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

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

Simple LP example

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

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

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).

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

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

Duality theorem of LP

Dualization Recipe

What is the D of this P?

Bilevel optimization revisited Engineering Biology

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

Dual of the “Biological Problem”

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

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

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

Example: Adaptive evolution of E. coli

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

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

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

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

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

Project discussion