Unifying metabolic networks, regulatory constraints, and resource allocation Dissertation Zur Erlangung des Grades Doctor rerum naturalium Fachbereich Mathematik und Informatik Freie Universitat¨ Berlin Lin Liu Berlin, 2020 Betreuer: Prof. Dr. Alexander Bockmayr Zweitgutachter: Dr. Hidde de Jong Tag der Disputation: 16. April 2020 Contents Abstract vii 1 Introduction1 1.1 Metabolic networks.........................3 1.2 Computational modeling of metabolism..............4 1.2.1 Kinetic modeling......................4 1.2.2 Constraint-based modeling.................7 1.2.3 Resource allocation modeling...............9 1.3 Modeling gene regulation...................... 10 1.3.1 Hill functions........................ 10 1.3.2 Piecewise linear differential equations........... 11 1.3.3 Boolean logical networks.................. 12 1.4 Integration of metabolism and regulation.............. 13 1.4.1 Hybrid modeling...................... 13 1.4.2 Constraint-based modeling with genetic regulation.... 14 1.5 Structure of the thesis........................ 15 2 Extensions of classic FBA 17 2.1 Dynamic flux balance analysis................... 17 2.1.1 Updating biomass and external metabolites........ 18 2.1.2 Updating constraints on uptake fluxes........... 19 2.1.3 Static optimization problem................ 19 2.2 Regulatory flux balance analysis.................. 20 2.3 Resource allocation analysis.................... 22 i Contents 2.3.1 Metabolic-genetic networks................ 22 2.3.2 Steady-state assumption during growth phase....... 24 2.3.3 Resource allocation constraints............... 25 2.3.4 Non-linear optimization problem.............. 27 2.3.5 Discussion......................... 28 2.4 Dynamic enzyme-cost flux balance analysis............ 29 2.4.1 Metabolic constraints and objective............ 29 2.4.2 Formulation of deFBA................... 32 2.4.3 Discretization of the variables in time........... 32 2.5 Conclusions............................. 33 3 Exploring the optimal solution space in rFBA 35 3.1 Introduction............................. 35 3.2 Analytic pipeline.......................... 36 3.2.1 FVA............................. 37 3.2.2 Characterizing the optimal solution space......... 37 3.2.3 Decomposing vertices into EFMs............. 38 3.2.4 Analytic pipeline...................... 38 3.3 Case study on a core carbon network................ 39 3.3.1 Network description.................... 39 3.3.2 Results........................... 41 3.3.3 Conclusions......................... 45 3.4 Case study on the central metabolic network of E. coli ...... 45 3.4.1 Network description.................... 45 3.4.2 Results........................... 46 3.4.3 Conclusions......................... 52 3.5 Discussion............................. 53 4 Iterating RBA incorporating regulatory rules 59 4.1 Introduction............................. 59 4.2 iRBA................................ 61 4.2.1 Dynamics of biomass and external metabolites...... 61 ii Contents 4.2.2 Updating uptake fluxes of nutrients............ 61 4.2.3 Static non-linear optimization problem.......... 62 4.2.4 Discussion......................... 63 4.3 riRBA................................ 64 4.3.1 Regulatory constraints................... 64 4.3.2 Static non-linear optimization problem with regulatory con- straints........................... 65 4.4 Comparison between DFBA, rFBA, iRBA and riRBA....... 66 4.4.1 A core carbon metabolic-genetic network and regulatory rules............................. 66 4.4.2 Investigating the relationship between fixed quota com- pound and maximal growth rate.............. 69 4.4.3 Predictions by riRBA, iRBA, rFBA and DFBA...... 69 4.5 Conclusions and discussion..................... 72 5 Formalizing metabolic-regulatory networks by hybrid automata 75 5.1 Introduction............................. 75 5.2 Construction of the MRN model.................. 77 5.3 Hybrid discrete-continuous dynamics............... 78 5.3.1 Continuous variables.................... 78 5.3.2 Discrete states....................... 79 5.4 Combining discrete and continuous dynamics in a hybrid automaton 79 5.5 Biological application........................ 81 5.5.1 MRN model of the diauxic shift.............. 81 5.5.2 Hybrid automaton model of the diauxic shift....... 82 5.5.3 Exploring the dynamics of Hdiaux ............. 84 5.6 Conclusion............................. 88 6 Regulatory dynamic enzyme-cost flux balance analysis 91 6.1 Introduction............................. 91 6.2 Hybrid dynamics of metabolic-regulatory networks........ 93 6.2.1 Continuous dynamics.................... 93 6.2.2 Discrete control....................... 93 iii Contents 6.2.3 Hybrid discrete-continuous system............. 94 6.3 Formalization of r-deFBA...................... 94 6.3.1 Metabolic constraints.................... 94 6.3.2 Regulatory logical control constraints........... 95 6.3.3 Formulating r-deFBA as a dynamic optimization problem 96 6.4 Numerically solving r-deFBA as a MILP.............. 97 6.4.1 Transforming logical functions into linear inequalities.. 97 6.4.2 Discretizing the variables in time to solve r-deFBA.... 99 6.5 Biological Application 1 on CCR model.............. 101 6.5.1 r-deFBA and deFBA model of CCR............ 101 6.5.2 Comparing r-deFBA, deFBA, and the hybrid automaton. 103 6.6 Biological Application 2: core carbon metabolism......... 105 6.6.1 MRN model of the core carbon network.......... 105 6.6.2 r-deFBA vs deFBA model of core carbon network.... 107 6.6.3 Comparing r-deFBA and deFBA.............. 107 6.7 Conclusion............................. 114 7 Perspectives: Formalizing metabolic-regulatory networks at population- level by product automata 115 7.1 Introduction............................. 115 7.2 Hybrid system of the composition of MRNs............ 117 7.2.1 Continuous variables.................... 117 7.2.2 Discrete states....................... 118 7.3 Combining discrete and continuous dynamics in a product automaton119 7.4 Biological applications....................... 121 7.4.1 Modeling competitiveness of cells having different βR .. 121 7.4.2 A community consisting of activator and repressor strains 124 7.5 Conclusion and discussion..................... 131 8 Conclusion 133 Bibliography 146 iv Contents Zusammenfassung 147 Acknowledgements 149 Declaration 151 v Abstract Metabolic and gene regulatory networks are two classic models of systems biology. Biologically, gene regulatory networks are the control system of protein expression while metabolic networks, especially the genome-scale reconstructions consist of thousands of enzymatic reactions breaking down nutrients into precursors and en- ergy to support the cellular survival. Metabolic-genetic networks, in addition, in- clude the translational processes as an integrated model of classical metabolic net- works and the gene expression machinery. Conversely, genetic regulation is also affected by the metabolic activities that provide feedbacks and precursors to the regulatory system. Thus, the two systems are highly interactive and depend on each other. Up to now, various efforts have been made to bridge the two network types. Yet, the dynamic integration of metabolic networks and genetic regulation remains a major challenge in computational systems biology. This PhD thesis is a contribution to mathematical modeling approaches for study- ing metabolic-regulatory systems. Inspired by regulatory flux balance analysis (rFBA), we first propose an analytic pipeline to explore the optimal solution space in rFBA. Then, our efforts focus on the dynamic combination of metabolic net- works together with enzyme production costs and genetic regulation. For this purpose, we first explore the intuitive idea that incorporates Boolean regulatory rules while iterating resource balance analysis. However, with the iterative strat- egy, the gene expression states are only updated in discrete time steps. Further- more, formalizing the metabolic-regulatory networks (MRNs) by hybrid automata provides a new mathematical framework that allows the quantitative integration of the metabolic-genetic network with the genetic regulation in a hybrid discrete- continuous system. For the application of this theoretical formalization, we de- velop a constraint-based approach regulatory dynamic enzyme-cost flux balance analysis (r-deFBA) as an optimal control strategy for the hybrid automata repre- senting MRNs. This allows the prediction of optimal regulatory state transitions, dynamics of metabolism, and resource allocation capable of achieving a maximal biomass production over a time interval. Finally, this PhD project ends with a chapter on perspectives; we apply the theory of product automata to model the dynamics at population-level, integrating continuous metabolism and discrete reg- ulatory states. vii Chapter 1 Introduction Since the days of Claude Shannon and Norbert Wiener, who introduced informa- tion theory (Shannon, 1948) and cybernetics (Wiener, 1948), the system-level un- derstanding of biology using information and communication has gained much at- tention. People noticed that a biological system is not merely composed of different kinds of chemical molecules with a certain structure. Organisms, including human beings, operate like a machine based on the processing of information between dif- ferent components such as genes and proteins (Quastler, 1953; Simon, 1991). This involves the interconnection of multiple biological parts and an enormous amount of information communicated as a huge network (Hartwell et al.,