Supporting Information Liu et al. 10.1073/pnas.1215508110 8 SI Text > _ = − <> x1 k1x1x2 _ = − − 2 x2 k1x1x2 k2x2x3 [S6] I. Algebraic Observability > _ = − 2 :> x3 k1x1x2 2k2x2x3 The mathematical description of a control system, which re- _ = 2 : x4 k2x2x3 sponds to external inputs uðtÞ ∈ ℝK and provides specific outputs yðtÞ ∈ ℝM , is best described in the state-space form If we consider an open system, we should introduce in-flux for ( pure reactants (that never act as products) and out-flux for pure x_ = f ; x ; u ðtÞ ðt ðtÞ ðtÞÞ [S1] products (that never act as reactants) as follows: y = h ; x ; u ; ðtÞ ðt ðtÞ ðtÞÞ 8 > _ = − + <> x1 k1x1x2 C1 ∈ ℝN f · h · _ = − − 2 + where xðtÞ is the state vector of the system; ð Þ and ð Þ are x2 k1x1x2 k2x2x3 C2 [S7] > _ = − − 2 in general nonlinear functions. :> x3 k1x1x2 2k2x2x3 x _ = 2 − ; Assume that we have no knowledge of the initial state ð0Þ of x4 k2x2x3 C4x4 the system, but we can monitor yðtÞ perfectly in some interval so t = that all their time derivatives at time 0 can be calculated. The where for pure reactants A and B, we introduce constant in-flux C1 observability problem concerns the existence of relationships be- and C ; and for pure product D, we introduce x-dependent out- y 2 tween the outputs ðtÞ and their time derivatives, the state vector flux C4x4. With the extra terms due to in- and out-flux, the in- xðtÞ,andtheinputsuðtÞ such that the system’s initial state xð0Þ can ference diagram changes slightly—there will be self-edges for be deduced (1–5). From the differential algebraic point of view pure products. However, this will not change the prediction made (6–8), the observability of a rational system is determined by the by GA, because a pure product with self-edge is still a root dimension of the space spanned by gradients of the Lie derivatives strongly connected component (SCC) of size 1 in the inference diagram; hence, it has to be measured to yield observability. For XN X XK ∂ ∂ + ∂ simplicity we only consider closed systems here. We also assume L := + f + uð j 1Þ [S2] ∂ i∂ l ð jÞ that there are no external inputs uðtÞ and we consider the sim- t = xi ∈ℕ = ∂u i 1 j l 1 l plest measurement scheme, i.e., we directly measure a subset of state variables (e.g., the concentrations of some chemical species) of its output functions hðt; xðtÞ; uðtÞÞ. The observability problem S1 can be further reduced to the so-called rank test: the system is yðtÞ = ð⋯; x ðtÞ; ⋯ÞT: [S8] algebraically observable if and only if the NM × N Jacobian matrix i 2 3 S5 ∂L0h ∂L0h ∂L0h Now we show that the reaction system is algebraically ob- f 1 f 1 ⋯ f 1 servable if we measure the concentration of the pure product D, 6 ∂ ∂ ∂ 7 6 x1 x2 xN 7 = 6 7 i.e., y x4. 6 ⋯⋯⋯⋯7 Proof: 6 7 We calculate the Lie derivatives of the output function: 6 0 0 0 7 6 ∂L hM ∂L hM ∂L hM 7 6 f f ⋯ f 7 ð0Þ = 0 = ; [S9] 6 ∂ ∂ ∂ 7 Y Lf y x4 6 x1 x2 xN 7 6 7 J = 6 ⋮⋮⋮⋮7 [S3] 6 7 ð1Þ = 1 = 2; [S10] 6 7 Y L y k2x2x 6 ∂ N−1 ∂ N−1 ∂ N−1 7 f 3 6 Lf h1 Lf h1 Lf h1 7 6 ⋯ 7 6 ∂x ∂x ∂x 7 6 1 2 N 7 Y ð2Þ = L2y 6 ⋯⋯⋯⋯7 f 6 7 = _ 2 + _ [S11] 4 N−1 N−1 N−1 5 k2x2x3 k2x22x3x3 ∂L hM ∂L hM ∂L hM À Á À Á f f ⋯ f = − − 2 2 + − 2 ; k2 k1x1x2 k2x2x3 x3 k2x22x3 k1x1x2 2k2x2x3 ∂x1 ∂x2 ∂xN has full rank (6, 7), ð3Þ = 3 Y Lf y = : [S4] = € 2 + _ _ + _ _ + _2 + € rank J N k2x2x3 k2x22x3x3 k2x22x3x3 k2x22x3 k2x22x3x3 À Á2 = 2k x k x x − 2k x x2 For example, we can perform the rank test for chemical re- 2 2 À1 1 2 2 2 3 ÁÀ Á + 4k x k x x − 2k x x2 −k x x − k x x2 action systems with mass-action kinetics. Consider a reaction 2 3 À1 1 2 2 2 3 À 1 1 2 2 2 3 Á [S12] system with four chemical species {A,B,C,D} involved in two 2 2 2 + 2k2x2x3 −k x1x − 4k2x2x3 k1x1x2 − 2k2x2x reactions (Fig. S1D): À 1 2 Á À 3 ÁÁ + k x −k x x − k x x2 − 2k x2 −k x x − k x x2 ( 1 1À 1 1 2 2 2 3À 2 3 1 1 Á2 2 2 3 k1 2 2 2 2 R1 : A + B ! C + k x k x x − 2k x x k x x − 2k x x [S5] 2 3 À 1 1 2 2 2 3 Á 1 1 2 À 2 2 3 ÁÁ : + k2 : − − − 2 − 2 − − 2 : R2 B 2C ! D k1x1 k1x1x2 k2x2x3 k2x3 k1x1x2 k2x2x3 Using mass-action kinetics, the balance equations for the closed Then the Jacobian matrix can be calculated: system can be written as Liu et al. www.pnas.org/cgi/content/short/1215508110 1of10 2 3 the GA-selected sensor nodes. Consider the simplest reaction ∂L0y ∂L0y ∂L0y ∂L0y 6 f f f f 7 system: A → B (as shown in Fig. S1A). If we just measure A’s 6 ∂ ∂ ∂ ∂ 7 6 x1 x2 x3 x4 7 concentration as a function of time x1ðtÞ, we will never infer any 6 7 information about the initial state of B, i.e., x ð0Þ. Similarly, if we 6 ∂L1y ∂L1y ∂L1y ∂L1y 7 2 3 2 6 f f f f 7 0001 do not measure x in Fig. S1E we can never infer it, because x 6 7 6 7 4 4 6 ∂x1 ∂x2 ∂x3 ∂x4 7 does not appear in any other node’s balance equation. = 6 7 = 6 0 J22 J23 0 7; J 6 7 4 5 6 ∂L2y ∂L2y ∂L2y ∂L2y 7 J31 J32 J33 0 6 f f f f 7 B. Sufficiency. If all of the GA-selected sensor nodes are measured, 6 ∂ ∂ ∂ ∂ 7 J41 J42 J43 0 6 x1 x2 x3 x4 7 then the Jacobian matrix does not contain any zero columns (e.g., 6 7 Fig. S1 C, F,andI, Left). Because all nonzero elements in the 4 ∂L3y ∂L3y ∂L3y ∂L3y 5 f f f f Jacobian matrix are complicated polynomials of the state varia- ∂x1 ∂x2 ∂x3 ∂x4 bles, the probability of having dependent columns in the Jacobian matrix will be very low, if not zero. In the following we show with that measuring the GA-selected sensor nodes will very likely be sufficient to yield observability of biochemical reaction systems. = 2 We also show the exceptional cases and argue that they are rare. J22 k2x3 We first give intuitive explanations about the sufficiency of monitoring the GA-selected sensor nodes for the observability of = J23 2k2x2x3 biochemical reaction systems. Our argument is based on struc- tural control theory (9, 10). We linearize the right-hand side of x J31 = k1k2x2ð2x2 − x3Þ x3 the balance equations at an arbitrary state , obtaining the lin- earized system x_ðtÞ = Ax, which can be associated with a weighted À Á directed network GðAÞ with its weighted adjacency matrix = − − + + 2 + ∂ J32 k2x3 k1x1ð 4x2 x3Þ k2x ð8x2 x3Þ fi 3 a = jx. In other words, if a ≠ 0 then it gives the strength of ij ∂xj ij À Á weight that node j can affect node i. Positive (or negative) values = − − + + 2 + → J33 2k2x2 k1x1ð x2 x3Þ 2k2x3ð3x2 x3Þ of aij suggest that the edge ( j i) is excitatory (or inhibitory). The weighted directed network is also called the system digraph À À Á in structural control theory (11–13). J = k k x 2k x2 −8x2 + 2x x + x2 41 1 2À 2 2 3 2 2 3À 3 ÁÁÁ For biochemical reaction systems, a simple rule to get the + − + + 2 − + 2 directed network GðAÞ directly from the reactions is the fol- k1 x2x3ð 2x2 x3Þ 2x1 2x2 6x2x3 x3 lowing: For each reaction (i) draw a directed edge from each À À Á reactant to each product; (ii) draw a self-edge for each reactant; J = k 2k k x x2 −24x2 + 4x x + x2 (iii) if there are more than two reactants, draw a bidirectional 42 2 1À 2 1 3 2 Á2 3 3 À edge between every two reactants. For example, consider a re- + k2x4 72x2 + 32x x + x2 + k2x 2x x ð−3x + x Þ 2À3 2 2 3 ÁÁÁ3 1 1 2 3 2 3 action system with four chemical species {A,B,C,D} involved in + 2 − + 2 x1 6x2 12x2x3 x3 two reactions (as shown in Fig. S1G): 8 > k À À Á < R : A 1! B + C = − 2 + + 2 1 J43 2k2x2 2k1k2x1x3 8x2 3x2x3 2x3 [S13] À Á > k2 : R : B ↽ * D : + k2x3 48x2 + 40x x + 3x2 + k2x ðx ð−3x + x Þ 2 2 3 2 Á 2 3 3 1 1 1 2 3 k3 + x2ð−x2 + x3ÞÞ : Using mass-action kinetics, the balance equations for the closed system can be written as It can be shown via symbolic calculation that J has full rank. 8 Thus, the system is algebraically observable. > _ = − <> x1 k1x1 Note that in general we are not going to explicitly calculate _ =+ − + the initial state from Eqs.