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The problems of modeling and intervention of biological phenomena have captured the interest of many researchers in the past few decades. The aim of the therapeutic intervention strategies is to move an undesirable state of a diseased network towards a more desirable one. Such an objective can be achieved by the application of drugs to act on some genes/metabolites that experience the undesirable behavior. For the purpose of design and analysis of intervention strategies, mathematical models that can capture the complex dynamics of the biological systems are needed. S-systems, which offer a good compromise between accuracy and mathematical flexibility, are a promising framework for modeling the dynamical behavior of biological phenomena. Due to the complex nonlinear dynamics of the biological phenomena represented by S-systems, nonlinear intervention schemes are needed to cope with the complexity of the nonlinear S-system models. Here, we present an intervention technique based on feedback linearization for biological phenomena modeled by S-systems. This technique is based on perfect knowledge of the S-system model. The proposed intervention technique is applied to the glycolytic-glycogenolytic pathway, and simulation results presented demonstrate the effectiveness of the proposed technique.

Biological systems are complex processes with nonlinear dynamics. S-systems are proposed in [

Recently, the authors in [

A basic problem in control theory is how to use feedback in order to modify the original internal dynamics of nonlinear systems to achieve some prescribed behavior [

Hence, in this paper we consider the problem designing a nonlinear intervention strategy based on feedback linearization for biological phenomena modeled by S-systems. In this proposed algorithm, the control variables are designed such that an integral action is added to the system. The main advantage of the integral action is in improving the steady state performance of the closed-loop system. As a case study, the proposed intervention strategy is applied to a glycolytic-glycogenolytic pathway model. The glycolytic-glycogenolytic pathway model is selected as it plays an important role in cellular energy generation when the level of glucose in the blood is low (fasting state) and glycogen has to be broken down to provide the substrate to run glycolysis. By controlling the glycogenolytic reaction, one can exert control over whether glycolysis will run or not under low-glucose conditions.

This paper is organized as follows. In Section

Consider the following S-system model [

Figure

S-system with integral control architecture.

Suppose that the outputs of the S-system (

Here, we show how feedback linearization can be utilized to design a nonlinear intervention strategy to control biological phenomena modeled by S-systems. Feedback linearization can be used to obtain a linear relationship between the output vector

Let the vector function

Let

Thus, the Lie derivative

Let

Repeated Lie brackets can then be defined recursively by

Consider the S-system model (

The system (

If a system has well-defined vector relative degree, then (

Diagram block of the linearizable system.

Feedback linearization transforms the system into a linear system where linear control approaches can be applied. Here,

In the case the system has vector relative degree, where

Suppose that the matrix

The proof of this proposition can be found in [

The new control vector

Closed loop of the linearizable system.

In this section, we demonstrate the efficacy of the feedback linearizable intervention approach described in this paper by applying it to a well-studied biological pathway model representing the glycolytic-glycogenolytic pathway shown in Figure

Glycolytic-glycogenolytic pathway [

In this case,

For this model, the metabolites

Hence, the overall system can be expressed in the form of (

Based on the S-system model describing the glycolytic-glycogenolytic pathway, it can be verified that the outputs need to be differentiated twice with respect to time so that the input variables (

The matrix form of the system of differential equations presented in (

Hence, the control laws based on (

Using (

Figures

Closed-loop outputs for constant reference signals.

Control signals for constant reference signals.

Output response for closed-loop tracking.

Control signals for closed-loop tracking.

To study the robustness properties of the feedback linearizable controller, similar simulation studies have been conducted when the parameters

In this paper, feedback linearizable control has been applied for intervention of biological phenomena modeled in the S-system framework. As a case study, the glycogenolytic-glycolytic pathway model has been used to demonstrate the efficacy of feedback linearization in controlling biological phenomena modeled by S-system. One main drawback of this approach is that it assumes full knowledge of the biological system model. Usually, the S-system model does not perfectly represent the actual dynamics of the biological phenomena. Hence, one future research direction is to develop an adaptive intervention strategy that is capable of controlling the biological system even in the presence of model uncertainties. Another future research direction is to develop intervention techniques that take into account additional constraints due to the nature of the drug injection process. Definitely, incorporating such knowledge from medical practitioners would require imposing constraints on the magnitude, duration, and possibly the rate of change of the injected drug into the design of intervention technique.

This work was made possible by NPRP Grant NPRP08-148-3-051 from the Qatar National Research Fund (a Member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.

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