Identification and robustness analysis of nonlinear hybrid dynamical system of genetic regulation in continuous culture

Qi Yang; Lei Wang; Enmin Feng; Hongchao Yin; Zhilong Xiu

doi:10.3934/jimo.2018168

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2020, Volume 16, Issue 2: 579-599. Doi: 10.3934/jimo.2018168

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Identification and robustness analysis of nonlinear hybrid dynamical system of genetic regulation in continuous culture

1.
School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, China
2.
School of Energy and Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China
3.
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
4.
School of Life Science and Biotechnology, Dalian University of Technology, Dalian, Liaoning 116024, China

* Corresponding author. E-mail address: wanglei@dlut.edu.cn
* Corresponding author. E-mail address: wanglei@dlut.edu.cn

Received: February 28, 2018

Revised: June 30, 2018

Early access: November 2018

Published: February 2020

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771008, 11171050 and 11371164), the National Science Foundation for the Youth of China (Grant Nos. 11301051, 11301081 and 11401073), the Provincial Natural Science Foundation of Fujian (Grant Nos. 2014J05001), the Fundamental Research Funds for Central Universities in China (Grant DUT15LK25) and the China Scholorship Council (CSC, No. 201506060121). The authors acknowledge the Supercomputer Center of Dalian University of Technology for providing computing resources

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Abstract

In this paper, we present a framework to infer the possible transmembrane transport of intracellular substances. Considering four key enzymes, a modified fourteen-dimensional nonlinear hybrid dynamic system is established to describe the microbial continuous culture with enzyme-catalytic and genetic regulation. A novel quantitative definition of biological robustness is proposed to characterize the system's resilience when system parameters were perturbed. It not only considers the expectation of system output data after parameter disturbance but also considers the influence of the variance of these data. In this way, the definition can be used as an objective function of the system identification model due to the lack of data on the concentration of intracellular substances. Then, we design a parallel computing method to solve the system identification model. Numerical results indicate that the most likely transmembrane mode of transport is active transport coupling with passive diffusion for glycerol and 1, 3-propanediol.

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Mathematics Subject Classification: Primary: 90B30, 65C20; Secondary: 35B20.

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Figure 1. Main pathway of 1, 3-propanediol biosynthesis in K. pneumoniae. [28]. Abbreviations: GDH-glycerol dehydrogenase; GDHt-glycerol dehydratase; DHAK Ⅰ-dihydroxyacetone kinases (ATP dependent); DHAK Ⅱ-dihydroxyacetone kinases (PEP dependent); PDOR-1, 3-propanediol oxydoreductase; HOR-hypothetical oxydoreductase; PDH-pyruvate dehydrogenase; PFL-pyruvate format lyase; DHA-dihydroxyacetone; DHAP-dihydroxyacetone phosphate; PEP-phosphoenolpyruvate; Pyr-pyruvate; HAc-acetate; EtOH-ethanol; 3-HPA-3-hydroxypropionaldehyde; 1, 3-PD-1, 3-propanediol; RP-regulatory protein

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Figure 2. The simulated results of three extracellular concentrations ($x_{1}$-biomass, $x_2$-glycerol, $x_3$-1, 3-PD) under three different initial conditions, i.e. $(D, C_{S0}) = (0.08, 435), ~(D, C_{S0}) = (0.08, 152)$ and $(D, C_{S0}) = (0.23, 375.7)$

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Figure 3. The simulated results of first three intracellular concentrations ($x_{6}$-glycerol, $x_7$-1, 3-PD, $x_8$-3-HPA) under three different initial conditions, i.e. $(D, C_{S0}) = (0.08, 435), ~(D, C_{S0}) = (0.08, 152)$ and $(D, C_{S0}) = (0.23, 375.7)$

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Figure 4. The simulated results of six intracellular concentrations ($x_9$-mR, $x_{10}$-R, $x_{11}$-mGDHt, $x_{12}$-DGHt, $x_{13}$-mPDOR, $x_{14}$-PDOR) under three different initial conditions, i.e. $(D, C_{S0}) = (0.08, 435), ~(D, C_{S0}) = (0.08, 152)$ and $(D, C_{S0}) = (0.23, 375.7)$

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Table 1. Transport mechanisms of glycerol and 1, 3-PD of metabolic system $NHDS(l_k)$, corresponding to parameter vector $l_k , k\in I_9$. Abbreviations: A, active transport; P, passive diffusion; AP, passive diffusion coupled with active transport

NHDS$(l_k)\rightarrow l_k$	Glycerol	1, 3-PD
NHDS$(l_1)\rightarrow l_1:(0, 1, 0, 1)$	P	P
NHDS$(l_2)\rightarrow l_2:(0, 1, 1, 0)$	P	A
NHDS$(l_3)\rightarrow l_1:(0, 1, 1, 1)$	P	AP
NHDS$(l_4)\rightarrow l_4:(1, 0, 0, 1)$	A	P
NHDS$(l_5)\rightarrow l_5:(1, 0, 1, 0)$	A	A
NHDS$(l_6)\rightarrow l_6:(1, 0, 1, 1)$	A	AP
NHDS$(l_7)\rightarrow l_7:(1, 1, 0, 1)$	AP	P
NHDS$(l_8)\rightarrow l_8:(1, 1, 1, 0)$	AP	A
NHDS$(l_9)\rightarrow l_9:(1, 1, 1, 1)$	AP	AP

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Table 2. The Parameters in Kinetic Models (16)-(18)

$i$	$m_i$	$Y_i$	$\Delta q_i$	$K_i^*$
2	2.20	0.0082	28.58	11.43
4	-0.97	33.07	5.74	85.71
5	5.26	11.66	-	-

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Table 3. The computational values of robustness index for system $NHDS$ corresponding to parameter vector $l_k , k\in I_9$

$k$	1	2	3	4	5	6	7	8	9
$J(p^*, j, l_k)$	1.748	$+\infty$	15.267	$+\infty$	3.6521	5.689	$+\infty$	0.657	0.1258

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Table 4. The computational values of the optimal parameters under three experiments

$(D, C_{s0})$	$p^*(l_9)$
(0.08, 435)	(81.7276, 4.97131, 1075.54, 99.9598, 1.89429, 23.9368, 0.362061,
	28.1892, 8.95164, $7.79472\times10^{-4}$, 40.415, 41.1295, 27.8876, 1.12982,
	1.50452, 10.0365, $4.69745\times10^{-5}$, 949.377, 9.47578, 25.4648, 32.6392,
	12.7944, 0.1133, 0.529596, 191.37, 1.75482, 20.1505, 9.64496, 14.0072,
	0.8037, 13.9279$)^T$
(0.08, 152)	(89.7339, 2.31116, 680.762, 77.858, 2.64026, 68.5132, 0.296838,
	18.2558, 9.80988, $9.56009\times10^{-4}$, 40.1752, 49.5181, 23.1997, 1.22681,
	1.95972, 17.3442, $8.18616\times10^{-5}$, 998.73, 13.4516, 23.1863, 42.6794,
	8.60437, 0.058844, 0.550928, 262.521, 1.69241, 25.8673, 5.83124, 9.62714,
	22.5502, 11.171$)^T$
(0.23, 375.7)	(87.3566, 4.11554, 599.756, 86.7367, 1.47777, 94.5825, 0.269305,
	34.3741, 9.01509, $8.30467\times10^{-4}$, 33.6429, 48.385, 27.4457, 2.66168,
	1.97522, 10.6856, $4.51274\times10^{-5}$, 1012.51, 16.0382, 25.8691, 40.1752,
	8.52086, 0.196155, 0.790479, 315.66, 1.84771, 18.6017, 6.09894, 8.43732,
	34.01, 15.35$)^T$

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Table 5. The experimental data y and the numerical results $x^*$ for three experiments

$(D, C_{s0})$	$y$	$x^*$
(0.08, 435)	(2.6, 0.22, 136,	(2.66078, 0.199219, 151.787, 56.1728, 204.64,
	55, 188$)^T$	0.189992, 48.8683, 0.0906137, 3.18097, 0.00879915,
		0.335116, 0.694374, 0.043648, 0.0648233$)^T$
(0.08, 152)	(1.1, 0.09, 53,	(1.15502, 0.0720033, 53.1286, 24.2607, 88.8432,
	28, 63$)^T$	0.0605281, 47.8053, 0.130965, 5.17053, 0.0108471,
		0.405015, 1.77233, 0.073951, 0.148219$)^T$
(0.23, 375.7)	(2.86, 0.91, 148.5,	(2.858, 0.893436, 150.942, 83.1966, 98.1893
	65.7, 101.6$)^T$	0.869124, 58.9767, 0.240652, 1.59739, 0.00399712,
		0.327636, 0.962971, 0.0785403, 0.171447 $)^T$

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