Systems Evolutionary Biology: Biological Network Evolution Theory, Stochastic Evolutionary Game Strategies, and Applications to Systems Synthetic Biology

By Bor-Sen Chen

Systems Evolutionary Biology: Biological Network Evolution Theory, Stochastic Evolutionary Game Strategies, and Applications to Systems Synthetic Biology discusses the evolutionary game theory and strategies of nonlinear stochastic biological networks under random genetic variations and environmental disturbances and their application to systematic synthetic biology design. The book provides more realistic stochastic biological system models to mimic the real biological systems in evolutionary process and then introduces network evolvability, stochastic evolutionary game theory and strategy based on nonlinear stochastic networks in evolution. Readers will find remarkable, revolutionary information on genetic evolutionary biology that be applied to economics, engineering and bioscience.

• Explains network fitness, network evolvability and network robustness of biological networks from the systematic perspective
• Discusses the evolutionary noncooperative and cooperative game strategies of biological networks
• Offers detailed diagrams to help readers understand biological networks, their systematic behaviors and the simulational results of evolutionary biological networks
• Includes examples in every chapter with computational simulation to illustrate the solution procedure of evolutionary theory, strategy and results

• Language: English
• Publisher: Academic Press
• Release date: Feb 3, 2018
• ISBN: 9780128140734

Bor-Sen Chen

Bor-Sen Chen received B.S. degree of electrical Engineering from Tatung Institute of Technology in 1970, M.S. degree of Geophysics from National Central University in 1973, and PhD in Electrical Engineering from University of Southern California in 1982. He is an expert on the topic of nonlinear robust control and filter designs based on stochastic Nash game theory to override the influence of intrinsic random fluctuations and attenuate the effect of environmental disturbances, which can be applied to evolutionary game strategies of biological networks under natural selection to respond to random genetic variations and environmental disturbances in the evolutionary process. Prof. Chen had audited more than 10 courses of biology before his research in systems biology. He has published about 100 papers in bioinformatics and systems biology. Further, he have published more than 100 papers in system theory and control, and more than 80 papers of signal processing and communication. In the last three years, he has also published 7 monographs. He was elected to an IEEE Fellow in 2001 and became an IEEE Life Fellow in 2014.

Book preview

Systems Evolutionary Biology

Biological Network Evolution Theory, Stochastic Evolutionary Game Strategies, and Applications to Systems Synthetic Biology

First Edition

Bor-Sen Chen

Lab of Control and Systems Biology, Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan

Department of Electrical Engineering, Yuan Ze University, Chungli, Taiwan

Table of Contents

Cover image

Title page

Copyright

Dedication

Preface

Part I: General Theory of Stochastic Evolutionary Biological Network

Chapter 1: Introduction to Systems Evolutionary Biology

Abstract

1.1 Introduction to Evolutionary Biology

1.2 Review of Current Systems Biology and Evolutionary Theory

1.3 Systems Evolutionary Biology as a Powerful Combination of Evolutionary Genetics With Systems Biology

1.4 The Scope of the Book

Chapter 2: Stochastic Dynamics Systems and Stochastic Nash Game in Evolutionary Biological Networks

Abstract

2.1 Introduction to the Robust Stability of Stochastic Dynamic Systems

2.2 Evolutionary Computation Algorithms

2.3 Stochastic Nash Evolutionary Game in Stochastic Biological Systems

2.4 Conclusion

2.5 Appendix

Chapter 3: Evolutionary Gene Regulatory Networks and Biochemical Networks

Abstract

3.1 Introduction to Evolutionary Biological Systems

3.2 On the Interplay Between the Evolvability and Network Robustness of the Linear Stochastic Gene Regulatory Network

3.3 On the Interplay Between the Network Evolvability and Network Robustness of a Nonlinear Stochastic Gene Regulatory Network in Evolution

3.4 On the Interplay Between the Network Evolvability and Robustness of Biochemical Networks in Evolution

3.5 On the Interplay Between the Evolvability and Network Robustness of High-Level Biological Networks in Evolution

3.6 Discussion and Conclusion

3.7 Appendix

Chapter 4: Evolutionary Ecological Networks

Abstract

4.1 Introduction to Intrinsic Robustness, Environmental Robustness, and Network Robustness in the Evolution of the Ecological System

4.2 Global Linearization and Finite Difference Methods for the Evolutionary Ecological Network

4.3 Computer Simulation Example

4.4 Conclusion

4.5 Appendix

Part II: Applications of Network Evolution to Systems Synthetic Biology

Chapter 5: Robust Design for Evolutionary Synthetic Gene Networks Under Genetic Mutations and Environmental Disturbances: Genetic Algorithm (GA) Approach in Genotype Space

Abstract

5.1 Introduction

5.2 Tradeoff Between Genetic Robustness, Environmental Robustness, and Network Robustness in Synthetic Biology

5.3 Robust Synthetic Gene Network Design via Network Evolution Through a GA Algorithm

5.4 Robust Synthetic Gene Network Design via Library-Based Network Evolution Through a GA Searching Algorithm

5.5 Computer Simulation Example

5.6 Conclusion

5.7 Appendix

Chapter 6: Robust Design of Genetic Networks: Evolutionary Systems Biology Approach via an Evolutionary Algorithm (EA) in Phenotype Space

Abstract

6.1 Stochastic Model for Biological Systems in vivo Under Intrinsic Genetic Mutation and External Noise

6.2 Robust Design of a Biological Circuit via Evolutionary Systems Biology Through the EA Searching Algorithm

6.3 Design Example In Silico

6.4 Discussion and Conclusion

Chapter 7: On the Adaptive Design Rules of Biochemical Networks in Evolution

Abstract

7.1 Introduction of Adaptive Evolution of Biochemical Networks

7.2 Mathematical Rules for Natural Selection in Biochemical Network Evolution

7.3 Computational Examples

7.4 Conclusion

Part III: Stochastic Evolutionary Game Strategies

Chapter 8: Stochastic Nash Evolutionary Game as a Natural Selection Strategy in a Population of Biological Networks

Abstract

8.1 Introduction to Biological Network Robustness and Evolvability

8.2 Stochastic Evolutionary Game in a Linear Biological Network

8.3 Stochastic Game in the Nonlinear Biological Network

8.4 Global Linearization Approach to the Nonlinear Stochastic Evolutionary Game

8.5 Computer Simulation Example

8.6 Conclusion

8.7 Appendix

Chapter 9: Stochastic Noncooperative and Cooperative Evolutionary Game Strategies of a Population of Biological Networks Under Natural Selection

Abstract

9.1 Review of Evolutionary Game Strategies of Stochastic Biological Networks

9.2 Noncooperative Evolutionary Game Strategy of Stochastic Biological Networks Under Natural Selection

9.3 Cooperative Evolutionary Game Strategy of Stochastic Biological Networks Under Natural Selection

9.4 Simulation Examples

9.5 Discussions and Conclusions

9.6 Appendix

Chapter 10: Evolutionary Game Strategy of an Evolutionary Biological Network of Somatic Cells in the Organ Carcinogenesis and Aging Process

Abstract

10.1 Introduction to an Evolutionary Somatic Cells Network in the Organ Carcinogenesis and Aging

10.2 Stochastic Evolutionary Biological Network of an Organ in Carcinogenesis

10.3 Natural Selection in Carcinogenesis and Aging

10.4 In Silico Example

10.5 Discussion

10.6 Conclusion

10.7 Appendix

Part IV: Evolution Measurements of Biological Networks

Chapter 11: On the System Entropy of Nonlinear Stochastic Biological Networks and Its Relationship to Network Evolution

Abstract

11.1 Introduction to System Entropy and Network Evolution of Biological Networks

11.2 Measuring the System Entropy of Biological Networks

11.3 Example of Calculating System Entropy of Biological Networks

11.4 Conclusion

11.5 Appendix

Chapter 12: On the Evolution Measurement of Somatic Networks by the Changes of Their Robustness and Response Ability in the Aging Process via Microarray Data

Abstract

12.1 Introduction to the Evolutionary Gene Regulatory Network (GRN) in the Aging Process

12.2 Measuring Network Evolution in the Aging Process by the Systems Biological Method via Microarray Data

12.3 Measurements of Network Evolution and Discussion of Evolutionary Network Robustness and Response Ability in the Aging Process

12.4 Conclusion

12.5 Appendix

Chapter 13: Evolution of Network Biomarkers Measured by Microarray Data From Early to Late Stage Bladder Cancer Samples

Abstract

13.1 Introduction to Network Biomarkers of Cancer

13.2 Materials and Evolution Measurement Methods of Network Biomarkers

13.3 Results and Discussion on Evolutionary Network Biomarkers

13.4 Conclusions

13.5 Appendix

References

Index

Copyright

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Dedication

Dedicated to my wife Shiow Fen.

Preface

The famous geneticist Dobzhansky has noted that, nothing in biology makes sense except in light of evolution. Later, Lynch expressed the idea that, nothing in evolution makes sense except in light of population genetics. Recently, Loewe has claimed that, nothing in population genetics makes sense except in light of systems biology. Evidentially, systems evolutionary biology will play an important role in biology in the future.

The recent accumulation of genome-wide data on various facets of gene expression, function, and evolution has stimulated the emergence of a new field: systems evolutionary biology. Systems evolutionary biology is a rapidly growing integrative approach with the core aim of generating mechanistic and evolutionary understanding of genotype-phenotype relationships at multiple levels. Systems evolutionary biology is a novel framework that brings together evolutionary theory and current approaches in systems biology to quantify the small effects of mutations, environmental disturbances, and their epistatic network interactions in silico. Central to systems evolutionary biology are the definitions of network fitness and network robustness that can be computed by some of the current systems biology models by employing rigorous algorithms that are core to many endeavors in computational systems biology. By combining stochastic system models, evolutionary game strategies, and molecular and cellular approaches to evolution, systems evolutionary biology provides new insights and systematic methods to modern evolutionary synthesis while offering ways to enhance its explanatory and predictive capacities.

In this book, a population of evolutionary gene regulatory networks is modeled as a nonlinear stochastic network with random intrinsic (genetic) fluctuation and environmental disturbance. The phenotype of the gene regulatory network is represented by an equilibrium point (attractor) favored by natural selection. Evolvability is defined as the greatest effect of environmental disturbance on the phenotype of the gene network due to the unpredictability of environmental disturbance. Therefore, we can measure the evolvability and fitness of the gene regulatory network via the minimax game method in the evolutionary process. Further, we can obtain the phenotype robustness criterion via network evolutionary game theory. Based on the phenotype robustness criterion, we also discuss the tradeoff between environmental robustness in response to environmental stimuli and genetic robustness in tolerating parameter variations in evolution from the systems biology perspective.

Since biochemical networks are the backbone of the physiological systems of organisms, a biochemical network must be sufficiently robust to tolerate random parametric variations and environmental changes that are due to genetic variation and external noise, respectively, to maintain its functional properties in the evolutionary process. Then, based on the S-system model, we discuss the tradeoff between the evolvability and robustness of a biochemical network in evolution from the systems biology perspective. To exploit systems evolutionary biology in high-level bionetworks, we discuss the tradeoff between network evolvability and the robustness of evolutionary ecological networks, based on nonlinear stochastic partial differential systems.

Because environmental disturbances are unpredictable, based on stochastic evolution game theory, some genetic variations need to be picked up through natural selection to enhance the network’s robustness in tolerating neutral genetic mutations and resisting environmental disturbances in evolution to maintain network phenotype through the minimax game strategy. Therefore, we also discuss the stochastic evolution game in a population of biological networks for natural selection schemes from the minimax strategy perspective in the evolutionary process. As for the application of evolutionary systems biology, we also introduce robust genetic circuit design and robust synthetic gene network based on network evolution methods to systems synthetic biology. In order to tolerate the accumulation of somatic genetic and epigenetic variations in the aging process, a genetic gene network needs to increase its network robustness to maintain its network function. We also discuss the evolution of network robustness in the aging process, that is the increase in network robustness during the aging process with regards to robustness changes of aging-related networks, based on microarray data from mice at different aging stages. Further, cancer is derived by the somatic evolution of cell lineages: molecular biologists have long recognized carcinogenesis and aging as evolutionary processes that involve natural selection. In this book, we derive the natural selection scheme of an evolutionary biological network of carcinogenesis from a robust negative feedback scheme based on the stochastic Nash game strategy. Aging and carcinogenesis are caused by the declining processes of the cellular protection mechanisms of robust negative feedback schemes in the aging and carcinogenesis processes. Biological networks are open systems that can utilize nutrients and energy from their environment in their metabolic process and produce metabolic products. System entropy is an indicator for living or nonliving biological systems, as biological systems can maintain or decrease their system entropy. Network evolvability has been found to be related to system entropy in biological systems. Therefore, we introduce and measure system entropy in biological networks. Finally, we introduce noncooperative and cooperative evolutionary game strategies from the perspective of n-person noncooperative and cooperative Nash stochastic games. We would like to thank Ms. Chiung-Lan Yang and Ms. Chih-Ying Wang for their careful typing of this book.

To Praise the Evolution of Nature

Take a look at the beautiful and bright river!

Entangling and embracing the wonderful land.

It pours its vast love source into a charming bay.

The blue sea is laying down the beach to accept peacefully.

One is penetrating and another is enjoying!

The green land looks like the father of nature, and the blue sea like the mother.

One is responsible for rearing all lives on the land, and another is responsible for bringing up the fishes in the sea.

As the sea plans the clouds and rain to freshen and nourish the land, the land returns the grace, to well up springs and converge streams to the river, giving a gift to fishes in the sea.

Please listen to the singing and shouting from young men on the high mountain!

Fishes enjoyably meet and cheer under the crests of waves too!

Take a look at the graceful scenery with endless mountains and rivers!

The idyllic crisscrossed paths on the far farmland, like a fascinating landscape painting in a good poem!

While all lives are imbibed in the arms of nature, the parents of nature still calmly manipulate and foster there.

Who will drive the evolution of nature?

If not this conflicting but harmonious force between the parents.

Bor-Sen Chen, Tsing Hua Distinguished Chair Professor

Part I

General Theory of Stochastic Evolutionary Biological Network

Chapter 1

Introduction to Systems Evolutionary Biology

Abstract

Systems evolutionary biology is a novel framework to bring evolutionary theories and current systems biology approaches together as a dynamic and increasingly interdisciplinary enterprise. Before the introduction of systems evolutionary biology, several famous geneticists highlighted its important role in biology. Nothing in biology makes sense except in light of evolution, was Dobzhansky's way of highlighting the unifying power of the modern theory of evolution. Nothing in evolution makes sense except in light of population genetics, by Lynch expresses the need for rigorous mechanistic models of evolution, a need that is felt by many evolutionary geneticists. In order to survive, an organism or a biological system needs to adapt to its environment by using the information of itself and its environment to modify behaviors and to solve the life history problem. The ability to adapt involves detecting, responding, and reacting to innumerable perturbations of daily life. This capacity is achieved via complex integration of internal and external stimuli and interactions between multiple control systems from the systems biology perspective. Will it be the case in the future that, nothing in population genetics makes sense except in light of systems biology, as claimed by Loewe? From a more general viewpoint, systems evolutionary biology focuses on one of the core problems of all biology: the evolutionary interplay between the genotype and phenotype. Therefore, systems evolutionary biology is considered a powerful combination of evolutionary genetics with systems biology.

Keywords

Systems evolutionary biology; Stochastic evolutionary network; Evolutionary synthetic network; Stochastic evolutionary game strategy; Evolution measurement; Evolution of network marker; Microarray data

1.1 Introduction to Evolutionary Biology

According to Darwinian natural selection, a biological system needs to select beneficial genetic variations to evolve in response to stochastic environmental disturbances to survive. However, current evolutionary biology theories have rarely considered random genetic variations and stochastic environmental disturbances in their model. Further, there is still no efficient theory or method of natural selection for biological systems to select beneficial genetic variations to adapt with the changes of the environment. Recently, we have gleaned some results on network evolvability, network fitness, and stochastic evolutionary game strategies for the gene regulatory network, the biochemical network, and the ecological network based on nonlinear stochastic game strategies. More recently, genetic algorithm (GA) and evolutionary algorithm (EA), which are inspired by biological evolution to mimic the real natural selection scheme, have been widely applied to efficiently solve complex optimization problems in science and engineering designs. These evolutionary algorithms could be also applied to systems synthetic biology. At present, high throughput microarray data has been widely used to measure system characteristics in systems biology; therefore, this could be employed to measure the evolvability of somatic cells in the aging process. Therefore, it is a good time to introduce evolutionary biology from the systems biology perspective. The ultimate goal of systems biology is to achieve an integrated understanding of lifeforms with all their characteristic complexity of interactions at multiple system levels. However, such an understanding is unimaginable without an essential evolutionary component; that is, deciphering the ways in which biological systems change over time, which changes are affected by selection and which changes are neutral, and how changes at one level of a system reflect on evolution at another level.

The significant role of evolution in biology has been enhanced by several evolutionary biologists [1–3]. In genetics and genomics, mutations with weak effects on fitness that interact with each other are of great interest, as their long-term consequences are much harder to predict than those of mutations with large effects [4]. Recently, systems biology has accumulated much data on genetic mutations with relatively large effects by using experimental methods and theoretical tools such as flux balance analysis, which could analyze the flux of metabolites in biochemical reaction networks [5–14]. However, mutations with small effects are not easily analyzed with stoichiometric modeling techniques such as flux balance analysis, and wet lab observations are usually not sensitive enough to analyze many effects of interest for questions of long-term stability [4]. Thus, kinetic modeling techniques are required, which are frequently based on ordinary differential equations. Recently, new techniques, hybrid approaches, and equivalences between existing techniques have been constantly developed to combine flux balance analysis and metabolic control theory [15] or to translate between stochastic simulations and ordinary differential equations. The pace of theoretical and experimental developments in systems biology raises the possibility that realistic quantitative models of many subsystems of living organisms might become available for systems biology in the future [4]. At the heart of such effort in systems biology is the construction of realistic and reliable systematic models of the mechanistic realities of life. This new systematic approach is well suited for investigating mutations with very small effects, which are particularly difficult to quantify by other methods.

1.2 Review of Current Systems Biology and Evolutionary Theory

Before the introduction of evolutionary systems biology, we will quickly review progress in molecular biology and evolution theory separately. Molecular biology has a strong tradition of inferring molecular interactions from well-designed experiments that produce clear-cut results and require little or no quantitative analyses. The success of molecular cell biology and related disciplines has led to the accumulation of so much knowledge that further progress increasingly depends on detailed quantitative models [16, 17]. Therefore, some experimentalists have started to collaborate with theoreticians to develop some models to lead to the emergence of current systems biology [18–20]. These models aim to capture the essence of important intercellular interactions of the system under investigation. An important goal using the tools that are needed for analyzing models of these biological interactions was highlighted in the evolution of microbiology [4, 12]. Some ambitious long-term goals such as building a virtual cell [21], a virtual plant [22], or even a virtual human [23], which could then help with designing drugs by predicting undesirable interactions in silico [24], will go further by working toward them in future. We could propose utilizing the momentum in current systems biology to lay the foundations for building such high-level models [4]. This work should not be prohibitively complicated in systems, where most of the hard work will be done independently by current systems biology approaches. This hard work will be to produce accurate mechanistic models of the molecular machinery of some interesting aspect of life [24]. In order to make such work fruitful for evolutionary systems biology, we propose to extend these models so that network fitness can be computed. A successful synthesis of knowledge from systems biology and evolutionary approaches is in Refs. [25–31].

Evolutionary theory has a long tradition of using mathematical models in population genetic theory that frequently abstract biological details [25–31]. For example [32–35], the concept of fitness is a powerful and widely used abstraction that reduces all molecular, developmental, biochemical, cellular, neuronal, behavioral, physiological, and other intraindividual biology to a single number, which can usually be defined as the average number of offspring that will effectively reach the next generation in a certain environment. As for the concept of a selection coefficient, it is a similarly successful abstraction as it simply summarizes the effects of a new mutation on fitness [4]. Hence this allows the classification of mutations according to their long-term evolutionary behavior: deleterious mutations will be selected against and thus never get fixed in a population; effectively neutral mutations will be accumulated by random drift as if they had no effect on fitness; and advantageous mutations will be accumulated faster than neutral ones due to positive selection [4]. When this simple caricature omits the transitions between these extremes, the related mathematical computation theory exists to compute the relevant details. The corresponding population genetics work is one of the scientific successes of the 20th century [29, 30, 35]. Further, the rigorous nature of many population genetics models and their extensive analysis has led to key insights into analyzing genomic sequences [36–38]. Additionally, experimental evolution approaches have contributed much to our understanding of evolution, especially in microbes and RNA, to allow experimental paleontology for going back in time and dissecting evolutionary events in detail [39–42]. Recently, however, some works have also shown that the simple molecular biological assumptions behind many of the evolutionary analysis methods limit their applicability. For instance, not all synonymous mutations are effective neutral [43–45], gene order is not random [46], back-mutations and compensatory mutations can be important [46, 47], and epistatic interactions between mutations are frequent. If the distribution of deleterious and advantageous mutational effects was known along with the distribution of epistatic effects, many analyses could be more rigorous; further progress in analyzing evolution will require increasingly realistic models of the underlying molecular interactions.

1.3 Systems Evolutionary Biology as a Powerful Combination of Evolutionary Genetics With Systems Biology

As evolutionary genetics and molecular biology have been very successful in furthering our understanding of the natural world, we propose that combining them even more closely with the help of current systems biology models will significantly improve their power. Also, the enthusiasm for the quantitative description of mechanistic processes in current systems biology could benefit from and contribute to the evolutionary biology objective to understand the force shaping the existing diversity of life [4]. Therefore, a functional synthesis of experimental molecular biology, evolutionary biology, and systems biology models is combined as systems evolutionary biology. In this combined systems evolutionary biology, systems biology can provide maps from genotypes to phenotypes that are much closer to reality than the simple models often used in evolutionary genetics. These maps in the form of computational models can allow the quick assessment of the effects of the change in the biological system. Further, evolutionary genetics can help analyze effects that are important in the long term but too small for observation in any laboratory [48, 49]. It could also help identify genetic structures that are no longer optimal due to a relaxation of purifying selection. Because the molecular functionality of such structures can no longer be assumed to be optimal, deleterious mutations may have compromised functional integrity [2, 3].

Mathematical models at the molecular level can be used to compute the probabilities of accessing particular genotypes by mutation and recombination. Systems biology models then might compute the key phenotypic properties of the corresponding genotypes. Then these models could together predict how far in genotype space an offspring individual can move from its parents and the functional consequences of a given phenotype move [4]. Therefore, they might also allow the computation of an approximate fitness function to determine the effect of any particular genotypic change on fitness-related properties. Predicting some phenotypic properties from genetic information and a current systems biology model has been shown to be possible for some biological systems [50–52]. Models at the population level could then be used to predict population sizes, population structures, and consequences of resulting genetic drift, migration, and selection in the evolution process. Natural selection might be linked to the phenotypic properties computed by molecular models by identifying their ecological meaning in terms of survival probabilities and rates of reproduction in specified environments. These models could then compute the fate of new mutations and as a consequence they might predict long-term evolutionary changes for a whole range of biological systems between single populations and whole ecosystems [4]. If molecular and population models are combined at a very high level, one can envision the formulation of an entirely new mechanistic evolutionary hypothesis. The central role of computational systems biology models is to extend the applicability of this framework beyond experimental molecular biology and evolution to elsewhere. We propose utilizing the momentum in current systems biology to lay the foundation for building such high-level models to produce reasonably accurate mechanistic models of molecular machinery of some interesting aspect of life. In order to make such work fruitful for systems evolutionary biology, we need to extend these models so that we can compute network fitness in the evolutionary process.

1.4 The Scope of the Book

This book provides a perspective on a new framework that can help bring together evolutionary theory and current systems biology, which have much to offer each other. This book is divided into four parts. In Part I, a general theory of stochastic evolutionary network is introduced from the systems biology perspective. In Part II, applications of network evolution to systematic designs of gene regulatory networks and biochemical networks in synthetic biology are given. In Part III, stochastic evolutionary game strategies of a population of biological networks are introduced from the noncooperative and cooperative from the systems biology perspective. In Part IV, the evolvability of biological network is measured by system entropy and microarray data.

Part I of this book investigates and calculates fitness and evolvability of biological networks. In Chapter 2, based on the nonlinear stochastic system models of biological networks, the evolvability of a biological network is defined as the worst-case response to all possible environmental disturbances; the fitness of a biological network is just an inverse proportion to the evolvability. The calculation of the evolvability (or fitness) of a linear gene regulatory network needs to solve a linear matrix inequality (LMI)-constrained optimization problem. For the calculation of the evolvability (or fitness) of a nonlinear stochastic gene regulatory network, we need to solve a Hamilton-Jacobi inequality (HJI)-constrained optimization problem. By the global linearization technique to interpolate several local linear stochastic systems to approach a nonlinear stochastic gene regulation network, HJI could be approached by a set of LMIs. Therefore, the evolvability (or fitness) of nonlinear stochastic gene regulatory systems in evolution could be calculated by solving an LMIs-constrained optimization problem. In general, the phenotype of nonlinear stochastic gene regulatory networks is defined at a stable equilibrium point (attractor) in the phenotype landscape. Based on the robust stability theory, the phenotype robust criterion for a gene regulatory network in evolution is derived in Chapter 3. From the phenotype robust criterion, the interplay between the network evolvability and network robustness is also investigated from the systems biology perspective [53].

Further, based on the S-system model, the interplay between the evolvability and robustness of biochemical networks in evolution is also discussed in Chapter 3, based on the robust stability theory of biochemical networks at the steady state [53]. We found that if the accumulation of intrinsic parametric perturbation is still less than the system matrix, then the phenotype of a biochemical network is still robust. Otherwise, it will shift to another equilibrium point (phenotype). In general, it is difficult to analyze the evolvability or fitness of ecological systems because ecological networks are complex nonlinear partial differential stochastic systems. In Chapter 4, based on global linearization, central difference, and the spatial state space method, a unifying mathematical framework is introduced for investigating the principles of both robust stabilization and environmental disturbance sensitivity (evolvability) in ecological networks. From the unifying mathematical framework [54–56], we found that the phenotype robustness criterion for ecological networks is that if intrinsic robustness + environmental network robustness ≤ network robustness, then the phenotype robustness of an ecological network can be maintained in spite of intrinsic parameter fluctuations and environmental disturbances in evolution.

Recently, evolutionary theory and natural selection have facilitated a deeper understanding of the adaptive landscape in evolutionary biology. This may motivate exchanges with the other two fields that investigate adaptive landscapes and provide wide applications [4]: evolutionary computation and artificial life. The designs of genetic circuits and biochemical networks with high-dimensional functional landscapes with the help of molecular systems biological models can be demonstrated by an artificial EA and GA. These consist of artificial reproduction, crossover, and mutation operators to search for the maximum fitness of a biological network; they are motivated by evolutionary theory and natural selection. After discussing the general theory of systems evolutionary biology, the robust design of a biological network or synthetic gene circuit with a desired phenotype or behavior under genetic variations and environmental disturbances can be applied by a network evolution method through the EA and GA searching (selecting) algorithm. In Part II, this book provides several genetic and chemical design examples in Chapters 5–7 to highlight the applications of EA and GA algorithms to robust synthetic gene networks under genetic mutations and environmental disturbances in vivo [57–59].

A population of evolutionary biological networks can be described by a stochastic dynamic system with intrinsic random parameter fluctuations due to genetic variations and external disturbances caused by environmental changes in the evolutionary process. Because information on environmental changes is unavailable and their occurrence is unpredictable, they can be considered as a game player with the potential to destroy phenotypic stability as much as possible. Therefore, natural selection can be considered as another game player in the evolutionary process, that is, a stochastic Nash game of minimizing the maximum network evolution level caused by the worst environmental disturbances. Based on the nonlinear stochastic game strategy, we find that some genetic variations can be used in natural selection to construct negative feedback loops, efficiently improving network robustness. This provides larger genetic robustness as a buffer against neutral genetic variations as well as larger environmental robustness to resist environmental disturbances and maintain a network phenotypic trait in the evolutionary process. In this situation, the robust phenotypic traits of stochastic biological networks can be more frequently selected by natural selection in evolution. However, if the harbored neutral genetic variations accumulate to a sufficiently large amount, and environmental disturbances are strong enough that the network robustness can no longer confer enough genetic robustness and environmental robustness, then the phenotype robustness criterion might break down. In this case, a network phenotypic trait may be pushed from one equilibrium point to another, changing the phenotypic trait and starting a new phase of network evolution through the hidden neutral genetic variations harbored in network robustness by adaptive evolution. Therefore, Part III of this book will discuss the stochastic evolutionary game for a population of biological networks under natural selection in Chapters 8–10.

Evolution by natural selection is an evolution game in the sense that it has players, strategies, and payoffs. The players are the individuals to select their more beneficial strategies with better payoffs. Noncooperative and cooperative game strategies of a nonlinear stochastic biological network with Poisson genetic variations and environmental disturbances are introduced in Chapter 9 as n-person noncooperative and cooperative Nash stochastic game problems, respectively. Noncooperative evolutionary game strategy is transformed to a multiobjective optimization problem, which could be easily solved by the multiobjective evolutionary algorithm (MOEA). Cooperative evolutionary game strategy is transformed to a single-objective optimization problem, which can be easily solved by the EA.

Cancer and aging are driven by somatic evolution of cell lineages. In order to maintain the phenotype of an evolutionary biological network against random intrinsic fluctuations and environmental disturbances, a network robustness scheme that incorporates natural selection needs to be developed in the somatic cell evolution process. This can be accomplished by selecting certain genetic and epigenetic variations to modify the network structure to attenuate intrinsic fluctuations efficiently, and to resist environmental disturbances consistently in order to maintain the phenotype of the evolutionary biological network at an equilibrium point. In Chapter 10, a natural selection scheme of an evolutionary biological network of somatic cells is derived from a robust negative feedback scheme to investigate carcinogenesis in the aging process, based on the nonlinear stochastic Nash game strategy.

In the previous three parts, we focused more on the dynamic system theoretical analysis and application of network evolution. In Part IV, we will focus on the measurement of network evolution through system entropy and microarray data of biological systems. Because the somatic evolution of cell lineages is related to aging and carcinogenesis and can be measured from the changes of systematic characters via microarray data, Part IV of this book will introduce some measurement techniques of network evolution.

Entropy is a measure of randomness or disorder in a physical system due to intrinsic random fluctuations and environmental disturbances. Biosystems are open irreversible stochastic thermodynamic systems. It is difficult to measure the entropy of biological systems. In Chapter 11, we will emphasize that system entropy is a systematic characteristic of biological systems and then investigate the relationship between system entropy and the evolvability of biological networks. We found that the system entropy is proportional to the evolvability of biological networks, which are both inversely proportional to their robustness and stability, and that intrinsic random fluctuations can increase both the system entropy and evolvability of biological networks.

Aging or carcinogenesis is driven by somatic evolution of cell lineages that have escaped controls on replication and by the population-level evolution of genes that influence aging or carcinogenesis risk [60, 61]. Molecular biologists have long recognized aging and carcinogenesis as an evolutionary process involving natural selection among renegade cells [62, 63]. In order to tolerate the accumulated intrinsic somatic genetic variations and environmental stress to maintain their normal functions in the aging process, genetic networks must increase their network robustness in the aging process. By the network robustness measurement through microarray data, we found that the genetic network increases its network robustness in the aging process [62]. Similarly, cancer cells need to tolerate many genetic variations and resist environmental stresses in the carcinogenesis process, the network robustness of cancer cells was found larger than that of the normal cell [63]. In Chapter 12, evolutions of network robustness in aging and carcinogenesis processes are estimated and discussed from their time profile microarray data.

Because network biomarkers are very important systematic characters of biological systems for investigating the corresponding molecular mechanisms and discovering therapeutic drug targets from the systems biology perspective, in Chapter 13 the evolution of network markers will be measured by microarray data of different stages of bladder cancer to discuss the evolutionary network biomarker of somatic cells in a carcinogenic process.

References☆

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