White Box Function Estimationusing Convenience Kinetics

COMP 150GA – Class ProjectFall 2011

Tufts University

Prof. Soha Hassoun, soha@cs.tufts.edu

Goal

• Develop approximate expressions for rate change of concentrations in metabolic networks, when given measurement data (data sets that correspond to various operating conditions).

“White Box” approximationAssume you have *all* measurements!

d[D-Glucose]/dt = -vGlkd[G6P]/dt =vGlk - vPgid[F6P]/dt = vPgi -vPfkAd[FDP]/dt = vPfkAd[ATP]/dt = -vGlk -vPfkAd[ADP]/dt = vGlk + vPfkA

ATP

ADP

FDP

d[D-Glucose]/dt = ?d[G6P]/dt =vGlk - vPgid[F6P]/dt = vPgi -vPfkAd[FDP]/dt = ?d[ATP]/dt = ?d[ADP]/dt = ?

Each rate equation maybe a function is of the concentration of one or more of compound on

the boundaries or inside the cell

ATP

ADP

FDP

• GOAL: – What is the best form that describes these concentrations?– Minimize the number of variables in the rate equations

d[D-Glucose]/dt = ?d[FDP]/dt = ?d[ATP]/dt = ?d[ADP]/dt = ?

Shape of theconcentrations for some data sets we have

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 961010

0.5

1

1.5

2

2.5

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97-0.5

0

0.5

1

1.5

2

2.5

3

3.5

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 970

0.5

1

1.5

2

2.5Glucose

FDP

ATP

ADP

FDPATP

SET1 SET2

SET3 SET4

Convenience Kinetics*

• A systematic way of generating rate equations.• Apply convenience kinetics to determine rate for each input and

output metabolite.• Each metabolite can influence one or more other metabolites

either as:(a) a reactant or product(b) an activator

(c) an inhibitor• Narrowing down parameter choices:

– We can look up Vmax for each reaction using Brenda. Vmax may dominate and can be used to seed guesses.

– We can use Gibbs free energy to constraint the solution space*http://www.tbiomed.com/content/pdf/1742-4682-3-41.pdf

Convenience Kinetics – reactant/product relationship example

• Given equation: A + X ↔ B + Y

• We have 6 parameters for this equation: those involved with ~a, ~x, ~b, ~y, and k+, and k-

=

Convenience Kinetics – reactant/product relationship (general)

Assume that α and β (stoichiometric coefficients) are knownEach kM is a parameter represents binding energy known as dissociation constant as

If irreversible, then kM goes to infinity… ~b goes to zero and products disappear from the equations

Convenience Kinetics – Activators/Inhibitors

• Activators modify the rate equation by a factor, where d is the concentration of the modifier and kA is the activation (binding) constant

• Inhibitors modify the rate equation by a factor, where d is the concentration of the modifier and kI is the inhibition (binding) constant

Convenience Kinetics – General Form• For an equation l with m metabolites,

Activator pre-factorsInhibitor pre-factors

Set of all activators

Set of all inhibitors

Metabolites in the reactant set Metabolites in

the product set

* El does not need to be an independent parameter, assume k+cat and k-cat are modified to account for E

Thermodynamic Constraints limit parameter solution space.We can re-write the rate equation

Is kiG known for a particular metabolite or does it need to be estimated?

Can klV be obtained from a data base (??)

Genetic Programming• Each solution has a list of reactants (R), products (P), inhibitors (I),

activators (A), and associated parameters.• Capture information using three matrices: each column is a reaction

and each row is a compound. – N matrix: Each entry shows how a metabolite participates in a particular

reaction (Given)– Regulation matrix: Each entry is +1, or -1 for activation or inhibition (guess!)– K matrix: Each entry describes an AFFINITY, if that relationship exists An order of

magnitude estimate is useful (.1, 1, 10).

• To evolve form:– Mutations: add/remove R, P, I, A– Crossover: move from R/P to I/A ????

• Should be able to perform parameter estimation for convenience kinetics format (an independent parameter estimation problem).

Test Case #1

• D-Glucose + 2 ATP ↔ 2 ADP + FDP

ATP

ADP

FDP

Test Case #2• S ↔ P

Test Case #3

• GLUC + 2NAD + NADP + ADP ↔ 2Ethanol + 2NADH + NADPH + ATP

• GLUC6P + 2NAD + NADP + 2ADP ↔ 2Ethanol + 2NADH + NADPH + 2ATP

• GAP + NAD + 2ADP ↔ 2Ethanol + NADH + 2ATP

GAP

GLUC6P

END

• Old slides follow – please ignore

Test Case #1

Test Case #1Equation Rates

Network

Parameters Used in the Paper (Data generated used similar K and n values)

Test case #2

Input concentrations are P and M2, E2 We simplified the equationsP

Test Case #3