Modeling and observer design for recombinant Escherichia coli strain

Abstract

A mathematical model for recombinant bacteria which includes foreign protein production is developed. The experimental system consists of an Escherichia Coli strain and plasmid pIT34 containing genes for bioluminescence and production of a protein, β-galactosidase. This recombinant strain is constructed to facilitate on-line estimation and control in a complex bioprocess. Several batch experiments are designed and performed to validate the developed model. The design of a model structure, the identification of the model parameters and the estimation problem are three parts of a joint design problem. A nonlinear observer is designed and an experimental evaluation is performed on a batch fermentation process to estimate the substrate consumption.

Appendix 1

We will show that the estimation error $e(t)=\hat{x}(t)-x(t)$ exponentially converges to zero. Using (13), we obtain:

$$\dot{e}(t) = F(\hat x,u)-F(x,u) -\rho \Delta_\theta R^{-1} C^T(C\hat{x}-y)$$

(17)

We will use the following decomposition:

$$ F(\hat x,\,u) - F(x,\,u) = \left[ {\begin{array}{*{20}c} {F_1 (\hat x_1 ,\,\hat x_2 ,\,u)} \\ 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}l} {F_1 (\hat x_1 ,\,x_2 ,\,u) - F_1 (x_1 ,\,x_2 ,\,u)} \hfill \\ {F_2 (\hat x_1 ,\,\hat x_2 ,\,u) - F_1 (x_1 ,\,x_2 ,\,u)} \hfill \\ \end{array} } \right] $$

(18)

From the mean value theorem we get:

$$ F(\hat x,\,u) - F(x,\,u) = A(\hat x,\,x)e + B(\hat x,\,x)e $$

(19)

where $A(\hat x,\,x) = \left[ {\begin{array}{*{20}c} 0 & {\frac{{\partial F_1 }} {{\partial x_2 }}(x_1 ,\,\hat x_2 + \tau e_2 ,\,u)} \\ 0 & 0 \\ \end{array} } \right],$ and τ∈[0, 1] is a number which may depend on $(u,\hat x,x).$ Similarly, from the mean value theorem we have:

$$ B(\hat x,\,x) = \left[ {\begin{array}{*{20}c} {\frac{{\partial F_1 }} {{\partial x_1 }}(\hat x_1 + \tau _1 e_1 ,\,x_2 ,\,u)} & 0 \\ {\frac{{\partial F_2 }} {{\partial x_1 }}(\hat x_1 + \tau _2 e_1 ,\,\hat x_2 ,\,u)} & {\frac{{\partial F_2 }} {{\partial x_2 }}(\hat x_1 ,\,\hat x_2 + \tau _3 e_2 ,\,u)} \\ \end{array} } \right] $$

where τ _i ∈[0, 1], 1,2. Combining (18) and (19), we deduce:

$$\dot{e}(t)=A(\hat x,x)e+B(\hat x,x)e-\rho \Delta_\theta R^{-1}C^TC e$$

(20)

Now setting $\bar e =\Delta_\theta ^{-1}e,$ then, it suffices to show that $\|\bar e(t)\|$ exponentially converges to 0 as t→ +∞. To do so, we can easily check that:

$$\dot{\bar e}=\theta[A(\hat x,x)-\rho R^{-1} C^TC] \bar e+\Delta^{-1}_\theta B(\hat x,x) \bar e$$

(21)

Now, consider the following positive definite quadratic function: $V=\bar{e}^TR\bar{e}.$ We only need to show that $\dot{V}\leq -\gamma V$ for some constant γ >0.

Using (21) we obtain:

$$\dot{V}= \theta \bar{e}^T(A(t)^TR+R A(t))\bar{e}-2\theta \rho \|C\bar{e}\|^2+2R\bar{e}^T\|\Delta_\theta ^{-1} B(\hat x,x,u)e\|$$

Since A(t) is of the form ${\left[ {\begin{array}{*{20}l} {0 \hfill} & {{a(t)} \hfill} \\ {0 \hfill} & {0 \hfill} \\ \end{array} } \right]},$ where $\alpha_1\leq a(t)={\partial F_1}/ {\partial x_2}(x_1,\hat x_2+\tau e_2,u) \leq \alpha_2,$ and α₁, α₂ are fixed constants given by (H2).

Obviously R is an SPD matrix. Indeed, let x∈IR ⁿ, x≠ 0, then

$$\begin{aligned} x^TRx &= R_{11}x_1^2+2 R_{12}x_1x_2+R_{22}x_2 \\ &\quad \geq R_{11}x_1^2-2|R_{12}||x_1x_2|+R_{22}x_2^2\\ & \quad \geq (1-\frac{|R_{12}|}{\sqrt{R_{11}R_{22}}})(R_{11}x_1^2+R_{22}x_2^2)\\ &\quad \geq 0 \forall x\neq 0 (\hbox{from} (14)), \end{aligned} $$

(22)

Now let us set P=A ^T R+RA−ρ C ^T C.

Using a similar argument as above, we obtain:

$$\begin{aligned} x^TPx &= -\rho x_1^2+2aR_{11}x_1x_2+2aR_{12}x_2^2 \\ &\quad \leq -\rho x_1^2+2\alpha_2R_{11}|x_1x_2|-2\alpha_1|R_{12}|x_2^2\\ &\quad \leq-(1-\frac{{\alpha_2R_{11}}}{ {\sqrt{2\rho \alpha_1|R_{12}|}}})(\rho x_1^2+2\alpha_1|R_{12}|x_2^2) \end{aligned} $$

From inequalities (14), we deduce $1-\alpha_2 R_{11}/{\sqrt{2\rho \alpha_1|R_{12}|}}>0.$ Hence,

$$ \left\{ {\begin{array}{*{20}l} {{x^{T} Px \leqslant - \eta {\left\| x \right\|}^{2} } \hfill} \\ {{{\text{where}}\,\eta = {\left( {1 - \frac{{R_{{11}} \alpha _{2} }} {{{\sqrt {2\rho \alpha _{1} |R_{{12}} } }}}} \right)}\,{\text{min}}\{ \rho ,\,2\alpha _{1} {\left| {R_{{12}} } \right|}\} } \hfill} \\ \end{array} } \right. $$

(23)

From (23), we get:

$$\begin{aligned} \dot V&\leq -\theta [\eta \|\bar{e}\|^2 -\rho \|C\bar{e}\|^2]-2\theta \rho \|C\bar{e}\|^2 +2\|R\|\|e\|\|\Delta_\theta ^{-1} B(\hat x,x,u)e\|\\ & \leq -\eta\theta \|\bar{e}\|^2+ 2\|R\|\|e\|\|\Delta_\theta ^{-1} B(\hat x,x,u)e\|\\ \end{aligned} $$

Now, using the triangular structure of $B(\hat x,x,u)$ and the fact that F is a global Lipschitz function (assumption (H1)), we can easily see that for every θ ≥ 1 we have: $\|\Delta_\theta^{-1} B(\hat x,x) \bar e\| \leq \sqrt{2}c \|\bar e\|,$ where c is the Lipschitz constant given by (H1). Hence,

$$\begin{aligned} \dot V & \leq (-\eta\theta +2\sqrt{2}c\|R\|)\|\bar{e}\|^2\\ &\leq -(\eta\theta -2\sqrt{2}c\|R\|)\lambda_{\text{min}}(R)V\\ \end{aligned} $$

where λ_min(R) denotes the smallest eigenvalue of R. To end the proof, it suffices to chose θ such that $\theta\geq \theta_0={2\sqrt{2}c\|R\|}/ {\eta}.$