#### Abstract

This paper considers the problem of designing a genetic circuit which is robust to noise effect. To achieve this goal, a mixed and Integral Quadratic Constraints (IQC) approach is proposed. In order to minimize the effects of external noise on the genetic regulatory network in terms of norm, a design procedure of Hill coefficients in the promoters is presented. The IQC approach is introduced to analyze and guarantee the stability of the designed circuit.

#### 1. Introduction

Genetic regulatory network (GRN) is subjected to noise disturbances that may occur at transcription, translation, transport, chromatin remodeling, and pathway specific regulation. The GRN diagrams that resemble complex electrical circuits are generated by the connectivity of mRNAs and proteins [1]. Mathematical and computational tools have been utilized to develop the genetic circuits and systems using biotechnological design principles of synthetic GRN, which involves new kinds of integrated circuits like neurochips inspired by the biological neural networks [2]. This method leads to a large-scale system composed of several interconnected subsystems. The previous work [3] performs a hierarchical analysis by propagating the IQC characterization of each uncertain subsystem through their interaction channels. More specifically, both plant states and the IQC dynamic states are used as feedback information in the closed-loop system model, and then the robust stability analysis is performed via dynamic IQCs. Thereby, the synthesis conditions for the proposed full-information feedback controller are derived for the linear matrix inequality (LMI) systems [4].

Therefore, stability analysis of uncertain GRN is a prerequisite for any design issue. From the perspectives of control engineering, is a key performance index to evaluate the noise rejection/attenuation capability. Unlike the external control inputs used in the conventional robust control theory [5], the feedback regulation mechanism is embedded in the GRN. We construct a genetic circuit by introducing a Hill function type feedback loop from proteins (mostly from transcription process) to regulate the expression of target genes. By binding to promoter domain, the GRN is mean square asymptotically stable with a given noise attenuation level .

The paper is organized as follows. Section 2 introduces the mathematical model of GRN. A design procedure for the Hill coefficients is proposed in Section 3. We provide an example to illustrate the developed design method in Section 4. The concluding remark is given in Section 5.

#### 2. Problem Formulation

The activities of a gene are regulated by other genes through their interactions, that is, the transcription and translation factors [6, 7]. The underlying dynamics can be modeled as a gene ,where are concentrations of mRNA and protein of the th gene at time , respectively, are the degradation rates of the mRNA and protein, is the translation rate, is the external noise, and is a monotonically increasing function [8] in which is the Hill coefficient, is a positive constant, and is the apparent dissociation constant derived from the law of mass action, which equals the ratio of the dissociation rate of the ligand-receptor complex to its association rate. The family of positive Hill functions is shown in Figure 1. In this paper, the Hill function assumes that protein is an activator of gene [6, 7]. The matrix is the coupling matrix of the GRN and is defined as a base rate. System (1) can be written into the compact matrix form:where , , , , , , and To simplify our exposition, we use a more general set of notations and shift the equilibrium point of the noiseless system to ; then model (3) can be expressed aswherewith being an arbitrary matrix such that , In this model, the system states of mRNAs and proteins play different roles in regulation, for example, activators, repressers, or other factors. We name as the deviation of concentration from the equilibrium point of (3). The rate of change in denoted by represents the concentration changes of the variables due to production or degradation. represents the regulation function on the th variable, which is generally a nonlinear or linear function on the variables , but has a form of monotonicity with each variable. The degradation parameters matrix has zero elements on its nondiagonal plane; the matrix defines the coupling topology, direction, and the transcriptional rate of the GRN. When the input is , the system model (4) can be rewritten aswhere is the vector of zero-mean white Gaussian noise.

In this paper, we aim to address the following problem.

*Problem 1. *Given the system represented by model (7) and parameters , , the purpose of robust genetic circuit design is to determine the parameters in such that (i)the whole system is stable;(ii) norm of the noise in the measurement channel is minimized (we assume full observation).

In the following section, we propose a mixed and IQC approach to tackle Problem 1. The objective of this approach is to promote method in the design of Hill function for GRN. We will give the theoretical analysis to underpin this technique.

#### 3. Analysis and Design

We take point as the equilibrium position due to the special shape of Hill functions in Figure 1. The intuitive way to address such an issue would be firstly to design a static feedback by which the closed-loop system can achieve the minimum . Then we prove the stability of the system with the nonlinearity involved by analysis methods [9]. However, the optimal performance with for the linear system does not necessarily guarantee the optimal performance for the nonlinear system with . In some cases, the nonlinearity of might even worsen the system performance to a degree which is far from optimal. Therefore, robust performance has rarely been considered for the nonlinear system. In the following subsections, we resolve this difficulty.

##### 3.1. Preliminaries

We first recall some preliminary results in the system analysis via IQCs from [10]. Let be the set of real proper rational function matrices without right-half plane poles and let be the set of functions that have finite energy on the interval , ; that is,The Fourier transform for an -valued function is denoted as . Consider the feedback configuration in Figure 2,where , , and and are two causal operators. Note that is stable and is bounded but could be nonlinear, time-varying, or uncertain. is said to satisfy the IQC defined by if the two vectors of signal , fulfillin which can be any measurable Hermitian-valued function defined on the imaginary axis and the superscript denotes the complex conjugate transpose.

Lemma 2 (see [10]). *Let and be a bounded causal operator. If the following assumptions hold, *(i)*for every , the interconnection of and is well-posed;*(ii)*for every , the IQC defined by is satisfied by ;*(iii)*there exists such that**then the feedback interconnection of and is stable.*

*Remark 3. *From Lemma 2, an important conclusion on IQC stability analysis can be drawn: if satisfies several IQCs, it also satisfies all nonnegative linear combinations of IQCs. For example, if , satisfy the IQC defined by and the IQC defined by , they will also satisfy the IQC defined by for all . In other words, the set of IQCs that satisfies forms a “description” for the block , and the more IQCs we know that satisfies, the more precisely that we can describe the uncertainty of .

Lemma 4 (KYP lemma, [11]). *Suppose Assume is stabilizable and has no eigenvalues on the imaginary axis. Then the following conditions are equivalent: *(i)*The system is stable, and .*(ii)*We have ,*(iii)*There exists a symmetric matrix , such that*

Lemma 5 (see [11]). *By using linear fraction transformation, we convert the system configuration from Figure 2 to Figure 3. Let , with corresponding dimension, and let the system -gain be the performance measurement.**Assume that satisfies an IQC defined by having the following block structure:then the system in Figure 3 is stable and has robust -gain , if *(i)*for every , the interconnection of and is well-posed;*(ii)*for every , the IQC defined by is satisfied by ;*(iii)*the frequency domain inequality,* *holds for all , where *

##### 3.2. Design Procedure

The algorithm for the design of Hill function considering both the stability and performance of the nonlinear system will be proposed.

Proposition 6. *After shifting the equilibrium point (Figure 1) to the origin, Hill function takes the following form:and there exist real scalars , such that satisfies the IQCs defined bywhere .*

*Proof. * is from a sector bound condition ; to see this we notice that in the time domainThe derivation of involves an approximation, in which with a scalar function with ; thenThis ends the proof.

*Remark 7. *From Proposition 6, we should know that the set of IQCs is a sufficient condition for stability; therefore, it is conservative. Yet the challenge for nonlinear system stability is significant, because all the existing methods, including Lyapunov theory and IQCs, are conservative.

In Proposition 6, and are for the case of scalar and . Here we consider the case of vectors and .

Corollary 8. *When and , then IQCs for take the following forms:where where with . Combining and , we obtain the following IQCs for :for any nonnegative diagonal matrices and .*

*Example 9. *Take , , , and as an example; therefore,Assume that the evolution of is within a limited a range ; then a sector condition of , with can form a good bound on , which is shown in Figure 4.

Proposition 10. *Consider the system configuration in Figure 5, let , be the genetic network, and and represent the designed feedback Hill function. We denote with the uncertainty derived from , which satisfies the IQCs in (24).**Then the system in Figure 5 is stable and has robust -gain , if *(i)*for every , the interconnection of and is well-posed;*(ii)*for every , the IQC defined by is satisfied by ;*(iii)*the frequency domain inequality,* *holds for all , where * *and take the form in (24).*

*Proof. *The result follows from applying the IQCs defined by (24) in Corollary 8.

Proposition 11. *Let the Hill coefficients be the set of proper values of for a Hill function; given any and , a Hill function can be uniquely defined and hence a set of IQCs in the form of (24) can be formulated. Then the design of Hill function feedback can be reformulated into the following linear matrix inequality (LMI) problem: such that (26) is satisfied.*

The frequency-dependent inequality in (26) can be resolved by using YALMIP (https://users.isy.liu.se/johanl/yalmip/). Since the linear matrix inequality (28) has to be solved simultaneously for all frequencies in the frequency domain, the basic strategy is to start with a small set of points and then check whether the solution satisfies all . If it does not, add one or more frequency points to the previous set and solve LMI again. Another more efficient way to solve the LMI is to transform (26) into the time domain by using the KYP lemma of Lemma 4 if the state-space realizations of and have been obtained.

#### 4. Biologically Inspired Example

In this section, we provide an example to illustrate the algorithm and design procedures developed in Section 3. Consider the following GRN:in which where , , and are randomly generated in MATLAB and , , and the elements in matrix follow the even distribution on the interval . In this simulation, the generated data are , , The system gain of the original system is 94.2653. Although the system is originally stable, its performance is far from satisfactory. In order to further investigate the effects of noise on , the procedure of designing is slightly modified in this example. Specifically, it is an undesirable result if a small causes a large . Therefore, rather than minimizing norm from to , we minimize norm from to , where is a constant. With ([8]) and , , we solve the LMI problem stated in (28), resulting in the feedback Hill function with , . As a result, norm of the closed-loop system is upper bounded by . Compared with , a distinct improvement has been made for the system performance in terms of norm.

#### 5. Conclusion

This paper tackles the problem of designing robust genetic circuit via constructing parameters of the Hill function. Targeting to minimize the noise effect in the genetic regulatory network, a design procedure is proposed. With the help of IQC approach, stability and performance of the designed circuit are guaranteed.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.