1. 1 Prof. Bor-Sen Chen 陳博現 Lab of Control and System Biology Department of Electrical Engineering National Tsing Hua University A new measure of biochemical network robustness

2. 2 Motivation The robustness of a biochemical network is defined as the tolerance of variations in kinetic parameters with respect to the maintenance of steady state. (phenotype) Robustness also plays an important role in the fail- safe mechanism in the evolutionary process of biochemical networks. The purposes of this paper are to use the synergism and saturation system (S-system) representation to describe a biochemical network and to develop a robustness measure of a biochemical network subject to variations in kinetic parameters. Since most biochemical networks in nature operate close to the steady state, only the robustness measurement of a biochemical network at the steady state is considered.

3. 3 Results We show that the upper bound of the tolerated parameter variations is related to the system matrix of a biochemical network at the steady state. Using this upper bound, we can calculate the tolerance of a biochemical network without testing all possible parametric perturbations and gain much insight into the robustness of a biochemical network. We find that a biochemical network with a large tolerance can also better attenuate the effects of variations in rate parameters and environments.

4. 4 Compensatory parameter variations and network redundancy are found to play important roles in the robustness of biochemical networks. Finally, four biochemical networks, i.e., a cascaded biochemical network, the glycolytic-glycogenolytic pathway in a perfused rat liver, the tricarboxylic acid (TCA) cycle in Dictyostellium discoideum and the cAMP oscillation network in bacterial chemotaxis, are used to illustrate the usefulness of the proposed robustness measure.

5. 5 METHODS and Results Model of a biochemical network The following S-system model has been an efficient model for describing the dynamic system of a biochemical network in the last three decades (Savageau1976; Irvine and Savageau, 1985a,b; Heinrich and Schuster,1996, 1998; Voit, 2000) where are the metabolites such as substrates, enzymes, factors and products of a biochemical network, in which denote n dependent variables and denote the independent variables.

6. 6 Robustness measure of a biochemical system Consider the steady state of a biochemical network in (1), i.e., inputs and outputs are in balance (Voit, 2000). Assume none of rate constants and variables in (2) is zero. Taking the logarithm on both sides of (2), we get

7. 7 Then, after some rearrangements, Introduce new variables and coefficients as follows The steady state of a biochemical system consists of n linear equations in n+m variables

8. 8 In the above equations, the dependent (unknown) variables are separated from the independent (known) variables.

9. 9 Let us denote where A D denotes the system matrix among the interactions of dependent variables and A I indicates the interactions between the dependent variables and independent variables Y I.

10. 10 We obtain the steady state equation In the nominal parameter case, we assume that the inverse of A D exists, and then get the steady state of a biochemical system as follows

11. 11 where denotes the ith singular value and denote the corresponding left and right singular vectors, respectively.

12. 12 Then with UU T =I, VV T =I and we obtain (Gill et al., 1991; Press et al., 1992)

13. 13 Suppose the parameter perturbation due to mutation or disease can alter the kinetic properties of the steady state of a biochemical system in (6) as follows where the parameter perturbations of biochemical network are defined by

14. 14 △ A D denotes the parameter perturbations due to the kinetic parameter variations △ g ij and △ h ij of dependent variables, △ b denotes the parameter perturbations due to rate constant variations and △ A I denotes the parameter perturbations due to the kinetic parameter variations of independent variables. In general, the effect of △ b and △ A I can not influence the existence of steady state. Their influences on the magnitude of Y D + △ Y D can be discussed by sensitivity matrices and, respectively (Savageau, 1969a, 1969b, 1970; Voit, 2000). The robustness is mainly to check the tolerance for △ A D with respect to the maintenance of steady state of the perturbed biochemical network.

15. 15 From (10), we obtain If the following robustness condition holds (Noble and Daniel, 1988; Gill et al., 1991; Weinmann, 1991) then the singular values of are free of zero and the inverse exists.

16. 16 Therefore, the steady state of the biochemical network in (11) is uniquely solved as The above analysis says that if the robustness condition in (12) holds, then the steady state of a biochemical system is preserved under parameter variations, i.e., in (11) has a small difference from the nominal in (7) under small parameter perturbations. However, if condition (12) does not hold, some singular values of may be zero and the inverse may not exist, and the steady state may cease to exist under parameter perturbation.

17. 17 The physical meaning of (12) is that if the norm of the normalized perturbation of kinetic parameters is less than one or is contractive, the effect of the kinetic parameter perturbation can be tolerated by the biochemical network and the steady state of the biochemical network is preserved. Therefore, the inequality in (12) can be used to test the robustness of the biochemical system under kinetic parameter perturbation due to mutation or disease. Equivalently, (12) can be rewritten as a more intuitive robustness condition as follows (Noble and Daniel, 1988; Gill et al., 1991; Weinmann, 1991) i.e., is the upper bound of.

18. 18 If the robustness condition (14) holds, the steady state of the perturbed biochemical network still exists. Let us denote the robustness condition in (14) as follows If the robustness matrix R is a symmetric positive definite matrix, the steady state of biochemical network is still preserved, under parameter perturbation. This is a simple criterion to check whether the parameter variation is tolerated or not. It has been shown that the robustness matrix R is positive definite if and only if all its eigenvalues are real and strictly positive (Gill et al., 1991; Noble and Daniel, 1988).

19. 19 Let us denote as the ith eigenvalue of R. Then the following inequalities are the robustness conditions for the biochemical network to tolerate. From the singular value decomposition in (8), we have Therefore, if a parameter variation is specified as follows

20. 20 Then Obviously, the inverse does not exist under the parameter perturbations in (18).

21. 21 Moreover, which is not positive definite (Gill et al., 1991) and violates the robustness condition in (15).

22. 22 Obviously, the perturbation with a magnitude at the direction will destroy the steady state of a biochemical network. That is, at the direction, the biochemical network has a weak structure, and a kinetic parameter perturbation at the direction will disrupt the steady state. If the perturbation is at the direction, then the biochemical system will tolerate the parameter perturbation to the value less than, i.e., the biochemical network is most weak at the direction, in which a smallest perturbation will destroy the steady state of the biochemical network.

23. 23 Relation between robustness and sensitivity From (6), it is easy to know the effects of variations △ b and △ Y I of rate constant and the environment on the output variation △ Y D. The sensitivity from △ b to △ Y D is given by (Savageau, 1970; Voit, 2000) Obviously, from (14), the sensitivity from △ b to △ Y D is inverse to the robustness of biochemical network, i.e., if the biochemical network is robust, it is less sensitive to the variation of rate constant, and vice versa.

24. 24 Similarly, the sensitivity from environment variation △ Y I to △ Y D output variation is given by (Savageau, 1970; Voit, 2000) Obviously, from (14), the sensitivity is also inverse to robustness and a biochemical network with strong robustness will be more resistant to the effect of environmental variation △ Y I. In the last two decades, system control theory has addressed the robustness problem about the effect of the environment on the system output at some operation points. How to design a feedback control to minimize the sensitivity to achieve an optimal robustness design is an important topic in modern control theory for the last two decades.

25. 25 From (14), i.e., it is seen that there are two mechanisms for the robustness improvement of biochemical networks. One mechanism is to increase in order to tolerate large parameter perturbations in. Another mechanism is to prevent the occurrence of large parameter perturbations in △ A D so that the robustness condition in (14) can not be easily violated. The redundancy and compensatory parameter variation (i.e., △ g ij = △ h ij so that △ A D =0 in (10)) may be two major sources of this kind of robustness. Compensatory parameter variations make △ A D small. Network redundancy will be a buffer to prevent possible violent kinetic perturbations in △ A D, which may violate the robustness condition in (14). Therefore, network redundancy and compensatory parameter variation may be two efficient mechanisms to attenuate the parameter perturbation to prevent violating the robustness condition in (14), which will be discussed in the following examples.

26. 26 Experimental Simulations Simulation experiment 1 The role of the cascaded system in Figure 1 has been investigated as an amplifier for biochemical signals (Savageau, 1976; Voit, 2000). Cascaded mechanisms are found in diverse areas of biochemistry and physiology, including hormonal control, gene regulation, immunology, blood clotting, and visual excitation.

27. 27 The cascaded network can be represented as follows (Voit, 2000) In this case, the system matrix of the cascaded network is as follows

28. 28 Suppose the network suffers parameter perturbations due to gene mutation as follows In this kinetic parameter perturbation case,

29. 29 and, which can be checked by the following strictly positive eigenvalues of R (0.073, 0.3595, 0.7276). From the simulation in Figure 2(b), the steady state is preserved. However, since the parameters of the cascaded network are perturbed, the steady state Y D + △ Y D has some changes even when the characteristics of the steady state are preserved after some parameter perturbations.

30. 30 Suppose the cascaded biochemical network suffers the following kinetic parameter variations In this case, we have (28)

31. 31 and, which is not a positive definite matrix because its eigenvalues are not all strictly positive (i.e., -0.0176, 0.3877, 0.7544). Obviously, the robustness condition in (14) or (15) is violated and the existence of the steady state of the cascaded biochemical network is not guaranteed. From the simulation result in Figure 2(c), the steady state of the cascaded biochemical network ceases to exist under this parameter perturbation. These computational results confirm the claim of our robustness condition.

32. 32 From the control system point of view (Weinmann, 1991; Qu, 1998), feedback inhibition plays an important role in the robustness of biochemical networks. In the cascaded biochemical network in (24), the kinetic parameters g 12 and g 13 model the feedback inhibition. Even with small changes, they have much influence on the robustness of the cascaded biochemical network, especially with a sign change. Suppose the negative feedback of the cascaded network is perturbed into positive feedback; for example, g 12 changes from –0.1 to 0.3 and g 13 from – 0.05 to 0.15,respectively. In this situation, the cascaded network becomes

33. 33 In this case,, with the eigenvalues of R as follows (-0.0432, 0.3325, 0.7731), and the robustness condition (15) is violated. The dynamic response of this cascaded network is shown in Figure 2(d), in which the steady state is not preserved. So we find that adequate negative feedback inhibition has the robustness property as in control theory, which contributes significantly to the robustness of the biochemical system.

34. 34 Furthermore, suppose the negative feedback inhibition from X 2 to X 1 in the cascaded network of Figure 1 consists of duplicated pathways but with same flux, then the first equation in (24) is modified as. In this situation, a failure of one redundancy will lead to, which will cause a smaller than that in the failure of the feedback loop from X 2 to X 1 without redundancy. Therefore, redundancy is a source of robustness via the mechanism of decreasing.

35. 35

36. 36 Simulation experiment 2 Consider the glycolytic-glycogenolytic pathway in perfused rat liver (Scrutton and Utter, 1968; Torres, 1994a, b, c).

37. 37 The kinetic properties of the pathway are obtained as follows (Voit, 2000), which are also shown in Figure 3 (29) where

38. 38 In this case, the system matrix A D is obtained as The dynamic response of the nominal glycolytic- glycogenolytic pathway in (29) is given in Figure 4(a) and the upper bound of the tolerance is given by That is, if the parameter perturbation measure is less than, i.e. the robustness matrix R > 0, the characteristics of the steady state will be preserved.

39. 39 Suppose the glycolytic-glycogenolytic pathway suffers from a kinetic perturbation as follows (30) in which the parameter perturbation is of the following form

40. 40 In this perturbed case, and, which is a positive definite matrix with its eigenvalues all positive (i.e., 0.0204, 2.8261, 56.3208). From our computational result and the perturbed dynamic response shown in Figure 4(b), the robustness of steady state is preserved.

41. 41 Suppose another parameter perturbation occurs such that the glycolytic-glycogenolytic pathway is perturbed as (31) In this case, we have,.

42. 42 is not positive definite, because its eigenvlaues are not all positive (i.e., -0.0915, 2.7967, 56.2427). Therefore, the robustness condition is violated. From the dynamic response shown in Figure 4(c), we can see that the steady state of the biochemical network ceases to exist. In order to confirm the compensatory parameter variations (i.e., for all i,j such that ) in Remark 2, we let △ g 22 = △ h 22 =5

43. 43

44. 44 Simulation experiment 3 The tricarboxylic acid (TCA) cycle in Dictyostelium, a soil-living amoeba, produces ATP very efficiently while decomposing pyruvate to water and CO2 via acetyl-CoA. Under a nutrient-rich condition, the cycle is fed by ingested proteins that are broken down into amino acids.

45. 45

46. 46 The S-system model is shown as follows (Voit, 2000)

47. 47 The TCA network in the above equation is shown in Figure 5, and the dynamic response is shown in Figure 6(a). The system matrix A D is obtained as

48. 48 From, we can calculate the upper bound of the perturbation tolerance. That means when the network is perturbed by such that the robustness condition (14) is violated, and the steady state of the biochemical network may not exist. Suppose there is a perturbation as follows (34)

49. 49 where,

50. 50 In the perturbed case, the robustness matrix R is not a positive definite matrix with its eigenvalues (-0.006, 0.0218, 0.0955, 0.3666, 0.5360, 0.8629, 0.9511, 1.3141, 2.2116, 2.4675, 2.9605, 3.9098, 13.7447) and the dynamic response of the perturbed TCA cycle is shown in Figure 6(b). Obviously, the steady state of the perturbed TCA cycle ceases to exist.

51. 51

52. 52 Simulation experiment 4 Periodic responses are often encountered in organisms ranging from bacteria to mammals. A periodic oscillation can be considered as one kind of steady state phenomenon from the system point of view.

53. 53 A periodic network in Figure 7 is modeled as follows to produce the spontaneous oscillations in cAMP observed during the early development of D. discoideum (Laub and Loomis, 1998) and account for the synchronization of the cells necessary for chemotaxis (Yi et al., 2000; Ma and Iglesia, 2002).

54. 54 From (14) or (15), we can find that if the perturbation measure is large such that the robustness condition is violated, the robustness of the steady state may not be preserved. Suppose the parameter perturbation as follows In this case, all eigenvalues of R are greater than 0, and the oscillation still exists as shown in Figure 8(b).

55. 55 the robustness condition R > 0 is violated, i.e., the smallest eigenvalue of R is less than zero. The oscillation disappears and the steady state of the biochemical network ceases to exist as shown in Figure 8(c). Obviously, the proposed robustness measure is an important indicator for the robustness of biochemical networks under parameter perturbations.

56. 56

57. 57 Discussion Mutations and diseases are unavoidable in biosystems, and they can permanently alter the kinetic properties and capabilities of a biochemical network. Such alterations are reflected in dynamic models as numerical changes in one or some of the system parameters. The effects of permanent changes in parameters of a biochemical network have been examined in this paper using robustness analysis. We found that the biochemical network can tolerate the parameter variations with respect to existence of steady state if the robustness condition in (14) or (15) is not violated. Obviously, the robustness analysis is an important research topic for biochemical networks.

58. 58 This study proposed an efficient method for measuring the robustness of biochemical networks using the S-system model. The proposed robustness measure scheme can tell how much parameter variation a biochemical network can tolerate in order to preserve the steady state of the network. We found that if the parameter variation is less than the upper bound or the robustness matrix R is positive definite, the network is robust at the steady state. The proposed robustness measure scheme gives an efficient method for computing the upper bound to tolerate the parameter variations of biochemical networks.

59. 59 As seen in the simulation examples, adequate negative feedback and redundancy may contribute significantly to the robustness of biochemical networks. We also found that in the compensatory parameter variation case, the biochemical network could tolerate very large parameter perturbation only with a small change in the transient period. It is also a good way to preserve the steady state for biochemical networks. In real biochemical networks, they must have enough robustness to tolerate the parameter and environmental variations or else they can not respond immediately to small but persistent parameter and environmental perturbations. Therefore, if a model of a biochemical network has a small robustness measure, i.e., a model lacking robustness, it is often a sign of structural inadequacies of the model and provides a tool for identification of inconsistencies in data. Thus, the proposed robustness measure scheme is also a good way to validate the models of biochemical networks.

60. 60 Conventionally, sensitivity analyses are used to assess the robustness of biochemical networks. That is, the sensitivity of the steady state concentration of a metabolite with respect to a change in a parameter is used to indicate the robustness of biochemical networks. Because conventional sensitivity analyses in biochemical systems pay more attention to the effects of and on but not on, it is not easy to use parameter sensitivity to assess quantitatively whether the steady state of a biochemical network is preserved or not. Furthermore, from (22) and (23) we found that if the biochemical network is more robust, it is also less sensitive to the variations of rate constant and environment, i.e., and.

61. 61 The robustness measure is confirmed by four metabolite networks with several numerical simulations. From the simulation examples, we found that if the parameter variation measure violates the upper bound of the tolerance, then the steady state of the biochemical network may cease to exist. Therefore, in the evolutionary process of the biochemical network, parameter perturbations due to DNA mutation in genes should be less than or the steady state of the biochemical network may cease to exist and may be eliminated by natural selection. It has been claimed (Nijhout, 2002) that the nonlinearity is the nature of robustness. In general, it is not true. The nonlinearity is only the nature of biochemical network. Robustness of biochemical networks should be more related to increasing the upper bound of the tolerance (i.e., ) or attenuating the perturbation (i.e., ).

62. 62 From the above experimental simulation examples, we also found that an adequate negative feedback pathway may increase and redundancy as well as compensatory parameter perturbation can attenuate the parameter variation, i.e., adequate negative feedback, compensatory parameter perturbation and redundancy may be three mechanisms providing the robustness of a biochemical network, which is consistent with the recent robustness results of system theory (Weinmann, 1991; Qu, 1998) and system biology (Yi et al., 2000; Hood et al., 2004). In this situation, we claim that adequate negative feedback, compensatory parameter perturbation and redundancy may be the nature of biochemical network robustness. According to the robustness analysis, understanding the diseases by studying the biochemical pathways or networks is helpful for the design of potential drugs from the viewpoint of improving the robustness of biochemical pathways or networks (Hood et al., 2004).