Non‐linear feedback control of the p 53 protein–mdm 2 inhibitor system using the derivative‐free non‐linear Kalman filter

Abstract

It is proven that the model of the p 53–mdm 2 protein synthesis loop is a differentially flat one and using a diffeomorphism (change of state variables) that is proposed by differential flatness theory it is shown that the protein synthesis model can be transformed into the canonical (Brunovsky) form. This enables the design of a feedback control law that maintains the concentration of the p 53 protein at the desirable levels. To estimate the non‐measurable elements of the state vector describing the p 53–mdm 2 system dynamics, the derivative‐free non‐linear Kalman filter is used. Moreover, to compensate for modelling uncertainties and external disturbances that affect the p 53–mdm 2 system, the derivative‐free non‐linear Kalman filter is re‐designed as a disturbance observer. The derivative‐free non‐linear Kalman filter consists of the Kalman filter recursion applied on the linearised equivalent of the protein synthesis model together with an inverse transformation based on differential flatness theory that enables to retrieve estimates for the state variables of the initial non‐linear model. The proposed non‐linear feedback control and perturbations compensation method for the p 53–mdm 2 system can result in more efficient chemotherapy schemes where the infusion of medication will be better administered.

1 Introduction

In the recent years, the problem of non‐linear control of complex biological and biochemical systems has gained much interest [1 –4 ]. In particular, the interaction of the p 53 protein and the mdm 2 inhibitor has been targeted as a therapeutic strategy for the treatment of cancer. The P 53 protein has been identified as a key factor in the abatement of tumours since it enhances cell‐cycle arrest and apoptosis. The concentration of the P 53 protein in the cytoplasm is primarily controlled by another protein, known as inhibitor protein mdm 2 within a feedback loop. When the concentration of the MDM 2 protein increases, the concentration of the P 53 protein is reduced (downregulation). The MDM 2 protein binds ubiquitin molecules to P 53 which result to the dissociation of the P 53 protein. On the other side, the increase of the concentration of P 53 enhances the transcription procedure of mdm 2 and consequently the produced MDM 2 protein downregulates P 53. In this manner, the p 53–mdm 2 feedback loop converges to an equilibrium [5 –10 ]. There are chemotherapy drugs that work by binding the MDM2 protein and consequently by preventing the MDM 2 protein from disintegrating the P 53 protein (ubiquitination) [11 –14 ]. This is a promising approach to the treatment of cancer. It is based on the infusion of MDM 2 antagonists which are called Nutlins. By deactivating MDM 2, these drugs restore the levels of concentration of the P 53 protein and consequently contribute to the fighting against cancer cells [15 –18 ].

In this paper, it is shown that it is possible to control the levels of the concentration of the P 53 protein through non‐linear feedback control, where the control input is the infusion rate of the chemotherapy drug. Previous results on non‐linear feedback control of biological oscillators and on control of protein synthesis processes can be found in [19 –21 ]. The pharmacokinetics–pharmacodynamics model of the P 53 protein is described by a complicated set of non‐linear differential equations. It is shown that with the use of differential flatness theory it is possible to transform this complicated model into the canonical Brunovsky form [22 –29 ]. In this latter form, a single‐input single‐output (SISO) description between the output (P 53 protein) and the input (drug's infusion rate) is obtained and this facilitates the design of a feedback control law that can make the P 53 protein concentration converge to the desirable levels.

However, for the implementation of a robust feedback control scheme for the P 53 protein still one comes against additional difficulties. These are: (i) there is no direct measurement for all state variables of the SISO decoupled model and (ii) one cannot be sure that the values of the model's parameters are exact. Indeed, despite the application of non‐linear identification and filtering methods for either estimating the parameters of protein synthesis models or for estimating the parameters of gene networks, one cannot be sure about the precision of the provided estimation or cannot assure that the estimated parameters are time invariant [30 –38 ]. To cope efficiently with (i) and (ii), it is proposed to use a disturbance estimator that is based on the derivative‐free non‐linear Kalman filter [19, 39 ]. The state‐space model of the p 53 protein synthesis system is extended by considering as additional state variables the disturbance and modelling uncertainty terms. For the extended state‐space description of the system, state estimation can be performed using the Kalman filter recursion while it is possible to obtain estimates for the initial state variables of the non‐linear model using an inverse transformation (diffeomorphism) that is provided by differential flatness theory. In this manner, one can obtain simultaneously estimates of the non‐measurable state variables of the p 53–mdm 2 protein model and estimates of the disturbance terms affecting it. By knowing perturbation variables, their compensation is possible with the inclusion on an additional term in the control input.

The structure of the paper is as follows. In Section 2, the dynamic model of the p 53 protein–mdm 2 inhibitor system is analysed. In Section 3, non‐linear feedback control of the p 53 protein system using differential flatness theory is explained. In Section 4, robust disturbances compensation for the p 53 protein–mdm 2 inhibitor system is proposed with the use of derivative‐free non‐linear Kalman filter. In Section 5, the performance of the proposed control scheme is evaluated through simulation experiments. Finally, in Section 6 concluding remarks are stated.

2 Dynamic model of the p 53 protein–mdm2 inhibitor system

2.1 Feedback control loops in the p53 protein–mdm2 inhibitor system

As mentioned the concentration of the p 53 protein is mainly controlled by the levels of the mdm 2 protein within a negative feedback loop. The synthesis of the P 53 protein is also affected by the ATM, ARF and E 2F 1 proteins through secondary feedback loops. The dynamic model of the p 53 protein–mdm 2 inhibitor system is described in Fig. 1. The meaning of the variables that appear in the p 53 protein–mdm 2 inhibitor dynamical system is summarised in Table 1 and is as follows [5, 6, 11, 15 ].

Fig 1.

Feedback control loop of the p53 protein–mdm2 inhibitor system

Table 1.

Variables of the p 53–mdm 2 protein synthesis

Notation	Variable's name
p 53	mRNA concentration of the p 53 gene after transcription
P 53	concentration of the P 53 protein in the cytoplasm after translation
P 53*	active form of the P 53 protein that is produced after phosphorylation of P 53
mdm 2	mRNA concentration of the inhibitor protein mdm 2 after transcription
MDM 2	concentration of the MDM 2 protein in the cytoplasm after translation
N	concentration of the chemotherapeutic drug
λ	infusion rate of the chemotherapeutic drug
ATM	protein that contributes to the phosphorylation of the P 53 protein
ATM *	concentration of the active form of the ATM protein
e 2f 1	mRNA concentration of the gene e 2f 1 after transcription
E 2F 1	concentration of the protein E 2F 1 after translation
E 2F 1*	active form of the E 2F 1 protein
arf	mRNA concentration of the gene arf after transcription
ARF	concentration of the ARF protein after translation

p 53 is the mRNA concentration of the p 53 gene after transcription, P 53 is the concentration of the P 53 protein in the cytoplasm after translation, P 53* is the active form of the P 53 protein that is produced after phosphorylation of P 53, mdm 2 is the mRNA concentration of the inhibitor protein mdm 2 after transcription, MDM 2 is the concentration of the MDM 2 protein in the cytoplasm after translation, N is the concentration of the chemotherapeutic drug, ATM is a protein that identifies the transcription of p 53 and contributes to the phosphorylation of the P 53 protein, ATM* is the concentration of the active form of the ATM protein. It contributes both to the phosphorylation of protein P 53 and of protein MDM 2, e 2f 1 is the mRNA concentration of the gene e 2f 1 after transcription, E 2F 1 is the concentration of the protein E 2F 1 after translation, E 2F 1* is the active form of the E 2F 1 protein, arf is the mRNA concentration of the gene arf after transcription, ARF is the concentration of the ARF protein after translation.

The basic feedback loop is that of the synthesis of the p 53 protein under the inhibitor protein mdm 2. When the concentration of the mdm 2 protein increases, the concentration of the p 53 protein is reduced (downregulation). This process is also known as proteolytic degradation. The mdm 2 protein binds ubiquitin molecules to p 53 which result to the dissociation of the P 53 protein. On the other side, the increase of P 53 enhances the transcription procedure of mdm 2 and consequently the produced MDM 2 protein downregulates P 53. In this manner, the p 53–mdm 2 feedback loop converges to an equilibrium.

The role of the ATM protein is explained as follows. ATM is a protein that plays a sensor–detector role in the p 53 network. ATM undergoes auto‐phosphorylation which leads to its transformation to the active form ATM *. This process can be accelerated by the exposure of the cell to radiation. In its turn ATM *, through phosphorylation, contributes to the synthesis of the proteins E 2F 1, MDM 2 and P 53. The transformation of ATM * through phosphorylation into MDM 2 and P 53 changes the equilibrium points of the p 53–mdm 2 loop. In particular, it enhances the levels of the P 53 protein and attenuates the effects of MDM 2 in the dissociation of the P 53 protein. With the raise of the concentration of P 53, cell‐cycle arrest is also enhanced while the apoptosis rate is also increased.

Another loop, one can distinguish in the p 53 network is between proteins E 2F 1 and ARF. As mentioned above, the ATM * protein through its phosphorylation contributes to the synthesis of E 2F 1. In turn, the E 2F 1 protein contributes to the transcription into mRNA of the arf gene and consequently to the synthesis (translation) of the ARF protein. The increased concentration of the ARF protein results in the downregulation of E 2F 1 and in this manner the E 2F 1–ARF loop closes and an equilibrium is reached. Moreover, ARF results in the downregulation of the MDM 2 and causes raise in the levels of the p 53 protein concentration. This also results to improved treatment against cancer cells. It has been confirmed that the removal of the ARF protein from human tissues is responsible for the appearance of breast, mind and lung tumours.

There are chemotherapy drugs that work by binding the MDM 2 protein and consequently by preventing the MDM 2 protein from dissociation of the P 53 protein (ubiquitination). This is based on the infusion of MDM 2 antagonists (Nutlins). By deactivating MDM 2, these drugs restore the levels of concentration of the P 53 protein and consequently contribute to the fighting against cancer cells. More results on the inhibition of the mdm 2 protein through chemotherapy medication having as a result the raise of the levels of p 53 protein and the more efficient tumours abatement can be found in [35 ].

The effect of Nutlins on the MDM 2 protein (P 53 inhibitor) is defined by the following dynamics

$$\dot{N}={\lambda }_N-{\mu }_{N}N-k_6\cdot N\cdot MDM2$$ (1)

where N is the drug's concentration in the cytoplasm, λ _N is the drug's infusion rate, μ _N is the drug's degradation rate and −k ₆ · N · MDM 2 is the binding of the drug by the MDM 2 protein. The variables of the p 53–mdm 2 protein synthesis model are defined in Table 1.

2.2 State‐space model of the p53 protein–mdm2 inhibitor system

The following state variables are defined for the dynamic model of the p 53 protein–mdm 2 inhibitor system

$$\begin{array}{rl}& x_1=p53,x_2=P53,x_3=P{53}^{\ast },x_4=mdm2,x_5=MDM2,x_6=N\\ & x_7=e2f1,x_8=E2F1,x_9=E2F{1}^{\ast },x_{10}=\mathrm{arf},x_{11}=\mathrm{ARF}\end{array}$$ (2)

The system can be described using the following state‐space equations [5 ]

$$\begin{array}{rl}{\dot{x}}_1& ={\lambda }_{p53}-{\mu }_{p53}{x}_1\\ {\dot{x}}_2& =a_{p53}{x}_1-{\mu }_{53}{x}_2-v_{p53}{x}_3-\frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-\frac{K_{cat}{x}_5{x}_2}{a{K}_{13}+x_2}\\ {\dot{x}}_3& =\frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-v_{\mathrm{p}53}{x}_3-\frac{K_{cat}^{\ast }{x}_5{x}_3}{a{K}_{13}+x_3}\\ {\dot{x}}_4& ={\lambda }_{mdm2}-{\mu }_{mdm2}{x}_4+{\varphi }_{mdm2}\frac{x_3{(t-r_1)}^{n_1}}{x_2{(0)}^{n_1}+x_3{(t-r_1)}^{n_1}}\\ {\dot{x}}_5& =a_{MDM2}{x}_4-{\mu }_{MDM2}{x}_5-\frac{K_{2}AT{M}^{\ast }{x}_5}{K_{M_2}+x_5}-K_4{x}_{11}{x}_5-K_6{x}_6{x}_5\\ {\dot{x}}_6& ={\lambda }_N-{\mu }_N{x}_6-K_6{x}_6{x}_5\\ {\dot{x}}_7& ={\lambda }_{e2f1}-{\mu }_{e2f1}{x}_7\\ {\dot{x}}_8& =a_{E2F1}{x}_7-{\mu }_{E2F1}{x}_8+v_{E2F1}{x}_9-\frac{K_{2}AT{M}^{\ast }{x}_8}{K_{M_3}+x_8}\\ {\dot{x}}_9& =\frac{K_{3}AT{M}^{\ast }{x}_8}{K_{M_3}+x_8}-v_{E2F1}{x}_9-K_5{x}_{11}{x}_9\\ {\dot{x}}_{10}& ={\lambda }_{\mathrm{arf}}-{\mu }_{\mathrm{arf}}{x}_{10}+{\varphi }_{\mathrm{arf}}\frac{x_9{(t-r_2)}^{n_2}}{x_8{(0)}^{n_2}+x_9{(t-r_2)}^{n_2}}\\ {\dot{x}}_{11}& =a_{\mathrm{ARF}}{x}_{10}-{\mu }_{\mathrm{ARF}}{x}_{11}-K_4{x}_{11}{x}_5-K_5{x}_{11}{x}_9\end{array}$$ (3)

In matrix form, the state‐space description of the system becomes

$$\left(\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\\ {\dot{x}}_4\\ {\dot{x}}_5\\ {\dot{x}}_6\\ {\dot{x}}_7\\ {\dot{x}}_8\\ {\dot{x}}_9\\ {\dot{x}}_{10}\\ {\dot{x}}_{11}\end{array}\right)=\left(\begin{array}{c}{\lambda }_{p53}-{\mu }_{p53}{x}_1\\ a_{p53}{x}_1-{\mu }_{53}{x}_2-v_{p53}{x}_3-\frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-\frac{K_{cat}{x}_5{x}_2}{a{K}_{13}+x_2}\\ \frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-v_{p53}{x}_3-\frac{K_{cat}^{\ast }{x}_5{x}_3}{a{K}_{13+x_3}}\\ {\lambda }_{mdm2}-{\mu }_{mdm2}{x}_4+{\varphi }_{mdm2}\frac{x_3{(t-r_1)}^{n_1}}{x_2{(0)}^{n_1}+x_3{(t-r_1)}^{n_1}}\\ a_{MDM2}{x}_4-{\mu }_{MDM2}{x}_5-\frac{K_{2}AT{M}^{\ast }{x}_5}{K_{M_2}+x_5}-K_4{x}_{11}{x}_5-K_6{x}_6{x}_5\\ -{\mu }_N{x}_6-K_6{x}_6{x}_5\\ {\lambda }_{e2f1}-{\mu }_{e2f1}{x}_7\\ a_{E2F1}{x}_7-{\mu }_{E2F1}{x}_8+v_{E2F1}{x}_9\frac{-K_{2}AT{M}^{\ast }{x}_8}{K_{M_3}+x_8}\\ \frac{K_{3}AT{M}^{\ast }{x}_8}{K_{M_3}+x_8}-v_{E2F1}{x}_9-K_5{x}_{11}{x}_9\\ {\lambda }_{arf}-{\mu }_{arf}{x}_{10}+{\varphi }_{arf}\frac{x_9{(t-r_2)}^{n_2}}{x_8{(0)}^{n_2}+x_9{(t-r_2)}^{n_2}}\\ a_{ARF}{x}_{10}-{\mu }_{ARF}{x}_{11}-K_4{x}_{11}{x}_5-K_5{x}_{11}{x}_9\end{array}\right)+\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right){\lambda }_N$$ (4)

which is also written in the form

$$\dot{x}=\mathit{f}(x)+\mathit{g}(x)u$$ (5)

where u = λ _N is the control input, and f (x ) ∈ R ^11×1, g (x ) ∈ R ^11×1 are vector fields. It will be shown that the considered model of the p 53 protein–mdm 2 inhibitor system is a differentially flat one. The following flat output is defined $Y=[P_{53}^{\ast },N,E2F{1}^{\ast },ARF]$ or y = [x ₃, x ₆, x ₉, x ₁₁ ]. Thus, one has Y = [y ₁, y ₂, y ₃, y ₄ ]^T.

3 Non‐linear feedback control of the p 53 protein system using differential flatness theory

3.1 Definition of differentially flat systems

Differential flatness theory will be used for implementing feedback control and synchronisation of the p 53 protein–mdm 2 inhibitor system. The main principles of differential flatness theory are as follows [25, 26 ]. A finite dimensional system is considered. This can be written in the general form of a set of ordinary differential equation (ODE), that is, $S_i(w,\dot{w},\ddot{w},\dots ,w^{(i)}),i=1,2,\dots ,q$ . The term w denotes the system variables (these variables are for instance the elements of the system's state vector and the control input) while w ^{(i )}, i = 1, 2, …, q are the associated derivatives. Such a system is said to be differentially flat if there is a collection of m functions y = (y ₁, …, y _m ) of the system variables and of their time derivatives, that is, $y_i=\varphi (w,\dot{w},\ddot{w},\dots ,w^{({\alpha }_i)}),i=1,\dots ,m$ satisfying the following two conditions [25, 26, 39 ]. (i) There does not exist any differential relation of the form $R(y,\dot{y},\dots ,y^{(\beta )})=0$ which implies that the derivatives of the flat output are not coupled in the sense of an ODE, or equivalently it can be said that the flat output is differentially independent. (ii) All system variables (i.e. the elements of the system's state vector w and the control input) can be expressed using only the flat output y and its time derivatives $w_i={\psi }_i(y,\dot{y},\dots ,y^{({\gamma }_i)}),i=1,\dots ,s$.

3.2 Differential flatness of the p53 protein–mdm 2 inhibitor dynamical system

From the sixth row of (3 ) and by solving with respect to x ₅ one obtains

$$\begin{array}{rl}{x}_5& ={\dot{x}}_6+{\mu }_N{x}_6-K_6{x}_6\Rightarrow x_5={\dot{y}}_2+{\mu }_N{y}_2-K_6{y}_2\Rightarrow \\ x_5& =\frac{\left[\begin{array}{cccc}0& 1& 0& 0\end{array}\right]\dot{y}+{\mu }_N\left[\begin{array}{cccc}0& 1& 0& 0\end{array}\right]y}{-K_6\left[\begin{array}{cccc}0& 1& 0& 0\end{array}\right]y}\Rightarrow x_5=f_5(y,\dot{y})\end{array}$$ (6)

From the third row of (3 ) and by solving with respect to x ₂ one obtains

$$\begin{array}{rl}{K}_{M_1}{\dot{x}}_3+{\dot{x}}_3{x}_2& =K_{1}AT{M}^{\ast }{x}_2-v_{p53}{K}_{M_1}{x}_3-v_{p53}{x}_2{x}_3\\ & -K_{M_1}\frac{K_{cat}^{\ast }{x}_5{x}_3}{a{K}_{13}+x_3}-\frac{K_{cat}^{\ast }{x}_5{x}_3}{a{K}_{13}+x_3}{x}_2\Rightarrow \\ x_2& =\frac{K_{M_1}{\dot{x}}_3-v_{p53}{K}_{M_1}{x}_3+K_{M_1}(K_{cat}^{\ast }{x}_5{x}_3/a{K}_{13}+x_3)}{K_{1}AT{M}^{\ast }+v_{p53}{x}_3+(K_{cat}^{\ast }{x}_5{x}_3/a{K}_{13}+x_3)-{\dot{x}}_3}\Rightarrow \\ x_2& =\frac{K_{M_1}{\dot{y}}_1-v_{p53}{K}_{M_1}{y}_1+K_{M_1}(K_{cat}^{\ast }{f}_5(y,\dot{y})y_1/a{K}_{13}+y_1)}{K_{1}AT{M}^{\ast }+v_{p53}{y}_1+(K_{cat}^{\ast }{f}_5(y,\dot{y})y_1/a{K}_{13}+y_1)-{\dot{y}}_1}\Rightarrow \\ x_2& =f_2(y,\dot{y})\end{array}$$ (7)

Equivalently, the second row of (3 ) is solved with respect to x ₁. This gives

$$\begin{array}{rl}& x_1={\dot{x}}_2+{\mu }_{p53}{x}_2+v_{p53}{x}_3+\frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-\frac{K_{cat}{x}_2{x}_5}{a{K}_{13}+x_2}\Rightarrow \\ & x_1=f_1(y,\dot{y})\end{array}$$ (8)

The fifth row of (3 ) is solved with respect to x ₄. Thus, one obtains

$$\begin{array}{rl}{x}_4& =\frac{{\dot{x}}_5+{\mu }_{MDM2}{x}_5+(K_{2}AT{M}^{\ast }{x}_5/K_{M_2}+X_5)+K_4{x}_{11}{x}_5+K_6{x}_6{x}_5}{a_{MDM2}}\Rightarrow \\ x_4& =f_4(y,\dot{y})\end{array}$$ (9)

The ninth row of (3 ) is solved with respect to x ₈. Thus, one obtains

$$\begin{array}{rl}{K}_{M_3}{\dot{x}}_9+{\dot{x}}_9{x}_8& =K_{3}AT{M}^{\ast }{x}_8-v_{E2F1}{K}_{M_3}{x}_9\\ & -v_{E2F1}{x}_8{x}_9-K_5{K}_{M_3}{x}_{11}{x}_9-K_5{x}_{11}{x}_9{x}_8\Rightarrow \\ x_8& =\frac{K_{M_3}{\dot{x}}_9+v_{E2F1}{K}_{M_3}{x}_3+K_5{K}_{M_3}{x}_{11}{x}_9}{K_{3}AT{M}^{\ast }-{\dot{x}}_9-v_{E2F1}{x}_9-K_9{x}_{11}{x}_9}\Rightarrow \\ x_8& =f_8(y,\dot{y})\end{array}$$ (10)

The eighth row of (3 ) is solved with respect to x ₇. Thus, one obtains

$$\begin{array}{rl}{x}_7& =\frac{{\dot{x}}_8+{\mu }_{E2F1}{x}_8-v_{E2F1}{x}_9+(K_{2}AT{M}^{\ast }{x}_8/K_{M_3}+x_8)}{a_{E2F1}}\Rightarrow \\ x_7& =f_7(y,\dot{y})\end{array}$$ (11)

The 11th row of (3 ) is solved with respect to x ₁₀. It holds

$$\begin{array}{rl}{x}_{10}& =\frac{{\dot{x}}_{11}+{\mu }_{ARF}{x}_{11}+K_4{x}_{11}{x}_5+K_5{x}_{11}{x}_9}{a_{ARF}}\Rightarrow \\ x_{10}& =f_{10}(y,\dot{y})\end{array}$$ (12)

Moreover, from the sixth row of (3 ) and using $x_5=f_5(y,\dot{y})$ and x ₆ = y ₂, one obtains the control input u = λ _N

$$\begin{array}{rl}u& ={\lambda }_N={\dot{x}}_6+{\mu }_N{x}_6+K_6{x}_6{x}_5\Rightarrow \\ {\lambda }_N& =f_u(y,\dot{y})\end{array}$$ (13)

Thus, one has that all state variables and the control input of the p 53 protein–mdm 2 inhibitor system are functions of the flat output y and of its derivatives. Consequently, the dynamical system of P 53 is a differentially flat one.

3.3 Flatness‐based control of the p53 protein–mdm2 inhibitor system

It will be shown that using the differentially flat description of the p 53 protein–mdm 2 inhibitor system it is possible to transform it to the canonical Brunovsky form.

It holds that y ₁ = x ₃, therefore

$${\dot{y}}_1={\dot{x}}_3\Rightarrow {\dot{y}}_1=\frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-v_{P53}{x}_3-\frac{K_{cat}^{\ast }{x}_5{x}_3}{a{K}_{13}+x_3}$$ (14)

Consequently, the second derivative of y ₁ is found to be

$$\begin{array}{rl}{\ddot{y}}_1& =\frac{(K_{1}AT{M}^{\ast }{\dot{x}}_2)(K_{M_1}+x_2)-(K_{1}AT{M}^{\ast }{x}_2){\dot{x}}_2}{{(K_{M_1}+x_2)}^2}-v_{p53}{\dot{x}}_3\\ & -\frac{K_{cat}^{\ast }({\dot{x}}_5{x}_3+x_5{\dot{x}}_3)(a{K}_{13}+x_3)-(K_{cat}^{\ast }{x}_5{x}_3){\dot{x}}_3}{{(a{K}_{13}+x_3)}^2}\end{array}$$ (15)

After intermediate operations one obtains

$$\begin{array}{rl}{\ddot{y}}_1& =\frac{K_{1}AT{M}^{\ast }{K}_{M_1}}{{(K_{M_1}+x_2)}^2}{\dot{x}}_2-v_{p53}{\dot{x}}_3-\frac{K_{cat}^{\ast }a{K}_{13}{x}_5{\dot{x}}_3}{{(a{K}_{13}+x_3)}^2}-\frac{K_{cat}^{\ast }{x}_3}{(a{K}_{13}+x_3)}{\dot{x}}_5\end{array}$$ (16)

and after substituting the derivatives of x ₃ and x ₅ one gets

$$\begin{array}{rl}{\ddot{y}}_1& =\frac{K_{1}AT{M}^{\ast }{K}_{M_1}}{{(K_{M_1}+x_2)}^2}\left[a_{p53}{x}_1-{\mu }_{p53}{x}_2-v_{p53}{x}_3-\frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-\frac{K_{cat}{x}_5{x}_2}{{(a{K}_{13}+x_2)}^2}\right]\\ & -\left[v_{p53}+\frac{K_{cat}^{\ast }a{K}_{13}{x}_5}{{(a{K}_{13}+x_3)}^2}\right]\cdot \left[\frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-v_{p53}{x}_3-\frac{K_{cat}^{\ast }{x}_5{x}_3}{(a{K}_{13}+x_3)}\right]\\ & -\frac{K_{cat}^{\ast }{x}_3}{(a{K}_{13}+x_3)}\left[a_{MDM2}{x}_4-{\mu }_{MDM2}{x}_5-\frac{K_{2}AT{M}^{\ast }{x}_5}{K_{M_2}+x_5}-K_4{x}_{11}{x}_5-K_6{x}_6{x}_5\right]\end{array}$$ (17)

By differentiating once more with respect to time, one obtains

$$y_1^{(3)}=f(y,\dot{y})+g(y,\dot{y})u$$ (18)

where the control input u = λ _N is the input rate of the chemotherapy drug, while functions $f(y,\dot{y})$ and $g(y,\dot{y})$ are defined as follows:

• (i) function $f(y,\dot{y})$

  $$\begin{array}{rl}f(y,\dot{y})& =\frac{-2(K_{M_1}+x_2){\dot{x}}_2{K}_{1}ATM{K}_{M_1}}{{(K_{M_1}+x_2)}^4}\left[a_{p53}{\dot{x}}_1-{\mu }_{p53}{\dot{x}}_2-v_{p53}{\dot{x}}_3-\frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-\frac{K_{cat}{x}_5{x}_2}{a{K}_{13}+x_2}\right]+\\ & +\frac{K_{1}AT{M}^{\ast }{K}_{M_1}}{{(K_{M_1}+x_2)}^2}\left[\phantom{-\frac{K_{1}ATM{\dot{x}}_2(K_{M_1}+x_2)-K_{1}AT{M}^{\ast }{\dot{x}}_2}{{(K_{M_1}+x_2)}^2}-\frac{K_{cat}({\dot{x}}_5{x}_2+x_5{\dot{x}}_2)(a{K}_{13}+x_2)-K_{cat}{x}_5{x}_2{\dot{x}}_2}{{(a{K}_{13}+x_2)}^2}}{a}_{p53}{\dot{x}}_1-{\mu }_{p53}{\dot{x}}_2-v_{p53}{\dot{x}}_3\right.\\ & \left.-\frac{K_{1}ATM{\dot{x}}_2(K_{M_1}+x_2)-K_{1}AT{M}^{\ast }{\dot{x}}_2}{{(K_{M_1}+x_2)}^2}-\frac{K_{cat}({\dot{x}}_5{x}_2+x_5{\dot{x}}_2)(a{K}_{13}+x_2)-K_{cat}{x}_5{x}_2{\dot{x}}_2}{{(a{K}_{13}+x_2)}^2}\right]\\ & \frac{-K_{cat}^{\ast }a{K}_{13}{\dot{x}}_5{(a{K}_{13}+x_3)}^2-K_{cat}^{\ast }a{K}_{13}{x}_{5}2(a{K}_{13}+x_3){\dot{x}}_3}{{(a{K}_{13}+x_3)}^4}\cdot \left[\frac{K_{1}AT{M}^{\ast }{x}_2}{K_{M_1}+x_2}-v_{p53}{x}_3-\frac{K_c{{}_{at}}^{\ast }{x}_5{x}_3}{a{K}_{13}+x_3}\right]\\ & -\left[v_{p53}+\frac{K_{cat}^{\ast }a{K}_{13}{x}_5}{{(a{K}_{13}+x_3)}^2}\right]\cdot \left[\frac{K_{1}AT{M}^{\ast }{\dot{x}}_2(K_{M_1}+x_2)-K_{1}AT{M}^{\ast }{x}_2{\dot{x}}_2}{K_{M_1}+{x_2}^2}\right.\\ & \left.\phantom{\frac{K_{1}AT{M}^{\ast }{\dot{x}}_2(K_{M_1}+x_2)-K_{1}AT{M}^{\ast }{x}_2{\dot{x}}_2}{K_{M_1}+{x_2}^2}}-v_{p53}{\dot{x}}_3-\frac{K_{cat}^{\ast }({\dot{x}}_5{x}_3+x_5{\dot{x}}_3)(a{K}_{13}+x_3)-K_{cat}^{\ast }{x}_5{x}_3(a{K}_{13}+x_3)}{{(a{K}_{13}+x_3)}^2}\right]-\frac{K_{cat}^{\ast }{\dot{x}}_3(a{K}_{13}+x_3)-K_c{{}_{at}}^{\ast }{x}_3{\dot{x}}_3}{{(a{K}_{13}+x_3)}^2}\cdot \\ & \left[a_{MDM2}{x}_4-{\mu }_{MDM2}{x}_5-\frac{K_{2}AT{M}^{\ast }{x}_5}{K_{M_2}+x_5}-K_4{x}_{11}{x}_5-K_6{x}_6{x}_5\right]\\ & -\frac{K_{cat}^{\ast }{x}_3}{(a{K}_{13}+x_3)}\cdot [a_{MDM2}{\dot{x}}_4]-{\mu }_{MDM2}{\dot{x}}_5-\frac{K_{2}AT{M}^{\ast }{x}_5-K_{M_2}AT{M}^{\ast }{x}_5{\dot{x}}_5}{K_{M_2}+{x_5}^2}\\ & -K_4[({\dot{x}}_{11}{x}_5+x_{11}{\dot{x}}_5)-K_6{x}_6{\dot{x}}_5]-\frac{K_{cat}^{\ast }{x}_3}{(a{K}_{13}+x_3)}[-{\mu }_N{x}_6-K_6{x}_6{x}_5](-K_6{x}_5)\end{array}$$ (19)
• (ii) function $g(y,\dot{y})$

  $$g(y,\dot{y})=-\frac{K_{cat}^{\ast }{x}_3}{a{K}_{13}+x_3}(-K_6{x}_5)$$ (20)

By defining the new control input $v=f(y,\dot{y})+g(y,\dot{y})u$, the dynamics of the active P 53 protein can be written in the form

$$y_1^{(3)}=f(y,\dot{y})+g(y,\dot{y})u\Rightarrow y^{(3)}=v$$ (21)

The Brunovsky canonical form of the system is defined in (21 ) and in (28 ). The system $y_1^{(3)}=f(y,\dot{y},\dots )+g(y,\dot{y},\dots )u$ takes the form of a chain of integrators, which means that state variable y _i is related to state variable y _{i +1} through the relation ${\dot{y}}_i=y_{i+1}$, while for the last element of the state vector holds ${\dot{y}}_n=v$ where $v=f(y.\dot{y},\dots )+g(y.\dot{y},\dots )u$ is the aggregate control input and u is the control input that is finally exerted on the system.

A suitable feedback control law for the system of (21 ) is

$$v=y_d^{(3)}-k_1(\ddot{y}-{\ddot{y}}_d)-k_2(\dot{y}-{\dot{y}}_d)-k_3(y-y_d)$$ (22)

where the gains k ₁, k ₂ and k ₃ are chosen such that the characteristic polynomial of the closed‐loop system to be a Hurwitz‐stable one. The dynamics of the tracking error $e=y-y_d=P{53}^{\ast }-P{53}_d^{\ast }$ is given by

$$e^{(3)}+k_1\ddot{e}+k_2\dot{e}+k_{3}e=0$$ (23)

and finally results into lim _{t →∞} e (t ) = 0. The control input that actually applied to the p 53 protein–mdm 2 inhibitor system is given by

$$u=g(y,\dot{y}){\hspace{0.17em}}^{-1}[v-f(y,\dot{y})]$$ (24)

The primary output of the system is taken to be the active p 53 protein which is denoted as P 53*. Moreover, to arrive at the input–output linearised description of the system's dynamics the flat output of the system is defined and this consists of four variables, namely P 53*, the chemotherapy drug's concentration N in the cytoplasm, the concentration of the E 2F 1 and the concentration of the ARF protein. The differential flatness properties of the system show that all its state variables can be expressed as functions of the flat output and its derivatives. Therefore, out of the 11 state variables of the model there is need to measure at each sampling instant only four. This facilitates the real‐time implementation of the proposed non‐linear control scheme.

The p 53–mdm 2 system is a differentially flat one. To remain consistent with the condition that in multiple‐input multiple‐output differentially flat systems the number of the control inputs should be equal to the number of the flat outputs (which finally enables to arrive at a linearised and decoupled form), one can define virtual control inputs. This can be done by considering as control inputs specific parameters appearing in the 3rd, 6th, 9th and 11th row of the state‐space model of (3 ) (e.g. $K_{\mathrm{cat}}^{\ast }$, K ₅ and K ₄ ).

4 Disturbances compensation using the derivative‐free non‐linear Kalman filter

To apply the feedback control law of (24 ) and (22 ) to the system of the p 53 protein synthesis it is possible to use measurements of the concentration of the active P 53* protein at the cytoplasm, however, the derivatives of P 53* with respect to time are missing. Moreover, the dynamic model of (18 ) is subjected to modelling uncertainties and external disturbances which are denoted by the aggregate term $\tilde{d}$ in the following equation

$$y_1^{(3)}=f(y,\dot{y})+g(y,\dot{y})u+\tilde{d}$$ (25)

The dynamics of the additive disturbance term $\tilde{d}$ can be equivalently represented through the n th order derivative and the associated initial conditions. However, due to the fact that the disturbance term will be finally estimated with the use of the Kalman filter, and the outcome of the filtering procedure is not dependant on initial conditions, the requirement about knowing a‐priori initial conditions becomes obsolete.

Next, without loss of generality it is considered that the dynamics of the disturbance term is described as the derivative of order n = 3, and thus one has ${\tilde{d}}^{(3)}=f_d$ . Furthermore, the system's state vector is extended so as to include the disturbance term's dynamics. The extended state vector contains the following state variables

$$\begin{array}{rl}& z_1=y,z_2=\dot{y},z_3=\ddot{y}\\ & z_4=\tilde{d},z_5=\dot{\tilde{d}},z_6=\ddot{\tilde{d}}\end{array}$$ (26)

Then, the dynamics of the p 53 protein–mdm 2 inhibitor system, including the modelling uncertainty and external disturbances terms is written in the following canonical Brunovsky form:

$$\begin{array}{rl}\dot{z}& =Az+Bv\\ z_m& =Cz\end{array}$$ (27)

or equivalently

$$\begin{array}{r}\left(\begin{array}{c}{\dot{z}}_1\\ {\dot{z}}_2\\ {\dot{z}}_3\\ {\dot{z}}_4\\ {\dot{z}}_5\\ {\dot{z}}_6\end{array}\right)=\left(\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\\ z_5\\ z_6\end{array}\right)+\left(\begin{array}{cc}0& 0\\ 0& 0\\ 1& 0\\ 0& 0\\ 0& 0\\ 0& 1\end{array}\right)\left(\begin{array}{c}v\\ f_d\end{array}\right)\end{array}$$ (28)

with measurement equation given by

$$z_m=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& 0\end{array}\right)z$$ (29)

For the dynamics of the p 53 protein–mdm 2 inhibitor that is described by (28 ) and (29 ) it is possible to perform simultaneous estimation of the non‐measurable state variables as well as of the external disturbances using the Kalman filter recursion. The application of Kalman filtering on the linearised equivalent of the system and the use of an inverse transformation based on the expression of the initial state variables as functions of the flat output (see (6 )–(10 )) enables also to obtain estimates for the state variables of the initial non‐linear dynamical system of (4 ). This recursive estimation and inverse transformation procedure constitutes the derivative‐free non‐linear Kalman filter.

The disturbance estimator is

$$\begin{array}{rl}\dot{\hat{z}}& ={\mathit{A}}_o\hat{z}+{\mathit{B}}_{o}u+K(z_m-{\hat{z}}_m)\\ {\hat{z}}_m& ={\mathit{C}}_o\hat{z}\end{array}$$ (30)

where A _o = A, C _o = C and

$${\mathit{B}}_o^{\mathrm{T}}=\left(\begin{array}{cccccc}0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right)$$ (31)

In the design of the associated disturbances’ estimator one has the dynamics defined in (30 ), where K ∈ R ^6×1 is the state estimator's gain and matrices A _o, B _o and C _o have been defined in (28 ) and (29 ). The discrete‐time equivalents of matrices A _o , B _o and C _o are denoted as ${\tilde{A}}_d$, ${\tilde{B}}_d$ and ${\tilde{C}}_d$ respectively, and are computed with the use of common discretisation methods [40 –43 ]. Next, a derivative‐free non‐linear Kalman filter can be designed for the aforementioned representation of the system dynamics [39, 40 ]. The associated Kalman filter‐based disturbance estimator is given by the following recursion [41 –43 ].

Measurement update

$$\begin{array}{rl}K(k)& =P^{-}(k){\tilde{C}}_d^T[{\tilde{C}}_d\cdot P^{-}(k){\tilde{C}}_d^T+R]{\hspace{0.17em}}^{-1}\\ \hat{z}(k)& ={\hat{z}}^{-}(k)+K(k)[{\tilde{C}}_{d}z(k)-{\tilde{C}}_d{\hat{z}}^{-}(k)]\\ P(k)& =P^{-}(k)-K(k){\tilde{C}}_d{P}^{-}(k)\end{array}$$ (32)

Time update

$$\begin{array}{rl}{P}^{-}(k+1)& ={\tilde{A}}_d(k)P(k){\tilde{A}}_d^T(k)+Q(k)\\ {\hat{z}}^{-}(k+1)& ={\tilde{A}}_d(k)\hat{z}(k)+{\tilde{B}}_d(k)\tilde{v}(k)\end{array}$$ (33)

There are specific advantages in the proposed control method for the p 53 protein synthesis model, which facilitate also the implementation of the control scheme in practice. These are: (i ) the proposed control scheme turns the complicated non‐linear model of the p 53 protein synthesis procedure into an equivalent linear one for which the design of a feedback controller becomes easy. Otherwise, the solution of the control problem would be highly complicated and cumbersome in its computer‐based implementation, (ii ) the proposed state estimation scheme can compensate for modelling uncertainties and external perturbations that affect the p 53 protein synthesis dynamics. Otherwise, it would be impossible to implement a control scheme based on the assumption that the dynamics of the protein synthesis is precisely known and that the external disturbances that affect this model are also known or that they can be measured.

The design of the feedback controller for the p 53–mdm 2 protein synthesis model has taken into account the difficulty in measuring specific parameters in protein synthesis networks, for example, intracellular or in‐the‐nucleus concentrations of proteins. The parameters that are considered as measurable in the implementation of the non‐linear feedback controller are the elements of the vector that constitute the flat output Y = [x ₃, x ₆, x ₉, x ₁₁ ] = [P 53*, N, E 2F 1*, ARF ], where P 53* is the active form of the P 53 protein that is produced after phosphorylation of p 53, N is the concentration of the chemotherapeutic drug, E 2F 1* is the active form of the E 2F 1 protein and ARF is the concentration of the ARF protein after translation. The rest 7 elements of the state vector of the 11th order model of the p 53–mdm 2 protein synthesis need not be measured, but can be estimated through filtering. Once one obtains measurements of the elements of the flat output vector it is possible to express all other state variables in the state‐space model of the p 53–mdm 2 system as functions of the flat output elements and their derivatives. For the implementation of the feedback control scheme it is also necessary to estimate the derivatives of P 53* up to order two, and this is achieved through Kalman filtering. About parametric uncertainties affecting the dynamical model of the p 53–mdm 2 system these are compensated by redesigning the Kalman filter as a disturbance observer.

It is pointed out that the control input for the p 53 protein synthesis model is not applied at the single cell level but is applied to populations of cells. Therefore, one has finally a mean value (average) control that is anticipated to lead the mean value properties of the considered cells population to the desirable levels. Therefore, the control input is computed considering that the p 53 protein synthesis dynamics for the majority of the cells within the targeted cells population behaves according to a certain dynamical model and the control algorithm is designed to be robust to parametric variations of this model. It is noted that, apart from chemotherapy there are several examples about controlled drug infusion, for example, in the case of blood glucose levels control, hormones levels control, in the case of anaesthesia depth control and so on.

After writing the non‐linear protein synthesis model into the equivalent input–output linearised form, all perturbation and uncertainty terms can be represented as additive disturbance inputs. These inputs can be due to uncertainty and variations in the parameters of the model, external perturbation terms (e.g. other drugs that affect the protein synthesis loop dynamics), or finally due to measurement noise. Therefore, the inclusion of such disturbance terms in the protein's linearised model is quite generic and covers a wide range of uncertainty cases.

The stability of the considered control loop is not dependent on prior knowledge of the n th order derivative of the disturbance's dynamics. Actually, there is no need to know the analytical function of the n th order derivative. The redesign of the Kalman filter as a disturbance observer assures that finally the dynamics of the disturbance term will be reconstructed without the need to know beforehand neither initial values of the disturbance nor the analytical function of its derivatives. The robustness features of the disturbance estimator‐based feedback control scheme are similar to those of linear quadratic Gaussian (LQG) control. The extended state‐space description of the system stands for a linear model with multiple poles at the origin of the complex plane (poles at 0). Such a system can be stabilised by pole placement methods. The stability margins of the controlled system dynamics can also be confirmed through Bode diagrams or through the Nyquist diagram.

5 Simulation tests

Through simulation tests it has been shown that the control loop for the p 53 protein–mdm 2 inhibitor system succeeds convergence of the P 53* protein's concentration to the designated reference levels. The concentration of the active P 53* protein in the cytoplasm can be controlled by varying the infusion rate of the chemotherapy drug as designated by the control law given in (22 ) and in (24 ). The protein concentration state variables of the p 53 model were measured in micro‐Mol (μM). Indicative values of the parameters of the p 53–mdm 2 protein synthesis model are given in Table 2.

Table 2.

Parameters of the p 53–mdm 2 protein synthesis model

Notation and value	Notation and value	Notation and value	Notation and value
λ p 53 = 2.1 μM h−1	μ p 53 = 0.2 h−1	a p 53 = 5.3 h−1	v p 53 = 0.2 h−1
K 1 = 2.1 h−1	K 2 = 0.2 h−1	K 3 = 2.3 h−1	K 4 = 0.2 μM−1 h−1
K 5 = 0.1 μM−1 h−1	K 6 = 0.001 μM−1 h−1	K 13 = 3.2 μM	ATM s = 0.005 μM
a = 0.001	$K_{M_1}=0.1\mu \mathrm{M}$	$K_{M_2}=0.2\mu \mathrm{M}$	a E 2F 1 = 0.5 h−1
K cat = 0.31 h−1	$K_{\mathrm{cat}}^s=2.10{\mathrm{h}}^{-1}$	λ mdmd 2 = 0.4 μM h−1	μ mdm 2 = 0.6 h−1
ϕ mdm 2 = 0.7 μ Mh−1	a MDM 2 = 0.8 h−1	μ MDM 2 = 0.9 h−1	μ N = 0.05 h−1
λ e 2f 1 = 0.3 μ Mh−1	μ e 2f 1 = 0.4 h−1	a E 2F 1 = 0.5 h−1	μ E 2F 1 = 0.6 h−1
v E 2F 1 = 0.7 h−1	λ arf = 0.4 μM h−1	μ arf = 0.5 h−1	ϕ arf = 10.6 μM h−1
a ARF = 0.7 h−1	μ ARF = 0.8 h−1

It is noted that the performance of the control scheme does not depend on the nominal values of parameters of the p 53 protein synthesis model. This is because the control input is computed according to an inversion of the aforementioned model. Thus, the dynamics which are due to the parameters’ values are internally annihilated by the control input. Moreover, the control law is computed to be robust to modelling uncertainties and external perturbations. Therefore, even if the model's parameters are not precisely known the control input succeeds convergence of the model's state variables to the desirable setpoints while it also assures that the transients in the variation of the state variables will remain satisfactory (avoidance of excessive overshoots, avoidance of oscillations and fast convergence to the reference setpoints).

The feedback gains k _i , i = 1, 2, 3 are chosen such that the roots of the characteristic polynomial associated with the tracking error dynamics of (23 ) are all found in the left complex semi‐plane. Indicative values are k ₁ = 6, k ₂ = 11 and k ₃ = 6, which results in closed‐loop system poles p ₁ = −1, p ₂ = −2 and p ₃ = −3. Tuning of the gains k _i , i = 1, 2, 3 determines the characteristics of the transient response of the control loop. The estimation error covariance matrix used by the Kalman filter algorithm was initialised to the value P ⁻ (0) = 10e⁻³ I _6×6, while the process and measurement noise covariance matrices were Q = 10e⁻³ I _6×6 and R = 10e⁻² I _1×1. The infusion of the chemotherapy drug was considered to be completed in 20 h.

The performance of the control loop for the p 53 protein synthesis model is tested in the case of tracking of various setpoints. The associated results for two test cases are depicted in Figs. 2 –4 and in Figs. 7 –9, respectively. In the first test case, the P 53* protein concentration exhibits a piecewise constant variation while in the second test case the P 53* protein concentration exhibits a sinusoidal variation. It can be observed that the proposed non‐linear feedback control scheme enables accurate tracking of the concentration of the P 53* protein to the desirable concentration levels.

Fig 2.

p53 protein model with disturbances for test case 1

a Non‐linear feedback control of the P 53* protein concentration (blue line) and convergence to the associated setpoints (red lines)

b Infusion rate as control input

Fig 4.

p53 protein model with disturbances for test case 1

a Variation of the e 2f 1 mRNA concentration, E 2F 1 concentration in the cytoplasm and active E 2F 1* concentration

b Variation of the arf mRNA concentration, ARF concentration in the cytoplasm

Fig 7.

p53 protein model with disturbances for test case 2

a Non‐linear feedback control of the P 53* protein concentration (dark black line) and convergence to the associated setpoints (light black lines)

b Infusion rate as control input

Fig 9.

p53 protein model with disturbances for test case 2

a Variation of the e 2f 1 mRNA concentration, E 2F 1 concentration in the cytoplasm and active E 2F 1* concentration

b Variation of the arf mRNA concentration, ARF concentration in the cytoplasm

Fig 5.

p53 protein model with disturbances for test case 1

a Convergence of the estimates of P 53* concentration and of its derivatives (grey lines) to the associated real parameter values (black lines)

b Estimation of disturbance terms (grey lines) that affect the model and convergence to the associated real parameter values (black lines)

In the simulation experiments, it is considered that there are model uncertainties and external disturbances that affect the p 53 protein–mdm 2 inhibitor system. The use of the derivative‐free non‐linear Kalman filter enables to perform simultaneous estimation of the non‐measurable elements of the system's state vector as well as estimation of the disturbance terms. By identifying the perturbation parameters their compensation becomes possible. It suffices to include an additional control input that compensates for the disturbances effects. Thus, the new control input becomes $v_1=v-{\hat{z}}_4$, where ${\hat{z}}_4$ is the fourth element of the extended state vector and is an estimate of disturbance term $z_4=\tilde{d}$ . Despite the existence of external perturbations the proposed non‐linear feedback control scheme enabled accurate tracking of the concentration of the P 53* protein to the desirable concentration levels. Finally, the convergence of the estimation error for the concentration of the P 53* protein and its derivatives is depicted in Figs. 6 a and 11 a while the convergence of the estimation error for the disturbance input $\tilde{d}$ and its derivatives is depicted in Fig.s 6 b and 11 b, respectively.

Fig 6.

Dynamics of the estimation error for test case 1

a Concentration of the P 53* protein and its derivatives

b Unknown disturbance input $\tilde{d}$ and its derivatives

Fig 11.

Dynamics of the estimation error for test case 2

a Concentration of the P 53* protein and its derivatives

b Unknown disturbance input $\tilde{d}$ and its derivatives

Fig 8.

p53 protein model with disturbances for test case 2

a Variation of the p 53 mRNA concentration, P 53 concentration in the cytoplasm and active P 53* concentration

b Variation of the mdm 2 mRNA concentration, MDM 2 concentration in the cytoplasm and active MDM 2* concentration

Fig 10.

p53 protein model with disturbances for test case 2

a Convergence of the estimates of P 53* concentration and of its derivatives (grey lines) to the associated real parameter values (dark black lines)

b Estimation of disturbance terms (grey lines) that affect the model and convergence to the associated real parameter values (dark black lines)

The p 53 protein–mdm 2 inhibitor system exhibits the so‐called zero dynamics [44 ]. This means that the model contains internal state variables which do not appear as outputs in the linearised equivalent of the p 53–mdm 2 model given in (18 ). From the simulation experiments, it has been confirmed that even these internal state variables remain bounded. Since the internal state variables describe also proteins concentration they are expected to vary within specific intervals. The boundedness of the internal state variables implies also boundedness of functions $f(y,\dot{y})$ and $g(y,\dot{y})$ which are given in (19 ) and (20 ), respectively, and boundedness of the control input, thus finally enabling state variable P 53* (i.e. the concentration of the active p 53 protein after phosphorylation of p 53) to converge to the desirable setpoints. Some recent results on asymptotic stability features for state estimation‐based control of non‐linear systems exhibiting zero dynamics and described in canonical forms have been given in [44 –46 ].

As explained, uncertainty about the parameters of the p 53–mdm 2 system and uncertainty about the external perturbation terms that affect this model are described through the cumulative disturbance input $\tilde{d}$ . By estimating the disturbance term with the use of the derivative‐free non‐linear Kalman filter, under the assumption of a Gaussian measurement noise, it is possible to compensate for its effects by introducing an additional element to the feedback control input. Another problem that arises is about how one obtains accurate estimates of the parameters of the model of the p 53–mdm 2 system. This can be also solved with the use of non‐linear Kalman filtering, or equivalent non‐linear least squares methods [30 –38 ]. In place of the state vector what is estimated now is the parameters vector which is considered to remain time invariant subject only to noise. The measurement equation to be used in the Kalman filter‐based scheme is the same as in the case of estimation of state vector of the p 53–mdm 2 system and consists of measurements of the flat output (i.e. y = [ x ₃, x ₆, x ₉, x ₁₁ ] = [P 53*, N, E 2F 1*, ARF ]).

About the implementation of the proposed control and estimation scheme of the p 53–mdm 2 loop the following hold: (i) the measured variables are the elements of the flat output vector x ₃, x ₆, x ₉, x ₁₁, (ii) the rest of the state variables of the model are expressed as functions of the elements of the flat output vector and consequently can be computed by using the estimates of x ₃, x ₆, x ₉, x ₁₁ and (iii) by applying differential flatness theory it was shown that the system's dynamics can be decomposed into linear and decoupled subsystems. Such a subsystem is the one having as output y ₁ = P 53*. Thus, one finally arrives at the equivalent model of (21 ) which is also a canonical form for the system. For the model of (21 ) state estimation can be performed using the Kalman filter recursion of (32 ) and (33 ).

The article is based on differential flatness theory. This is a global linearisation method (valid throughout the entire state space) which results in an equivalent input–output linearised description of the p 53–mdm 2 model. For the linearised equivalent model of this protein synthesis model one can solve the control problem by applying state feedback methods and thus can make the p 53 protein levels track specific reference values. Besides one can solve the state estimation problem by applying the derivative‐free non‐linear Kalman filter and thus can implement feedback control without the need to measure a large number of the system's state variables. Moreover, by designing the Kalman filter as a disturbance observer, estimation becomes possible for model uncertainty terms and external perturbations that affect the p 53–mdm 2 model.

Comparing with the extended Kalman filter, it has been proven that the derivative‐free non‐linear Kalman filter is a much more efficient non‐linear estimation method. In terms of accuracy the derivative‐free non‐linear Kalman filter gives more precise estimates comparing to the extended Kalman filter. This is because it uses the exactly linearised model of the system instead of the approximately linearised model that is used by the extended Kalman filter. Conditionally, the derivative‐free non‐linear Kalman filter is an optimal non‐linear estimator, and retains the optimality properties (minimal trace of the estimation error covariance matrix) that hold for the linear Kalman filter. On the other hand, the extended Kalman filter is neither an optimal estimator nor its convergence properties can be proven (this is because it is based on an approximate model of the system that is obtained from truncation of higher order terms in the associated Taylor series expansion). In terms of computation speed, the derivative‐free non‐linear Kalman filter performs better (is faster) than the extended Kalman filter (it is about 30% faster) because it does not need to compute online the Jacobian matrices used by the extended Kalman filter. The significantly improved performance of the derivative‐free non‐linear Kalman filter, comparing with the extended Kalman filter or other non‐linear filtering approaches, has been extensively analysed in [47 ].

Rapid immunoassay techniques have already been developed, enabling the fast measurement of the concentration of the P 53* concentration in patients’ sera. Consequently, sufficient sampling of the p 53–mdm 2 system is achieved and the implementation of a feedback control scheme for this system in real‐time becomes possible. As analysed in [48 ], in rapid immunomagnetic‐electrochemiluminescent (ECL) methods, taking a sample from the patient's serum can be completed in seconds. Moreover, the rest of the processing which is based on amplification and counting of photon emissions due to the ECL reaction of the antibodies bound on the P 53* protein can be completed in minutes. Consequently, the concentration of the P 53* protein can be completed in a much shorter time interval than common immunoassay methods. Electrochemical immunosensors combine portable electrochemical measurement systems with specific and sensitive immunoassay procedures and thus represent a promising bioanalytical approach in clinical analyses. It has intrinsic advantages of simple instrumentation, easy signal quantification, low cost of the entire assay and rapid completion of the biomarkers’ measuring procedure.

The evolution of the controlled system in time takes place in N iterations, where the sampling interval between successive iterations is chosen according to the so‐called Nyquist theorem and in a manner that permits to collect sufficient information about the temporal variation of the system's output (avoidance of undersampling). Since the dynamics of the p 53–mdm 2 system is a slow one (in the scale of hours or days), its sampling can be performed using a larger sampling period (this will also enable to exploit existing fast immunoassay methods). If one opts to use a larger sampling period than the one presented in the existing simulation results, then to complete the infusion of the chemotherapy drug in a prescribed time he has to implement the control loop using a smaller number of iterations (measurement samples). Thus, if the sampling period is increased by multiplying it by a factor of a > 1 (while always assuring that the Nyquist sampling theorem holds), then it suffices to implement the control loop in a number of iterations that should be equal to N /a. For instance, if 100 sampling intervals are needed to achieve convergence of the serum proteins to the desirable levels, and considering a sampling interval equal to 12 min, then one can achieve the targeted therapeutic results from the infusion again within 20 h. This confirms that the implementation of the proposed p 53–mdm 2 control loop, at a significantly higher sampling period, is still feasible.

In the simulation results presented in Figs. 3 –11 it has been assumed that the infusion procedure lasted for 20 h and that in this time interval 2000 samplings were performed at a sampling period equal to 36 s. However, from the above noted diagrams, it becomes clear that the settling time for the infusion, that is, the time needed for the concentration of the P 53* protein to reach its final value, is about 1–1.5 h (suitable tuning of the controller's gains could enable very fast convergence to the desirable setpoints). Equivalently, this means that the desirable effects from the infusion are reached within 100 sampling periods. This allows to increase the sampling period by a factor of 20 (provided always that for the new sampling period Nyquist's theorem will still hold and that the system will not be undersampled). Therefore, one can achieve the desirable results from the treatment if a sampling period of 720 s or 12 min is used. The latter sampling period indicates that the controlled infusion of the chemotherapy medication can be performed at a sampling period which is feasible when using fast immunoassay methods.

Fig 3.

p53 protein model with disturbances for test case 1

a Variation of the p 53 mRNA concentration, P 53 concentration in the cytoplasm and active P 53* concentration

b Variation of the mdm 2 mRNA concentration, MDM 2 concentration in the cytoplasm and active MDM 2* concentration

6 Conclusions

A non‐linear feedback control method has been proposed for the p 53 protein–mdm 2 inhibitor system. The control scheme is based on differential flatness theory and the derivative‐free non‐linear Kalman filter. The p 53 protein plays a key role in the fight against cancer cells. Increased levels of the concentration of the p 53 protein result in cell‐cycle arrest and apoptosis. The concentration of the p 53 protein is affected by another protein known as mdm 2 which actually downregulates P 53 within a negative feedback loop.

The first stage for the design of the control scheme was the transformation of the initial description of the system dynamics from a set of complex coupled non‐linear differential equations into a SISO model of the canonical Brunovsky form. The transformation was based on differential flatness theory. The latter model connected the infusion rate of the chemotherapy drug (control input) to the concentration of the active P 53* protein (system's output). For the transformed model the design of state feedback control was possible. Under complete knowledge of the initial non‐linear system's dynamics and assuming no external disturbances, this feedback controller enabled convergence of the concentration of the active P 53* protein to the desirable setpoints.

Moreover, to make the control scheme robust to modelling uncertainty and external disturbances and to cope with the non‐measurable elements of the state vector (derivatives of the P 53* protein concentration), a disturbance estimator was designed with the use of the derivative‐free non‐linear Kalman filter. The state‐space model of the system was extended by considering as additional state variables the disturbance and modelling uncertainty terms. For the extended state‐space description of the system, state estimation was performed using the standard Kalman filter recursion while estimates for the initial state variables of the non‐linear model were obtained using an inverse transformation (diffeomorphism) that was provided by differential flatness theory. By identifying the perturbation variables, their compensation was possible with the inclusion on an additional term in the control input.

The efficiency of the proposed control scheme was evaluated through simulation experiments. It was shown that the derivative‐free non‐linear Kalman filter provided accurate estimates of both the non‐measurable elements of the state vector and of the disturbance terms that affected the p 53 protein–mdm 2 inhibitor system. Moreover, the non‐linear control scheme achieved the convergence of the active P 53* protein's concentration to the desirable setpoints.