Robust filtering circuit design for stochastic gene networks under intrinsic and extrinsic molecular noises

An Approximate Method of System Entropy in Discrete-Time Nonlinear Biological Networks

This study discusses the calculation of entropy of discrete-time stochastic biological systems. First, measurement methods of the system entropy of discrete-time linear stochastic networks are introduced. The system entropy is found to be characterized by system matrices of the discrete-time biological systems. Secondly, the system entropy of nonlinear discrete-time stochastic biological systems is discussed and is calculated based on a global linearization method. The approximation of the values of system entropy of nonlinear stochastic systems needs to solve an optimization problem that is constrained by a kind of linear matrix inequality (LMI). Finally, a practical biochemical system is provided to verify the effectiveness of the proposed calculation method.

Robust Gene Circuit Control Design for Time-Delayed Genetic Regulatory Networks Without SUM Regulatory Logic

This paper investigates the gene circuit control design problem of time-delayed genetic regulatory networks. In the genetic regulatory networks, the time delays are unknown constants, and the genetic regulatory is not conventional SUM regulatory logic and can be modeled to be an unknown nonlinear function of the time-delayed states of the other genes in a cell. By Lyapunov stability, a novel adaptive gene circuit control design approach is proposed for the genetic regulatory networks, where the unknown time delays are estimated online by adaptive algorithms and the unknown regulatory functions are approximated by neural networks. The design approach in this paper is delay-dependent and has less conservatism than the delay-independent approach. From theoretical analysis, the closed-loop system is asymptotically stable and all the signals in the system converge to an adjustable neighborhood of the origin. Finally, a numerical example is given to show the effectiveness of the new design approach.

S-system-based analysis of the robust properties common to many biochemical network models

Robustness is a key feature to characterize the adaptation of organisms to changes in their internal and external environments. A broad range of kinetic or dynamic models of biochemical systems have been developed. Robustness analyses are attractive for exploring some common properties of many biochemical models. To reveal such features, we transform different types of mathematical equations into a standard or intelligible formula and use the multiple parameter sensitivity (MPS) to identify some factors critically responsible for the total robustness to many perturbations. The MPS would be determined by the top quarter of the highly sensitive parameters rather than the single parameter with the maximum sensitivity. The MPS did not show any correlation to the network size. The MPS is closely related to the standard deviation of the sensitivity profile. A decrease in the standard deviation enhanced the total robustness, which shows the hallmark of distributed robustness that many factors (pathways) involve the total robustness.

A Unifying Mathematical Framework for Genetic Robustness, Environmental Robustness, Network Robustness and their Trade-offs on Phenotype Robustness in Biological Networks. Part III: Synthetic Gene Networks in Synthetic Biology

Robust stabilization and environmental disturbance attenuation are ubiquitous systematic properties that are observed in biological systems at many different levels. The underlying principles for robust stabilization and environmental disturbance attenuation are universal to both complex biological systems and sophisticated engineering systems. In many biological networks, network robustness should be large enough to confer: intrinsic robustness for tolerating intrinsic parameter fluctuations; genetic robustness for buffering genetic variations; and environmental robustness for resisting environmental disturbances. Network robustness is needed so phenotype stability of biological network can be maintained, guaranteeing phenotype robustness. Synthetic biology is foreseen to have important applications in biotechnology and medicine; it is expected to contribute significantly to a better understanding of functioning of complex biological systems. This paper presents a unifying mathematical framework for investigating the principles of both robust stabilization and environmental disturbance attenuation for synthetic gene networks in synthetic biology. Further, from the unifying mathematical framework, we found that the phenotype robustness criterion for synthetic gene networks is the following: if intrinsic robustness + genetic robustness + environmental robustness ≦ network robustness, then the phenotype robustness can be maintained in spite of intrinsic parameter fluctuations, genetic variations, and environmental disturbances. Therefore, the trade-offs between intrinsic robustness, genetic robustness, environmental robustness, and network robustness in synthetic biology can also be investigated through corresponding phenotype robustness criteria from the systematic point of view. Finally, a robust synthetic design that involves network evolution algorithms with desired behavior under intrinsic parameter fluctuations, genetic variations, and environmental disturbances, is also proposed, together with a simulation example.

A Unifying Mathematical Framework for Genetic Robustness, Environmental Robustness, Network Robustness and their Trade-off on Phenotype Robustness in Biological Networks Part I: Gene Regulatory Networks in Systems and Evolutionary Biology

Robust stabilization and environmental disturbance attenuation are ubiquitous systematic properties observed in biological systems at different levels. The underlying principles for robust stabilization and environmental disturbance attenuation are universal to both complex biological systems and sophisticated engineering systems. In many biological networks, network robustness should be enough to confer intrinsic robustness in order to tolerate intrinsic parameter fluctuations, genetic robustness for buffering genetic variations, and environmental robustness for resisting environmental disturbances. With this, the phenotypic stability of biological network can be maintained, thus guaranteeing phenotype robustness. This paper presents a survey on biological systems and then develops a unifying mathematical framework for investigating the principles of both robust stabilization and environmental disturbance attenuation in systems and evolutionary biology. Further, from the unifying mathematical framework, it was discovered that the phenotype robustness criterion for biological networks at different levels relies upon intrinsic robustness + genetic robustness + environmental robustness ≦ network robustness. When this is true, the phenotype robustness can be maintained in spite of intrinsic parameter fluctuations, genetic variations, and environmental disturbances. Therefore, the trade-offs between intrinsic robustness, genetic robustness, environmental robustness, and network robustness in systems and evolutionary biology can also be investigated through their corresponding phenotype robustness criterion from the systematic point of view.

A Unifying Mathematical Framework for Genetic Robustness, Environmental Robustness, Network Robustness and their Tradeoff on Phenotype Robustness in Biological Networks Part II: Ecological Networks

In ecological networks, network robustness should be large enough to confer intrinsic robustness for tolerating intrinsic parameter fluctuations, as well as environmental robustness for resisting environmental disturbances, so that the phenotype stability of ecological networks can be maintained, thus guaranteeing phenotype robustness. However, it is difficult to analyze the network robustness of ecological systems because they are complex nonlinear partial differential stochastic systems. This paper develops a unifying mathematical framework for investigating the principles of both robust stabilization and environmental disturbance sensitivity in ecological networks. We found that the phenotype robustness criterion for ecological networks is that if intrinsic robustness + environmental robustness ≦ network robustness, then the phenotype robustness can be maintained in spite of intrinsic parameter fluctuations and environmental disturbances. These results in robust ecological networks are similar to that in robust gene regulatory networks and evolutionary networks even they have different spatial-time scales.

Biochemical Systems Theory: A Review

Biochemical systems theory (BST) is the foundation for a set of analytical andmodeling tools that facilitate the analysis of dynamic biological systems. This paper depicts major developments in BST up to the current state of the art in 2012. It discusses its rationale, describes the typical strategies and methods of designing, diagnosing, analyzing, and utilizing BST models, and reviews areas of application. The paper is intended as a guide for investigators entering the fascinating field of biological systems analysis and as a resource for practitioners and experts.

Robust synchronization control scheme of a population of nonlinear stochastic synthetic genetic oscillators under intrinsic and extrinsic molecular noise via quorum sensing

Background

Collective rhythms of gene regulatory networks have been a subject of considerable interest for biologists and theoreticians, in particular the synchronization of dynamic cells mediated by intercellular communication. Synchronization of a population of synthetic genetic oscillators is an important design in practical applications, because such a population distributed over different host cells needs to exploit molecular phenomena simultaneously in order to emerge a biological phenomenon. However, this synchronization may be corrupted by intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. Therefore, robust synchronization is an important design topic in nonlinear stochastic coupled synthetic genetic oscillators with intrinsic kinetic parameter fluctuations and extrinsic molecular noise.

Results

Initially, the condition for robust synchronization of synthetic genetic oscillators was derived based on Hamilton Jacobi inequality (HJI). We found that if the synchronization robustness can confer enough intrinsic robustness to tolerate intrinsic parameter fluctuation and extrinsic robustness to filter the environmental noise, then robust synchronization of coupled synthetic genetic oscillators is guaranteed. If the synchronization robustness of a population of nonlinear stochastic coupled synthetic genetic oscillators distributed over different host cells could not be maintained, then robust synchronization could be enhanced by external control input through quorum sensing molecules. In order to simplify the analysis and design of robust synchronization of nonlinear stochastic synthetic genetic oscillators, the fuzzy interpolation method was employed to interpolate several local linear stochastic coupled systems to approximate the nonlinear stochastic coupled system so that the HJI-based synchronization design problem could be replaced by a simple linear matrix inequality (LMI)-based design problem, which could be solved with the help of LMI toolbox in MATLAB easily.

Conclusion

If the synchronization robustness criterion, i.e. the synchronization robustness ≥ intrinsic robustness + extrinsic robustness, then the stochastic coupled synthetic oscillators can be robustly synchronized in spite of intrinsic parameter fluctuation and extrinsic noise. If the synchronization robustness criterion is violated, external control scheme by adding inducer can be designed to improve synchronization robustness of coupled synthetic genetic oscillators. The investigated robust synchronization criteria and proposed external control method are useful for a population of coupled synthetic networks with emergent synchronization behavior, especially for multi-cellular, engineered networks.

On the Interplay between the Evolvability and Network Robustness in an Evolutionary Biological Network: A Systems Biology Approach

In the evolutionary process, the random transmission and mutation of genes provide biological diversities for natural selection. In order to preserve functional phenotypes between generations, gene networks need to evolve robustly under the influence of random perturbations. Therefore, the robustness of the phenotype, in the evolutionary process, exerts a selection force on gene networks to keep network functions. However, gene networks need to adjust, by variations in genetic content, to generate phenotypes for new challenges in the network's evolution, ie, the evolvability. Hence, there should be some interplay between the evolvability and network robustness in evolutionary gene networks. In this study, the interplay between the evolvability and network robustness of a gene network and a biochemical network is discussed from a nonlinear stochastic system point of view. It was found that if the genetic robustness plus environmental robustness is less than the network robustness, the phenotype of the biological network is robust in evolution. The tradeoff between the genetic robustness and environmental robustness in evolution is discussed from the stochastic stability robustness and sensitivity of the nonlinear stochastic biological network, which may be relevant to the statistical tradeoff between bias and variance, the so-called bias/variance dilemma. Further, the tradeoff could be considered as an antagonistic pleiotropic action of a gene network and discussed from the systems biology perspective.

On Robust State Estimation of Gene Networks

Gene networks in biological systems are not only nonlinear but also stochastic due to noise corruption. How to accurately estimate the internal states of the noisy gene networks is an attractive issue to researchers. However, the internal states of biological systems are mostly inaccessible by direct measurement. This paper intends to develop a robust extended Kalman filter for state and parameter estimation of a class of gene network systems with uncertain process noises. Quantitative analysis of the estimation performance is conducted and some representative examples are provided for demonstration.

Robust synthetic biology design: stochastic game theory approach

MOTIVATION

Synthetic biology is to engineer artificial biological systems to investigate natural biological phenomena and for a variety of applications. However, the development of synthetic gene networks is still difficult and most newly created gene networks are non-functioning due to uncertain initial conditions and disturbances of extra-cellular environments on the host cell. At present, how to design a robust synthetic gene network to work properly under these uncertain factors is the most important topic of synthetic biology.

RESULTS

A robust regulation design is proposed for a stochastic synthetic gene network to achieve the prescribed steady states under these uncertain factors from the minimax regulation perspective. This minimax regulation design problem can be transformed to an equivalent stochastic game problem. Since it is not easy to solve the robust regulation design problem of synthetic gene networks by non-linear stochastic game method directly, the Takagi-Sugeno (T-S) fuzzy model is proposed to approximate the non-linear synthetic gene network via the linear matrix inequality (LMI) technique through the Robust Control Toolbox in Matlab. Finally, an in silico example is given to illustrate the design procedure and to confirm the efficiency and efficacy of the proposed robust gene design method.

AVAILABILITY

http://www.ee.nthu.edu.tw/bschen/SyntheticBioDesign_supplement.pdf.

A systematic molecular circuit design method for gene networks under biochemical time delays and molecular noises

Background

Gene networks in nanoscale are of nonlinear stochastic process. Time delays are common and substantial in these biochemical processes due to gene transcription, translation, posttranslation protein modification and diffusion. Molecular noises in gene networks come from intrinsic fluctuations, transmitted noise from upstream genes, and the global noise affecting all genes. Knowledge of molecular noise filtering and biochemical process delay compensation in gene networks is crucial to understand the signal processing in gene networks and the design of noise-tolerant and delay-robust gene circuits for synthetic biology.

Results

A nonlinear stochastic dynamic model with multiple time delays is proposed for describing a gene network under process delays, intrinsic molecular fluctuations, and extrinsic molecular noises. Then, the stochastic biochemical processing scheme of gene regulatory networks for attenuating these molecular noises and compensating process delays is investigated from the nonlinear signal processing perspective. In order to improve the robust stability for delay toleration and noise filtering, a robust gene circuit for nonlinear stochastic time-delay gene networks is engineered based on the nonlinear robust H_∞ stochastic filtering scheme. Further, in order to avoid solving these complicated noise-tolerant and delay-robust design problems, based on Takagi-Sugeno (T-S) fuzzy time-delay model and linear matrix inequalities (LMIs) technique, a systematic gene circuit design method is proposed to simplify the design procedure.

Conclusion

The proposed gene circuit design method has much potential for application to systems biology, synthetic biology and drug design when a gene regulatory network has to be designed for improving its robust stability and filtering ability of disease-perturbed gene network or when a synthetic gene network needs to perform robustly under process delays and molecular noises.

Robust engineered circuit design principles for stochastic biochemical networks with parameter uncertainties and disturbances

Biochemical regulatory networks including genes, proteins and other regulatory molecules suffer from internal parametrical fluctuations (thermal, transcriptional, and splicing) as well as external noises (environmental and intercellular). Robustness is an essential property of intracellular biochemical regulatory networks to attenuate the effects of internal fluctuation and external noise. In this study, several system control schemes are proposed for the robust circuit control design of stochastic linear and nonlinear biochemical regulatory networks. First, the robust stability of genetic and proteomic regulatory networks is discussed under internal fluctuations. Then, the filtering ability of external noises is analyzed for stochastic biochemical regulatory networks. For the case where a biochemical regulatory network is not sufficiently robust to tolerate internal fluctuation and does not have enough filtering ability to filter the external noise, how to improve the robustness and noise filtering ability of stochastic biochemical regulatory networks by engineered control mechanisms is also proposed via biochemical circuit design. The proposed robust gene circuit design principles have potential applications for robust biosynthetic network design. Finally, two design examples are given in-silico to illustrate the design procedure and to confirm the performance of the proposed robust circuit design method.