# Systems Biology as an Integrated Platform for Bioinformatics, Systems Synthetic Biology, and Systems Metabolic Engineering

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## Abstract

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## 1. Introduction

**Figure 1.**The role of systems biology as an integrated platform in modern biological research Systems biology integrates information on genetics, proteins, DNA-protein binding, and metabolism with system dynamics modeling and system identification technology to develop models and mechanisms for the interpretation of phenotypes or behaviors of cellular physiology. Since large-scale data sets need to be mined, powerful computational tools are necessary. Based on system models and mechanisms in systems biology, synthetic genetic circuits are designed to investigate specific desired cellular behaviors of cellular physiology. Discrepancies between real and desired cellular behaviors are used as feedback to adjust system models and mechanisms. Systems biology is thus positioned to play the role of integrated platform for bioinformatics, systems synthetic biology, and systems metabolic engineering.

## 2. Systems Biology Approach to GRNs and PPI Networks via Bioinformatics Methodology

#### 2.1. Construction of GRNs via Microarray Data

_{i}(t) denote the gene expression of the i-th target gene in the potential GRN. The dynamic equation for the regulation of the i-th gene is then modeled as [21,22]

_{i}(t) represents the mRNA expression level of target gene i at time t, x

_{j}(t) represents the regulation function of the j-th TF binding to the target gene i, a

_{ij}denotes the regulatory ability from the j-th regulatory gene to the i-th target gene (the positive sign indicates activation and the negative sign indicates repression), λ

_{i}indicates the degradation effect of the present time point on the next time point t+1, k

_{i}represents the basal level, and n

_{i}(t) is the stochastic noise due to model uncertainty and fluctuation of microarray data of the target gene. Expression of the i-th gene in (2.1) can be represented by the following regression equation [77]

_{i}denotes the regression vector, which can be obtained from microarray data. θ

_{i}is the regulatory parameter vector of target gene i, which is to be estimated.

_{i}are estimated, the system order (i.e., the number of regulatory genes) is determined by model comparison (e.g., Akaike’s information criterion (AIC)), and the P-value statistical method is employed to determine the significant regulatory genes for target gene i. This is done by pruning false-positive regulations in the potential GRN. That is, some a

_{ij}is pruned because of false positive deletion. Based on the dynamic model in (2.1), the true GRN can then be constructed one target gene at a time through microarray data. Using similar methods, GRNs for yeast cell cycles [18,23,24], cancer cell cycles [78], stress response [44], and inflammation [41] can be constructed.

#### 2.2. Construction of PPI Networks

_{i}(t) represents the protein activity level of the target protein i at time t, b

_{ij}denotes the interaction ability of the j-th interactive protein with the i-th protein, y

_{j}(t) represents the activity level of the j-th protein interacting with the target protein i, β

_{i}denotes the degradation effect of the protein, h

_{i}represents the basal activity level, and ν

_{i}(t) is the stochastic noise. The rate of PPI is proportional to the product of the concentrations of both proteins [7], that is, it is proportional to the probability of molecular collisions between them. The physical interpretation of Equation (2.3) is that the activity of target protein i at time t+1 is the combination of present protein activity, regulatory interactions with M interactive proteins, levels of basal protein from other sources and M interactive proteins in the cell, and stochastic noise, less the protein degradation of the present state. The PPI dynamic equation of target protein i in the potential PPI network can be represented by the following regression equation [77]:

_{i}can be estimated from protein profile microarray data by least-squares or maximum-likelihood parameter estimation [77] (if protein profile microarray data are unavailable, ten mRNA microarray data could be used to replace them, with some modification [19,20]). By using AIC to prune false positive interactions, the real PPI network can then be constructed one target protein at a time by following the above two-step procedure. Some dynamic metabolic pathways [20] and PPI networks of cancer [39] and inflammation [41] have recently been constructed by using the microarray data and AIC method. Comparison of PPI networks between healthy and cancer cells can provide network-based biomarkers for molecular investigation and diagnosis of cancer [70].

#### 2.3. Construction of Integrated GRN and PPI Cellular Networks

_{j}(t) is modeled as the sigmoid regulation function of y

_{j}(t), that is, for the protein activity profiles of transcription factor j [71],

_{j}(y

_{j}(t)) denotes the sigmoid function, u

_{j}and σ

_{j}represent the mean and deviation of protein activity level of TF j, and α

_{n}denotes the translation effect from mRNA x

_{n}(t) to protein y

_{n}(t).

**Figure 2.**Schematic diagram of the integrated cellular network. The integrated cellular network consists of two subnetworks. The signaling regulatory pathway (green) contains protein–protein interaction (PPIs). The gene regulatory network (yellow) contains transcription regulations. The transcription factors serve as the interface between the two subnetworks.

_{j}(t) at the end of the signal regulatory pathway regulate their target genes through the regulation function z

_{j}(t) = f

_{j}(y

_{j}(t)) in (2.6), then the regulatory genes influence their corresponding proteins in signal and metabolic pathways through translation effect α

_{n}x

_{n}(t). The interplay between genes and proteins can be seen from their coupling dynamic equations in (2.5) and

**Figure 2**. Here, the TFs serve as the interface between the signaling regulatory pathway and gene regulatory network. In other words, the interplay between transcription regulation and PPIs constitutes the framework of the integrated cellular network.

_{ij}between genes and their possible regulatory TFs and through the translation parameter α

_{n}for gene expression to protein expression. The potential signaling or metabolic pathways can be linked through the interaction parameter b

_{nm}between possible interaction proteins. Since omics data on the potential gene regulatory network and potential signaling or metabolic pathway only indicate possible TF-gene regulation and protein interactions, they should be confirmed using microarray data of gene and protein expressions. In particular, values of a

_{ij}and b

_{nm}in (2.5) should be identified and validated by least-squares estimation via microarray data in a specific biological condition or phenotype [71]. Significant regulations and interactions between genes and proteins were detected using model selection methods such as AIC and statistical assessments like such as Student’s t-test [77]. Based on the interface between gene regulatory and signal/metabolic networks (i.e., transcription factors) in a specific biological condition, the two networks are coupled together to form the integrated cellular network. The integrated cellular network for S. cerevisiae under hyperosmotic stress is shown in Figure 3. After the construction of GRN and PPI network from microarray data and bioinformatic method, various system characteristics of the biological network are estimated or measured using systems biology methods in the following sections so that these systematic methodologies can be applied the systematic design of systems synthetic biology and systems metabolic engineering.

**Figure 3.**The S. cerevisiae integrated cellular network under hyperosmotic stress. By following the schematic diagram of an integrated genetic regulatory network (GRN) and PPI cellular network in Figure S1, the integrated cellular network of signaling regulatory pathway and GRN for hyperosmotic stress in S. cerevisiae is identified by dynamic modeling and data mining. Receptor proteins in the plasma membrane, signal regulatory pathways in the cytoplasm, and transcription factors and GRNs in the nucleus are used to construct an integrated cellular network for S. cerevisiae under hyperosmotic stress.

#### 2.4. Network Robustness and Sensitivity Estimation via Microarray Data Using a Systems Biology Approach

_{1}(t) ⋯ x

_{N}(t)]

^{T}stands for the discrete-time mRNA expression levels of total N genes at times t = 1, 2, ..., K. The system matrix Â denotes gene interactions in the gene network estimated by microarray data, that is, â

_{ij}denotes the estimated interaction of gene j with gene i. If i ≠ j, then denotes the estimated basal level of the i-th gene. n(t) denotes the model residual and measurement noise. The steady state x

_{s}of x(t) is obtained as t→∞:

_{s}, that is, x(t) = (t) + x

_{s}. This shift allows the following shifted dynamic system to be achieved by subtracting Equation (2.7) from Equation (2.8) [34]:

^{T}(t)P (t) > 0, with V(0) = 0 for a positive symmetric matrix, P = P

^{T}> 0. The perturbative GRN in Equation (2.10) is robustly stable if ΔV( ) = V( (t+1)) − V( (t)) ≤ 0, i.e., the energy of the gene network is not increased by intrinsic perturbations [79]. Based on this idea of robust stability, if the following inequality has a positive definite solution P = P

^{T}> 0 [34],

_{1}(t) ⋯ u

_{N}(t)]

^{T}represents external stimuli and Y(t) denotes the output signal response of specific genes of interest. For example, if the output signal response of the gene i to external stimuli U(t) is to be analyzed, then C = diag(0, 0, ..., 0, 1, 0, 0, ..., 0), i.e., all elements of C are zero except for a single element at the i-th diagonal component. For example, C = diag(0, 0, 1, 0, ..., 0) describes the gene response of the P53 gene. If the network response ability of the entire GRN to external stimuli is analyzed, then C = I (identity matrix).

**Figure 4.**Multiple loops of a gene regulatory network associated with aging-related pathophyiological phenotypes. This aging-related gene regulatory network includes 16 genes: FOXO

_{s}, NF-kB, P53, SIRT1, HIC1, Mdm2, Arf1, PTEN, P13K, Akt, JNK, IKK, IkB, BTG3, E3F1, and ATM. Dashed red lines and black arrows indicate negative and positive parameters of regulated interaction, respectively.

^{T}> 0 solution to the following LMI [34]:

**Table 1.**The network robustness (η°) and network response ability (δ°) of a gene regulatory network with 16 aging-related genes (Figure 4) across different tissues at young, aged, and calorie-restrictive (CR) stages.

Tissue | Young | Aged | CR |
---|---|---|---|

Thymas | |||

η° | 0.2310 | 0.3750 | 0.2050 |

δ° | 1.1653 | 0.9463 | 1.2279 |

Spinal Cord | |||

η° | 0.2410 | 0.6270 | 0.1400 |

δ° | 1.1630 | 0.9367 | 1.2645 |

Eye | |||

η° | 0.1600 | 0.2910 | |

δ° | 1.2376 | 0.8798 |

#### 2.5. Network-Based Biomarker Determination via Sample Microarray Data Using a Systems Biology Approach

_{i}(n) represents the gene expression level of the target protein i for the sample n, and α

_{ik}denotes the association ability between the target protein y

_{i}(n) and protein y

_{k}(n) for sample n. N

_{i}represents the number of proteins interacting with the target protein i; it can be obtained from the rough PPI network. ε

_{i}(n) denotes stochastic noise associated with other factors or model uncertainty. Equation (2.17) states that, biologically, the expression level of the target protein i is associated with the expression levels of interacting proteins.

_{ik}in Equation (2.17) is identified through maximum likelihood estimation [77] on microarray data. AIC and Student’s t-test were employed for model order selection and for tests on the statistical significance of protein associations. Based on α

_{ik}, two matrices are established to represent the cancer protein association network (CPAN) and the non-cancer protein association network (NPAN) as follows [77]

_{ij,C}and α

_{ij,N}indicate the quantitative protein association ability between protein i and protein j for CPAN and NPAN, respectively, and K is the number of proteins in the protein association network. The resulting CPAN and NPAN constitute the network-based biomarker used for identifying significant proteins in lung carcinogenesis through the diagnostic evaluation

_{C}= [y

_{1,C(n)}⋯ y

_{K}

_{,C(n)}]

^{T}, Y

_{N}= [y

_{1,N(n)}⋯ y

_{K}

_{,N(n)}]

^{T}denotes the vectors of expression levels, and E

_{C}and E

_{N}indicate the noise vectors in cancer and non-cancer cases, respectively. A matrix indicating the difference between two protein association networks is defined as C − N, i.e.,

_{ij}denotes the difference in protein association ability between CPAN and NPAN among proteins i and j. Using matrix D to represent the difference in network structure between CPAN and NPAN, a carcinogenesis relevance value (CRV) was derived to quantify the correlation of each protein significant to lung carcinogenesis. To identify significant proteins, two important issues are taken into consideration. First, the magnitude of the association ability α

_{ij}denotes the significance of association of one protein to another. A higher absolute value of α

_{ij}implies that the two proteins are more tightly associated. Second, if a protein plays a more crucial role in lung carcinogenesis, then the difference in association numbers linked to the protein for CPAN and NPAN would be larger. For example, if one protein shares a strong association with many proteins in CPAN, but has weaker associations (no protein) in NPAN, then the protein in question is more likely to be involved in lung carcinogenesis. As a result, CRV is determined based on the difference in protein association abilities using the following equation [70]:

**Figure 5.**The constructed network-based biomarker. (

**A**) Cancer protein association network (CPAN) obtained from C in (2.18) by maximum likelihood estimation, Akaike’s information criterion (AIC) selection, and Student’s t-test. (

**B**) Non-cancer protein association network (NPAN) obtained from N in Equation (2.18) using the same criteria.

**Figure 6.**The difference between CPAN and NPAN obtained from Equation (2.20) for network-based biomarkers for lung cancer. The significance of proteins (indicated by circle size) to the network-based marker is dependent on their CRVs in Equation (2.21), which are listed in Table S1.

#### 2.6. On the Network Robustness and Filtering Ability versus Molecular Noise in GRNs Using a Stochastic System Approach

_{i}(t) denotes the concentration of the i-th gene, and N

_{ij}denotes the interaction between genes j and i.

_{ij}denotes the random parametric fluctuation of N

_{ij}; M

_{ij}denotes the deterministic part (amplitude) of fluctuation; and n(t) is white Gaussian noise with zero mean and unit variance, and denotes the stochastic part of fluctuation, i.e., the stochastic part of fluctuation is absorbed to n(t).

**Figure 7.**The linear n genes GRN with interaction N

_{ij}, intrinsic fluctuation ΔN

_{ij}, gene expression x

_{i}(t), and extrinsic fluctuation υ

_{i}(t).

^{T}(t)Px(t) for some positive definite matrix P, the following result is derived.

**Proposition 1**[25]:

**Remark 1**: (i) In the intrinsic noise-free case, the stable condition in (2.26) is reduced to PN + N

^{T}P ≤ 0, i.e., the eigenvalues of system matrix N should be on the left-hand side of the complex domain. Obviously, if the LMI in Equation (2.26) has a positive solution P > 0, then the eigenvalues of N should be located on the far left-hand side of the complex domain with more negative real values in order to overcome the additional term M

^{T}PM due to intrinsic random noise. (ii) If some eigenvalues of system interaction matrix N are near the jω axis, then intrinsic random molecular fluctuations across the jω axis perturb these modes more easily such that the linear GRN becomes unstable. The LMI in Equation (2.26) can be rearranged to

^{T}P) in Equation (2.27) can be taken as a measure of network robustness, and M

^{T}PM due to the random parametric fluctuation can be taken as a measure of intrinsic robustness. The physical interpretation of Equation (2.27) is that if the network robustness can confer enough intrinsic robustness for tolerating intrinsic random parameter fluctuation, then the phenotype of the GRN is maintained.

**Proposition 2**[25]:

_{1}(t) ⋯ v

_{n}(t)]

^{T}outside the network (see Figure 7), then

_{i}(t)), then we let C = diag(0…010…0). That is, every element of C is zero except for the i-th element. To investigate the effect of molecular noises on the whole GRN, then C = I, the identity matrix. The positive value ρ in the following inequality is then called the effect of environmental noises (or signals) on Z(t) in the stochastic GRN in Equation (2.30) with x(0) = 0

**Proposition 3**[25]:

_{0}of the linear stochastic GRN (Equation (2.30)) can be obtained by solving the following constrained optimization

**Remark 2**: (i) If ρ

_{0}< 1, then environmental molecular noises υ(t) are attenuated by the GRN and ρ

_{0}is called the filtering ability of the GRN, i.e., the GRN is less sensitive to environmental noises. If ρ

_{0}> 1, then environmental molecular noises are amplified by the GRN, i.e., the GRN is more sensitive to environmental noises. (ii) Substituting ρ

_{0}for ρ in Equation (2.32) and rearranging, the following phenotype robustness criterion of the stochastic GRN is derived

_{0}).

_{i}(z), i = 1,⋯, L denotes the interpolation functions of global linearization or fuzzy bases with h

_{i}(z) ≥ 0, h

_{i}(z) = 1, and dx = (N

_{i}x + H

_{i}v)dt + M

_{i}xdω is the i-th local linearized GRN.

## 3. Systems Synthetic Biology

#### 3.1. Design of Specifications-Based Systems Synthetic Biology

_{a}and x

_{b}denote the concentrations of proteins A and B, respectively. k and γ are the kinetic parameter and decay rate, respectively. r is the regulation function capturing the regulator effect of a transcriptional protein on gene expression, and has a smooth sigmoidal form (e.g., Hill function) [88].

**Figure 8.**A simple two-gene cross-inhibition network. The network’s regulation functions are given in Equations (3.1) and (3.2).

_{i}and Δγ

_{i}denote the amplitudes of fluctuations of the stochastic parameters and decay rates, respectively; and n

_{i}is a random white noise with zero mean and unit variance.

_{a}x

_{b}]

^{T}and ν = [ν

_{a}ν

_{b}]

^{T}denote the state vector and environmental disturbance of the synthetic gene network in the host cell, respectively. These intrinsic parameter fluctuations and environmental disturbances may cause the engineered synthetic gene network to be dysfunctional in the host cell.

_{1}⋯x

_{n}]

^{T}denotes the concentrations of proteins in the synthetic gene network, N denotes the corresponding stoichiometric matrix of the n-gene network, M

_{i}'; i = 1,..., m denotes fluctuation matrices associated with independent random noise sources n

_{i}, i = 1,..., m; and the elements of M

_{i}denote the standard deviations of the corresponding parameter fluctuations. ν = [ν

_{1}⋯ν

_{n}]

^{T}denotes the vector of external disturbances. This stochastic system is used to mimic the realistic dynamic behavior of a synthetic gene network of n genes in the host cell. As the network is subject to intrinsic parameter fluctuations and environmental disturbances in the context of the host cell, a robust synthetic gene network should be designed with the ability to not only tolerate parameter fluctuations and attenuate external disturbances, but also to achieve desired steady-state behaviors.

_{i}(t) is a standard Wiener process with dW

_{i}(t) = n

_{i}dt.

_{i}in the state-dependent noise term

_{∞}filtering ability)

_{d}should be less than the attenuation level ρ from the average energy perspective.

_{d}and for convenience of design, the origin of the nonlinear stochastic system in Equation (3.7) should be shifted to x

_{d}. Stabilizing the shifted nonlinear stochastic system at the origin would then also achieve x

_{d}, simplifying the design procedure. Let = x – x

_{d}. The following shifted stochastic synthetic genetic system is then derived [79]:

_{d}of the original stochastic system in Equation (3.7). N ∈ [N

_{1},N

_{2}] is then specified to tolerate the stochastic parameter fluctuation M

_{i}g

_{i}( + x

_{d})dW

_{i}and efficiently attenuate the environmental disturbance v to the prescribed level

**Proposition 4**[53]:

_{1},N

_{2}] are chosen such that the following HJI has a positive solution V( ) > 0

_{d}in probability, i.e., design specification (iii) is achieved.

_{1},N

_{2}] to solve HJI in Equation (3.11) for V( ) > 0 using a systematic method. If all global linearizations are bound by a polytope consisting of M vertices

_{j}( ) satisfies 0 ≤ α

_{j}( ) ≤ 1 and ∑i = 1M α

_{j}( ) = 1. The trajectory of the nonlinear stochastic synthetic gene network in Equation (3.9) could thus be represented by the interpolated synthetic gene network in Equation (3.13). This yields the following result.

**Proposition 5**[53]:

_{1}, N

_{2}] are chosen such that the following M LMIs have a common symmetric positive definite solution P > 0

**supplementary example 1**.)

**Remark 3**: (i) Gene circuit design can now be implemented using recombination technology to insert or delete TF binding sites in the promoter region of a regulated gene with the aim of increasing or decreasing the value of the kinetic parameter κ

_{i}(i.e., different levels of affinity) of the regulated gene [89]. By inserting strong or weak binding sites, large or small values can be obtained. For example, the binding site of κ

_{i}= 1 is 10 times larger than that of κ

_{i}= 0.1 at the promoter region of target gene i. Changes to the decay rate γ

_{i}can be implemented by shortening the 3' polyadenylate tail (referred to as deadenylation), which primarily triggers decapping, resulting in 5' to 3' exonucleolysis. Alternatively, removal of the 3' polyadenylate tail can increase γ

_{i}[53]. Therefore, by shortening or elongating the gene’s 3' polyadenylate tail, we can increase or decrease the decay rate γ

_{i}of gene i. Directed evolution methods are also useful in changing the elasticity (kinetic property of κ

_{i}) and in designing biochemical circuits [53]. From a systems biology perspective, these advances in implementation techniques of κ

_{i}and γ

_{i}enable engineering of synthetic gene networks in the near future.

_{i}using evolutionary (or genetic) algorithms such that the network can achieve the following optimal tracking

_{p}denotes the present time. Let

_{i}∈ [k

_{i}

^{L},k

_{i}

^{U}] by EA or GA to minimize J(k) for the desired network behavior tracking of the synthetic gene network is equivalent to maximizing the fitness function in Equation (3.18) to meet the natural selection. A robust biological network design with a desired output behavior y

_{d}(t) is therefore equivalent to as solution to the following fitness maximization problem using an evolutionary network method [54,58]

**Figure 9.**Block diagram of the optimal tracking scheme for synthetic biological circuit design using an evolutionary systems biology algorithm. Based on a network algorithm mimicking natural selection in an evolutionary process, the design parameters k of a synthetic biological circuit are tuned to minimize the tracking error between the desired logic circuit and the stochastic synthetic biological circuit, and to achieve the desired behavior tracking.

#### 3.2. Robust Synthetic Gene Network Design via Library-Based Search Methods

_{yEGFP}denote the degradation rates of mRNA and protein yEGFP, respectively, and α denotes the translation rate. The promoter regulation function P

_{TetR}(c,r), which is dependent on repressor activity γ and the choice of promoter c, has the form

_{r}and c

_{s}(the minimum and maximum values of the promoter regulation function P

_{TetR}(c, r), respectively) for the TetR-regulated promoter library Lib

_{TetR}. That is, c = (c

_{r}, c

_{s}) ∈ Lib

_{TetR}. K

_{TetR}and n

_{TetR}denote the TetR–DNA binding affinity and binding cooperativity of regulatory protein TetR and DNA, respectively. H

_{TetR}(r) is a Hill function capturing the effect of a regulatory protein.

_{r}, c

_{s}) via maximum and minimum values of the steady state of protein expression data, some promoter libraries can be redefined in such a way that they can be efficiently selected from the design of the synthetic gene network (Table 2) [56,57]. Since a synthetic gene network always consists of a set of promoter-regulation gene circuits (Figure 10), the design of complex synthetic gene network addresses how to select a set of promoters from the corresponding promoter libraries that have promoter activities adequate for achieving the design specifications. A well-known gene circuit topology, the simple toggle switch, is shown for illustration purposes in Figure 11. The toggle switch has two distinct stable states and can be reversibly switched between them by changing the inducers ATc and IPTG. Let x

_{1}(c

_{1}, t), x

_{2}(c

_{2}, t), and x

_{3}(c

_{3}, t) denote the concentrations of mRNAs tetR, lacI, and yegfp, respectively; and let X

_{1}(c

_{1}, t), X

_{2}(c

_{2}, t), and X

_{3}(c

_{3}, t) denote the concentrations of proteins TetR, LacI, and yEGFP, respectively. The dynamic model of the toggle switch gene network in Figure 11 is then modeled as

_{LacI}(c

_{1}, r

_{1}), p

_{TetR}(c

_{2}, r

_{2}), and p

_{TetR}(c

_{3}, r

_{3}) are dependent on the selection of promoters c

_{1}, c

_{2}, and c

_{3}from the corresponding promoter libraries in Table 2. The output of interest y(c, t) is dependent of the selected promoter set c = [c

_{1}, c

_{2}, c

_{3}] with adequate promoter activities from the corresponding promoter libraries in Table 2. This dynamic toggle switch gene network model consists of three interactive dynamic models of promoter-regulation gene parts, as shown in Equation (3.20).

**Table 2.**Redefined TetR- and LacI-regulated promoter libraries. The redefined TetR- and LacI-regulated promoter libraries (i.e., Lib

_{TetR}and Lib

_{LacI}) comprise different promoters (i.e., T

_{k}and L

_{k}, k = 0,…,20) with their corresponding activities of c

_{s}and c

_{r}obtained from previous libraries of experimental data.

TetR-regulated promoter library (Lib_{TetR}) | LacI-regulated promoter library (Lib_{LacI}) | ||||
---|---|---|---|---|---|

Promoter | Promoter activity | Promoter | Promoter activity | ||

c_{s} | c_{r} | c_{s} | c_{r} | ||

T_{0} | 2121 | 0.1724 | L_{0} | 1657.5 | 0.3018 |

T_{1} | 1604 | 0.7576 | L_{1} | 923.97 | 0.2567 |

T_{2} | 1376.6 | 0.1936 | L_{2} | 860.87 | 0.2244 |

T_{3} | 1169.8 | 0.4672 | L_{3} | 674.92 | 1.9189 |

T_{4} | 974.52 | 0.0753 | L_{4} | 651.58 | 1.1680 |

T_{5} | 942.77 | 0.2281 | L_{5} | 570.07 | 3.5062 |

T_{6} | 967.17 | 0.1493 | L_{6} | 527.83 | 0.5497 |

T_{7} | 738.57 | 0.0702 | L_{7} | 323.45 | 0.1248 |

T_{8} | 641.74 | 0.7135 | L_{8} | 327.77 | 0.1772 |

T_{9} | 564.24 | 0.2620 | L_{9} | 309.74 | 0.5439 |

T_{10} | 501.35 | 0.0756 | L_{10} | 298.35 | 0.1146 |

T_{11} | 469.35 | 0.0788 | L_{11} | 250.16 | 0.1326 |

T_{12} | 466.16 | 0.1636 | L_{12} | 248.03 | 0.1171 |

T_{13} | 356.88 | 0.0927 | L_{13} | 239.32 | 0.1010 |

T_{14} | 348.95 | 0.1483 | L_{14} | 190.2 | 0.0959 |

T_{15} | 274.79 | 0.1067 | L_{15} | 163.84 | 0.4813 |

T_{16} | 250.04 | 0.0857 | L_{16} | 166.42 | 0.0989 |

T_{17} | 188.77 | 0.1366 | L_{17} | 131.63 | 0.1190 |

T_{18} | 119.57 | 0.0753 | L_{18} | 108.96 | 0.0903 |

T_{19} | 111.57 | 0.1185 | L_{19} | 101.89 | 0.0982 |

T_{20} | 70.909 | 0.1606 | L_{20} | 85.673 | 0.2174 |

**Figure 10.**Single schematic diagram of the synthetic promoter-regulation gene circuit. The existing TetR-regulated promoter library contains the minimum and maximum values of fluorescence [y

_{imin}, y

_{imax}] corresponding to with and without TetR (repressor) binding. Based on the promoter regulation function (3.21) and these values, the promoter library is redefined for the design of synthetic gene networks (Table 2).

_{2}/H

_{∞}synthetic gene network based on promoter libraries selects an adequate promoter set c = [c

_{1}, c

_{2}, c

_{3}] from corresponding promoter libraries such that the following two design objectives are achieved simultaneously [56]:

_{∞}desired noise attenuation level ρ

_{d}:

_{2}optimal reference tracking:

_{1}, c

_{2}, c

_{3}] [L

_{9}, T

_{2}, L

_{8}] is selected from promoter libraries in Table 2 to achieve the multi-objective H

_{2}/H

_{∞}reference tracking specified in Equations (3.23) and (3.24). Simulation results with ν(t) = 10 × [n

_{1}⋯n

_{6}]

^{T}are shown in Figure 12.

**Figure 11.**Synthetic gene circuit topology: simple toggle switch. The regulatory protein TetR, which is induced by ATc, inhibits the transcription of lacI by binding promoter c

_{2}. TetR also inhibits transcription of yegfp by binding promoter c

_{3}to repress the expression of the fluorescent protein yEGFP. The protein LacI, which is induced by the inducer IPTG, inhibits the transcription of tetR by binding promoter c

_{1}. The gene circuit has two distinct stable states, and can reversibly switch between them by changing the inducers ATc and IPTG. If an adequate promoter set c = [c

_{1}, c

_{2}, c

_{3}] is selected from corresponding promoter libraries, then yEGFP can be used to track the desired behaviors generated by a reference model. In the reference model, c

_{1}is selected from the LacI-regulated promoter library, and c

_{2}and c

_{3}are selected from the TetR-regulated promoter library in Table 2 (i.e., c

_{1}∈ Lib

_{LacI}, c

_{2}, c

_{3}∈ Lib

_{TetR}).

**Figure 12.**Simulation of toggle switch. By solving the LMI-constrained optimization problem of the H

_{2}/H

_{∞}design objective Equations (3.23) and (3.24) for the synthetic gene network in Figure 11 through the library searching method, an adequate promoter set c = [c

_{1}, c

_{2}, c

_{3}] = [L

_{9}, T

_{2}, L

_{8}] is selected from the corresponding promoter libraries. The inducer ATc is added to the synthetic gene network at 80–160 hours to induce the gene network, and then the inducer IPTG is added at 160–240 hours. The output y(c, t) clearly produces a robust track with the desired reference output y

_{r}(t).

**Remark 4**: Collective rhythms of GRNs, especially the synchronization of dynamic cells mediated by intercellular communication, have become a subject of considerable interest to biologists and theoreticians [90]. Synchronization of a population of synthetic genetic oscillators is an important consideration in practical applications, because a population distributed over different host cells needs to exploit molecular phenomena in a simultaneous manner in order to function as a biological entity. However, this synchronization of synthetic gene networks in different host cells may be corrupted by intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. Therefore, robust synchronization is an important design topic in nonlinear stochastic coupled synthetic genetic oscillators with intrinsic kinetic parametric fluctuations and extrinsic molecular noise. A systems biology approach indicates [59,60] that if the synchronization robustness criterion is greater than or equal to the sum of the intrinsic robustness and extrinsic robustness, then the stochastic coupled synthetic oscillators can be robustly synchronized in spite of intrinsic parameter fluctuation and extrinsic noise. If the criterion for synchronization robustness is violated, then an external control scheme can be designed to improve robustness by adding inducers to the coupled synthetic genetic network. These robust synchronization criteria and control methods are useful for a population of coupled synthetic networks with emergent synchronization behavior, especially for multicellular engineered gene networks [60].

## 4. Systems Metabolic Engineering

_{1},⋯,X

_{n}

_{+m}are metabolites, such as substrates, enzymes, factors, or products of a biochemical network, in which X

_{1},⋯,X

_{n}denote n dependent variables and X

_{n+}

_{1},⋯,X

_{n}

_{+m}denote the independent variables. In a metabolic network, intermediate metabolites and products are dependent variables, whereas substrates and enzymes are independent variables. The rate of change in X

_{i}, Ẋ

_{i}is equal to the difference between two terms, one for production or accumulation, and the other for degradation or clearance. Each term is the product of a positive rate constant α

_{i}or β

_{i}and all dependent and independent variables that directly affect production or degradation, respectively. Each variable X

_{j}is raised to the power of a kinetic parameter g

_{ij}and h

_{ij}, which represents an activating effect of X

_{j}on X

_{i}when its value is positive, and an inhibitive effect when its value is negative. V

_{i}and V

_{–}

_{i}represent aggregate flux into and out of the X

_{i}pool.

#### 4.1. Robust Biochemical Circuit Design in Metabolic Networks

_{D}denotes the system matrix of the interactions between dependent variables Y

_{D}, and A

_{I}indicates the interactions between dependent variables Y

_{D}and independent variables Y

_{I}. In the nominal parameter case, it is assumed that the inverse of A

_{D}exists so that Y

_{D}can be solved uniquely, i.e., the metabolic network results in only one steady state (phenotype). Therefore, the steady state of the biochemical system is given by

_{D}denotes perturbations due to kinetic parameter variations, Δb denotes perturbations due to rate constant variations, and ΔA

_{I}denotes perturbations due to kinetic parameter variations between dependent and independent variables. ΔA

_{D}can influence the existence of the steady state of the metabolic network.

_{D}

^{−1}ΔA

_{D}are free of zero and the inverse (I + A

_{D}

^{−1}ΔA

_{D})

^{−1}exists. Therefore, the steady state of the perturbative metabolic network in Equation (4.8) is uniquely solved: as

_{D}, i.e., Y

_{D}+ ΔY

_{D}in Equation (4.10) has a small difference ΔY

_{D}from the nominal in Equation (4.6) under small perturbation. However, if the condition in Equation (4.9) does not hold, then individual values of I + A

_{D}

^{−1}ΔA

_{D}may be zero, the inverse (I + A

_{D}

^{−1}ΔA

_{D})

^{−1}may not exist, and the steady state Y

_{D}+ ΔY

_{D}may cease to exist under the parameter perturbation ΔA

_{D}. As an example, consider the singular value decomposition:

_{i}denotes the i-th singular value and u

_{i}, v

_{i}∈R

^{n}denote the corresponding left and right singular vectors, respectively. Therefore, if a parameter variation is specified as follows:

_{D}

^{−1}ΔA

_{D})

^{−1}or (I+A

_{D}

^{−1}ΔA

_{D})

^{−1}does not exist under the parameter perturbation in Equation (4.12).

**Remark 5**: The parameter perturbations in the direction of singular vectors like Equation (4.12) are the network’s points of fragility. The robustness prevents this kind of parameter variation and guarantees the existence of the steady state of the metabolic networks. When unexpected perturbations like Equation (4.12) are encountered, a catastrophic failure of the network follows. Robust circuit design is a necessary fail-safe mechanism in such situation. For example, the trehalose pathway in yeast consists of only a few metabolites that form a substrate cycle. It is governed by a surprisingly complex control system that is composed of several inhibiting or activating signaling mechanisms [7].

_{D}A

_{D}

^{T}is the upper bound of ΔA

_{D}ΔA

_{D}

^{T}without violation of robust stability at steady state. If the network robustness criterion in Equation (4.14) holds, then the steady state of the perturbative metabolic network still exists.

_{k}regulating the production of X

_{i}by the kinetic parameter f

_{ik}. denotes a new biochemical control circuit with X

_{k}regulating the degradation of X

_{i}by the kinetic parameter l

_{ik}. The choice of regulating objects, X

_{k}and X

_{i}, and the specification of the kinetic parameters, f

_{ik}and l

_{ik}, are designed according to the feasibility of biochemical circuit linkage to achieve a desired robustness to tolerate ΔA

_{D}within the prescribed range of kinetic parameter perturbations in a metabolic network. Since f

_{ik}and l

_{ik}represent the elasticities of the corresponding enzymes in the designed control circuits, the implementation of control circuits is heavily dependent on the elasticity specification of these enzymes.

_{ij}and l

_{il}are the kinetic parameters in F of the biochemical control circuit to be specified in Equation (4.15).

_{D}+ F + ΔA

_{D}) exists. Equation (4.16) is equivalent then to

_{D}. The phenotype (i.e., the steady state of the controlled metabolic network in Equation (4.16)) is then given by

**supplementary example 2**.

#### 4.2. Multipurpose Circuit Control Design of Metabolic Networks

_{D}. The effects of rate constant variations Δb and of environmental changes or upstream regulatory changes ΔY

_{I}on output variations ΔY

_{D}should also be considered in the circuit design of metabolic networks to guarantee robustness against both intrinsic parameter variations and extrinsic environmental perturbations. The sensitivity of ΔY

_{D}to Δb in the designed metabolic network of Equation (4.16) is given by [7]

_{D}to ΔY

_{I}is given by

_{1}and s

_{2}are prescribed in advance by the biochemical circuit designer. From Equations (4.20)–(4.22), the equivalent sensitivity criteria for Equation (4.22) are obtained as [27]

_{22}in supplementary example 2 to tolerate kinetic parameter variation ΔA

_{D}and satisfy the network sensitivity in Equation (4.22) or (4.23) with prescribed sensitivities of ‖ΔY

_{D}/Δb‖

_{2}< ‖ΔY

_{D}/Δb‖

_{2,nominal}≡ s

_{1}= ‖A

_{D}

^{−1}‖

_{2}= 3.42 and ‖ΔY

_{D}/ΔY

_{I}‖

_{2}< ‖ΔY

_{D}/ΔY

_{I}‖

_{2,nominal}≡ s

_{2}= ‖A

_{D}

^{−1}A

_{I}‖

_{2}= 2.66. Then f

_{22}should be specified to satisfy the robust circuit design and the following inequalities:

_{22}necessary to tolerate ΔA

_{D}in and satisfy the desired network sensitivity in Equation (4.24) is found to be [−1, −0.081]. We choose f

_{22}= −0.407 as a design example, which is a negative self-regulation. It has been found to efficiently eliminate the effect of parameter variations by negative compensation. About 10% of yeast genes encoding regulators are negatively self-regulating; thus, this mechanism seems to be important for maintaining robustness in yeast [27]. The metabolic network and time responses are shown in Figure S5(d) and (e), respectively.

_{12}and l

_{22}to satisfy equations and (4.24), i.e.,

_{1}and s

_{2}are the same as above. Similarly, the ranges of f

_{12}and l

_{22}necessary to tolerate ΔA

_{D}in equation () and satisfy the desired sensitivity criteria in Equation (4.25) are found to be [−1, 0] and [0,1], respectively. f

_{12}= −0.08 and l

_{22}= 0.31 are chosen as a design example. The metabolic network and time responses are shown in Figure S5(f) and (g), respectively. Another example of a TCA cycle is given in

**supplementary example 3**.

_{D}and to achieve the desired sensitivities s

_{1}and s

_{2}in Equation (4.23), the robust control circuit design problem can be reduced to specifying F to satisfy the following multipurpose control circuit design derived from Equations (4.18) and (4.23):

_{1}= ‖A

_{D}

^{−1}‖

_{2}= 8.3685 and s

_{2}= ‖A

_{D}

^{−1}A

_{I}‖

_{2}= 7.5464.

_{12}is designed to satisfy the multi-objective design criteria in Equation (4.26), then the range of f

_{12}is found to be within [−0.8, −0.1]. If f

_{12}is chosen as −0.2 (Figure S6(a), blue line), then the time responses of the designed TCA cycle network shown in Figure S6(d), which match the desired properties of the proposed design method, are obtained. That is, the robust controlled biochemical network not only can tolerate ΔA

_{D}(to preserve its phenotype under parameter perturbations) but also retains sensitivity to environmental molecules in the nominal case. If dynamic circuit design is employed to implement the biochemical circuit f

_{12}, then an enzyme capable of catalyzing the reaction X

_{2}→X

_{1}is required. Additionally, a TF (Z) has to be found such that oxaloacetate2 (X

_{2}) can bind to the promoter of the enzyme’s inhibitor gene, as f

_{12}is negative. The concentration of X

_{2}could therefore regulate X

_{1}through the kinetic parameter f

_{12}. The elasticity of the enzyme inhibitor’s gene sequence has to be modulated then to the specified performance by rational design or directed evolution. This allows the construction of the biochemical control circuit f

_{12}.

#### 4.3. Robust Control Circuit Design of Stochastic Dynamic Metabolic Networks

_{i}(t), i = 1,⋯, n are molecular concentrations of mRNAs, proteins, and other complexes in the biochemical network.

**Figure 13.**Linear metabolic network of n molecules with intrinsic parameter fluctuation Δa

_{ij}and extrinsic noise ν

_{i }f

_{ij}denotes the biochemical circuit design from x

_{j}to x

_{i}to improve network robustness stability and noise-filtering ability.

_{i}n

_{i}(t) denotes the intrinsic parameter fluctuations due to an L random fluctuation source (e.g., thermal fluctuation, alternative splicing, molecular diffusion, etc.), and n

_{i}(t) denotes the i-th random noise with the statistics E[n

_{i}(t)] = 0 and E[n

_{i}

^{2}(t)] = σ

_{i}

^{2}, i = 1,⋯, L. ν(t) denotes the environmental disturbance.

**Proposition 6**[28]:

^{T}> 0 such that the following matrix inequality holds for a desired disturbance attenuation level ρ

**Remark 6**: (i) The physical interpretation of the phenotype robustness criterion of the metabolic network in Equation (4.31) is that if intrinsic robustness allowing tolerance of intrinsic parameter fluctuation and environmental robustness, as well as attenuation of environmental disturbance are simultaneously conferred by the network robustness, then the phenotype of the metabolic network is maintained. It can be shown that if the eigenvalues of A are closer to the origin, then the network robustness is commensurately larger.

_{0}(i.e., the minimum ρ) can be obtained by solving the following constrained optimization problem:

_{P}(i.e., ρ

_{P}< ρ

_{0}) specified for therapeutic or biotechnological purposes, then a biochemical circuit design using state feedback is necessary:

_{P}< ρ

_{0}. In general, the interactions of a biochemical regulatory network in metabolic processes are nonlinear in real biosystems. In this situation, a nonlinear biochemical regulatory network of metabolic pathways under intrinsic stochastic parameter perturbation and environmental disturbance can be represented based on the stochastic dynamic model of systems biology in Section 2:

_{0}on ν(k) at y(k) in the stochastic nonlinear metabolic network in Equation (4.36) is discussed.

**Proposition 7**[28]:

_{0}of the nonlinear stochastic metabolic network in Equation (4.36) that attenuates environmental disturbance can be obtained by solving the following constrained optimization problem:

_{i}(X) are approximated by α

_{j}(X)A

_{j}X(t), G

_{j}X(t), and A

_{ij}X(t), respectively. In this situation, the following robust chemical circuit design for a metabolic network with a prescribed noise attenuation level ρ

_{P}is derived.

**Proposition 8:**

_{P}is achieved by the biochemical circuit design.

_{0}in Equation (4.44) is the disturbance (noise)-filtering ability of the optimally controlled metabolic network in Equation (4.40) or (4.42) (see

**supplementary example 4**).

**Figure 14.**Engineered synthetic metabolic pathway for isobutanol production in E. coli. (

**A**) Schematic representation of engineered isobutanol production pathway. (

**B**) Engineered synthetic genetic circuit to generate the enzymes necessary for pathway in (

**A**) for isobutanol production in E. coli.

## 5. Future Challenges in Systems Biology

## Supplementary Materials

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Chen, B.-S.; Wu, C.-C.
Systems Biology as an Integrated Platform for Bioinformatics, Systems Synthetic Biology, and Systems Metabolic Engineering. *Cells* **2013**, *2*, 635-688.
https://doi.org/10.3390/cells2040635

**AMA Style**

Chen B-S, Wu C-C.
Systems Biology as an Integrated Platform for Bioinformatics, Systems Synthetic Biology, and Systems Metabolic Engineering. *Cells*. 2013; 2(4):635-688.
https://doi.org/10.3390/cells2040635

**Chicago/Turabian Style**

Chen, Bor-Sen, and Chia-Chou Wu.
2013. "Systems Biology as an Integrated Platform for Bioinformatics, Systems Synthetic Biology, and Systems Metabolic Engineering" *Cells* 2, no. 4: 635-688.
https://doi.org/10.3390/cells2040635