DOTcvpSB, a software toolbox for dynamic optimization in systems biology
 Tomáš Hirmajer^{1, 2},
 Eva BalsaCanto^{1} and
 Julio R Banga^{1}Email author
DOI: 10.1186/1471210510199
© Hirmajer et al; licensee BioMed Central Ltd. 2009
Received: 28 January 2009
Accepted: 29 June 2009
Published: 29 June 2009
Abstract
Background
Mathematical optimization aims to make a system or design as effective or functional as possible, computing the quality of the different alternatives using a mathematical model. Most models in systems biology have a dynamic nature, usually described by sets of differential equations. Dynamic optimization addresses this class of systems, seeking the computation of the optimal timevarying conditions (control variables) to minimize or maximize a certain performance index. Dynamic optimization can solve many important problems in systems biology, including optimal control for obtaining a desired biological performance, the analysis of network designs and computer aided design of biological units.
Results
Here, we present a software toolbox, DOTcvpSB, which uses a rich ensemble of stateoftheart numerical methods for solving continuous and mixedinteger dynamic optimization (MIDO) problems. The toolbox has been written in MATLAB and provides an easy and user friendly environment, including a graphical user interface, while ensuring a good numerical performance. Problems are easily stated thanks to the compact input definition. The toolbox also offers the possibility of importing SBML models, thus enabling it as a powerful optimization companion to modelling packages in systems biology. It serves as a means of handling generic blackbox models as well.
Conclusion
Here we illustrate the capabilities and performance of DOTcvpSB by solving several challenging optimization problems related with bioreactor optimization, optimal drug infusion to a patient and the minimization of intracellular oscillations. The results illustrate how the suite of solvers available allows the efficient solution of a wide class of dynamic optimization problems, including challenging multimodal ones. The toolbox is freely available for academic use.
Background
Optimization plays a key role in computational biology and bioinformatics [1, 2]. Dynamic optimization, also known as openloop optimal control, seeks the maximization or minimization of a suitable performance index (which characterizes the solution quality) of a dynamic system taking into account possible equality or inequality constraints. The solution is represented by the optimal decision variables, which can be continuous (real numbers), discrete (integer numbers), or both. Continuous variables can be used to encode timevarying stimuli, while discrete variables usually represent events (like an on/off switch) or configurations. An overview of optimization in the context of computational systems biology was given by [3] and more recently by [4], the latter highlighting the need of robust and efficient dynamic optimization methods. Examples of relevant problems covered there include optimal control for modification of selforganized dynamics, optimal experimental design, dynamic flux balance analysis, the discovery of biological network design strategies and computational design of integrated biological circuits (synthetic biology).
A popular numerical approach for solving dynamic optimization problems is the control vector parameterization (CVP) method [5], which transforms the original problem into an outer nonlinear programming (NLP) or mixedinteger nonlinear programming (MINLP) problem, with an inner initial value problem (IVP). Solving the outer problem requires a suitable (MI)NLP solver. Since most biological systems are nonlinear, the resulting optimization problems are frequently multimodal and very challenging to solve, so it is necessary to use proper global optimization methods [6].
This work presents DOTcvpSB, a user friendly MATLAB dynamic optimization toolbox based on the CVP method, which provides an easy to use environment while ensuring a good numerical performance. Users only need to define their dynamic optimization problems via a simple and compact input file which is close to the standard mathematical notation. Advanced users can tweak many configuration options for the different solvers in order to finetune the solution process. Although other existing toolboxes and software packages allow the definition and solution of optimization problems in systems biology (e.g. COPASI [7], PottersWheel [8] or SBtoolbox2 [9], to name a few), they are restricted to problems where the decision variables are static (timeindependent). DOTcvpSB allows the definition and solution of dynamic optimization problems where decision variables are timedependent, thus reaching a much broader class of optimization problems.
Implementation
In this section, we first describe the class of problems considered and the framework chosen for its numerical solution. Next, we describe the organization and capabilities of the toolbox, highlighting its key features and modules.
Mixedinteger Optimal Control Problem
The mixedinteger optimal control problem, also called mixedinteger dynamic optimization (MIDO) problem, considers the computation of time dependent operating conditions (controls), discrete – binary or integer decisions and timeindependent parameters so as to minimize (or maximize) a performance index (or cost function) while keeping a set of constraints coming from safety and/or quality demands and environmental regulations. Mathematically this is formulated as follows:
where is the vector of state variables, is the vector of real valued control variables, is the vector of integer control variables, is the vector of timeindependent parameters, t_{ f }is the final time of the process, m_{ e }, m_{ i }represent the number of equality and inequality constraints, respectively and g collects all state constraints, pathway, pointwise and final time constraints and u_{ L }, i_{ L }, p_{ L }, u_{ U }, i_{ U }, p_{ U }correspond to the lower and upper bounds for the control variables and the timeindependent parameters.
Control Vector Parameterization
DOTcvpSB is based on the control vector parameterization (CVP) framework to solve the class of problems stated above. The CVP methodology proceeds dividing the control variables (u(t) and i(t)) into a number of elements and then approximating each element by means of different basis functions, usually low order polynomials. In this way the control variables are parameterized using w_{ u }∈ R^{ ρ }and w_{ i }∈ Z^{ ρ }, which become decision variables. This parameterization transforms the original infinite dimensional problem into a finite dimension (mixedinteger) nonlinear programming problem that may be solved by a suitable (MI)NLP solver. Note that the evaluation of the objective function and constraints requires the solution of the system dynamics by solving an inner initial value problem (IVP).
If the outer (MI)NLP problem is convex, deterministic (gradientbased) local methods are the best alternative to efficiently solve it. In this regard, (mixedinteger) sequential quadratic programming methods, such as MISQP [10], can be considered the stateoftheart. Nevertheless, in presence of nonconvexities, local methods usually present convergence to local minima, thus requiring the use of global optimization methods.
Global optimization methods can be roughly classified in two major groups: deterministic and stochastic methods. Certain deterministic global methods can guarantee global optimality for particular classes of problems, although the computational cost becomes infeasible for problems of realistic size. They have been recently applied for the solution of MIDO problems [11, 12]. Regarding stochastic methods, several works, as reviewed by [6], have illustrated their potential for dynamic optimization (DO) and, more recently, for mixedinteger dynamic optimization (MIDO) [13]. Stochastic methods usually locate the vicinity of global solutions with reasonable efficiency, but the cost to pay is that global optimality can not be guaranteed. Alternatives such as globallocal hybrid methods have been presented both for DO [14] and MIDO [15], significantly improving the computational efficiency. Thus, we could summarize the current stateoftheart in this domain by concluding that there is no silver bullet for global optimization of arbitrary MIDO problems. And this is why DOTcvpSB includes a suite of optimization solvers, following a "Swiss Army knife" approach.
Many of these optimization methods require the computation of the gradient of the objective and/or constraints with respect to the decision variables. Vassiliadis [5] proposed the use of first order parametric sensitivities to compute such information. The sensitivity equations result from a chain rule differentiation applied to the system defined in Eqns. 2 with respect to the decision variables and may be solved in combination with the original system. For this purpose, the use of Backward Differentiation Formulas (BDF) methods is very attractive since they are able to exploit the fact that the original system and the sensitivities share the same Jacobian.
Toolbox description
Key Features
The core capabilities of the toolbox can be summarized as follows:

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 stateoftheart methods available in SUNDIALS [16];

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 reoptimization [17], and hybrid strategies [14];

a graphical user interface (GUI) which makes the definition and edition of a problem more easy and clear;

possibility of importing SBML models [18];

many output options for the results, including detailed figures.
Description of main modules
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 SBMLimport modules;

optimization modules: offering several optimization strategies, namely single optimization, multistart, successive reoptimization, and hybrid optimization modules.
Utility 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 wizardlike 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 userdefined 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 errorprone. 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 [18, 19]. 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.
Optimization modules
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 nonconvex problems which are cheap to evaluate. In any case, it is recommended as the first strategy to try with any new problem.

Multistart 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 reoptimization module: Sucessive reoptimization 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).
Numerical optimization methods (NLP and MINLP solvers)
The toolbox provides interfaces to several optimization stateoftheart solvers:

local deterministic
 1.
IPOPT [20] implements a primaldual interior point method, and uses line searches based on Filter methods;
 2.
FMINCON [21] is a part of the MATLAB optimization toolbox which uses sequential quadratic programming (SQP);
 3.
MISQP [10] solves mixedinteger nonlinear programming problems by a modified sequential quadratic programming method;

stochastic global

and hybrid metaheuristics
where the deterministic MISQP solver and all hybrid solvers are able to handle mixedinteger problems directly. Users can change solvers by simply changing an option in the input data structure, thus requiring no problem reformulation.
Numerical simulation method (IVP solvers)
Forward integration of the ODE, Jacobian, and sensitivities (when needed) is ensured by CVODES, a part of SUNDIALS [16], which is also able to perform simultaneous or staggered sensitivity analysis. 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 nonstiff 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.
Recommended operating procedure
It should be noted that, for a general MIDO formulation, there is no a priori way to distinguish if the resulting MINLP will be convex or not inside the search space considered, so the user has no clue on which optimization strategy should be using. Thus, we recommend that, for any new problem, the user follows this protocol:

Step 1: try solving the problem with the single optimization strategy and a local deterministic method, such as FMINCON or IPOPT for DO problems, or MISQP for MIDO problems, using a rather crude control discretization (e.g. 10 elements). After obtaining a solution, repeat changing the initial guess for the control variable. If a rather different solution is obtained, suspect multimodality and go to step 2 below. If not, solve the problem again using a finer discretization. For faster and more satisfactory results regarding control discretization, use the successive reoptimization module.

Step 2: solve the problem using the multistart optimization module. In general 100 runs is a sensible number for this task, but for costly problems the user might want to reduce this. Plotting an histogram of the resulting set of solutions will give a good view of the problem multimodality. For clearly multimodal problems, go to step 3. If not, stop, or go back to step 1 if e.g. more refined control levels are desired.

Step 3: use the single optimization strategy as in step 1, but use a global stochastic method, like DE or SRES for DO problems, or ACOmi or MITS for MIDO problems. If satisfactory results are obtained in reasonable computation times, stop. If the computational cost is excessive, go to step 4.

Step 4: use a hybrid globallocal strategy. More advanced users can tweak the different options to increase efficiency and/or robustness.
This protocol is especially recommended for novel users who are not familiar with numerical optimization methods. Advanced users can tweak the hybrid strategy options, or even create their own strategies combining calls to the different solvers in a MATLAB script.
Results and discussion
This section illustrates the usage and performance of the different modules of DOTcvpSB considering several illustrative examples.
Importing and checking a SBML dynamic model
For illustrative purposes, a dynamic model of the cell cycle [24] was chosen and imported into the DOTcvpSB toolbox. The problem is marked as BIOMD0000000005, Tyson1991_CellCycle_6var, 1831270 can be downloaded as an '.xml' file from the Biomodels database web page: http://www.ebi.ac.uk/biomodels/.
Single optimization
Here we solve a relatively simple problem to illustrate the usage of the single optimization strategy with a local deterministic solver.
Drug displacement problem
DOTcvpSB typical input data structure for the drug displacement problem.
% Example of the DOTcvp simple input file for the drug displacement problem  

data.name  = 'DrugDisplacement';  % name of the problem 
data.odes.parameters(1)  = {'A = 232'};  % constant parameters before ODE 
data.odes.parameters(2)  = {'B = 46.4'};  
data.odes.parameters(3)  = {'C = 2152.96'};  
data.odes.res(1)  = {'((1+0.2*(y(1)+y(2)))^2/(((1+0.2*(y(1)+y(2)))^2+A+B*y(2))*((1+0.2*(y(1)+y(2)))^2+A+B*y(1))C*y(1)*y(2)))*(((1+0.2*(y(1)+y(2)))^2+A+B*y(1))*(0.02y(1))+B*y(1)*(u(1)2*y(2)))'};  
data.odes.res(2)  = {'((1+0.2*(y(1)+y(2)))^2/(((1+0.2*(y(1)+y(2)))^2+A+B*y(2))*((1+0.2*(y(1)+y(2)))^2+A+B*y(1))C*y(1)*y(2)))*(((1+0.2*(y(1)+y(2)))^2+A+B*y(2))*(u(1)2*y(2))+46.4*(0.02y(1)))'};  
data.odes.res(3)  = {'1'};  
data.odes.ic  = [0.02 0.0 0.0];  % vector of initial conditions 
data.odes.tf  = 300.0;  % final time 
data.nlp.RHO  = 5;  % CVP discretization level 
data.nlp.J0  = 'y(3)';  % performance index, minmax(performance index) 
data.nlp.u0  = 4.0;  % initial guess for control values 
data.nlp.lb  = 0.0;  % lower bounds for control values 
data.nlp.ub  = 8.0;  % upper bounds for control values 
data.nlp.solver  = 'IPOPT';  % ['FMINCON''IPOPT''SRES''DE''ACOMI''MISQP''MITS'] 
data.nlp.FreeTime  = 'on';  % ['on''off'] set 'on' if free time is considered 
data.nlp.eq.status  = 'on';  % ['on''off'] switch on/off of the equality constraints 
data.nlp.eq.NEC  = 2;  % number of active equality constraints 
data.nlp.eq.eq(1)  = {'y(1)0.02'};  % first equality constraint 
data.nlp.eq.eq(2)  = {'y(2)2.0'};  % second equality constraint 
data.nlp.eq.time(1)  = data.nlp.RHO;  % to indicate that it is an endpoint constraint 
data.nlp.eq.time(2)  = data.nlp.RHO;  % to indicate that it is an endpoint constraint 
data.options.trajectories  = size(data.odes.res,2)1;  % how many state trajectories will be displayed 
where the decision variable (control (u)) is constrained with lower and upper bounds set at values of 0.00 and 8.00. The desired concentrations of the drugs in the blood at final time should be equal to 0.02 and 2.00, respectively.
Successive reoptimization
Here we show how to use the successive reoptimization module in order to obtain refined optimal control profiles.
LeeRamirez bioreactor
where the state variables represent the reactor volume (x_{1}), the cell density (x_{2}), the nutrient concentration (x_{3}), the foreign protein concentration (x_{4}), the inducer concentration (x_{5}), the inducer shock factor on cell growth rate (x_{6}), and the inducer recovery factor on cell growth rate (x_{7}). The final time is specified as 10 h. The additional constrains at the decision variables are lower and upper bounds set at the value of 0.00 and 1.00.
Hybrid optimization
Here we solve a multimodal problem using the powerful hybrid strategy, where the adequate combination of an stochastic global and a deterministic local solver allows reaching the vicinity of global solution in a reasonable computation time.
Drug displacement problem with path constraint
Multistart and single optimization with a global method
The multistart strategy is a good way of checking the possible nonconvexity of problems. When the multimodality of a problem has been confirmed, users can choose a global or a hybrid strategy to find a solution in the close vicinity of the global one. We illustrate all this here considering a challenging MIDO problem.
Phase resetting of a calcium oscillator problem: a mixedinteger dynamic optimization problem
Parameter values for the calcium oscillator problem
Model Parameters (Reaction Coefficients)  Weighted Coefficients  Initial Values  Desired Values  

k_{1} = 0.09  k_{8} = 32.24  K_{15} = 0.16  w_{1} = 5.0  x_{1}(0) = 0.03966  = 6.78677 
k_{2} = 2.30066  K_{9} = 29.09  k_{16} = 4.85  w_{2} = 5.0  x_{2}(0) = 1.09799  = 22.65836 
k_{3} = 0.64  k_{10} = 5.0  K_{17} = 0.05  w_{3} = 15.0  x_{3}(0) = 0.00142  = 0.38431 
K_{4} = 0.19  K_{11} = 2.67  t_{ F }= 22.0  w_{4} = 25.0  x_{4}(0) = 1.65431  = 0.28977 
k_{5} = 4.88  k_{12} = 0.7  w_{5} = 50.0  
k_{6} = 1.18  k_{13} = 13.58  w_{6} = 5.0  
k_{7} = 2.08  k_{14} = 153.0 
and the timeindependent parameter: 1 ≤ p_{1} ≤ 1.3, where state variables represent the concentration of activated Gprotein (x_{1}), active phospholipase C (x_{2}), intracellular calcium (x_{3}), and intraER calcium (x_{4}). The timefixed parameters together with the initial concentrations, desired values of the state variables and weighted coefficients are described in detail in the Table 2. The control variables are chosen binaries (i_{1}, i_{2}), which refer to the concentrations of an uncompetitive inhibitor of the PMCA (plasma membrane Ca^{2+}) ion pump and the inhibitor of PLC activation by the Gprotein. The influence of the first inhibitor is modeled according to MichaelisMenten kinetics while that of the second inhibitor is modelled with the help of the term (1  i_{2}), where i_{2} = 1 corresponds with the maximum amount of the inhibitor. An additional equality constraint was added to fix the final time at the fixed value (t_{ F }). The best performance index reported in [29] was 1538.00, where this reported cost function corresponds to the term . These authors reported that the system is extremely sensitive to small perturbations in the stimulus.
Conclusion
Here we have presented DOTcvpSB, a MATLAB toolbox for solving dynamic optimization problems from the domain of systems biology. This toolbox is able to handle very general mixedinteger dynamic optimization formulations, thus providing the opportunity to state and solve complex problems, such as e.g. optimal control for obtaining a desired biological performance, dynamic analysis of network designs or computer aided design of biological units. Problems are easily defined via a compact input structure, or optionally using a graphical user interface.
This toolbox has been developed placing particular care in providing stateoftheart solvers in order to ensure a good compromise between computational robustness and efficiency. DOTcvpSB offers two key and unique advantages:

It incorporates a suite of local and global optimization solvers so as to handle a wide range of problems, including nonconvex (multimodal) ones.

It offers several optimization strategies, including single, multistart, sucessive reoptimization and hybrid methods. These strategies can be effectively used to enhance the solution of difficult multimodal problems.
The capabilities and performance of DOTcvpSB were successfully tested using several challenging benchmarks problems taken from the open literature. The results confirmed that the toolbox was able to get excellent results in reasonable computation times, showing a good compromise between robustness and efficiency.
Availability and requirements
Project name: DOTcvpSB, a Software Toolbox for Dynamic Optimization in Systems Biology
Project homepage: The toolbox can be downloaded from the following website, which also offers documentation (installation instructions, manual, tutorial and video demos): http://www.iim.csic.es/~dotcvpsb/
Operating system(s): Windows. A Linux version is planned for the near future.
Programming language: MATLAB versions 7.0–7.6 (2008a) is required, and the MATLAB Optimisation Toolbox and Symbolic Math Toolbox are highly recommended.
Other requirements: The toolbox distribution includes most of the needed external solvers: IVP solver CVODE (part of SUNDIALS suite), and (MI)NLP solvers ACOmi, DE, IPOPT, MISQP, MITS and SRES. The Optimization Toolbox is needed if the user wants to use FMINCON as a NLP solver. FORTRAN compilation to speedup computations is secured by a combination of gnumex and MinGW, packages which are distributed with the toolbox as well. On the other hand, the Symbolic Math Toolbox is needed if automatic generation of sensitivities and Jacobian are desired (recommended). Users must install the SBML and libSMBL toolboxes in order to be able to import SBML models.
License: The toolbox can be obtained and used for free for academic purposes, and is under the creative commons license. The conditions of the license can be found on: http://creativecommons.org/licenses/byncnd/3.0/
Any restrictions to use by nonacademics: Following the previous license.
Abbreviations
 ACOmi:

Ant Colony Optimization for Mixed Integer nonlinear programming problems
 ATP:

Adenosine TriPhosphate
 BDF:

Backward Differentiation Formula
 CVP:

Control Vector Parameterization
 DE:

Differential Evolution
 FMINCON:

Find MINimum of CONstrained nonlinear multivariable function
 MISQP:

MixedInteger Sequential Quadratic Programming
 GUI:

Graphical User Interface
 IPOPT:

Interior Point OPTimizer
 IVP:

Initial Value Problem
 LMM:

Linear Multistep Method
 MI:

MixedInteger
 MIDO:

MixedInteger Dynamic Optimization
 MINLP:

MixedInteger NonLinear Programming
 MITS:

MixedInteger Tabu Search algorithm
 NLP:

NonLinear Programming
 ODEs:

Ordinary Differential Equations
 SBML:

Systems Biology Markup Language
 SRES:

Stochastic Ranking Evolution Strategy.
Declarations
Acknowledgements
The authors acknowledge financial support from Spanish MICINN project MultiSysBio DPI200806880C0302. TH acknowledges a JAEdoc fellowship from Spanish Council for Scientific Research (CSIC), Spain.
Authors’ Affiliations
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