Mathematical Representation of Reconstructed Networks The Left Null space The Row and column spaces of S

Introduction The system biology paradigm: “components network in silico models phenotype” System biology focuses on the nature of the links and their associated functional states (~phenotypes). Cells must select useful ‘functional states’.

Functional states E coli: 1.Outside the body (low O2, low temp) 2.Inside the body (high temp) 3.Stomach (Low PH) 4.Small intestine (low O2) H pylori: 1.Human gastric (Low PH)

Reaction / links ‘key properties’ 1. Stoichiometry. The stoichiometry is fixed, invariant between organisms for the same reactions and condition independent (pressure, pH, temp..) 2. Relative rates. Fixed by basic thermodynamic properties which depend on conditions such as pressure, pH, temp.. 3. Absolute rates. In contrast to stoichiometry and thermodynamics, highly manipulated by the cell enzymes.

Reminder The dynamic mass balance equation: Where, X is a vector of m metabolites. V represent vector of n ‘reaction rates’. S is m x n matrix of stoichiometric coefficients, rows represent metabolites and columns represent reactions. The right null space 1. For most practical purposes metabolism is in a steady state. 2. The null space contains all the steady state flux distributions and is thus of special importance to us.

Reminder Constraint-based analysis Linear programming (Simplex)

Each one of the generating vectors corresponds to an extreme pathway which the cell could theoretically control to reach every point in the flux cone. A particular point within this flux cone corresponds to a given flux distribution which represents a particular metabolic phenotype.

The analysis of the left null space of S allow us to define the achievable states of the cell and their physiological relevance. We look for ‘metabolic pools’ that have physiological meaningful interpretation.

Definition: The Left Null space of S li Span the left null space of S sj All are in the column space. (rank= r)

The Time invariants A linear combination of individual metabolic concentrations that do not change over time is called a metabolic pool. A dynamic motion along a reaction vector in the column space do not change the total mass in the pool.

The concentrations space a is a vector that gives the total concentrations of the pools. i.e. is the conservation vector. The rows of L ( i.e. ) that span the left null space define a concentration space. The time invariant metabolic pools resides in this concentration space. defines an affine hyper plane. This plane does not go through the origin. The concentration vector x resides in this space.

Classifying the pools Co-Factors, carriers e.g. the carbon backbone in glycolysis

Reaction map Vs. Compound map Groupings of chemical elements that move together.

Classifying the pools Futile cycle Internal cycle Through flux pathways Cofactor conservation Primary& secondary moieties conservation Primary moieties conservation

Simple reversible state One Type A Pool: Comment: The pool ‘AP+PA’ is constant both in SS and dynamic.

Reference states We can choose that lie in the left null space. This reference state is orthogonal to x is not orthogonal to the left null space, whereas and are. Now we can span the concentration space using the reaction vectors

(1) (2) Two conditions:

Bilinear association The Pools Interpretation Type 1. A+AP : Total cofactor : A 2. P+AP : Total energy : B Ordered by: A, P, AP

Carrier coupled reaction The pools: 1.C+CP : conservation of the substrate C. 2.A+AP : conservation of the cofactor A. 3.CP+AP : occupancy of P / total energy 4.C+A : vacancy of P / low energy state The entries of x ordered by: (CP, C, AP, A)

Redox carrier coupled reaction The pools The pools interpretation Type 1. : Total R. : A 2. : Redox occupancy 1. : B 3. : Redox occupancy 2. : B 4. : Redox vacancy : B 5. : Total redox carrier : C 6. :Total redox carrier : C

Multiple redox coupled reaction The pools The pools interpretation Type 1. : Total R. : A 2. : Redox occupancy 1. : B 3. : Redox occupancy 2. : B 4. : Redox vacancy : B 5. : Total redox carrier 1 : C 6. :Total redox carrier 2 : C (1) (2) (3)

Glycolysis Type B pools: High energy Conservation of P Low energy Stand alone inorganic P

TCA cycle Exchanging carbon group. Recycled C4 moiety which ‘carries’ the two carbon group that is oxidized. H group that contains the redox inventory in the system. Redox vacancy. Total cofactor pool.

Summary: Left Null space of S Contains dynamic invariants. A convex basis for this space is biological meaningful and can be found. Three basic types of convex basis vectors can be defined. The metabolic pools can be displayed on the compound map – similar to pathways in a flux map. Integration of time derivatives leads to bounded affine space of concentrations. The affine space of concentrations 1. All the concentrations states, dynamic and steady, lie in this space. 2. A suitable reference state can be defined (parallel to the left null space and orthogonal to the column space). The shifted concentration space is spanned by the Si’s.

The column space Contain the time derivatives of the concentrations. Spanned by the reaction vectors. Change in the flux levels determine the location of in the column space. Fast reactions that quickly come to SS reduce the column space dimension on slower time scales. Reduction in the columns space dimension leads to effective additional dimension in the left null space. Constraints on the fluxes induce constraints on the ‘s. Hence the column space is a closed space.

Example 1 O H The left null space will be spanned by the elemental matrix LS = ES = 0 R is a group of concentrations changing over time

Example 2

Example 3 (Ignore the P for a moment) Note: If one reaction is fast compare to the other, we get ‘L’ shape

The row space The row space contain all the thermodynamic driving forces (i.e. fluxes). The individual reaction fluxes form an orthogonal basis for the raw space. Each reaction has a natural thermodynamic basis vector. Since the fluxes are constrained, All the fluxes are in a rectangle in the positive orthant. The null space lies within the rectangle and its orthogonal complement is the row space.

Constraints on the flux values The magnitude of the individual fluxes is constrained. These constraints are derived from: 1. The limitation on the concentration. 2. Upper limit on the kinetic constants. The turnover rate of an enzyme complex X: Where the total amount of enzyme ( ) present is limited to X + e. Bilinear association of substrate to an enzyme: The rate is: Where is the size of the most limiting conservation pool of which Xi is a member. The total amount of enzyme (alone) limits the flux through enzymatic pathway. The release step of a product from enzyme is often the rate limiting step in enzyme catalysis.

Thermodynamic driving forces If the fluxes are imbalanced, there will be a net generation or elimination of compounds in the network. Since the r’s are fixed and the V’s are bounded, the inner product is also bounded.

The column space is naturally spanned by the reaction vectors. The row space can be represented by an orthogonal basis formed by the individual fluxes with values only in the positive orthant. The magnitude of the individual fluxes is limited by kinetics and caps on concentration values. This limitation also limits the possible value of the time derivatives and thus the column space. The column and row spaces are closed. Summary: The row and column spaces of S

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