Optimization aims to make a system or design as effective or functional as possible. In systems biology, most models have a dynamic nature, usually consisting of sets of differential equations. Dynamic optimization seeks the computation of optimal time-varying conditions (called control variables) for this type of systems. These control variables allow us to stimulate a given biosystem so as to achieve a desired dynamic behavior, e.g. stimulate a cell signaling cascade to maximize amplitude strength and/or duration, stimulate an oscillatory system to synchronize oscillations, remove oscillations, etc.

In other words, DO allows the computation of optimal operating policies for processes or systems which maximize (or minimize) a pre-defined performance index subject to the system dynamics and other possible constraints. DO may be solved by several methods, including the control vector parameterization (CVP) method, which has demonstrated excellent performance for a wide variety of problems.

Here we present a software toolbox, DOTcvpSB, which uses the CVP approach for handling continuous and mixed integer DO problems. DOTcvpSB has been successfully applied to several problems in systems biology and bioprocess engineering. The toolbox is written in MATLAB and provides an easy to use environment while maintaining a quite good performance. DOTcvpSB is designed for the Windows operating systems. The toolbox also contains a function for importing SBML models.

Key features:

• handling of a wide class of dynamic optimization problems, including constrained, unconstrained, fixed, and free terminal time problems described by ordinary differential equations (ODEs), as well as continuous and mixed integer decision variables;
• the inner initial value problem (IVP) is solved using the state-of-the-art methods available in SUNDIALS;
• the outer (MI)NLP problem can be solved using a number of advanced solvers, including local deterministic methods, stochastic global optimization methods, and hybrid metaheuristics;
• in addition to the traditional single optimization approach, the toolbox also offers more sophisticated strategies, like multistart, sucessive re-optimization, and hybrid strategies;
• a graphical user interface (GUI) which makes the definition and edition of a problem more easy and clear;
• possibility of importing SBML models;
• many output options for the results, including detailed figures.

DOTcvpSB has been implemented in MATLAB. The original dynamic optimization or mixed-integer dynamic optimization problem is solved numerically by the use of a suitable optimizer (outer iteration) which requires the solution of an IVP (inner iteration) which will in general consist on a set of ODEs plus a set of sensitivities to compute gradient information. The solution of the inner IVP is accomplished by calls to tailored solvers from the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS), more specifically CVODES. Since these simulations are the most computationally demanding task in the CVP method, our toolbox can automatically create compiled dynamically linked subroutines (known as MEX files in MATLAB) for the ODEs, Jacobian, and sensitivities, thus ensuring high performance.

The toolbox contains a number of modules (implemented as MATLAB functions) which can be grouped in two categories:

• utility modules: graphical user interface (GUI), simulation, and SBML-import modules;
• optimization modules: offering several optimization strategies, namely single optimization, multi-start, successive re-optimization, and hybrid optimization modules.

The utility modules offer several facilities for the definition, checking, and handling of problems. The toolbox can be operated through two equivalent approaches: by the use of the GUI, or directly from the command line (from where scripts with problem definitions can be created and executed). It also offers a module to import dynamic models from SBML files, and the imported models can be checked by a simulation module.

• Graphical User Interface (GUI) module: this module was developed in order to help users in the definition and execution of problems. With the help of this module, which follows an intuitive wizard-like approach, problem definitions and modifications are guided in an easy and convenient stepwise manner, especially indicated for entry users.
• Simulation module: this module carries out the dynamic simulation of the user-defined dynamics (plus assigned initial conditions and controls) generating the corresponding state trajectories. This module is especially useful for checking the model correctness during the definition phase, which is particularly error-prone. Typical errors like those related with units inconsistencies can be readily identified with this procedure.
• SBML to DOTcvpSB module: this module allows DOTcvpSB to import the systems dynamics from SBML (Systems Biology Markup Language) models. Once a dynamic model is imported, it is necessary to check the model correctness by simulation (previous module). If everything works correctly, the user can proceed with the definition of the other terms of the dynamic optimization problem (performance index, constraints) and, finally, with its numerical solution.

The optimization modules offers a suite of four different optimization strategies, each one with different options for the optimization solvers, following the 'Swiss Army knife' approach mentioned previously. All these modules are described in more detail below.

• Single optimization module: this module makes a single call to one of the optimization solvers, which can be either a local deterministic or global stochastic method (see available solvers below). This procedure can be sufficient for well conditioned, convex problems, or non-convex problems which are cheap to evaluate. In any case, it is recommended as the first strategy to try with any new problem.
• Multi-start optimization module: this modules runs a selected optimization solver (typically a local one) repeatedly. The set of solutions (performance index values) obtained can then be analyzed (e.g. plotting a histogram) in order to check the multimodality of the problem.
• Sucessive re-optimization module: Sucessive re-optimization can be used to speed up the convergence for problems where a high discretization level is desired (e.g. those where the control profiles behave wildly). This procedure runs several successive single optimizations automatically increasing the control discretization, NLP, and IVP tolerances after each run.
• Hybrid optimization module: Hybrid optimization is characterized by the combination of a stochastic global method plus a deterministic local method. This procedures ensures a compromise between the robustness of global methods and the efficiency of local ones. This module is especially indicated for difficult multimodal problems. In any case, tweaking the hybrid method requires a deep knowledge of the solvers, and this approach will be almost always more costly (in CPU time) than the single optimization procedures using local methods (the price to pay for the increased robustness).

The toolbox provides interfaces to several optimization state-of-the-art solvers:

local deterministic

1. IPOPT [Interior Point OPTimizer] implements a primal-dual interior point method, and uses line searches based on Filter methods;
2. FMINCON [Find MINimum of CONstrained nonlinear multivariable function] is a part of the MATLAB optimization toolbox which uses sequential quadratic programming (SQP);
3. MISQP [Mixed-Integer Sequential Quadratic Programming] solves mixed-integer non-linear programming problems by a modified sequential quadratic programming method;

stochastic global

1. DE [Differential Evolution] uses population based approach for minimizing the performance index;
2. SRES [Stochastic Ranking Evolution Strategy] uses an evolution strategy combined with an approach to balance objective and penalty functions;

and hybrid metaheuristics (combination of stochastic and deterministic solvers)

1. ACOmi [Ant Colony Optimization for mixed integer nonlinear programming problems] is inspired by ants foraging behavior and for the local search it uses MISQP;
2. MITS [Mixed-Integer Tabu Search algorithm] is based on extensions of the metaheuristic Tabu Search algorithm and for local search it uses MISQP too;

All NLP and MINLP solvers besides FMINCON, which is a part of MATLAB Optimization Toolboxes are included directly into the DOTcvp package.

Forward integration of the ODE, Jacobian, and sensitivities if it is needed is ensured by CVODES, a part of SUNDIALS, which is able to perform the simultaneous or staggered sensitivity analysis too. The IVP problem can be solved with the Newton or Functional iteration module and with the Adams or BDF linear multistep method (LMM). The Adams method is recommended for solving of the non-stiff problems while BDF is recommended for solving of the stiff problems. Note that the sensitivity equations are provided analytically and the error control strategy for the sensitivity variables could be enabled.