1. School of Computer Science, Tel-Aviv University, Tel-Aviv, Israel
Constraint-Based Modeling of Metabolic Networks based on: “Genome-scale models of microbial cells: Evaluating the consequences of constraints”, Price, et. al (2004)
Tomer Shlomi
School of Computer Science, Tel-Aviv University, Tel-Aviv, Israel
January, 2006

2. Outline Metabolism and metabolic networks
Kinetic models vs. constraints-based modeling
Flux Balance Analysis
Exploring the solution space
Altering phenotypic potential: gene knockouts

3. Cellular Metabolism The essence of life.. Catabolism and anabolism
The metabolic core – production of energy – anaerobic and aerobic metabolism
Probably the best understood of all cellular networks: metabolic, PPI, regulatory, signaling
Tremendous importance in Medicine; antibiotics, metabolic disorders, liver disorders, heart disorders
Bioengineering; efficient production of biological products.

4. Metabolites and Biochemical Reactions
Metabolite: an organic substance, e.g. glucose, oxygen
Biochemical reaction: the process in which two or more molecules (reactants) interact, usually with the help of an enzyme, and produce a product
Glucose + ATP
Glucokinase
Glucose-6-Phosphate + ADP

6. Kinetic Models Dynamics of metabolic behavior over time
Metabolite concentrations
Enzyme concentrations
Enzyme activity rate – depends on enzyme concentrations and metabolite concentrations
Solved using a set of differential equations
Impossible to model large-scale networks
Requires specific enzyme rates data
Too complicated

7. Constraint Based Modeling
Provides a steady-state description of metabolic behavior
A single, constant flux rate for each reaction
Ignores metabolite concentrations
Independent of enzyme activity rates
Assume a set of constraints on reaction fluxes
Genome scale models
Flux rate:
μ-mol / (mg * h)

8. Constraint Based Modeling
Find a steady-state flux distribution through all
biochemical reactions
Under the constraints:
Mass balance: metabolite production and consumption rates are equal
Thermodynamic: irreversibility of reactions
Enzymatic capacity: bounds on enzyme rates
Availability of nutrients

9. Network Reconstruction
Metabolic Networks
Biochemistry
Cell
Physiology
Genome
Annotation
Inferred
Reactions
Network Reconstruction
Metabolic Network
Analytical Methods

10. Mathematical Representation
Stoichiometric matrix – network topology with stoichiometry of biochemical reactions
Glucokinase
Glucose + ATP
Glucokinase
Glucose-6-Phosphate + ADP
Glucose
ATP
G-6-P
ADP
Mass balance
S·v = 0
Subspace of R
Thermodynamic
vi > 0
Convex cone
Capacity
vi < vmax
Bounded convex cone n

11. Growth Medium Constraints
Exchange reactions enable the uptake of nutrients from the media and the secretion of waste products
Lower bound Upper bound
Glucose
Oxygen Inf
CO Inf
G-Ex O-Ex Co2-Ex
Glucose
Oxygen CO

12. Determination of Likely Physiological States
How to identify plausible physiological states?
Optimization methods
Maximal biomass production rate
Minimal ATP production rate
Minimal nutrient uptake rate
Exploring the solution space
Extreme pathways
Elementary modes
Optimization methods are used for several purposes,

13. Outline: Optimization Methods
Predicting the metabolic state of a wild-type strain
Flux Balance Analysis (FBA)
Predicting the metabolic state after a gene knockout
Minimization Of Metabolic Adjustment
Regulatory On/Off Minimization

14. Biomass Production Optimization
Metabolic demands of precursors and cofactors required for 1g of biomass of E. coli
Classes of macromolecules:
Amino Acids, Carbohydrates
Ribonucleotides, Deoxyribonucleotides
Lipids, Phospholipids
Sterol, Fatty acids
These precursors are removed from the
metabolic network in the corresponding ratios
We define a growth reaction
Z = VATP VNADH VNADPH + ….

15. Biomass Composition Issues
Varies across different organisms
Depends on the growth medium
Depends on the growth rate
The optimum does not change much with changes in composition within a class of macromolecules
The optimum does change if the relative composition of the major macromolecules changes

16. Flux Balance Analysis (FBA)
Successfully predicts:
Growth rates
Nutrient uptake rates
Byproduct secretion rates
Solved using Linear Programming (LP)
Finds flux distribution with maximal growth rate
Max vgro, maximize growth
s.t
S∙v = 0, mass balance constraints
vmin  v  vmax capacity constraints
growth
Fell, et al (1986), Varma and Palsson (1993)

17. FBA Example (1)

18. FBA Example (2)

19. FBA Example (2)

20. Linear Programming Basics (1)

21. Linear Programming Basics (2)

22. Linear Programming Basics (3)

23. Linear Programming: Types of Solutions (1)

24. Linear Programming: Types of Solutions (2)

25. Linear Programming Algorithms
Simplex
Used in practice
Does not guarantee polynomial running time
Interior point
Worse case running time is polynomial
growth

26. Phenotype Predictions: Evolving Growth Rate

28. Exploring the Convex Solution Space

29. Alternative Optima The optimal FBA solution is not unique
One solution Optimal solutions Near-optimal solutions
growth
growth
growth
Basic solutions enumeration – MILP (Lee, et. al, 2000)
Flux variability analysis (Mahadevan, et. al. 2003)
Hit and run sampling (Almaas, et. al, 2004)
Uniform random sampling (Wiback, et. al, 2004)

30. What Do Multiple Solutions Represent ?
Some of the solutions probably do not represent biologically meaningful metabolic behaviors as there are missing constraints
Previous studies tackled this problem by:
Incorporating additional constraints: regulatory constraints (Covert, et. al., 2004)
Looking for reactions for which new constraints may significantly reduce the solution space (Wiback, et. al., 2004)
FBA solution space
Meaningful
solutions

31. Interpretations of Metabolic Space
Effect of exogenous factors – the metabolic space corresponds to growth in a medium under various external conditions that are beyond the model’s scope such as stress or temperature
Heterogeneity within a population - the metabolic space represents heterogenous metabolic behaviors by individuals within a cell population (Mahadevan, et. al., 2003, Price, et. al., 2004)
Alternative evolutionary paths – the metabolic space represents different metabolic states attainable through different evolutionary paths (Mahadevan, et. al., 2003, Fong, et. al., 2004)
The three interpretations are obviously not mutually exclusive

32. Alternative Optima: Basic Solutions Enumeration
Lee, et. al, 2000
Basic solutions – metabolic states with minimal number of non-zero fluxes
Different solutions differ in at least a single zero flux
Use Mixed Integer Linear Programming
Formulate optimization as to identify new solutions that are different from the previous ones
Applicable only to small scale models
growth

33. Alternative Optima: Flux Variability Analysis
Mahadevan, et. al. 2003
Find metabolic states with extreme values of fluxes
Use linear programming to minimize and maximize the flux through each reaction while satisfying all constraints
Max / Min vi, maximize growth
s.t
S∙v = 0, mass balance constraints
vmin  v  vmax capacity constraints
Vgro = Vopt set maximal growth rate

34. Alternative Optima: Hit and Run Sampling
Almaas, et. al, 2004
Based on a random walk inside the solution space polytope
Choose an arbitrary solution
Iteratively make a step in a random direction
Bounce off the walls of the polytope in random directions

35. Alternative Optima: Uniform Random Sampling
Wiback, et. al, 2004
The problem of uniform sampling a high-dimensional polytope is NP-Hard
Find a tight parallelepiped object that binds the polytope
Randomly sample solutions from the parallelepiped
Can be used to estimate the volume of the polytope

36. Topological Methods Not biased by a statement of an objective
Network based pathways:
Extreme Pathways (Schilling, et. al., 1999)
Elementary Flux Modes (Schuster, el. al., 1999)
Decomposing flux distribution into extreme pathways
Extreme pathways defining phenotypic phase planes
Uniform random sampling

37. Extreme Pathways and Elementary Flux Modes
Unique set of vectors that spans a solution space
Consists of minimum number of reactions
Extreme Pathways are systematically independent (convex basis vectors)
A pathway is a metabolic state which satisfies stoichiometric and thermodynamic constrains.
Extreme pathways and elementary flux modes are both unique sets of pathways.
Both type of pathways are minimal – i.e. there is no other pathway with a subset of the reactions.
Only in extreme pathways the pathways are systematically independent. No EP can be described as a combination of the others. EP are the basic of the convex space.
Each metabolic state can be described as non-negative combination of EPs.
EP and EM are the same where all exchange fluxes are unidirectional.
Example network.
There was an ongoing debate on which method is better in describing the phenotypic potential of the network. In a recent (and not so convincing paper) of the main supporters of both methods it was agreed that:
The main advantage of EP is that there are far fewer of them in a typical network and that they have a mathematical justification being the basis of the convex space. EM are suitable for studying network properties such as redundancy as it includes all minimal pathways.

38. Extreme Pathways and Elementary Flux Modes
Inherent redundancy in metabolic networks (Price, et. al., 2002)
Robustness to gene deletion and changes in gene expression (Stelling, et. al., 2002)
Enzyme subsets (correlated reaction sets) in yeast (Papin, et. al., 2002)
Design strains (Carlson, et. al., 2002)
Assign functions to genes (Forster, et. al, 2002)
Both methods were used in numerous papers to study different network properties:
The redundancy of the network in producing different nutrients (amino acids) was studied by Price et al. They estimated the redundancy by the number of pathways that involved the synthesis of the nutrients.
Robustness of the network to gene deletion was studied by Stelling el al. They estimated robustness to gene knockouts as the percentage of EM that are still feasible after genes are knocked out. The percentage of EM shows high correlation with measurements of lethality. Furthermore, they have shown that the maximal growth is robust to reduction in the number of feasible pathways.
Papin have used EP to find sets of reaction that are always activated together. These sets are called Enzyme subsets.

39. Altering Phenotypic Potential: Gene Knockouts

40. Altering Phenotypic Potential: Gene Knockouts
Minimization Of Metabolic Adjustment (MOMA) (Segre et. al, 2002)
The flux distribution after a knockout is close to the wild-type’s state under the Euclidian norm
Regulatory On/Off Minimization (ROOM) (Shlomi et. al, 2005)
Minimize the number of Boolean flux changes from the wild-type’s state
Predicting the metabolic state of an organism that undergo gene knock in the lab is harder than predicting the metabolic state of wild-type strain.
Minimization Of Metabolic Adjustment (MOMA) developed by Segre is an optimization method which assumes that the metabolic state of the knocked-out organism should be close to that of the wild-type strain as the mutated organism did no evolve to maximize its growth. MOMA uses an Euclidian metric to measure the distance between the metabolic state of the wild-type and knocked-out strains.
Regulatory On/Off Minimization (ROOM) is based on the same assumption (that the metabolic state of the knocked-out strain should be close to that of the wild-type strain), but uses a different metric to measure the distance that is based on the number of Boolean flux changes from the metabolic state of the wild-type strain. w v

41. Altering Phenotypic Potential
Explaining gene dispensability (Papp, el. al., 2004)
Only 32% of yeast genes contribute to biomass production in rich media
Considered one arbitrary optimal growth solution
OptKnock – Identify gene deletions that generate desired phenotype (Burgard, et. al., 2003)
OptStrain – Identify strains which can generate desired phenotypes by adding/deleting genes (Pharkya, el., al., 2004)
FBA was used to study metabolic states after the knockouts of dispensable genes. (genes with no observable change in growth following their knockout). They found that only 32% of the genes had non-zero flux – contribute to biomass production under rich media. In a work we did we found that if you consider alternative non-optimal FBA solutions, 90% of the genes may contribute in rich media. Therefore, their function role can be found in rich media without the need to look at other medias.
OptKnock is an optimization methods developed by Burgard which identifies configurations of genes whose knockout may cause the overproduction of desired nutrients (amino acids). This method used FBA or MOMA to predict the metabolic state of the knocked-out strain.
OptStrain is a newer method which identifies a specific strain which along with a set of genes that can be added or removed from its genome for the same biotechnological purposes.

42. Modeling Gene Knockouts
Enzyme knockout
Reaction knockout

43. Cellular Adaptation to Genetic and Environmental Perturbations
Transient changes in expression levels in hundreds of genes (Gasch 2000, Ideker 2001)
Convergence to expression steady-state close to the wild-type (Gasch 2000, Daran 2004, Braun 2004)
Drop in growth rates followed by a gradual increase (Fong 2004)
minutes
There are various experiments showing that following a gene knockout or some kind of environmental perturbation there are large-scale changes in gene expression levels. For example Ideker..
However, these experiments and others have shown that following an adaptation period the cell converges to steady-state which is close to that of the wild-type strain. For example we see on the figure on the left (taken from Gasch et al) the average expression ratio following a perturbation in the from of stressful environment which is characterized by sharp changes in expression levels which is followed by a steady-state that is close to the wild-type.
A recent experiment by Fong et al have shown that the growth rate of the organism drops immediately following the gene knockout and the gradually increase and reaches a steady-state growth rate which is close or even higher to that of the wild-type.
growth
generations

44. Regulatory On/Off Minimization (ROOM)
Predicts the metabolic steady-state following the adaptation to the knockout
Assumes the organism adapts by minimizing the set of regulatory changes
Boolean Regulatory
Change
Boolean Flux
Change
Finds flux distribution with minimal number of Boolean flux changes
Based on these findings, we’ve developed the algorithm Regulatory On/Off Minimization to predict the metabolic steady-state following the adaptation of the knocked-out strain to the knockout.
ROOM is based on the assumption that an organism adapting to gene knockout, minimizes the set of regulatory changes that it makes. Assigning an equal “cost” to each regulatory change, ROOM tries to minimize the total “cost” of adapting to the knockout.
Now, since regulatory constraints are not explicitly incorporated into the metabolic model, ROOM identifies Boolean regulatory changes implicitly by identifying Boolean changes in flux. Therefore ROOM aims to find feasible flux distribution with a minimal number of Boolean flux changes from the flux distribution of the wild-type strain. w v

45. ROOM: Implementation
Solved using Mixed Integer Linear Programming (MILP)
Boolean variable yi
yi = 1
Flux vi change from wild-type
Min yi - minimize changes
s.t
v – y ( vmax - w)  w - distance constraints
v – y ( vmin - w)  w - distance constraints
S∙v = 0, - mass balance constraints
vj = 0, jG knockout constraints
ROOM is implemented using Mixed Integer Linear Programming (MILP), which is an optimization algorithm similar to LP that allows variables to be defined as integers.
We define integer Boolean variables yi that specify whether the i’th flux change from the wild-type’s flux distribution. In order to minimize the number of flux changes from the wild-type’s flux distribution, ROOM is formulated as to minimize the sum of yi’s.
To constrain the i’th flux to its wild-type value if and only if yi equals zero we use the two constraints labeled “distance constraints”. When yi equals zero then the distance constraints fix vi to its wild-type value wi, and when yi equals one, the distance constraints do not impose new constraints on vi.
MILP is NP-hard meaning that the running time of any solver would be exponentially dependent on the size of the problem. Since the size of the problem is determined by the number of constraints which is in the order of hundreds the problem is computationally intractable.
In our analysis we have used two relaxation methods that provide a reasonable running time:
1. One uses Linear Programming by relaxing the Boolean constraints and allowing the Boolean variables to take any value between zero and one.
2. The other relaxation is in proximity to the wild-type flux distribution, looking for a flux distribution that only minimizes the number of significant flux changes.
MILP is NP-Hard
Relax Boolean constraints - solve using LP
Relax strict constraint of proximity to wild-type

46. Example Network Wild-type’s solution. MOMA’s solution diverges flux.
ROOM’s solution finds a short alternative pathway.
ROOM preserves linearity of flow
ROOM’s solution is with maximal growth

47. ROOM’s Implicit Growth Rate Maximization
ROOM implicitly attempts to maintain the maximal possible growth rate of the wild-type organism
A change in growth requires numerous changes in fluxes M1 M2
Growth Reaction
To understand this, let’s go back to the pseudo growth reaction that represents the growth rate of the organism. The growth reaction drains out from the cell the metabolites that are required for growth. This number of such metabolites is in the order of tens. Any change in growth rate that is represented as a change in flux through this reaction requires many more changes to preserve mass balance constraint. So ROOM by using a metric that minimizes the number of flux changes implicitly prefers solutions that do not reduce the growth rate.
Therefore, we get that although the organism did not evolve to maximize its growth under all possible knockout configurations, the evolved regulatory mechanism that cope with the knockout by minimizing the number of regulatory changes may work to that effect. .
Biomass Mn

48. Intracellular Flux Measurements
Intracellular fluxes measurements in E. coli central carbon metabolism
Obtained using NMR spectroscopy in C labeling experiments
5 knockouts: pyk, pgi, zwf, gnd, ppc in Glycolysis and Pentose Phosphate pathways
Glucose limited and Ammonia limited medias
FBA wild-type predictions above 90% accuracy 13
To compare the flux predictions obtained by ROOM, MOMA and FBA on a real metabolic network, we searched the literature for experimental flux measurements in E. coli’s central carbon metabolism.
We’ve found experimental flux measurements in 4 different knocked-out strains on Glycolysis and Pentose-Phosphate pathway as shown in the figure. The flux measurements were done on different glucose-limited and ammonia-limited medias.
All measurements were obtained in experiments using NMR spectroscopy with isotope carbon sources.
For each experiment, we start by applying FBA to predict the flux distribution of the wild-type strain. The accuracy of the predictions was above 90% for all experiments. We note that the accuracy is calculated as the the Pearson correlation between the predicted and measured fluxes.
Emmerling, M. et al. (2002), Hua, Q. et al. (2003), Jiao, Z et al. (2003), Peng, et. al (2004)

49. Knockout Flux Predictions
ROOM flux predictions are significantly more accurate than MOMA and FBA in 5 out of 9 experiments
ROOM steady-state growth rate predictions are significantly more accurate than MOMA
Comparing the flux predictions obtained by ROOM with MOMA and FBA for the knocked-out strains we get that, in 4 out of 8 experiments, ROOM flux predictions were significantly more accurate than MOMA and FBA. This is shown in the left graph showing the accuracy of the predictions obtained by ROOM (red), MOMA (green) and FBA (blue). Only in 1 out of the 8 experiments, MOMA’s prediction is significantly more accurate than ROOM’s (we will discuss this later on).
Furthermore, we’ve found that the growth rate predictions obtained by ROOM and FBA are significantly more accurate than MOMA in 4 out of the 8 experiments. The graph on the right shows the error in the growth rate prediction as obtained by the different algorithms. We see that ROOM’s and FBA’s errors are less than 15% in all cases, while MOMA’s error (in green) reaches 50% and 90%.
Both ROOM and MOMA predicts a flux distribution of the knocked-out organism starting from a specific flux distribution of the wild-type strain. We note that the starting from alternative FBA solutions for the wild-type gives almost the same results that we present here.

50. ROOM vs. MOMA ROOM predicts metabolic steady-state after adaptation
Provides accurate flux predictions
Preserved flux linearity
Finds alternative pathways
Predicts steady-state growth rates
MOMA predicts transient metabolic states following the knockout
Provides more accurate transient growth rates

51. Additional Constraints
Transcriptional regulatory constraints (Covert, et. al., 2002)
Boolean representation of regulatory network
Used to predict growth, changes in expression levels, simulate courses of batch cultures
Energy balance analysis (Beard, et. al., 2002)
Loops are not feasible according to thermodynamic principles – resulting in a non-convex solution space

52. Additional Constraints: Slow Changes in the Environment
Timescales of cellular process are shorter than those of surrounding environment
Generate dynamic curves to simulate batch experiments (Varma, et. al., 1994)

53. Thank you for listening
Questions