. 2015 Feb 17;6:49. doi: 10.3389/fpls.2015.00049

Table 1.

Mathematical formalisms in computational biology used throughout this review.

Formalism	Mathematical formulation
Flux balance analysis—FBA (Varma and Palsson, 1994c)	max c^T · v
This computational approach predicts steady-state flux distributions that are thermodynamically feasible and mass-balanced. The underlying assumption of the method is that the organism under consideration operates under a certain optimality goal, e.g., the maximization of growth for microorganisms. The optimization problem formulation relies solely on the stoichiometry of the participating reactions (represented in the stoichiometric matrix S), lower and upper boundaries (v_min and v_max) for the respective fluxes and an optimization goal (captured in the vector c) as input. No knowledge of initial metabolite concentrations or kinetic parameters is required.	s.t. $\frac{d x}{d t}$ = S · v = 0
	v_min ≤ v ≤ v_max
Minimization of metabolic adjustment—MOMA (Segrè et al., 2002)	min (v − w)²
This FBA-based method was developed to identify a feasible flux distribution of a genetically perturbed system which is closest to the wild-type flux distribution. The rationale for this approach is the assumption that the metabolic network of the organism under consideration adjusts to the perturbation with a minimal rewiring of the flux profile. The set A contains all reactions which are switched off in the perturbed organism. Due to the introduction of the Euclidean norm the resulting optimization problem is quadratic.	s.t. $\frac{d x}{d t}$ = S · v = 0
	v_min ≤ v ≤ v_max
	v_k = 0, k ∈ A
Regulatory on/off minimization—ROOM (Shlomi et al., 2005)	min $\sum_{i = 1}^{m} y_{i}$
Similar to MOMA, this method aims at predicting steady-state flux distributions in a perturbed system which are closest to the wild-type flux distribution. However, ROOM relies on the minimization of the number of significant changes of fluxes (hence on/off) with respect to the wild type. The thresholds determining significant flux changes are given by w^U and w^L for upper and lower bounds, respectively. They are defined by relative (δ) and absolute (ε) ranges of tolerance from the wild type fluxes. For each flux v_i, 1 ≤ i ≤ N, y_i = 1 for a significant flux change in flux v_i and y_i = 0 otherwise. The introduced binary variables (y_i) render the problem a mixed-integer linear problem (see below).	s.t. $\frac{d x}{d t}$ = S · v = 0
	v_min ≤ v ≤ v_max
	v_k = 0, k ∈ A
	∀_i1 ≤ i ≤ m
	v_i − y_i (v_max,i − w^U_i) ≤ w^U_i
	v_i − y_i (v_min,i − w^L_i) ≥ w^L_i
	y_i ∈ {0, 1}
	w^U_i = w_i + δ\|w_i\| + ε
	w^L_i = w_i − δ\|w_i\| − ε
Dual formulation	Primal:
The duality theorem states that for every Primal optimization problem, there exists a Dual problem. In general, the solution to the Dual problem provides a lower bound to the solution of the Primal (minimization) problem. For convex optimization problems the value of an optimal solution of the Primal problem is given by the value of an optimal solution of the Dual problem (Boyd and Vandenberghe, 2004). Here, x and y is the vector of the Primal/Dual variable, respectively.	max c · x
	s.t. S · x = b
	x ≥ 0
	Dual:
	min b · y
	s.t. y · S = c
	x ≥ 0
Convex vs. non-convex optimization
For a convex optimization problem, if it is feasible, there can only be one optimal solution, which is globally optimal. Linear programming problems are always convex problems. Non-convex optimization may have multiple local optima. Hence, convex optimization problems can be much faster and more efficiently solved than non-convex optimization problems.
Linear programming (LP)	min c^T · x
Optimization problem in which the objective and all constraints are linear. If the vector of variables x includes entries which are only allowed to be integers the problem changes into a Mixed-integer linear programming (MILP) problem.	s.t. Ax ≤ b
	x_min ≤ x ≤ x_max
Quadratic programming (QP)	min $\frac{1}{2}$ x^TQ · x + c^T · x
Optimization problem in which the objective is a quadratic function and all constraints are linear.	s.t. Ax ≤ b
	x_min ≤ x ≤ x_max
Nonlinear programming (NLP)	min f(x)
Optimization problem in which the objective and/or constraints are nonlinear. If the vector of variables x includes entries which are only allowed to be integers the problem changes into a Mixed-integer nonlinear programming (MINLP) problem.	s.t. g(x) ≤ b
	x_min ≤ x ≤ x_max