Change Detection in the Dynamics of an Intracellular Protein Synthesis Model Using Nonlinear Kalman Filtering

A method for early diagnosis of parametric changes in intracellular protein synthesis models (e.g. the p53 protein-mdm2 inhibitor model) is developed with the use of a nonlinear Kalman Filtering approach (Derivative-free nonlinear Kalman Filter) and of statistical change detection methods. By applying a diffeomorphism that is based on differential flatness theory a linearized form of such models can be obtained. For the linearized equivalent models, state estimation can be performed using the Kalman Filter recursion. By comparing the output of the Kalman Filter (which is assumed to correspond to the undistorted protein synthesis model) with measurements obtained from the monitored protein synthesis system, a sequence of differences (residuals) is obtained. The statistical processing of the residuals with the use of χ 2 change detection tests, can provide indication within specific confidence intervals about parametric changes in the considered biological system and consequently indications about the appearance of specific diseases (e.g. malignancies).


INTRODUCTION
The paper studies the problem of parametric change detection in intracellular protein synthesis models, such as the model describing the p53 protein -mdm2 inhibitor dynamics (Jahoor Alam et al., 2012), (Elias et el., 2013), (Liu et al., 2011),(Wagner et al., 2005).The P 53 protein is of major importance for preventing the development of tumors since it enhances cell-cycle arrest and apoptosis (Lillaci et al., 2006), (Peirce and Findley, 2010), (Qi et al., 2007), (Xin and Jia, 2010).By applying nonlinear estimation (identification) methods it has become possible to obtain numerical values for the parameters of the p53 protein -mdm2 inhibitor system (Meskin et al., 2013), (Quach et al., 2007), (Wang et al., 2009).However, the parameters of such a model are subjected to uncertainties and parametric changes.Actually, the deviation of the protein synthesis model parameters from their nominal values is associated with deregulation of the cells population and is likely to provoke the appearance of malignancies.
To detect pathological symptoms in the p53-mdm2 protein loop, a model of the system's dynamics is generated with the use of nonlinear Kalman Filtering and this is parameterized with the nominal values which are associated with the normal protein synthesis conditions.The considered filtering approach is the Derivative-free nonlinear Kalman Filter (Rigatos, 2013).This consists of the application of the standard Kalman Filter recursion on the linearized equivalent of the protein synthesis system which has been obtained after applying differential flatness theory (Chien et al., 2008), (Fliess and Mounier, 1999).Moreover, using an inverse transformation based again on differential flatness theory one can obtain estimates of the state variable of the initial nonlinear model.
For detecting abnormalities in the p53-mdm2 protein loop, two sequences of data are generated.The first sequence consists of real measurements of the p53 protein concentration with are obtained at specific sampling instances.The second sequence is the Kalman Filter's output, again sampled at the same time instances.By comparing the two signals, a residuals (estimation error) sequence is generated.The processing of the residuals with the use of statistical decision making criteria provides an indication about the existence of parametric changes (damages) in the p53-mdm2 protein synthesis model, which otherwise could not have been detected (Basseville and Nikiforov, 1993), (Rigatos and Zhang, 2009).Thus, by applying fault detection tests based on the χ 2 distribution it can be concluded if the p53 protein-mdm2 inhibitor system remains healthy and if the nominal parameter values for its model still hold.Otherwise, abnormality can be detected.

DYNAMIC MODEL OF THE P53 PROTEIN -MDM2 INHIBITOR SYSTEM
The concentration of the P53 protein is mainly controlled by the levels of the mdm2 protein within a negative feed-

INTRODUCTION
The paper studies the problem of parametric change detection in intracellular protein synthesis models, such as the model describing the p53 protein -mdm2 inhibitor dynamics (Jahoor Alam et al., 2012),(Elias et el., 2013), (Liu et al., 2011),(Wagner et al., 2005).The P 53 protein is of major importance for preventing the development of tumors since it enhances cell-cycle arrest and apoptosis (Lillaci et al., 2006),(Peirce and Findley, 2010), (Qi et al., 2007), (Xin and Jia, 2010).By applying nonlinear estimation (identification) methods it has become possible to obtain numerical values for the parameters of the p53 protein -mdm2 inhibitor system (Meskin et al., 2013), (Quach et al., 2007),(Wang et al., 2009).However, the parameters of such a model are subjected to uncertainties and parametric changes.Actually, the deviation of the protein synthesis model parameters from their nominal values is associated with deregulation of the cells population and is likely to provoke the appearance of malignancies.
To detect pathological symptoms in the p53-mdm2 protein loop, a model of the system's dynamics is generated with the use of nonlinear Kalman Filtering and this is parameterized with the nominal values which are associated with the normal protein synthesis conditions.The considered filtering approach is the Derivative-free nonlinear Kalman Filter (Rigatos, 2013).This consists of the application of the standard Kalman Filter recursion on the linearized equivalent of the protein synthesis system which has been obtained after applying differential flatness theory (Chien et al., 2008), (Fliess and Mounier, 1999).Moreover, using an inverse transformation based again on differential flatness theory one can obtain estimates of the state variable of the initial nonlinear model.
For detecting abnormalities in the p53-mdm2 protein loop, two sequences of data are generated.The first sequence consists of real measurements of the p53 protein concentration with are obtained at specific sampling instances.
The second sequence is the Kalman Filter's output, again sampled at the same time instances.By comparing the two signals, a residuals (estimation error) sequence is generated.The processing of the residuals with the use of statistical decision making criteria provides an indication about the existence of parametric changes (damages) in the p53-mdm2 protein synthesis model, which otherwise could not have been detected (Basseville and Nikiforov, 1993), (Rigatos and Zhang, 2009).
Thus, by applying fault detection tests based on the χ 2 distribution it can be concluded if the p53 protein-mdm2 inhibitor system remains healthy and if the nominal parameter values for its model still hold.Otherwise, abnormality can be detected.

DYNAMIC MODEL OF THE P53 PROTEIN -MDM2 INHIBITOR SYSTEM
The concentration of the P53 protein is mainly controlled by the levels of the mdm2 protein within a negative feed-

INTRODUCTION
The paper studies the problem of parametric change detection in intracellular protein synthesis models, such as the model describing the p53 protein -mdm2 inhibitor dynamics (Jahoor Alam et al., 2012) , 2009).However, the parameters of such a model are subjected to uncertainties and parametric changes.Actually, the deviation of the protein synthesis model parameters from their nominal values is associated with deregulation of the cells population and is likely to provoke the appearance of malignancies.
To detect pathological symptoms in the p53-mdm2 protein loop, a model of the system's dynamics is generated with the use of nonlinear Kalman Filtering and this is parameterized with the nominal values which are associated with the normal protein synthesis conditions.The considered filtering approach is the Derivative-free nonlinear Kalman Filter (Rigatos, 2013).This consists of the application of the standard Kalman Filter recursion on the linearized equivalent of the protein synthesis system which has been obtained after applying differential flatness theory (Chien et al., 2008), (Fliess and Mounier, 1999).Moreover, using an inverse transformation based again on differential flatness theory one can obtain estimates of the state variable of the initial nonlinear model.
For detecting abnormalities in the p53-mdm2 protein loop, two sequences of data are generated.The first sequence consists of real measurements of the p53 protein concentration with are obtained at specific sampling instances.The second sequence is the Kalman Filter's output, again sampled at the same time instances.By comparing the two signals, a residuals (estimation error) sequence is generated.The processing of the residuals with the use of statistical decision making criteria provides an indication about the existence of parametric changes (damages) in the p53-mdm2 protein synthesis model, which otherwise could not have been detected (Basseville and Nikiforov, 1993), (Rigatos and Zhang, 2009).
Thus, by applying fault detection tests based on the χ 2 distribution it can be concluded if the p53 protein-mdm2 inhibitor system remains healthy and if the nominal parameter values for its model still hold.Otherwise, abnormality can be detected.

DYNAMIC MODEL OF THE P53 PROTEIN -MDM2 INHIBITOR SYSTEM
The concentration of the P53 protein is mainly controlled by the levels of the mdm2 protein within a negative feed-

INTRODUCTION
The paper studies the problem of parametric change detection in intracellular protein synthesis models, such as the model describing the p53 protein -mdm2 inhibitor dynamics (Jahoor Alam et al., 2012),(Elias et el., 2013), (Liu et al., 2011),(Wagner et al., 2005).The P 53 protein is of major importance for preventing the development of tumors since it enhances cell-cycle arrest and apoptosis (Lillaci et al., 2006),(Peirce and Findley, 2010),(Qi et al., 2007), (Xin and Jia, 2010).By applying nonlinear estimation (identification) methods it has become possible to obtain numerical values for the parameters of the p53 protein -mdm2 inhibitor system (Meskin et al., 2013), (Quach et al., 2007),(Wang et al., 2009).However, the parameters of such a model are subjected to uncertainties and parametric changes.Actually, the deviation of the protein synthesis model parameters from their nominal values is associated with deregulation of the cells population and is likely to provoke the appearance of malignancies.
To detect pathological symptoms in the p53-mdm2 protein loop, a model of the system's dynamics is generated with the use of nonlinear Kalman Filtering and this is parameterized with the nominal values which are associated with the normal protein synthesis conditions.The considered filtering approach is the Derivative-free nonlinear Kalman Filter (Rigatos, 2013).This consists of the application of the standard Kalman Filter recursion on the linearized equivalent of the protein synthesis system which has been obtained after applying differential flatness theory (Chien et al., 2008), (Fliess and Mounier, 1999).Moreover, using an inverse transformation based again on differential flatness theory one can obtain estimates of the state variable of the initial nonlinear model.
For detecting abnormalities in the p53-mdm2 protein loop, two sequences of data are generated.The first sequence consists of real measurements of the p53 protein concentration with are obtained at specific sampling instances.The second sequence is the Kalman Filter's output, again sampled at the same time instances.By comparing the two signals, a residuals (estimation error) sequence is generated.The processing of the residuals with the use of statistical decision making criteria provides an indication about the existence of parametric changes (damages) in the p53-mdm2 protein synthesis model, which otherwise could not have been detected (Basseville and Nikiforov, 1993), (Rigatos and Zhang, 2009).
Thus, by applying fault detection tests based on the χ 2 distribution it can be concluded if the p53 protein-mdm2 inhibitor system remains healthy and if the nominal parameter values for its model still hold.Otherwise, abnormality can be detected.

DYNAMIC MODEL OF THE P53 PROTEIN -MDM2 INHIBITOR SYSTEM
The concentration of the P53 protein is mainly controlled by the levels of the mdm2 protein within a negative feed-