1. BT8118 – Adv. Topics in Systems Biology
Prof. Eivind Almaas
Dept. of Biotechnology, NTNU

2. Overview of today Go through 2 homework problems
Duality of linear program
 Bilevel optimization revisited
Phenotype Phase Plane (PhPP) Analysis
Guest lecture: Per Bruheim on Metabolite and flux measurements
Discussion of individual projects

3. Simple LP example

4. Simple IP example
MatLab Task: 1. Change rhs value from 15 to 17. Find new maximal (x1, x2)
Use glpk function in MatLab to solve this as a MILP with
2. x1 and x2 as binary variables
3. x1 as binary variable
4. x1 as integer variable

5. FBA & MoMA: detailed comparison
Example reaction set: Corresponding network & WT solution from FBA:
Task:
1. Write down S matrix and constraints
2. Use MatLab “linprog” function to maximize R8
3. What happens with objective if R5 upper bound is reduced?
3. Knockout the “gene” R4
4. Find new fluxes using “linprog”
5. Find new fluxes using “quadprog” and the MoMA formulation
Maximize reaction R8

6. FBA & MoMA: detailed comparison
Example reaction set: Corresponding network & WT solution from FBA:
Using perturbed network:
(1) FBA optimal solution (2) MoMA solution
Maximize reaction R8
D. Segre et. al. OMICS 7, 301 (2003).

7. Duality of Linear Programming
Given linear program:
The dual program is:
How do we go from primal problem to dual problem?

8. Relationship between P and D
BUT: Will min(D) = max(P) ??

9. Duality theorem of LP

10. Dualization Recipe

11. What is the D of this P?

12. Bilevel optimization revisited
Engineering
Biology

13. Can you put together the pieces and state the
OptKnock formulation
Challenge:
Can you put together the pieces and state the
OptKnock LP problem?

14. Dual of the “Biological Problem”

16. Assessment of sensitivity of objective function
Phenotype Phase-Plane (PhPP) Analysis
Central tool for PhPP:
Assessment of sensitivity of objective function
What is response of objective function Z to
Changes in internal fluxes?  reduced costs
Changes in the bounds vector?  shadow price

17. Definitions Shadow price: Reduced cost:
PhPP defined using Shadow Prices

18. Example: Adaptive evolution of E. coli
Fong et al, J. Bact. 185, 6400 (2003)

19. Example: Adaptive evolution of E. coli

20. Geometrical interpretation of Shadow Price
Note: The shadow price = optimal value of dual variable!
Aucamp & Steinberg, J. Opl Res. Soc. 33, 557 (1982)

21. Optimal dual and shadow price
x*=(0.8,0)
π2=1.2

22. Optimal dual and shadow price
 Significant caveat for degenerate situations:
Aucamp & Steinberg, J. Opl Res. Soc. 33, 557 (1982)

23. Return to PhPP Analysis
Cellular phenotype in response to variation in uptake fluxes
PhPP has 2 separate regions: growth and non-growth
Growth region is further split into phases according to shadow price of the independent variables (variables along axes)
Within each phase, shadow prices are constant.
Each phase is consistent with different cellular phenotype

25. Challenge: Calculate the PhPP
Note: glpk will provide 1. optimal value of dual variable (extra.lambda)
2. reduced cost (extra.redcosts)

26. Project discussion