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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 5, MAY 2011

Intervention in Biological Phenomena Modeled by S-Systems Nader Meskin*, Member, IEEE, Hazem N. Nounou, Senior Member, IEEE, Mohamed Nounou, Senior Member, IEEE, Aniruddha Datta, Fellow, IEEE, and Edward R. Dougherty, Senior Member, IEEE

Abstract—Recent years have witnessed extensive research activity in modeling biological phenomena as well as in developing intervention strategies for such phenomena. S-systems, which offer a good compromise between accuracy and mathematical flexibility, are a promising framework for modeling the dynamical behavior of biological phenomena. In this paper, two different intervention strategies, namely direct and indirect, are proposed for the S-system model. In the indirect approach, the prespecified desired values for the target variables are used to compute the reference values for the control inputs, and two control algorithms, namely simple sampled-data control and model predictive control (MPC), are developed for transferring the control variables from their initial values to the computed reference ones. In the direct approach, a MPC algorithm is developed that directly guides the target variables to their desired values. The proposed intervention strategies are applied to the glycolytic–glycogenolytic pathway and the simulation results presented demonstrate the effectiveness of the proposed schemes. Index Terms—Genetic regulatory networks, intervention, model predictive control, S-systems.

I. INTRODUCTION -SYSTEMS are proposed in [1] and [2] as a canonical nonlinear model to capture the dynamical behavior of a large class of biological phenomena [3], [4]. They are characterized by a good tradeoff between accuracy and mathematical flexibility [5]. In this modeling approach, nonlinear systems are approximated by products of power-law functions which are derived

S

Manuscript received August 19, 2010; revised November 14, 2010; accepted December 5, 2010. Date of publication December 17, 2010; date of current version April 20, 2011. This paper was supported by the Qatar National Research Fund (a member of the Qatar Foundation) under Grant NPRP08-148-3-051. Asterisk indicates the corresponding author. *N. Meskin is with the Department of Electrical Engineering, Qatar University, Doha 2713, Qatar (e-mail: [email protected]). H. N. Nounou is with the Electrical and Computer Engineering Program, Texas A&M University at Qatar, Doha 23874, Qatar (e-mail: [email protected]). M. Nounou is with the Chemical Engineering Program, Texas A&M University at Qatar, Doha 23874, Qatar (e-mail: [email protected]). A. Datta is with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail: [email protected]). E. R. Dougherty is with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843 USA, and also with the Translational Genomics Research Institute, Phoenix, AZ 85004 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2010.2099658

from multivariate linearization in logarithmic coordinates. It has been shown that this type of representation is a valid description of biological processes in a variety of settings. S-systems have been proposed in the literature to mathematically capture the behavior of genetic regulatory networks [6]–[10]. Moreover, the problem of estimating the S-system model parameters, the rate coefficients, and the kinetic orders has been addressed by several researchers [10]–[13]. In [14], the authors studied the controllability of S-systems based on the feedback linearization approach. For a more elaborate discussion of S-systems, the interested reader is referred to [2] and [15]. In this paper, we develop intervention strategies applicable to biological systems and phenomena modeled by S-systems. These include genetic regulatory networks and metabolic pathways. Two different intervention strategies, namely indirect and direct, are proposed for biological phenomena modeled by Ssystems. The goal of these intervention strategies is to transfer the target variables from an initial steady-state level to desired final ones by manipulating the control variables. Toward this goal, the S-system model of biological phenomena is first augmented by integrators. This provides a mechanism for manipulating the control (independent) variables and guaranteeing zero steadystate error in accordance with the internal model principle of control theory [16]. In an indirect approach, the reference levels for the control variables are obtained based on the steady-state analysis of the S-system at the desired prespecified levels for the target variables. Then, two control algorithms, namely simple sampled-data control and model predictive control (MPC), are proposed for transferring the control variables from their initial values to their reference ones. Specifically, in the indirect approach, the reference values for the control variables are first calculated from the prespecified desired values for the target variables, and the control algorithms only take care of guiding the control variables to their calculated reference values. The S-system is assumed to be stable in the desired steady state, and based on this assumption, the target variables also converge to their prespecified desired values once the control variables reach their reference values. However, for the cases where the number of control variables is more than the number of target variables, the steady-state analysis will lead to an infinite number of solutions and it is not clear which one of the solutions is the optimal one to use. To remedy this problem, we propose the direct intervention approach in which the dynamics of all the variables in the S-system framework are used for developing a MPC algorithm to achieve the intervention objective without the need for any steady-state

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analysis. In the cases where the number of control variables is more than the number of target variables, the choice of the optimization parameters in the direct MPC approach determines the optimal values of the variables with no prespecified desired values. The main advantage of the MPC-based approach (both direct and indirect) is that it allows the incorporation of input and state constraints in the control design. This paper is organized as follows. In Section II, the control problem for S-systems is formulated and two control approaches, namely indirect and direct, are proposed. In Section III, the glycolytic–glycogenolytic pathway model is considered as a case study. Comments and possible future research directions are outlined in Section IV. II. CONTROL OF S-SYSTEMS Consider the following S-system dynamics: x˙ i = αi

N +m j =1

g

xj i j − βi

N +m

h

xj i j ,

i = 1, 2, . . . , N

(1)

j =1

where αi > 0 and βi > 0 are rate coefficients and gij and hij are kinetic orders and there exist N + m variables (genes/metabolites) where the first N variables are dependent and the remaining m independent variables can be manipulated to control the system. It is assumed that the m independent variables are differentiable and time dependent. In order to control the values (expressions/concentrations) of the dependent variables, we consider an integral control approach where the following m equations are added to the aforesaid S-system:

Fig. 1.

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S-system with integral control architecture.

Problem Formulation: Suppose that the S-system (1) is ini0 tially in the steady-state condition xss . Let us denote by Xt ⊂ {1, . . . , N } the set of indices corresponding to the target variables (genes/metabolites) whose desired final steady-state d values are specified as xss . Then, the control problem is to find the control inputs ui , i = 1, . . . , m, that can guide the target 0 variables from the initial steady-state condition xss to the final d one xss . It is assumed that the S-system (1) is locally stable at d the target steady-state xss . We first need to check if the control problem has a solution. To do so, we investigate the steady-state behavior of system (1). A. Steady-State Analysis One of the advantages of the S-system representation of nonlinear systems is that the calculation of the steady-state values is straightforward. This is due to the structural property of Ssystems. At steady state, we have αi

N +m j =1

x˙ i+N = ui ,

i = 1, . . . , m.

(2)

Fig. 1 shows the S-system (1) augmented by the integral control. The S-system with integral control [(1) and (2)] can be written as follows: x˙ = f (x) + g(x)u

(3)

where x = [x1 , . . . , xN +m ]T ∈ RN +m , u = [u1 , . . . , um ]T ∈ Rm , and ⎡  +m g 1 j  +m h 1 j ⎤ α1 N − β1 N j =1 xj j =1 xj ⎢ ⎥ .. ⎢ ⎥ . ⎢ N +m g N j N +m h N j ⎥ ⎢ ⎥ x − βN j =1 xj ⎥ α f (x) = ⎢ ⎢ N j =1 j ⎥ 0 ⎢ ⎥

⎢ ⎥ . .. ⎣ ⎦ m 0 0 g(x) = N ×m . Im ×m

gij xss j

= βi

N +m

xss j

h ij

,

i = 1, 2, . . . , N.

(5)

j =1

If none of the αi , βi terms equals zero, then using a simple logarithmic transformation, we obtain the following set of linear algebraic equations:   N +m βi (gij − hij ) log xss = log , i = 1, 2, . . . , N. j α i j =1 Let us define yj = log(xss j ) and aij = gij − hij . Then based on the set Xt , we have  

βi aij yj = log aij yj , i = 1, 2, . . . , N (6) − αi j ∈N

j ∈Xt

where N = {1, . . . , N + m} − Xt represents the indices corresponding to the set of control variables as well as dependent variables for which no desired values have been specified in the aforesaid control problem. Equation (6) can be rewritten as Ay = b

(7)

(4)

where A = [aij ], i = 1, . . . , N , j ∈ N , y = [yj ], j ∈ N , and b = [b1 , . . . , bN ]T with   βi bi = log aij yj . − αi

where F (x[k]) = x[k] + Ts f (x[k]), G(x[k]) = Ts g(x[k]), Ts is the sampling time, x[k] is defined as the sampled continuoustime state x(kTs ), and u[k] := u(kTs ).

Based on the cardinality of the set N , the set of N equations in (7) may have no solution (|N | < N ), unique solution (|N | = N and det(A) = 0), or an infinite number of solutions (|N | > N ).

Using the Euler approximation, we can write the discrete time representation of the (3) as x[k + 1] = F (x[k]) + G(x[k])u[k]

j ∈Xt

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Fig. 3.

Indirect control of the S-system: first approach.

It is clear that for the case where |Xt | = m, i.e., the number of dependent variables with prespecified desired values is equal to the number of independent variables (which means that we have enough degrees of freedom), the previous equations will have a unique solution if the matrix A is nonsingular. Thus, one can use the solvability of the set of linear equations in (7) to determine the feasibility of solving a given intervention problem. It is clear that if these equations do not have a solution, then the control problem as posed is infeasible and one needs to relax the demands placed on the intervention problem by reducing the cardinality of Xt . B. Controller Design In this paper, two different control design approaches to the S-system control problem are proposed, namely indirect control and direct control. 1) Indirect Control: In indirect control, it is assumed that (7) has a unique solution. Then, a controller is designed to transfer the independent variables xi , i = N + 1, . . . , N + m, from their initial values to their target values found from the solution of (7). Based on the stability assumption and uniqueness of the solution, all the target-dependent variables xi , i ∈ Xt , will converge to their prespecified desired values given that the initial values of the dependent variables are in the basin of d attraction of the desired steady state xss . Due to the fact that we only control the independent variables, this approach is called indirect control. d Let yiss , i = N + 1, . . . , N + m, denote the solution of (7) d

ssd

for yi , i = N + 1, . . . , N + m, and let xss = ey i , i = N + i 1, . . . , N + m, be the corresponding concentrations. Then, the following simple sampled-data controllers [17], shown in Fig. 2, can be used to guide the target variables (genes/metabolites) to their prespecified desired values: ui (t) = −Ki (xi+N [k] −

d xss i+N ),

kTs ≤ t < (k + 1)Ts

Indirect MPC control of the S-system.

Therefore, for assuring the asymptotic stability of the aforesaid closed-loop system, we need to have |1 − Ts Ki | < 1, which leads to the selection criteria 0 < Ki < T2s for the controller gain Ki . Remark 1: The indirect control scheme shown in Fig. 2 can be logically justified as follows. Clearly, the controller structure being used here is of the feedforward type and there is no feedback as far as the S-system is concerned. Consequently, since the exogenous signals here are constants and the gains Ki , i = 1, . . . , m, are positive, the signals xi+N , i = 1, . . . , m, will attain constant values in the steady state. Since the time derivative of a constant signal is zero, it follows that whenever the signal at the output of an integrator becomes constant, the input to the corresponding integrator must necessarily become zero. In this case, it means that, in the steady state, u1 (t) = u2 (t) . . . = um (t) = 0, which in turn implies that as k → ∞, xN +1 , xN +2 , . . . , xN +m approach the desired values, calculated from the solution of (7). Remark 2: The indirect control scheme shown in Fig. 2 could have also been implemented using a continuous-time controller without any sampling or zero-order hold operations. However, we prefer to consider sampled data control from the very outset since the MPC schemes to be introduced later have typically been considered in the literature in the sampled-data framework. An alternative approach is to use MPC [18] (see Fig. 3) for achieving the aforementioned goal. In this method, one can incorporate hard constraints on the input signals as well as on the target variables (gene expressions/metabolite concentrations) of interest. The control inputs are found by solving the following optimization problem:  N +m T −1

d 2 qi+1,j (xj [k + i] − xss min j ) u [k ],...,u [k +T −1]

+

m

i=0

ri+1,j u2j [k

j =N +1

+ i] +

j =1

for i = 1, . . . , m where Ts is the sampling time and xi+N [k] := xi+N (kTs ), k = 0, 1, . . . . By substituting the earlier feedback control in the discrete-time system (4), we have d

xi+N [k + 1] = (1 − Ts Ki )xi+N [k] + Ts Ki xss i+N .

m

 wi+1,j Δu2j [k

+ i]

j =1

subject to the following constraints: um in,i ≤ ui ≤ um ax,i , 0 ≤ xi ≤ xm ax,i ,

i = 1, . . . , m i = N + 1, . . . , N + m

(8)

MESKIN et al.: INTERVENTION IN BIOLOGICAL PHENOMENA MODELED BY S-SYSTEMS

Δui,m in ≤ Δui ≤ Δui,m ax , xi [k + 1] = xi [k] + Ts ui [k],

i = 1, . . . , m i = N + 1, . . . , N + m

where Δui [k] = ui [k] − ui [k − 1], T is the prediction horizon, and Ts is the sampling time of the model. In the cost function in (8), the weight variable q ≥ 0 penalizes the deviation of the independent variables from their desired steady-state values calculated from (7), while the weights r ≥ 0 and w ≥ 0 penalize the control input and changes in the control input, respectively. These weights affect the performance of the MPC controller in terms of response time and steady-state error. The previous optimization problem is solved at every controller sample point and the control input is selected as u(t) = u∗ [k],

kTs ≤ t < (k + 1)Ts



where u [k] is the solution of the previous optimization problem. The main advantage of the indirect MPC approach with respect to the simple sampled-data controller is that the constraints on the inputs, i.e., um ax,i and um in,i , as well as those on the rate of change of the inputs, i.e., Δum ax,i and Δum in,i , can be explicitly incorporated while calculating the control inputs through the solution of the optimization problem (8). 2) Direct Control: In this approach, we try to directly control the target variables (genes/metabolites) corresponding to the indices Xt , using an MPC approach. In other words, the control problem for system (1) reduces to finding a control policy by solving the following optimization problem:  T −1

d 2 qi+1,j (xj [k + i] − xss min j ) u [k ],...,u [k +T −1]

+

m

i=0

ri+1,j u2j [k

j ∈Xt

+ i] +

j =1

m

 wi+1,j Δu2j [k

+ i]

(9)

j =1

subject to the following constraints: um in,i ≤ u ≤ um ax,i , 0 ≤ xi ≤ xi,m ax ,

Fig. 4.

Direct MPC control of the S-system.

finding the solution for it is in general difficult. One way to solve (9) is to make it linear via local linearization [19]. In this approach, at any time step, the nonlinear system x[k + 1] = F (x[k]) + G(x[k])u[k] is linearized at the current state, and the local linear model x[k + 1] = Ax[k] + Bu[k] is used as the system constraint in the optimization problem. Since the optimization problem (9) is solved over a finite prediction window T , the local linear model at each time step can be considered as a good local approximation of the nonlinear system during the prediction window. The new local optimization problem can be solved using the quadratic programming approach. One of the advantages of the direct MPC approach is for the cases where the number of control variables is more than the number of target variables (m > |Xt |) and the indirect approach cannot be used. In this case, the direct MPC approach will select the optimal values of the variables with no prespecified desired target values based on the optimization parameters in (9).

i = 1, . . . , m i = 1, . . . , N + m

Δui,m in ≤ Δui ≤ Δui,m ax ,

i = 1, . . . , m

x[k + 1] = F (x[k]) + G(x[k])u[k] where T is the prediction horizon and Ts is the sampling time. The aforesaid optimization problem is solved at each controller sample point and the control input is selected as u(t) = u∗ [k],

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kTs ≤ t < (k + 1)Ts

where u∗ [k] is the solution of the aforesaid optimization problem. The main difference between the cost functions in (8) and (9) is that in (8), the deviations between the independent variables and their steady-state values [calculated from (7)] are being penalized while in (9), one directly penalizes the deviations between the target variables and their prespecified desired values. Fig. 4 shows a schematic diagram for the direct MPC control of an S-system. Due to the fact that the S-systems are inherently nonlinear, the optimization problem (9) is a complex nonlinear one and

III. CASE STUDY In this section, we demonstrate the efficacy of the intervention approaches developed in this paper by applying them to an S-system that can be used to model a well-studied biological pathway. Consider the following S-system which represents the glycolytic–glycogenolytic pathway, shown in Fig. 5 [14], [20]: x˙ 1 = 0.077884314x0.66 x6 − 1.06270825x1.53 x−0.59 x7 4 1 2 x˙ 2 = 0.585012402x0.95 x−0.41 x0.32 x0.62 x0.38 1 5 7 10 2 − 0.0007934561x3.97 x−3.06 x8 2 3 x˙ 3 = 0.0007934561x3.97 x−3.06 x8 − 1.05880847x0.3 2 3 x9 (10) 3 where N = 3, m = 7, and the independent variables have the values x4 = 10, x5 = 5, x6 = 3, x7 = 40, x8 = 136, x9 = 2.86, and x10 = 4. The steady-state concentrations are x1 = 0.067, x2 = 0.465, and x3 = 0.150. One can check using classical linearization that the aforesaid S-system is locally stable at these steady-state concentrations. We next consider two intervention scenarios.

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Fig. 5.

Fig. 6. Metabolite concentrations corresponding to scenario # 1 with indirect linear controller (x1 : “∗,” x2 : “,” x3 : “ ”).

Glycolytic-glycogenolytic pathway [20].

A. Scenario # 1 In the first scenario, we try to control x1 , x2 , and x3 by manipulating x4 , x5 , and x8 , i.e., x˙ 4 = u1 x˙ 5 = u2 x˙ 8 = u3

(11)

and all other xi ’s i = 6, 7, 9, 10 are kept fixed. Physically, this corresponds to the problem of using the glucose, inorganic phosphate ion, and phosphoglucose isomerase concentrations to control the concentrations of glucose-1-phosphate, glucose-6-phosphate, and fructose-6-phosphate. The target vald d = 0.2, xss = 0.5, ues for x1 , x2 , and x3 are selected as xss 1 2 ss d and x3 = 0.4. First, the feasibility of this control problem is checked for system (10). It is clear that N = {4, 5, 8} and we have three equations with three unknown variables. The solution of (7) for the previous system is y4 = 4.6903, d = y5 = −0.6248, and y8 = 7.9201, which corresponds to xss 4 d d ss ss 108.8806, x5 = 0.5354, and x8 = 2752.1. Next the local stability of the new steady-state concentration is verified by d classical linearization. Based on the steady-state values xss 4 , d d ss xss 5 , and x8 , the controllers d

kTs ≤ t < (k + 1)Ts

d

kTs ≤ t < (k + 1)Ts

d

kTs ≤ t < (k + 1)Ts

ui (t) = −K1 (x4 [k] − xss 4 ), u2 (t) = −K2 (x5 [k] − xss 5 ), u3 (t) = −K3 (x8 [k] − xss 8 ),

can be applied where 0 < Ki < T2s , i = 1, 2, 3, are the controller gains. Figs. 6 and 7 show the trajectory response and control input of the aforesaid system for Ki = 0.05, i = 1, 2, 3, and Ts = 5 min. As shown in Fig. 6, the states x1 , x2 , and x3 converge to their desired values. Next, the indirect MPC control approach is applied to the system (10) with integral control (11). The optimization parameters in (8) are selected as T = 10 (prediction horizon), ri+1,j = 0, wi+1,j = 0.01, i = 0, . . . , 9, j = 1, 2, 3 (control input weights), qi+1,j = 1, i = 0, . . . , 9, j = 1, 2, 3 (target state weights), um ax,i = 2, um in,i = −2, i = 1, 2, um ax,3 = 50, um in,3 = −50, Δum ax,i = 1, Δum in,i = −1, i = 1, 2, 3, and Ts = 3 min (sampling time). Figs. 8 and 9 depict the trajectory response and the control input, respectively, for the aforementioned system. It is clear from Fig. 8 that the target states x1 , x2 ,

Fig. 7. Control input signals u corresponding to scenario # 1 with indirect linear controller.

Fig. 8. Metabolite concentrations corresponding to scenario # 1 with indirect MPC-based controller (x1 : “∗,” x2 : “,” x3 : “ ”). d

d

and x3 converge to their desired values xss = 0.2, xss = 0.5, 1 2 ss d and x3 = 0.4, respectively. Finally, we try to apply the direct MPC control approach for guiding x1 , x2 , and x3 directly to their target values. The local linear model is used at any time step for solving the optimization problem (9). The prediction horizon is 10 and the sampling time Ts is selected as 3 min. The optimization parameters are selected similar to the aforementioned indirect MPC approach. Figs. 10 and 11 show the trajectory response and control input for the S-system (10) controlled by the direct MPC-based approach whose goal is to make the target states x1 , x2 , and x3 converge to their desired values. B. Scenario # 2 In this scenario, we consider the case where the number of control variables is more than the number of target variables.

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Fig. 12. Metabolite concentrations corresponding to scenario # 2 with direct MPC-based controller (x1 : “∗,” x2 : “,” x3 : “ ”). Fig. 9. Control input signals u corresponding to scenario # 1 with indirect MPC-based controller.

Fig. 10. Metabolite concentrations corresponding to scenario # 1 with direct MPC-based controller (x1 : “∗,” x2 : “,” x3 : “ ”).

Fig. 13. Control input signals u corresponding to scenario # 2 with direct MPC-based controller.

selects the optimal values for the control variables x4 , x5 , and x8 as well as the variable x3 , which does not have any prespecified desired value. Figs. 12 and 13 depict the trajectory response and control input for the system (10) corresponding to scenario # 2, controlled by the direct MPC approach, where the optimization parameters are selected as T = 10 (prediction horizon), ri+1,j = wi+1,j = 0.01, i = 0, . . . , 9, j = 1, 2, 3 (control input weights), qi+1,j = 1, i = 0, . . . , 9, j = 1, 2 (target state weights), and Ts = 3 min (controller sampling time). As shown in Fig. 12, for the previous choice of optimization parameters, the steady-state value of x3 is 0.178. Fig. 11. Control input signals u corresponding to scenario # 1 with direct MPC-based controller.

IV. CONCLUSION

For instance, we seek to control x1 and x2 by manipulating d d = 0.2 and xss = 0.5. Physically, x4 , x5 , and x8 where xss 1 2 this corresponds to the problem of using the glucose, inorganic phosphate ion, and phosphoglucose isomerase concentrations to control the concentrations of glucose-1-phosphate and glucose6-phosphate. In this case, it is clear that if we were to pursue the steady-state analysis in Section II-A, then (7) would have an infinite number of solutions. Since there exists no criterion for selecting the optimal values of the control variables from the infinite number of solutions of (7), the indirect approach cannot be applied in this case. However, the direct approach can still be used and based on the optimization parameters in (9), the controller automatically

In this paper, we have developed intervention strategies for biological phenomena modeled in the S-system framework. We proposed two different approaches. In the first approach, referred to as the indirect approach, the prespecified desired values for the target variables are used to compute the reference values for the control inputs and the control algorithms are designed to guide the inputs to their computed reference values. In the direct approach, a MPC algorithm is developed that directly guides the target variables to their desired values. The proposed control algorithms are applied to the glycolytic–glycogenolytic pathway and the simulation results look promising. One of the future research directions for this paper would be the development of robust control algorithms for biological phenomena modeled by S-systems. Such robust control algorithms are essential to

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ensure that a control design based on a theoretical model succeeds when applied to the actual biological phenomenon. The indirect control approaches in this paper have been developed under the assumption that the given S-system enjoys local stability about the desired steady-state values. When such local stability is missing one would either have to use the direct control approach or first stabilize the S-system about the desired steady-state value before applying the indirect approaches to the resulting stabilized S-system. This latter problem is beyond the scope of this paper and is a topic for further investigation. ACKNOWLEDGMENT The statements made herein are solely the responsibility of the authors. REFERENCES [1] M. A. Savageau, “Biochemical system analysis, I. some mathematical properties of the rate law for the component enzymatic reactions,” J. Theor. Biol., vol. 25, pp. 365–369, 1969. [2] E. O. Voit, Canonical Nonlinear Modeling : S-System Approach to Understanding Complexity. New York: Van Nostrand/Reinhold, 1991. [3] E. Voit, “A systems-theoretical framework for health and disease: Inflammation and preconditioning from an abstract modeling point of view,” Math. Biosci., vol. 217, pp. 11–18, 2009. [4] E. O. Voit, F. Alvarez-Vasquez, and Y. A. Hannun, “Computational analysis of sphingolipid pathways systems,” Adv. Exp. Med. Biol., vol. 688, pp. 264–275, 2010. [5] R. Gentilini, “Toward integration of systems biology formalism: the gene regulatory networks case,” Genome Inform., vol. 16, pp. 215–224, 2005. [6] E. O. Voit and J. Almeida, “Decoupling dynamical systems for pathway identification from metabolic profiles,” Bioinformatics, vol. 20, no. 11, pp. 1670–1681, 2004. [7] I.-C. Chou, H. Martens, and E. O. Voit, “Parameter estimation in biochemical systems models with alternating regression,” Theor. Biol. Med. Model., vol. 3, no. 25, 2006. [8] H. Wang, L. Qian, and E. Dougherty, “Inference of genetic regulatory networks by evolutionary algorithm and H∞ filtering,” in Proc. IEEE Statist. Signal Process. Workshop, 2007, pp. 21–25. [9] H. Wang, L. Qian, and E. R. Dougherty, “Steady-state analysis of genetic regulatory networks modeled by nonlinear ordinary differential equations,” in Proc. IEEE Comput. Intell. Bioinformatics Comput. Biol. (CIBCB), 2009, pp. 182–185. [10] H. Wang, L. Qian, and E. Dougherty, “Inference of gene regulatory networks using S-systems: a unified approach,” IET Syst. Biol., vol. 4, no. 2, pp. 145–156, Mar. 2010. [11] O. R. Gonzalez, C. Kuper, P. C. Naval Jr., and E. Mendoza, “Parameter estimation using simulated annealing for S-system models of biochemical networks,” Bioinformatics, vol. 23, no. 4, pp. 480–486, 2007. [12] Z. Kutalik, W. Tucker, and V. Moulton, “S-system parameter estimation for noisy metabolic profiles using newton-flow analysis,” IET Syst. Biol., vol. 1, no. 3, pp. 174–180, May 2007. [13] I.-C. Chou and E. Voit, “Recent developments in parameter estimation and structure identification of biochemical and genomic systems,” Math. Biosci., vol. 219, pp. 57–83, 2009. [14] A. Ervadi-Radhakrishnan and E. O. Voit, “Controllabilty of non-linear biochemical systems,” Math. Biosci., vol. 196, pp. 99–123, 2005. [15] E. O. Voit, Computational Analysis of Biochemical System. A Practice Guide for Biochemists and Molecular Biologists. Cambridge, U.K.: Cambridge Univ. Press, 2002. [16] B. A. Francis and W. M. Wonham, “The internal model principle of control theory,” Automatica, vol. 12, no. 5, pp. 457–465, 1976. [17] B. C. Kuo, Analysis and Synthesis of Sampled-Data Control Systems. Englewood Cliffs, NJ: Prentice-Hall, 1963. [18] J. M. Maciejowski, Predictive Control with Constraints. Englewood Cliffs, NJ: Prentice-Hall, 2002. [19] J. H. Lee and N. L. Ricker, “Extended Kalman filter based nonlinear model predictive control,” Ind. Eng. Chem. Res., vol. 33, no. 6, pp. 1530–1541, 1994.

[20] N. V. Torres, “Modelization and experimental studies on the control of the glycolytic-glucogenolytic pathway in rat liver,” Mol. Cellular Biochem., vol. 132, no. 2, pp. 117–126, 1994.

Nader Meskin (S’06–M’09) received the B.Sc. degree from the Sharif University of Technology, Tehran, Iran, in 1998, the M.Sc. degree from the University of Tehran, Tehran, in 2001, and the Ph.D. degree in electrical and computer engineering from Concordia University, Montreal, Canada, in 2008. He was a Postdoctoral Fellow at Texas A&M University at Qatar, Doha, Qatar, from January 2010 to December 2010. He is currently an Assistant Professor at Qatar University, Doha. He has published more than 25 refereed journal and conference papers. He is a coauthor (with K. Khorasani) of the book Fault Detection and Isolation: Multi-Vehicle Unmanned Systems (Berlin. Germany: Springer, 2011). His research interests include fault detection and isolation, system biology, time delay systems, and Markovian jump systems.

Hazem N. Nounou (SM’08) received the B.S. degree (magna cum laude) from Texas A&M, College Station, University in 1995, and the M.S. and Ph.D. degrees from Ohio State University, Columbus, in 1997 and 2000, respectively, all in electrical engineering. In 2001, he was a Development Engineer for PDF Solutions, a consulting firm for the semiconductor industry, in San Jose, CA. Then, in 2001, he joined the Department of Electrical Engineering at King Fahd University of Petroleum and Minerals in Dhahran, Saudi Arabia, as an Assistant Professor. In 2002, he moved to the Department of Electrical Engineering, United Arab Emirates University, Al-Ain, United Arab Emirates. In 2007, he joined the Electrical and Computer Engineering Program, Texas A&M University at Qatar, Doha, Qatar, where he is currently an Associate Professor. He published more than 40 refereed journal and conference papers. He served as an Associate Editor in technical committees of several international journals and conferencesHis research interests include intelligent and adaptive control, control of time-delay systems, system biology, and system identification and estimation.

Mohamed Nounou (SM’08) received the B.S. degree (magna cum laude) from Texas A&M University, College Station, in 1995, and the M.S. and Ph.D. degrees from the Ohio State University, Columbus, in 1997 and 2000, respectively, all in chemical engineering. From 2000 to 2002, he was with PDF Solutions, a consulting company for the semiconductor industry, in San Jose, CA. In 2002, he joined the Department of Chemical and Petroleum Engineering, United Arab Emirates University, as an Assistant Professor. In 2006, he joined the Chemical Engineering Program at Texas A&M University at Qatar, Doha, Qatar, where he is currently an Associate Professor. He has published more than 40 refereed journal and conference papers and book chapters. He also served as an Associate Editor in technical committees of several international journals and conferences. His research interests include process modeling and estimation, system biology, and intelligent control. Dr. Nounou is a member of the American Institute of Chemical Engineers (AIChE).

MESKIN et al.: INTERVENTION IN BIOLOGICAL PHENOMENA MODELED BY S-SYSTEMS

Aniruddha Datta (F’09) received the B.Tech. degree in electrical engineering from the Indian Institute of Technology, Kharagpur, India, in 1985, the M.S.E.E. degree from Southern Illinois University, Carbondale, in 1987, and the M.S. (in applied mathematics) and Ph.D. degrees from the University of Southern California, Los Angeles, in 1991. In August 1991, he joined the Department of Electrical and Computer Engineering at Texas A&M University, College Station, where he is currently a Professor and holder of the J. W. Runyon, Jr. ’35 Professorship II. From 2001 to 2003, he worked on cancer research as a postdoctoral trainee under a National Cancer Institute Training Grant. He is the author of the book Adaptive Internal Model Control (Berlin, Germany: Springer-Verlag, 1998), a coauthor (with M. T. Ho and S. P. Bhattacharyya) of the book Structure and Synthesis of PID Controllers (Berlin, Germany: Springer-Verlag, 2000), a coauthor (with G. J. Silva and S. P. Bhattacharyya) of the book PID Controllers for Time Delay Systems (Birkhauser, Boston, 2004), a coauthor (with E. R. Dougherty) of the book Introduction to Genomic Signal Processing With Control (Boca Raton, FL: CRC Press, 2007), and a coauthor (with S. P. Bhattacharyya and L. H. Keel) of the book Linear Control Theory: Structure, Robustness and Optimization (Boca Raton, FL: CRC Press, 2009). His research interests include adaptive control, robust control, PID control, and genomic signal processing. Dr. Datta has served as an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL from 2001 to 2003, the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS-PART B from 2005 to 2006, and is currently serving as an Associate Editor of the EURASIP Journal on Bioinformatics and Systems Biology.

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Edward R. Dougherty (M’05–SM’09) received the M.S. degree in Computer Science from the Stevens Institute of Technology, Hoboken, NJ, in 1986 and the Ph.D. degree in mathematics from Rutgers University, New Brunswick, NJ, in 1974. He is currently a Professor in the Department of Electrical and Computer Engineering, Texas A&M University, College Station, where he holds the Robert M. Kennedy ’26 Chair in Electrical Engineering and is the Director of the Genomic Signal Processing Laboratory. He is also the Co-Director of the Computational Biology Division of the Translational Genomics Research Institute, Phoenix, AZ, and is an Adjunct Professor in the Department of Bioinformatics and Computational Biology, University of Texas M. D. Anderson Cancer Center, Houston. He is the author of 15 books, an editor of five others, and the author of 250 journal papers. He has contributed extensively to the statistical design of nonlinear operators for image processing and the consequent application of pattern recognition theory to nonlinear image processing. His current research in genomic signal processing is aimed at diagnosis and prognosis based on genetic signatures and using gene regulatory networks to develop therapies based on the disruption or mitigation of aberrant gene function contributing to the pathology of a disease. Dr. Dougherty has been awarded the Doctor Honoris Causa by the Tampere University of Technology in Finland. He is a Fellow of the SPIE, has received the SPIE President’s Award, and served as the Editor of the SPIE/IS&T Journal of Electronic Imaging. At Texas A&M University, he has received the Association of Former Students Distinguished Achievement Award in Research, been named Fellow of the Texas Engineering Experiment Station, and named Halliburton Professor of the Dwight Look College of Engineering.