0.5; n 0; K 0 00; K 60 0; K 0 00; K 50 ( System of ifferential equations: [mrna i ]/t i synth - i egr, i,,,. Initial conitions. We assume that at t0, all four genes were inactie (the alytic constants of transcription rates an [mrna i ] equale zero. When t +0, the constants were assigne the alues gien in the Supplementary Table aboe, an the transition to an actie state began. Perturbations. The following parameters were perturbe in orer to calculate the appropriate row of Jacobian elements: row : s,, s, row : s,, s, row : s,, s, row : s,, s, The finite ifferences between the control an perturbe transitions were calculate accoring to Eq. 7 of the main text.

6 Supplementary Table. Rate expressions, ifferential equations an parameter alues for the MAPK cascae moel of Fig.. Concentrations an the Michaelis constants (Kij, i,, 5, 7, 9, ; j,, ; Kmp; Ki are gien in nm. The alytic rate constants (k i, i,, 5, 6, 9, 0 an the imal enzyme rates ( i, i,, 7, 8,, are expresse in s - an nm s -, respectiely Rate equation k [ RasGTP] [ MKKK] ( K + [ MKKK] + [ MKKKP] K K ( + [ MAPKPP] K k [ RasGTP] [ MKKKP] ( K + [ MKKK] + [ MKKKP] K K ( + [ MAPKPP] Ki [ MKKKPP] ( K + [ MKKKPP] + [ MKKKP] K K + [ MKKK] K K [ MKKKP] ( K + [ MKKKPP] + [ MKKKP] K K + [ MKKK] K K k5 [ MKK] [ MKKKPP] ( K5 + [ MKK] + [ MKKP] K5 K5 k6 [ MKKP] [ MKKKPP] ( K5 + [ MKK] + [ MKKP] K5 K5 7 [ MKKPP] ( + A [ MAPKPP] Kmp ( K7 + [ MKKPP] + [ MKKP] K7 K7 + [ MKK] K7 K7 + [ MAPKPP] 8 [ MKKP] ( + A [ MAPKPP] Kmp ( K7 + [ MKKPP] + [ MKKP] K7 K7 + [ MKK] K7 K7 + [ MAPKPP] k9 [ MKKPP] [ MAPK] ( K9 + [ MAPK] + [ MAPKP] K 9 K9 k0 [ MKKPP] [ MAPKP] ( K9 + [ MAPK] + [ MAPKP] K 9 K9 [ MAPKPP] ( K + [ MAPKPP] + [ MAPKP] K K + [ MAPK] K K [ MAPKP] ( K + [ MAPKPP] + [ MAPKP] K K + [ MAPK] K K i ( K mp ( K mp Parameter alues k ; [RasGTP]0; K 00; K 0; K i 00 k 5; [MKKK] total ; K ; K 8; K k 5 ; K 5 00; K 5 0 k 6 5; [MKK] total ; K 7 ; K 7 8; K 7 80; A5; K mp k 9 ; K 9 00; K 9 0 k 0 5; [MAPK] total 60 8.; K ; K 8; K

7 Supplementary Table continue Differential equation system: Concentrations of unphosphorylate forms: [ MKKKP] / t ν -ν + ν -ν [ MKKK] [ MKKK] total MKKKP] MKKKPP] [ MKKKPP] / t ν -ν [ MKKP]/ t ν -ν + ν -ν [ MKK] [ MKK] total MKKP] MKKPP] [ MKKPP] / t ν -ν 6 7 [ MAPKP] / t ν -ν + ν -ν 9 0 [ MAPK] [ MAPK] total MAPKP] MAPKPP] [ MAPKPP] / t ν -ν 0 Initial conitions. In all simulations, the initial conition (t0 correspone to the steay state of the MAPK pathway with a low Ras actiity, [RasGTP] 0. nm, [MKKKP] nm, [MKKKPP] 0.0 nm, [MKKP] 0.68 nm, [MKKPP] nm, [MAPKP].607 nm, [MAPKPP] 0.59 nm. When t +0, the [RasGTP] leel increase to a new high alue of 0 nm, an the transition from the steay state with a low actiity to a high actiity state was consiere. Note that the responses R ij (0 0 at time zero, since both perturbe an unperturbe solutions hae the same initial conition. Perturbations. The six following parameters were perturbe in orer to calculate the appropriate row of Jacobian elements: row : k 5, k 6, 7, 8, k 9, k 0 row : k,, k 5, k 6, 7, 8 row : k, k, k 9, k 0,, row : k, k,,, k 5, 8 row 5:,, k 5, k 6, 7, 8 row 6: k 5, k 6, 7, 8, k 9, The finite ifferences between the control an perturbe transitions were calculate for perturbation magnitues of 5, 5 an 50% accoring to Eq. 7 of the main text. 6

8 Supplementary Table. A snapshot of the retriee experimental (superscript a an known theoretical (b interaction strengths for the MAPK pathway moel. The Jacobian elements, F ij, are calculate using 5% perturbation of the parameters inie in Supplementary Table, an correspon to 0.75 min after a transition of the cascae from a resting state to an actie state began. 7

9 Supplementary Table. Rate expressions, ifferential equations an parameter alues for the oscillating MAPK cascae moel. A MAPK cascae moel, where the phosphorylation leels of component proteins exhibit sustaine oscillations, was reporte preiously []. It possesses a single negatie feeback from MAPK-PP to MKKK-P. A rigorous stuy of the emergence of oscillations in this moel was carrie out in [,5]. All concentrations an the Michaelis constants (K - K 0 are gien below in nm. The alytic rate constants (k, k, k 7, k 8 an the imal enzyme rates (,, 5, 6, 9, 0 are expresse in min - an nm min -, respectiely. Reaction Rate equation Parameter alues number * [MKKK]/(( + ([MAPK-PP]/K I n (K + 50; n; K I 9; K 0; [MKKK] [MKKK-P]/(K + [MKKK-P] 5; K 8; k [MKKK-P] [MKK]/(K + [MKK] k.5; K 5; k [MKKK-P] [MKK-P]/(K + [MKK-P] k.5; K 5; 5 5 [MKK-PP]/(K 5 + [MKK-PP] 5 5; K 5 5; 6 6 [MKK-P]/(K 6 + [MKK-P] 6 5; K 6 5; 7 k 7 [MKK-PP] [MAPK]/(K 7 + [MAPK] k 7.5; K 7 5; 8 k 8 [MKK-PP] [MAPK-P]/(K 8 + [MAPK-P] k 8.5; K 8 5; 9 9 [MAPK-PP]/(K 9 + [MAPK-PP] 9 0; K 9 5; 0 0 [MAPK-P]/(K 0 + [MAPK-P] 0 0; K 0 5; Total concentrations: [MKKK] total 00; [MKK] total 00; [MAPK] total 00 System of ifferential equations: Moiety conseration relations: [MKKK-P]/t - [MKKK] total [MKKK] + [MKKK-P] [MKK-P]/t [MKK-PP]/t 5 [MAPK-P]/t [MAPK-PP]/t 8 9 [MKK] total [MKK] + [MKK-P] + [MKK-PP] [MAPK] total [MAPK] + [MAPK-P] + [MAPK-PP] Perturbations. The following parameters were perturbe in orer to calculate the appropriate row of Jacobian elements: row : k, k, 5, 6, k 7 row :,, k 8, 9, 0 row :,, k, 6, k 7 row :,, k, k, 5 row 5: k, k, 5, 6, 0 There are aitional requirements on the experimental protocol applie to sustaine (limit cycle oscillations. In fact, we hae shown elsewhere that because of the phase ifferences between the original an perturbe trajectories, the sensitiities to parameter change ten to infinity when the time after perturbation infinitely increases [6]. Therefore, experimental setup shoul allow to effectiely restart perturbation responses following oscillatory behaior, e.g., applying perturbing agents to aliquots of unperturbe cells at the selecte time points. 8

10 References.. Kholoenko, B. N. (000 Negatie feeback an ultrasensitiity can bring about oscillations in the mitogen-actiate protein kinase cascaes. Eur J Biochem. 67, Sontag, E.D. (00 Asymptotic amplitues an Cauchy gains: A small-gain principle an an appliion to inhibitory biological feeback. Systems an Control Letters, 7, Angeli, D. & Sontag, E.D. (00 Monotone control systems. IEEE Trans. Autom. Control 8, Kholoenko, B. N., Demin, O.. & Westerhoff, H.. (997 Control analysis of perioic phenomena in biological systems, J. Phys. Chem. 0,

6 General properties of an autonomous system of two first order ODE

6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x