Modular decomposition of a metabolic network

Transcription

1 UNIVERSITY OF CALIFORNIA Santa Barbara Modular decomposition of a metabolic network A master s thesis submitted in partial satisfaction of the requirements for the degree of Master of Science in Electrical and Computer Engineering by Kaili Shen Committee in Charge: Professor João P. Hespanha, Chair Professor Brad Paden Professor Katie Byl September 2013

2 The master s thesis of Kaili Shen is approved: Professor Brad Paden Professor Katie Byl Professor João P. Hespanha, Committee Chair September 2013

3 Modular decomposition of a metabolic network Copyright c 2013 by Kaili Shen iii

4 Abstract The modular decomposition of a metabolic network is discussed. We show that a metabolic network can be decomposed into isolated functional modules using the spectral method with knowledge of the chemical reactions alone. Moreover, it can be seen that the modular decomposition can be improved by taking the reaction dynamics into account. We present our work using the metabolic network obtained from a cell simulation model called Whole-Cell Model.

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6 Contents Abstract iii 1. Introduction 1 2. Whole-Cell Model Introduction Simulation Algorithm Cellular Processes Metabolism Process Results Metabolic network Graph partition with equally weighted interconnections Supply of cofactors Production of substrates Transport of substrates Graph partition with inequally weighted interconnections Rate constants Equilibrium concentrations Conclusion and future work 23 A. Appendix 25 A.1. Flux-balance analysis A.2. Metabolic network A.2.1. Content of metabolic network A.2.2. Median rate constants

7 vi Contents A.2.3. Modules within the metabolic network Bibliography 39

8 List of Figures 2.1. Whole-Cell Model: Architecture WholeCell Model: Simulation Algorithm Whole-Cell Model: Cellular Processes Role of metabolites in cell Network example with equally weighing interconnections Graph partition with equally weighing interconnections Comparison: Graph partition by spectral method Histogram of rate constants Histogram of rate constants A.1. FBA: Network example

9 viii List of Figures

10 CHAPTER 1 Introduction Biological systems are inherently complex because they consist of several entities that interact in a nonlinear fashion [1]. Recent increase in the amount and availabiliy of biological data has placed a new focus on the complex networks embedded in biological systems [2]. These networks consist of components, that can be described by nodes, and interconnections, that represent the interactions between two components. In a mathematical sense, such networks can be considered as graphs [3] and are used to judge the functional behavior of interaction networks on the molecular level [4]. Hartwell et Al. suggested in 1999 that biological systems, where networks are embedded in, can be decomposed into several functionally isolated modules in order to reduce their complexity [5]. According to Sauro [6], an ideal module is one if the behavior of a composite network of two or more modules can be predicted from the input/output characteristics of the individual systems. A concept called retroactivity was originally presented by Saez-Rodriguez et Al. [7] and then expanded by Del Vecchio et Al. [8] to describe the change of input-output dynamics of a biological module upon its interconnection with other modules. Therefore, it can be concluded that it is not possible to predict global network behavior by using substrate concentrations as inputs and outputs. Sivakumar and Hespanha [1] show with an example of a simple gene regulatory network how to predict the global network behavior from the input-output relationships of the individual modules just by using production rates as inputs and outputs. However, since the individual modules in a gene regulatory network are easy to identify according to Sivakumar and Hespanha [1], there s still the problem how

11 2 1. Introduction to identify individual modules in a large biological network such as the metabolic network. Here, we obtain a metabolic network from the whole cell simulation model from Karr et Al. [9]. This whole cell model describes the life cycle of the bacterium Mycoplasma genitalium by simulating every cellular process there exists during a cellular life cycle. Our goal in this thesis is to show the modular decomposition of the metabolic network and gain insight into its functional modules. The decomposition is done by the spectral method [10]. This method considers a certain cost related to breaking an interconnection and tries to decompose the network while minimizing the cost. Multiple other methods have been used to find functional modules. For example, Schuster et Al. presented an algorithm in 2002 to decompose biological networks based on the degree of metabolites [11]. They labelled metabolites with degree k larger than some threshold value k max as external and then considered connected components of internal metabolites as modules [12]. Another method to compute the modular structure of networks was introduced in 2003 by Rives et Al., who applied it on protein-interaction networks [13]. Their method is based on constructing a network with equally important interconnections and then modularizing it by using the all-pairs-shortest-path algorithm, which finds a path between two nodes such that the sum of the weights of its constituent interconnections is minimized [14]. We construct the metabolic network by using metabolic reactions and their rate constants so that each species is assigned to one node and interconnections between two nodes represent chemical reactions [15]. After using the spectral and Louvain s method, which aims to achieve the highest possible modularity, which was defined by Newman ([16]) to be a scalar for quantifying the quality of partitioning a network. The remainder of this thesis is organized as follows. Chapter 2 explains how information can be extracted from the whole cell model and used to obtain the metabolic network. The resulting partition of the metabolic network is presented in Chapter 3, which is followed in Chapter 4 by a discussion about the quality of the partition and future work.

12 CHAPTER 2 Whole-Cell Model In this chapter the Whole-Cell Model is described. Subsequently, each cellular process within this model is analyzed in order to find a suitable process, which can be represented as a set of chemical reactions Introduction The Whole-Cell model is a simulation model in Matlab, which was developed by bioengineers from Stanford University [9]. It describes the life cycle of a single cell from the level of individual molecules and their interactions by simulating the cell growth of the bacterium Mycoplasma genitalium, which is a human urogenital parasite. Additionally, the model accounts for the specific function of every annotated gene product and also accurately predicts a wide range of observable cellular behaviors. That s why the Whole-Cell Model is mostly used for biological research to get insight into biological processes for which experimental assessment is not feasible. Our goal is to use the Whole-Cell Model to further our understanding about the underlying functional modules that exist within the various cellular processes that the cell undergoes.

13 4 2. Whole-Cell Model Simulation Algorithm Figure 2.1 illustrates the architecture of the Whole-Cell Model. Figure 2.1.: Architecture of the Whole-Cell Model The squares in Figure 2.1 represent classes, since the WholeCell Model is programmed using object-oriented programming. In total there exist approximately 200 classes that are defined in over code lines. The simulation class has two inputs. The first one comes from the so-called Knowledge Base, which is a database, where information about the cell such as chemical reactions and names and properties of occuring substrates is stored. The second input comes from a class called FitConstants [17]. Since the information from the database originate from different experimental settings and state of cells, certain parameters need to be fitted if one wants to use it in a quantitative model such as the Whole-Cell Model. FitConstants resolves this conflict. For closer details the reader is invited to read [17]. Figure 2.2 gives an overview how the different processes are linked to the cell variables.

14 2.1. Introduction 5 Figure 2.2.: Overview over the connection between cell variables and cellular processes and simulation algorithm In the left rectangle a set of 16 cell variables is listed. These 16 cell variables represent physiological properties such as the copy number of metabolites, cell-level properties like mass or volume and nascent polymers of DNA, RNA molecules and proteins. Additionally, the 28 cellular processes, that represent the cell s total functionality, are linked by this set of cell variables, which is shown in form of the arrows between the two rectangles in Figure 2.2. For example, we see that the cell variable Metabolite is linked to every process except of host interconnection, which shows the importance of the metabolites. Another example is the cell variable chromosome, which is only linked to the eight processes that affect DNA molecules. Moreover, Figure 2.2 illustrates the simulation algorithm. At first, the cell variables are initialized and used to parametrize each one of the processes by setting initial conditions. Then, each cellular process is simulated on a 1 second time scale. Afterwards, the resulting values of each process are used to update the set of cell variables and as initial conditions for the following simulation timestep. This makes sure that each one of the processes can be considered independently, because the processes don t need the variable values, that are computed from other processes on-time, for their own simulation. Finally, this procedure repeats for approx times until the cell divides.

15 6 2. Whole-Cell Model Cellular Processes The life cycle of the Mycoplasma genitalium has been modelled by 28 processes ranging from metabolism over protein folding to DNA replication. These processes are illustrated in Figure 2.3 in a different perspective compared to Figure 2.2. Figure 2.3.: Overview over cellular processes in the Whole-Cell Model Each process can be assigned to one of the following 6 main categories: Metabolism RNA synthesis and degradation Protein synthesis and degradation Chromosomal Replication and maintenance Cytokinesis Host Interaction The first category is called metabolism and is defined as the set of life-sustaining chemical transformations within cells of living organisms [18]. The processes in the second category take account of RNA synthesis and degradation by simulating RNA decay or RNA Modification, which benefits the gene encoding processes. Protein synthesis and degradation is similar to the second category with the difference that it contains more processes because protein are more complex than RNA and have to be folded a certain way in order to work properly. The fourth category Chromosomal replication and maintenance describes the processes, that

16 2.1. Introduction 7 are involved in the change of DNA molecules, which they undergo in terms of damage, repair and replication, for example. We can conclude that the first four categories affect the production of metabolites, proteines, RNA and DNA molecules that are essential to a functioning cell. The fifth category cytokinesis denotes the cell division process, whereas host interconnection models interactions between the host (human cell, for example) and the simulated bacterium, which results in secretion of certain substrates such as hydrogen peroxide. For a detailed description of every process the reader is advised to read [9] and [17]. One can state that there two types of processes. One process type influences cell geometry only. Examples for this type of process are Protein folding or FtsZ polymerization. Other processes such as Metabolism pertain to the second process type, which is the production of chemical substrates, that cells need to function properly.

17 8 2. Whole-Cell Model 2.2. Metabolism Process Despite the fact that the knowledge base of the Whole-Cell Model contains information about chemical reactions for every process, that is involved in the production of substrates, not every one of these processes can be mapped to a set of differential equations using the law of mass action kinetics ([1]) due to the knowledge base not capturing the chemical reaction dynamics as a continuous closed-form expression. Out of 28 processes, there are then three processes, for which we could obtain a system of differential equations that would describe the evolution of protein concentrations within that process. These are: Metabolism Protein Decay DNA Damage We chose to analyze the metabolic network for the following reasons: Firstly, metabolism is one of the three key cellular processes ([19]) because it describes the set of life-sustaining chemical transformations within cells of living organisms ([18]). Thus, without a fully functioning metabolism, cells are not able to survive because they would lack certain substrates. Figure 2.4 illustrates this by showing that the four main types of cell molecules metabolites, proteins, RNA and DNA molecules depend on each other in order. Because the arrows in Figure 2.4 start from and end in metabolites, we see that every process starts and ends with metabolites. Figure 2.4.: Overview about the role of metabolites within a cell Secondly, the metabolism process produces enzymes, that play a very important role during cell growth because they act as catalysts. Without enzymes, certain reactions cannot take place [20]. In this case, the cell might lack a certain metabolite, which it necessarily needs to produce a certain protein or RNA molecule. Since the production of enzymes is not only regulated by metabolites directly, but also by other regulation mechanisms such as the posttranscriptional regulation, that are themselves regulated by certain metabolites, metabolism has a great impact on enzyme production and regulation.

18 2.2. Metabolism Process 9 Thus, the production of enzymes appears to be an interesting part of the cell to analyze. We then do a decomposition of the metabolic network into modules to be able to judge the functional behavior of each module.

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20 CHAPTER 3 Results In this chapter the spectral method [10] is used to decompose the metabolic network with the assumption that every interconnection is equally important. Subsequently, the spectral method is used again in order to decompose the metabolic network under the assumption that every interconnection is not equally important Metabolic network In order to being able to partition the metabolic network, we have to construct it first. This can be done in accordance with the method used in [21], which states that we need the chemical reactions and rate constants. At this point it is worth noting that an interconnection between two or more modules happens when there exists a chemical reaction, which is common to all of those modules. From the knowledge base, the chemical reaction can be easily obtained. However, retrieving the rate constants is not trivial because the Whole-Cell Model uses the so-called flux-balance analysis, which is explained in appendix A.1, to compute the reaction rates. The flux-balance analysis assumes steady state, which prevents the metabolic model from predicting the concentrations of metabolites, which are internal to the metabolic network. Within the metabolic model these metabolites have zero net concentration change with respect to time. Therefore, only the reactions, which involve metabolites, whose concentration is not zero, can be used for the method proposed by Sivakumar et Al. [21].

21 12 3. Results That s why both graph partitions use a reduced metabolic network, which contains 92 species and 51 reactions. The chemical reactions are obtained from the knowledge base of the Whole-Cell Model. How the rate constants are computed, is shown in Chapter 3.3. Appendix A.2 lists the species, reactions and rate constants of the reduced metabolic network.

22 3.2. Graph partition with equally weighted interconnections Graph partition with equally weighted interconnections In this section, the spectral method is used to decompose the metabolic network with the assumption that every interconnection is assigned the weight of one, which means that every interconnection is considered to be equally important (see Figure 3.1). Figure 3.1.: Example for a network with equally weighing interconnections By setting the weight of every interconnection to one, we want to see the modules density of communication after the metabolic network is decomposed without any information other than the nodes and their interconnections, that are both captured by the chemical reactions. The structure seen in Figure 3.1 can be applied to the metabolic network. For reasons of clarity, the metabolic network with its 92 species, and therefore 92 nodes, is not shown. Because we have to decide into how many modules we want the spectral method to decompose our metabolic network, we first need to know the optimal number of partitions. According to Louvain s method ([22])the optimal number of partitions is 10. Optimal number means in this case that Louvain s method stops after yielding the highest possible modularity, which was defined by Newman ([16]) to be a scalar for quantifying the quality of partitioning. Figure 3.2 shows the ten resulting modules the metabolic network is partitioned into after using spectral method.

23 14 3. Results! Figure 3.2.: Overview over all modules within the metabolic network and their interconnections to each other after using spectral method assuming equally weighing interconnections These 10 modules can be structured into 3 functional classes: Supply of cofactors Production of substrates Transport of substrates Cofactors are chemical compounds that are required for certain enzymes to become active [20]. Examples can be seen in Chapter In the following subsections, each one of the 10 modules is explained in detail. For a complete list of species pertaining to each module please see appendix A.2.3. Moreover, the reader can refer to appendix A.2, where the full names and corresponding abbreviations of every substrate are listed Supply of cofactors This functional class contains only one module. It has 30 species, that are involved in 51 reactions, which is also the total number of reactions in the metabolic network. This means that the species in this module are involved in every reaction. Examples of the species in this module are ATP (Adenosintriphosphate) and H 2 O (Water). ATP serves as energy carrier in the cell and is therefore essential for cell growth. H 2 O, which is water, is also a vital part of the cell, since we know that

24 3.2. Graph partition with equally weighted interconnections 15 cell cannot survive without water. From these two examples, which can be seen as representatives for most of the remaining species, we can see that the species are used as cofactors in various chemical reactions. Hence, the function of this module is to supply cofactors for various chemical reactions Production of substrates Five different modules belong to this class. In some of the chemical reactions, that will be shown in the following sections, [c] or [e] can be found. [c] denotes that the reaction or substrate resides in the cytosol, which is the cell compartment, in which all metabolic reactions occur. [e] denotes that the reaction or substrate resides in extracellular space. Production of FMN The first one of the five modules is named Production of FMN. It contains three molecules RIBFLV[c], RIBFLV[e] and FMN[c], that are involved in two reactions, that can be seen in Equation (3.1). [c] : ATP + RIBFLV ADP + FMN + H RIBFLV[e] RIBFLV[c] (3.1) In the first reaction RIBFLV (Riboflavin) produces FMN (Flavinmononukleotid), ADP (Adenosindiphosphate) and H (Hydrogen) with the help of ATP, whereas ATP, ADP and H are parts of the module, which supplies cofactors. This reflects the interconnections between these two modules. FMN mainly serves as an oxidizing agent for several important cell reactions that involve one- and two-electron transfers. Production of NAD, NADP and NADPH Another module accounts for the production of NAD (nicotinamide adenine dinucleotide, oxidized), NADP (nicotinamide adenine dinucleotide phosphate, oxidized) and NADPH (nicotinamide adenine dinucleotide phosphate, reduced). The production of NADPH is shown in Equation (3.2). [c] : METTHF + NADP 2H + METHF + NADPH (3.2) NADP reacts with the help of METTHF (5,10-Methylenetetrahydrofolate) to H, METHF (5,10-methyltetrahydrofolate) and NADPH. In total, this module contains 9 species, that are involved in 5 other reactions. Since their structure is

25 16 3. Results similar to the one in Equation (3.2) they re not shown in detail. Furthermore, based on the fact that this module contains 9 species, we see that it is module is a little bit larger than the module from section NAD, NADP and NADPH serve as oxidizing or reducing agents for important cell reactions such as the production of the energy carrier ATP. Production of L-homocysteine HCYS (L-homocysteine) is an amino acid, which is crucial for energy transfer. It is produced by AHCYS (S-Adenosyl-L-homocysteine) and H 2 O. The according reaction, in which three species pertaining to this module are involved in, can be seen in Equation (3.3). [c] : AHCYS + H 2 O ADN + HCYS (3.3) Production of nucleotides Nucleotides are molecules, that contain nucleo bases to form the building blocks for DNA and RNA. In fact, there exist five different types of nucleotides: ATP, UTP, CTP, GTP and TTP. To be exact, these nucleotides would have to be called nucleoside triphosphates. In order to produce nucleoside triphosphates nucleoside diphosphates have to be produced first. After the graph partition two modules can be accounted for the production of nucleotides. One module produces CDP and UDP, which is presented by Equation (3.4) and Equation (3.5). [c] : ATP + CMP ADP + CDP (3.4) [c] : ATP + UMP ADP + UDP (3.5) CDP and UDP are components of RNA molecules. The other module produces deoxy-versions of CDP, GDP, UDP and TDP. The reactions are similar to Equation (3.6) with the difference that the respective nucleotide is interchanged. [c] : ATP + DGMP ADP + DGDP (3.6) Together with DADP, which resides in the cofactor-supplying module, DCDP, DGDP, DUDP and DTDP are the main components of DNA molecules.

26 3.2. Graph partition with equally weighted interconnections Transport of substrates There are 4 modules, who all have in common that their function is to transport substrates out of and into the cell. Cell transport mechanisms Metabolic reactions occur in a certain cell compartment, which is called cytosol. Therefore, every metabolite has to be available in the cytosol before it can be consumed. There are three main mechanisms in the cell, that transport substrates from extracellular space into the cytosol. Namely, these mechanisms are proton symport, proton antiport and transport via ABC system. Everyone of these three transport mechanisms is assigned an module of its own. In total, these 3 modules contain 32 species, that are involved in 27 reactions. One example for a proton symport transport reaction is shown in Equation (3.7). H[e] + TRP[e] H[c] + TRP[c] (3.7) Compared to Equation 3.8, which displays an example for a proton antiport transport reaction, we can see the difference between these two transport mechanisms. H[c] + Na[e] H[e] + Na[c] (3.8) If a substrate is transported by proton symport it means that it is accompanied by a proton H during its transport from extracellular space into cytosol. However the proton antiport leads to a proton H leaving the cytosol when a substrate is transported into the cytosol. The transport via ABC system means that the transport happens with the help of ATP and H 2 O, that both reside in the cytosol already (see Equation 3.9). ATP[c] + H 2 O[c] + SO 4 [e] ADP[c] + H[c] + PI[c] + SO 4 [c] (3.9) Ammonium transport During the metabolism process waste is produced. Ammonium is regarded as metabolic waste. The fourth module grouped into this functional class consists of 5 species, that are involved in 4 reactions. The main important reactions are ammonium production and ammonium transport. Both are shown in Equation (3.10).

27 18 3. Results [c] : H + NH 3 NH 4 NH 4 [c] NH 4 [e] (3.10)

28 3.3. Graph partition with inequally weighted interconnections Graph partition with inequally weighted interconnections Now, a metabolic network shall be considered, whose interconnections are not equally important. Figure 3.3 shows how the spectral method can lead to different results, if the interconnections are weighed inequally according to their importance. Figure 3.3.: Spectral method used for a network with equally weighed interconnections (left) and inequally weighed interconnections (right) (numbers: weight; letters: community) Sivakumar et Al. propose in [21] that the weights of each interconnection can be characterized by computing the H -norm over all frequencies of H 12 (I H 22 ) 1 H 21. In order to compute the H -norm we need to obtain rate constants of each chemical reaction and equilibrium concentrations of each species Rate constants The rate constant k of the chemical reaction shown in Equation (3.11) is computed by Equation (3.12). X + Y k Z (3.11) k = Reaction rate X Y (3.12) Equation (3.12) can be generalized in order to compute rate constants for each reaction pertaining to the metabolic network (see Equation 3.13).

29 20 3. Results Rate constant = Reaction rate Educt concentrations (3.13) Educt concentrations are the product of the concentration of each educt which occurs in a certain chemical reaction (for our example in Equation (3.12) it s X Y ). The reaction rate is obtained from the simulation results of the WholeCell Model, which uses the so-called flux-balance analysis. A detailed explanation of the flux-balance analysis can be found in appendix A.1. Since one complete simulation of the WholeCell Model takes approximately seconds, we obtain as many different rate constants for each one of the 51 reactions. In order to compute the H -norm, we need one representative rate constant for each reaction. Figures 3.4 and 3.5 show two histograms of the rate constants of two chemical reactions within the metabolic network, that were chosen arbitrarily. The first histogram refers to a reaction, which produces energy for the cell, and the second histogram refers to a reaction, which is important for the control of gene expression Rate constant x Figure 3.4.: Histogram of rate constants for the reaction [c] : AMP + ATP 2ADP

30 3.3. Graph partition with inequally weighted interconnections x Rate constant x 10 8 Figure 3.5.: Histogram of rate constants for the reaction [c] : ATP + H 2 O + MET AMET + PI + PPI As we can see, the median rate constant lies in an area, where most of the rate constants accumulate and thus, proves to be a good representative value for all rate constants. Due to simplicity reasons the histograms of the remaining 49 reactions are not shown here, but it is stated that those histograms look very much alike the two histograms shown in Figures 3.4 and 3.5. Each median of the rate constants of each reaction is listed in appendix A Equilibrium concentrations In order to approximate the species concentrations we used the Euler method as ODE solver [23]. In the following our method how to compute the equilibrium concentration of each species is presented. Species are known to have inherent birth and death rates shown in Equation (3.14): ϕ X u X γ ϕ. (3.14) By simply applying the law of mass action kinectics, we obtain differential equations that only describe the rate of change of concentrations, but don t take the birth and death rates into account since chemical reactions only caputure the interactions between particular species. Therefore, these differential equations need to be modified in order to prevent the system from going unstable or yielding a trivial equilibrium point of 0. We handled this problem by the following three cases that we observe. The first case refers to differential equations that look like Equation (3.15):

31 22 3. Results Ẋ = k X Y (3.15) This differential equation only caputures the decay of the concentration of species X, which is why we need to add a birth rate u to it (see Equation (3.16)). Ẋ = k X Y + u (3.16) Equation (3.17) shows another case, how the differential equations could look like. Ẋ = +k X Y (3.17) Here, the death rate γ X needs be to considered additionally for the rate of change of concentration of species X, which is shown in Equation (3.18). Ẋ = +k X Y γ X; (3.18) The third case describes differential equations that look like Ż = k X Y. (3.19) If we have such a differential equation, both a production rate u and a decay rate γ Z have to be added, which can be seen in Equation (3.20): Ż = k X Y γ Z + u; (3.20)

32 CHAPTER 4 Conclusion and future work We have shown that the metabolic network can be decomposed into ten different modules without taking reaction dynamics into account. Each one of those ten modules can be assigned a specific function to help cell growth. Furthermore, a graph partition can be considered as good if it decomposes the network into modules, that have as many as possible connections within each module and as little as possible connections between each other. Since our result gave us a main module, which communicates with every other module while every other module does not communicate with other modules than the main module, we can conclude that we got a very good result for the decomposition while not considering reaction dynamics. However, there is still room to improve the decomposition. There are two examples for this: Firstly, the ammonium transport module also contains two reactions involving the so-called coenzyme A, which serves as coenzyme for different cell reactions and does not contribute to ammonium production or transport. Therefore, a possible better decomposition would have had a new module, whose function is directly related to coenzyme A. Secondly, two of the ten modules regard the production of nucleotides. One module produces DCDP, DGDP, DUDP and DTDP. Together with DADP, which resides in the main module, these nucleotides are essential to produce DNA molecules. Thus, this module can be considered as complete because every necessary nucleotide is produced in this module. The other module produces UDP and CDP. These are two out of four nucleotides, that are needed to produce RNA molecules. Therefore, this module lacks two nucleotides. One of the lacking nucleotides is ADP, which

33 24 4. Conclusion and future work resides in the main module. This is sensible since ADP serves as an energy carrier for cell reactions and is therefore needed for many more reasons than the production of RNA and DNA molecules. The second lacking nucleotide is GDP, whose main use is to be part of the production of RNA and DNA molecules. This means that its use within the cell can be compared with UDP and CDP, which are part of one of the nucleotides production modules. We conclude that GDP could have been assigned to the same module as UDP and CDP as well in order to obtain an even more precise decomposition. According to our results we can identify three main areas, where future research might yield interesting findings. Firstly, we can use the spectral method to decompose the metabolic network after assigning different weights to each interconnection. Our hope is that the decomposition will be improved compared to the results in Chapter 3.2. The second area lies in multi-level graph partitioning. What we did was to decompose the metabolic network only on one level. However, we noticed that the main module is very big (30 species and 51 reactions). Therefore, it might be interesting to see how the main module itself could be decomposed further, because not every species residing in the main module is involved in the majority of metabolic reactions (as it is the case with ATP, for example). The third area lies in analyzing the exact impact of the production of enzymes in regard to other cellular processes such as posttranslational modification or gene expression. This connection might be modellable as control systems with feedback loops and then analyzed further regarding stability, for example.

34 APPENDIX A Appendix A.1. Flux-balance analysis The WholeCell Model uses a mathematical approach called flux-balance analysis to compute fluxes through a metabolic network [24]. Fluxes are defined as the rates of flow of metabolites along a metabolic pathway and therefore, can be seen equal as reaction rates. The only information it needs from the cell are the chemical reactions and the substrates stoichiometric coefficients. It is also possible to accurately represent biological limits of the system by defining upper and lower bound for reaction rates, for example. In the following the FBA algorithm shall be explained by computing the fluxes of the network shown in Figure A.1, where three different substrates A, B and C interact because of seven reactions denoted by R1,..., R7. R4 R2 R1 A B C R5 R3 R6 R7 Figure A.1.: Network example The first step is to represent metabolic reactions mathematically. A stoichiometric matrix S mxn is defined, where m denotes the number of substrates and n the

35 26 A. Appendix number of reactions. Additionally, every consumed substrate is considered to have a negative stoichiometric coefficient while every produced substrate is considered to have a positive stoichiometric coefficient. For our example (see Figure A.1) the stoichiometric matrix would look as follows: S = (A.1) Besides, vector v nx1 represents the fluxes of every reaction (see Equation (A.2)). v 1 v 2 v 3 v = v 4 v 5 v 6 v 7 (A.2) Then, we make the assumption of being in steady state. This means that the flux through each reaction is given by Equation (A.3). S v = 0 (A.3) This gives us a set of linear equations seen in Equation (A.4). v 1 + v 4 = 0 v 1 v 2 + v 3 v 5 = 0 v 2 v 3 v 6 + v 7 = 0 (A.4) Since we have more unknowns v than linear equations, this need to be solved by linear programming. For this, an objective function Z has to be defined (see Equation (A.5)). This objective function corresponds to a biological objective such as the maximization of biomass. Z = c T v (A.5) c is a vector of weights indicating how much each reaction contributes to the objective. Since in our example we chose as objective to maximize the fifth flux v 5, c is defined as follows:

36 A.1. Flux-balance analysis c = (A.6) Linear programming (see Equation (A.7)) yields a particular flux distribution v. maximize c T v subject to S v = 0 (A.7) In our example the solution for the fluxes is: v = (A.8) As we can see v 5 has the biggest value compared to the other fluxes, which corresponds to our chosen objective.

37 28 A. Appendix A.2. Metabolic network This appendix first lists the species and chemical reactions and their rate constants pertained to the metabolic network, which was used in this thesis. Subsequently, a table containing the decomposed modules and species pertained to them can be found. A.2.1. Content of metabolic network Table A.1 lists the names of the 92 species and their abbreviations pertaining to the metabolic network. For reasons of simplicity, the suffices [c] (cytosol) or [e] (extracellular space) are neglected because they only denote the place where each particular species resides in. Therefore, the table lists less than 92 species since some species are available both in cytosol and extracellular space. Abbreviation ACCOA ACTP ADN ADP AHCYS AMET AMP ATP CA CDP CL CMP COA CO CU DADP DAMP DATP DCDP DCMP DGDP DGMP DTDP DTMP FE 3 FMN FTHF10 Substrate name Acetyl-Coenzyme A Acetyl phosphate Adenosine Adenosine diphosphate S-Adenosyl-L-homocysteine S-Adenosyl-L-methionine Adenosine monophosphate Adenosine triphosphate Calcium ion Cytidine monophosphate Chloride ion Cytidine monophosphate Coenzyme A Cobalt ion Cupric ion Deoxyadenosine diphosphate Deoxyadenosine monophosphate Deoxyadenosine triphosphate Deoxycytidine diphosphate Deoxycytidine monophosphate Deoxyguanosine diphosphate Deoxyguanosine monophosphate Thymidine diphosphate Thymidine monophosphate Ferric ion Flavin mononucleotide, oxidized 10-formyl-tetrahydrofolate

38 A.2. Metabolic network 29 Abbreviation GDP GLY FOR GMP GN GTP H 2 O H HCYS K METTHF METHF MET MG MG 124 MONOMER ox MG 124 MONOMER MN MTHF NADH NADPH NADP NAD NA NH 3 NH 4 NI PHE PI PPGPP PPI PTRC RIBFLV SER SO 4 SPMD THF TRP TYR UDP UMP ZN Substrate name Guanosine diphosphate Glycine Formate Guanosine monophosphate Guanine Guanosine triphosphate Water Hydrogen ion L-homocysteine Potassium ion 5,10-Methylenetetrahydrofolate 5,10-methyltetrahydrofolate L-Methionine Magnesium ion Oxidized thioredoxin Thioredoxin Manganese ion 5-Methyltetrahydrofolate Nicotinamide adenine dinucleotide, reduced Nicotinamide adenine dinucleotide phosphate, reduced Nicotinamide adenine dinucleotide phosphate, oxidized Nicotinamide adenine dinucleotide, oxidized Sodium ion Ammonia Ammonium Nickel L-phenylalanine Phosphate Guanosine 3,5 -bis(diphosphate) Diphosphate Putrescine Riboflavin L-Serine Sulphate Spermidine 5,6,7,8-Tetrahydrofolate L-tryptophan L-tyrosine Uridine diphosphate Uridine monophosphate Zinc ion Table A.1.: List of all species and their abbreviations in the metabolic network

39 30 A. Appendix These 92 species are connected by 51 reactions, which are listed in the following: [c] : AMP + ATP 2 ADP [c] : AMP + GTP ADP + GDP [c] : ATP + DAMP ADP + DADP [c] : AHCYS + H 2 O ADN + HCYS (A.9) (A.10) (A.11) (A.12) ADP[c] + 4 H[e] + H[c] + PI[c] ATP + 4 H[c] + H 2 O[c] (A.13) [c] : ATP + DCMP ADP + DCDP [c] : ATP + CMP ADP + CDP [c] : ATP + UMP ADP + UDP (A.14) (A.15) (A.16) [c] : METTHF + NADP 2 H + METHF + NADPH (A.17) [c] : ATP + FTHF H ADP + METTHF + PI (A.18) [c] : ATP + FOR + THF ADP + FTHF10 + PI (A.19) [c] : 2 H + SER + THF GLY + H 2 O + METTHF (A.20) [c] : ATP + GMP ADP + GDP [c] : ATP + DGMP ADP + DGDP [c] : DATP + GMP DADP + GDP [c] : METTHF + NADH MTHF + NAD [c] : ATP + H 2 O + MET AMET + PI + PPI [c] : ATP + NAD ADP + NADP [c] : H + NH 3 NH 4 [c] : H 2 O + PPI H + 2 PI [c] : ACCOA + PI ACTP + COA [c] : ATP + GDP AMP + H + PPGPP [c] : ATP + RIBFLV ADP + FMN + H [c] : ATP + DTMP ADP + DTDP (A.21) (A.22) (A.23) (A.24) (A.25) (A.26) (A.27) (A.28) (A.29) (A.30) (A.31) (A.32) [c] : H + MG 124 MONOMER ox + NADPH MG 124 MONOMER + NADP (A.33) H[e] + TRP[e] H[c] + TRP[c] H[e] + PHE[e] H[c] + PHE[c] H[e] + TYR[e] H[c] + TYR[c] H[e] + TRP[e] H[c] + TRP[c] H[e] + PHE[e] H[c] + PHE[c] (A.34) (A.35) (A.36) (A.37) (A.38)

40 A.2. Metabolic network 31 H[e] + TYR[e] H[c] + TYR[c] (A.39) ATP[c] + CO[e] + H 2 O[c] ADP[c] + CO[c] + H[c] + PI[c] (A.40) ATP[c] + H 2 O[c] + NI[e] ADP[c] + H[c] + NI[c] + PI[c] (A.41) CL[e] + 2 H[c] CL[c] + 2 H[e] COA[e] COA[c] GN[e] + H[e] GN[c] + H[c] H 2 O[e] H 2 O[c] (A.42) (A.43) (A.44) (A.45) ATP[c] + CA[e] + H 2 O[c] ADP[c] + CA[c] + H[c] + PI[c] (A.46) ATP[c] + CU[e] + H 2 O[c] ADP[c] + CU[c] + H[c] + PI[c] (A.47) ATP[c] + FE 3 [e] + H 2 O[c] ADP[c] + FE 3 [c] + H[c] + PI[c] (A.48) ATP[c] + H 2 O[c] + MG[e] ADP[c] + H[c] + MG[c] + PI[c] (A.49) ATP[c] + H 2 O[c] + MN[e] ADP[c] + H[c] + MN[c] + PI[c] (A.50) ATP[c] + H 2 O[c] + ZN[e] ADP[c] + H[c] + PI[c] + ZN[c] (A.51) H[c] + NA[e] H[e] + NA[c] NH 4 [e] NH 4 [c] RIBFLV[e] RIBFLV[c] (A.52) (A.53) (A.54) ATP[c] + H 2 O[c] + SO 4 [e] ADP[c] + H[c] + PI[c] + SO 4 [c] (A.55) ATP[c] + H 2 O[c] + SPMD[e] ADP[c] + H[c] + PI[c] + SPMD[c] (A.56) ATP[c] + H 2 O[c] + PTRC[e] ADP[c] + H[c] + PI[c] + PTRC[c] (A.57) THF[e] THF[c] H[e] + K[e] H[c] + K[c] (A.58) (A.59) A.2.2. Median rate constants The method how to compute the rate constants of each reaction and the motivation why we decided to use median rate constants were presented in Chapter The following list shows the median rate constant k i of reaction i:

41 32 A. Appendix k 1 = k 2 = k 3 = k 4 = k 5 = k 6 = k 7 = k 8 = k 9 = k 10 = k 11 = k 12 = k 13 = k 14 = k 15 = k 16 = k 17 = k 18 = k 19 = k 20 = k 21 = k 22 = k 23 = k 24 = k 25 = k 26 = k 27 = k 28 = k 29 = k 30 =

42 A.2. Metabolic network 33 k 31 = k 32 = k 33 = k 34 = k 35 = k 36 = k 37 = k 38 = k 39 = k 40 = k 41 = k 42 = k 43 = k 44 = k 45 = k 46 = k 47 = k 48 = k 49 = k 50 = k 51 = A.2.3. Modules within the metabolic network Here, species pertaining to each one of the 10 modules (see Chapter 3.2) are listed. For reasons of simplicity, abbreviations of the species names are used. The module, which supplies cofactors, contains the following species: ACCOA[c] ADP[c] AMP[c] ATP[c]

43 34 A. Appendix CA[e] CO[e] CU[e] DADP[c] DAMP[c] DATP[c] FE 3 [e] FOR[c] FTHF10[c] GDP[c] GMP[c] GTP[c] H 2 O[c] H[c] MG[e] MN[e] NI[e] PI[c] PPGPP[c] PPI[c] PTRC[e] SER[c] SO 4 [e] SPMD[e] THF[c] ZN[e] The module, which produces FMN, contains the following species: RIBFLV[c] RIBFLV[e] FMN[c]

44 A.2. Metabolic network 35 The module, which produces NAD, NADP and NADPH, contains the following species: METTHF[c] MG 124 MONOMER ox[c] NADH[c] NADPH[c] NADP[c] NAD[c] METHF[c] MG 124 MONOMER[c] MTHF[c] The module, which produces L-homocysteine contains the following species: AHCYS[c] ADN[c] HCYS[c] The module, which produces CDP and UDP, contains the following species: CMP[c] UMP[c] CDP[c] UDP[c] The module, which produces DCDP, DGDP, DUDP and DTDP, contains the following species: DCMP[c] DGMP[c] DTMP[c] DCDP[c] DGDP[c] DTDP[c] The proton symport module contains the following species:

45 36 A. Appendix GN[e] H[e] K[e] PHE[e] TRP[e] TYR[e] GN[c] K[c] PHE[c] TRP[c] TYR[c] The proton antiport module contains the following species: CL[e] NA[e] CL[c] NA[c] The transport via ABC system module contains the following species: H 2 O[e] MET[c] THF[e] ACTP[c] AMET[c] CA[c] CO[c] CU[c] FE 3 [c] GLY[c] MG[c] MN[c]

46 A.2. Metabolic network 37 NI[c] PTRC[c] SO 4 [c] SPMD[c] ZN[c] The ammonium transport module contains the following species: COA[e] NH 3 [c] NH 4 [e] COA[c] NH 4 [c]

47

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