Optimization problem

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The general nonlinear optimization problem [1] can be formulated as follows: find a minimum of the objective function ϕ(x), where x lies in the intersection of the N-dimensional search space

Optimization formula 1.png

and the admissible region ℱ ⊆ ℝN defined by a set of equality and/or inequality constraints on x. Since the equality gs(x) = 0 can be replaced by two inequalities gs(x) ≤ 0 and –gs(x) ≤ 0, the admissible region can be defined without loss of generality as

Optimization formula 2.png

In order to get solution situated inside ℱ, we minimize the penalty function

Optimization formula 3.png

The problem could be solved by different optimization methods. We implemented the following of them in the BioUML software:

  • stochastic ranking evolution strategy (SRES) [1];
  • cellular genetic algorithm MOCell [2];
  • particle swarm optimization (PSO) [3];
  • deterministic method of global optimization glbSolve [4];
  • adaptive simulated annealing (ASA) [5].

The below table shows the generic scheme of the optimization process for these methods. SRES, MOCell, PSO and glbSolve run a predefined number of iterations Nit considering a sequence of sets (populations) Pi, i = 0,…,Nit − 1, of potential solutions (guesses). In the case of the first three methods, the size s ∈ ℕ+ of the population is fixed, whereas in glbSolve the initial population P0 consists of one guess, while the size sk+ 1 of the population Pk+1 is found during the iteration with the number k = 0,…, Nit − 1. The method ASA considers sequentially generated guesses xk ∈ Ω, k ∈ ℕ+, and stops if distance between xk and xk+1 defined as Euclidean norm

Optimization formula 4.png

becomes less than a predefined accuracy ε.

Optimization table 1.png

All methods, excepting glbSolve, are stochastic and seek global minimum of the function ϕ taking into account the admissible region ℱ. Thus, a guess x ∈ Ω is more preferable than a guess y ∈ Ω at some iteration of methods, if ψ(x) = 0 and ψ(y) ≠ 0 or ψ(x) < ψ(y). The method glbSolve is suited to solve only the problems with Ω ⊆ ℱ. Values of the function ψ are calculated but do not affect on the generation of potential solutions.

Application of non-linear optimization to systems biology

We assume that a mathematical model of some biological process consists of a set of chemical species S = {S1,…,Sm} associated with variables C(t) = (C1(t),…,Cm(t)) representing their concentrations, and a set of biochemical reactions ℛ = {R1,…,Rn} with rates v(t) = (v1(t),…,vn(t)) depending on a set of kinetic constants K. Reaction rates are modeled by standard laws of chemical kinetics. A Cauchy problem for ordinary differential equations representing a linear combination of reaction rates is used to describe the model behavior over time:

Optimization formula 5.png   (*)

Here N is a stoichiometric matrix of n by m. We say that Css is a steady state of the system (*) if

Optimization formula 6.png

Identification of parameters K and initial concentrations C0 is based on experimental data represented by a set of points Ciexp(ti,j) defining dynamics of variables C1(t),…,Cl(t), lm, at given times ti,j, j = 1,…,ri, where ri is the number of such points for the concentration Ci(t), i = 1,…,l. The problem of parameter identification consists in minimization of the function of deviations defined as the normalized sum of squares [6]:

Optimization formula 7.png

where normalization factors ωmini with ωmin = miniωi are used to make all concentration trajectories have similar importance. The weights ωi are calculated by one of the formulas on experimentally measured concentrations:

Optimization formula 8.png (mean square value), Optimization formula 9.png (mean value), Optimization formula 10.png (standard deviation).

If we want to consider additional constrains

Optimization formula 11.png

holding for concentrations C(t) and parameters K for some period of time Optimization formula 12.png, the penalty function is defined as

Optimization formula 13.png

References

  1. Runarsson T.P., Yao X. Stochastic ranking for constrained evolutionary optimization. IEEE Transactions on Evolutionary Computation. 2000. 4(3):284–294.
  2. Nebro A.J., Durillo J.J., Luna F., Dorronsoro B., Alba E. MOCell: A cellular genetic algorithm for multiobjective optimization. International Journal of Intelligent Systems. 2009. 24(7):726–746.
  3. Sierra M.R., Coello C.A. Improving PSO-Based Multi-objective Optimization Using Crowding, Mutation and ∈-Dominance. Evolutionary Multi-Criterion Optimization. Lecture Notes in Computer Scienc. 2005. 3410:505-519.
  4. Björkman M., Holmström K. Global Optimization Using the DIRECT Algorithm in Matlab. Advanced Modeling and Optimization. 1999. 1(2):17–37.
  5. Ingber L. Adaptive simulated annealing (ASA): Lessons learned. Control and Cybernetics. 1996. 25(1):33–54.
  6. Hoops S., Sahle S., Gauges R., Lee C., Pahle J., Simus N., Singhal M., Xu L., Mendes P., Kummer U. COPASI-a COmplex PAthway SImulator. Bioinformatics. 2006. 22(24):3067–3074.
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