Optimization Methods in Metabolic Networks

Provides a tutorial on the computational tools that use mathematical optimization concepts and representations for the curation, analysis and redesign of metabolic networks

• Organizes, for the first time, the fundamentals of mathematical optimization in the context of metabolic network analysis
• Reviews the fundamentals of different classes of optimization problems including LP, MILP, MLP and MINLP
• Explains the most efficient ways of formulating a biological problem using mathematical optimization
• Reviews a variety of relevant problems in metabolic network curation, analysis and redesign with an emphasis on details of optimization formulations
• Provides a detailed treatment of bilevel optimization techniques for computational strain design and other relevant problems

Preface xiii

1 Mathematical Optimization Fundamentals 1

1.1 Mathematical Optimization and Modeling 1

1.2 Basic Concepts and Definitions 7

1.2.1 Neighborhood of a Point 7

1.2.2 Interior of a Set 7

1.2.3 Open Set 8

1.2.4 Closure of a Set 8

1.2.5 Closed Set 8

1.2.6 Bounded Set 8

1.2.7 Compact Set 8

1.2.8 Continuous Functions 9

1.2.9 Global and Local Minima 9

1.2.10 Existence of an Optimal Solution 9

1.3 Convex Analysis 10

1.3.1 Convex Sets and Their Properties 10

1.3.2 Convex Functions and Their Properties 13

1.3.3 Convex Optimization Problems 19

1.3.4 Generalization of Convex Functions 20

Exercises 20

References 22

2 LP and Duality Theory 23

2.1 Canonical and Standard Forms of an LP Problem 23

2.1.1 Canonical Form 24

2.1.2 Standard Form 24

2.2 Geometric Interpretation of an LP Problem 26

2.3 Basic Feasible Solutions 28

2.4 Simplex Method 30

2.5 Duality in Linear Programming 35

2.5.1 Formulation of the Dual Problem 35

2.5.2 Primal‐Dual Relations 38

2.5.3 The Karush‐Kuhn‐Tucker (KKT) Optimality Conditions 39

2.5.4 Economic Interpretation of the Dual Variables 40

2.6 Nonlinear Optimization Problems that can be Transformed into LP Problems 45

2.6.1 Absolute Values in the Objective Function 45

2.6.2 Minmax Optimization Problems with Linear Constraints 46

2.6.3 Linear Fractional Programming 47

Exercises 49

References 50

3 Flux Balance Analysis and LP Problems 53

3.1 Mathematical Modeling of Metabolism 54

3.1.1 Kinetic Modeling of Metabolism 54

3.1.2 Stoichiometric-Based Modeling of Metabolism 54

3.2 Genome‐Scale Stoichiometric Models of Metabolism 55

3.2.1 Gene–Protein–Reaction Associations 55

3.2.2 The Biomass Reaction 56

3.2.3 Metabolite Compartments 57

3.2.4 Scope and Applications 57

3.3 Flux Balance Analysis (FBA) 57

3.3.1 Cellular Inputs, Outputs and Metabolic Sinks 58

3.3.2 Component Balances 59

3.3.3 Thermodynamic and Capacity Constraints 60

3.3.4 Objective Function 61

3.3.5 FBA Optimization Formulation 62

3.4 Simulating Gene Knockouts 67

3.5 Maximum Theoretical Yield 68

3.5.1 Maximum Theoretical Yield of Product Formation 68

3.5.2 Biomass vs. Product Trade‐Off 69

3.6 Flux Variability Analysis (Fva) 71

3.7 Flux Coupling Analysis 73

Exercises 77

References 78

4 Modeling with Binary Variables and MILP Fundamentals 81

4.1 Modeling with Binary Variables 83

4.1.1 Continuous Variable On/Off Switching 83

4.1.2 Condition‐Dependent Variable Switching 83

4.1.3 Condition‐Dependent Constraint Switching 84

4.1.4 Modeling AND Relations 84

4.1.5 Modeling OR Relations 86

4.1.6 Exact Linearization of the Product of a Continuous and a Binary Variable 86

4.1.7 Modeling Piecewise Linear Functions 87

4.2 Solving Milp Problems 89

4.2.1 Branch‐and‐Bound Procedure for Solving MILP Problems 90

4.2.2 Finding Alternative Optimal Integer Solutions 97

4.3 Efficient Formulation Strategies for Milp Problems 97

4.3.1 Using the Fewest Possible Binary Variables 97

4.3.2 Fix All Binary Variables that do not Affect the Optimal Solution 98

4.3.3 Group All Coupled Binary Variables 98

4.3.4 Segregate Binary Variables in Constraints Rather than in the Objective Function 98

4.3.5 Use Tight Bounds for All Continuous Variables 99

4.3.6 Introduce LP Relaxation Tightening Constraints 99

4.4 Identifying Minimal Reaction Sets Supporting Growth 102

Exercises 104

References 106

5 T hermodynamic Analysis of Metabolic Networks 107

5.1 Thermodynamic Assessment of Reaction Directionality 107

5.2 Eliminating Thermodynamically Infeasible Cycles (TICs) 109

5.2.1 Cycles in Cellular Metabolism 109

5.2.2 Thermodynamically Infeasible Cycles 110

5.2.3 Identifying Reactions Participating in TICs 111

5.2.4 Thermodynamics‐Based Metabolic Flux Analysis 111

5.2.5 Elimination of the TICs by Applying the Loop Law 113

5.2.6 Elimination of the TICs by Modifying the Metabolic Model 115

Exercises 116

References 117

6 Resolving Network Gaps and Growth Prediction Inconsistencies in Metabolic Networks 119

6.1 Finding and Filling Network Gaps in Metabolic Models 119

6.1.1 Categorization of Gaps in a Metabolic Model 119

6.1.2 Gap Finding 120

6.1.3 Gap Filling 123

6.2 Resolving Growth Prediction Inconsistencies 126

6.2.1 Quality Metrics for Quantifying the Accuracy of Metabolic Models 127

6.2.2 Automated Reconciliation of Growth Prediction Inconsistencies Using GrowMatch 127

6.2.3 Resolution of Higher‐Order Gene Deletion Inconsistencies 130

6.3 Verification of Model Correction Strategies 132

Exercise 133

References 133

7 Identification of Connected Paths to Target Metabolites 137

7.1 Using Milp to Identify Shortest Paths in Metabolic Graphs 137

7.2 Using Milp to Identify Non‐Native Reactions for the Production of a Target Metabolite 142

7.3 Designing Overall Stoichiometric Conversions 144

7.3.1 Determining the Stoichiometry of Overall Conversion 144

7.3.2 Identifying Reactions Steps Conforming to the Identified Overall Stoichiometry 146

Exercises 151

References 151

8 Computational Strain Design 155

8.1 Early Computational Treatment of Strain Design 156

8.2 Optknock 158

8.2.1 Solution Procedure for OptKnock 159

8.2.2 Improving the Computational Efficiency of OptKnock 164

8.2.3 Connecting Reaction Eliminations with Gene Knockouts 165

8.2.4 Impact of Knockouts on the Biomass vs. Product Trade‐Off 165

8.3 Optknock Modifications 167

8.3.1 RobustKnock 167

8.3.2 Tilting the Objective Function 168

8.4 Other Strain Design Algorithms 168

Exercises 170

References 171

9 N LP Fundamentals 173

9.1 Unconstrained Nonlinear Optimization 173

9.1.1 Optimality Conditions for Unconstrained Optimization Problems 174

9.1.2 An Overview of the Solution Methods for Unconstrained Optimization Problems 176

9.1.3 Steepest Descent (Cauchy or Gradient) Method 176

9.1.4 Newton’s Method 177

9.1.5 Quasi‐Newton Methods 178

9.1.6 Conjugate Gradients (CG) Methods 179

9.2 Constrained Nonlinear Optimization 180

9.2.1 Equality‐Constrained Nonlinear Problems 180

9.2.2 Nonlinear Problems with Equality and Inequality Constraints 186

9.2.3 Karush–Kuhn–Tucker Optimality Conditions 187

9.2.4 Sequential (Successive) Quadratic Programming 189

9.2.5 Generalized Reduced Gradient 192

9.3 Lagrangian Duality Theory 195

9.3.1 Relationships between the Primal and Dual Problems 196

Exercises 196

References 197

10 N LP Applications in Metabolic Networks 199

10.1 Minimization of the Metabolic Adjustment 199

10.2 Incorporation of Kinetic Expressions in Stoichiometric Models 203

10.3 Metabolic Flux Analysis (Mfa) 206

10.3.1 Definition of the Relevant Parameters and Variables 208

10.3.2 Isotopomer Mass Balance 214

10.3.3 Optimization Formulation for MFA 215

Exercises 218

References 220

11 Minlp Fundamentals and Applications 223

11.1 An Overview of the Minlp Solution Procedures 224

11.2 Generalized Benders Decomposition 224

11.2.1 The Primal Problem 225

11.2.2 The Master Problem 226

11.2.3 Steps of the GBD Algorithm 229

11.3 Outer Approximation 230

11.3.1 The Primal Problem 231

11.3.2 The Master Problem 231

11.3.3 Steps of the OA Algorithm 235

11.4 Outer Approximation With Equality Relaxation 236

11.4.1 The Master Problem 237

11.5 Kinetic Optknock 238

11.5.1 k‐OptKnock Formulation 239

11.5.2 Solution Procedure for k‐OptKnock 240

Exercises 242

References 243

Appendix A 245

Index 257

Costas D. Maranas is a Donald B. Broughton Professor in the Department of Chemical Engineering at Pennsylvania State University, USA. Dr. Maranas is a Fellow of the American Institute of Medical and Biological Engineering (AIMBE). In 2002 he was awarded by AIChE the Allan P. Colburn Award for Excellence in Publications by a Young Member of the Institute.

Ali R. Zomorrodi obtained his PhD in Chemical Engineering at Pennsylvania State University and is currently a Postdoctoral Research Associate at Boston University, USA. Dr. Zomorrodi's areas of expertise include optimization-based modeling and model-driven analysis of biological networks.

Provides a tutorial on the computational tools that use mathematical optimization concepts and representations for the curation, analysis and redesign of metabolic networks

Analysis and redesign of metabolic networks often requires the calculation of product yield, gene essentiality prediction, identification of network gaps in the model, resolution of discrepancies with experimental data and identification of network modifications that increase product yield. The main goal of Optimization Methods in Metabolic Networks is to apply the language and tools of mathematical programming to describe and solve such frequently occurring tasks.

Topics covered in this book start with a formal treatment of the relevant optimization problem class followed by relevant metabolic network applications. The class of optimization problems becomes progressively more complex, starting with linear programming (LP) and mixed-integer linear programming (MILP) problems and concluding with nonlinear programming (NLP) and mixed-integer nonlinear programming (MINLP) problems.

Optimization Methods in Metabolic Networks:

• Organizes, for the first time, the fundamentals of mathematical optimization in the context of metabolic network analysis
• Reviews the foundational principles of different classes of optimization problems including LP, MILP, NLP and MINLP
• Explains how to formulate a metabolic network analysis or redesign task using mathematical optimization
• Reviews a variety of problems in metabolic network curation, analysis and redesign with an emphasis on the details of the optimization formulations
• Provides a detailed treatment of bilevel optimization techniques for computational strain design and other relevant applications
• Provides input files for examples presented on an accompanying website
• Includes problems and exercises for helping to reinforce the introduced concepts

Optimization Methods in Metabolic Networks can be used to introduce students with the knowledge of metabolism to formal mathematical treatments of core computational tasks in metabolic networks or alternatively expose students with a mathematical programming background to metabolism. The hope is that this book will serve as a starting point for more in-depth investigations of relevant techniques and concepts found in the cited literature.