Advances in Mathematical Physics

Volume 2017, Article ID 5757943, 6 pages

https://doi.org/10.1155/2017/5757943

## Robust Genetic Circuit Design: A Mixed and IQC Analysis

^{1}School of Automation, Huazhong University of Science and Technology, Wuhan, China^{2}Centre for Synthetic Biology and Innovation, Imperial College London, London, UK^{3}School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, China

Correspondence should be addressed to Xin He; nc.ude.tsuh@hnix

Received 7 January 2017; Revised 12 March 2017; Accepted 26 March 2017; Published 6 April 2017

Academic Editor: Antonio Scarfone

Copyright © 2017 Ruijuan Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.