Dynamic sensitivity analysis of biological systems
 Wu Hsiung Wu^{1},
 Feng Sheng Wang^{2}Email author and
 Maw Shang Chang^{1}
https://doi.org/10.1186/147121059S12S17
© Wu et al; licensee BioMed Central Ltd. 2008
Published: 12 December 2008
Abstract
Background
A mathematical model to understand, predict, control, or even design a real biological system is a central theme in systems biology. A dynamic biological system is always modeled as a nonlinear ordinary differential equation (ODE) system. How to simulate the dynamic behavior and dynamic parameter sensitivities of systems described by ODEs efficiently and accurately is a critical job. In many practical applications, e.g., the fedbatch fermentation systems, the system admissible input (corresponding to independent variables of the system) can be timedependent. The main difficulty for investigating the dynamic log gains of these systems is the infinite dimension due to the timedependent input. The classical dynamic sensitivity analysis does not take into account this case for the dynamic log gains.
Results
We present an algorithm with an adaptive step size control that can be used for computing the solution and dynamic sensitivities of an autonomous ODE system simultaneously. Although our algorithm is one of the decouple direct methods in computing dynamic sensitivities of an ODE system, the step size determined by model equations can be used on the computations of the time profile and dynamic sensitivities with moderate accuracy even when sensitivity equations are more stiff than model equations. To show this algorithm can perform the dynamic sensitivity analysis on very stiff ODE systems with moderate accuracy, it is implemented and applied to two sets of chemical reactions: pyrolysis of ethane and oxidation of formaldehyde. The accuracy of this algorithm is demonstrated by comparing the dynamic parameter sensitivities obtained from this new algorithm and from the direct method with Rosenbrock stiff integrator based on the indirect method. The same dynamic sensitivity analysis was performed on an ethanol fedbatch fermentation system with a timevarying feed rate to evaluate the applicability of the algorithm to realistic models with timedependent admissible input.
Conclusion
By combining the accuracy we show with the efficiency of being a decouple direct method, our algorithm is an excellent method for computing dynamic parameter sensitivities in stiff problems. We extend the scope of classical dynamic sensitivity analysis to the investigation of dynamic log gains of models with timedependent admissible input.
Keywords
Background
A mathematical model to understand, predict, control, or even design a real biological system is a central theme in systems biology. The most often used mathematical models for dynamic biological systems are formulated as nonlinear ordinary differential equations (ODEs). The critical challenges to get an ODE model are structure identification and parameter estimation of the model. To identify the structure and parameters of a dynamic model, the most important and essential job is to find the solution of ODEs efficiently and accurately. This job can be treated by analytical and numerical methods. Analytical methods are limited to certain special forms of ODEs, but numerical methods have no such limitations. There are several numerical methods can be used to solve ODEs [1–3], e.g., Taylorseries methods, modified Euler methods, and RungeKutta methods with variable step size control. The Taylorseries method takes more computation time to solve ODEs if the various derivatives are complicated, and the error is difficult to determine for arbitrary functions. The modified Euler method is a special case of a secondorder RungeKutta method, and is more efficient compared to the Taylorseries method [4]. Fourthorder RungeKutta method is the most widely used ODE solver to meet requirements on both efficiency and accuracy. Collocation methods [5] are another common algorithms for solving ODEs and have been used for more than forty years. Wang [6] proposed a modified collocation method to transform ODEs into algebraic equations, and solved them by the NewtonRaphson method or an iteration method with step length restriction. The restricted step size is fixed and computed by trial and error. To overcome this drawback, we propose an adaptive step size control approach based on the Banach fixed point theorem for the modified collocation method in this paper.
There are different types of gains and dynamic sensitivities defined for sensitivity analysis [7–9]. The relative change of a dependent variable in response to a relative change in an independent variable is called a logarithmic gain, or a log gain. Log gains describe the change of dependent variables due to an environment change and are very useful for the assessment of the robustness and parameter estimation of a model. The change of a dependent variable in response to a change in a parameter is called a parameter sensitivity. In contrast to log gains, parameter sensitivities are the change of dependent variables correspond to a structure change in the model. The Biochemical Systems Theory (BST) [10] and Metabolic Control Analysis (MCA) [11–13] have achieved a great success in addressing the sensitivities at a steady state. However, the transient or periodic behavior is the primary interest in many systems (e.g., oscillation systems and fermentation systems that do not have a steady state). In these systems, the parameter sensitivities and log gains change with time, therefore the calculation methods for the steady state responses can not function. Dynamic sensitivity analysis is used in studying timevarying sensitivities in dynamic biological systems. A dynamic biological system can be characterized using logarithmic gains, sensitivities with respect to parameters and initial conditions. Several methods have been published to evaluate dynamic sensitivities [14–24]. They can be divided into the indirect methods (IDMs) and the direct methods (DMs). In the IDMs, the value of one dedicated parameter is varied infinitesimally while the values of other parameters are fixed. The model equations are solved anew for these sets of values of the parameters that differ in the value of the dedicated parameter only. The sensitivity of each variable with respect to this dedicated parameter is computed using the difference between the solutions of that variable for the two sets of values of the parameters divided by the infinitesimal difference of the dedicated parameter. In the DMs, using an ODE solver to solve the model equations and sensitivity equations simultaneously is the most used method for computing dynamic sensitivities. Shiraishi et al. [25] published a variableorder, variablestep Taylorseries method that can be used as an ODE solver providing a highly accurate calculation to compute dynamic sensitivities. This method is limited to general mass action (GMA) models described by powerlaw differential equations. RungeKutta methods with variable step size control can be used to compute dynamic sensitivities for most of the nonlinear differential equations, but is inefficient to determine the step size in a large dimensional system including the model differential equations and sensitivity differential equations. Due to the efficiency, Dunker [15] proposed the decoupled direct method (DDM), in which the sensitivity equations are solved separately from the model equations. He said: "the decoupled method has advantages in simplicity, stability, accuracy, efficiency, storage requirements, and program size over other methods". Although the DDM approach has so many advantages, the step size for the time profile determined by the error control of model equations is unable to be used for the sensitivities when the sensitivity equations are more stiff than the model equations and will generate inaccurate results.
Dynamic sensitivity analysis evaluates the influences on dependent variables due to variations of parameters, initial conditions and independent variables. In many practical applications, e.g., the fedbatch fermentation systems, the system admissible input (corresponding to independent variables of the system) can be timedependent. The main difficulty for investigating the dynamic log gains of these systems is the infinite dimension due to the timedependent input. Shiraishi et al. [26] proposed an efficient method, based on a combination of the recasting technique and the Taylorseries method, for calculating the time courses of log gains to investigate the dynamic behavior of log gains for oscillation models with a limit cycle. The method is limited to the computations of dynamic log gains with respect to constant independent variables. We extend the computations of dynamic log gains to a model with continuous timevarying admissible input based on the finite parameterization method (PM). The classical PM was created for numerical solutions of optimal control problems [27]. The central idea of the method relies on a simple approximation mechanism. The whole time domain of a continuous admissible input is partitioned into several subintervals, and the input for each subinterval is approximated by a piecewise constant function. The dynamic log gains with respect to the continuous admissible input can be computed based on the partial derivations of dependent variables with respect to the piecewise constant input [28–30].
In this paper, we present an algorithm with an adaptive step size control that can be used for computing the solution and dynamic sensitivities of an autonomous ODE system simultaneously. This algorithm is the modified collocation method, proposed by Wang [6], with an adaptive step size control approach. Although our algorithm is one of the decouple direct methods in computing dynamic sensitivities of an ODE system, the step size determined by model equations can be used on the computations of the time profile and dynamic sensitivities with moderate accuracy even when sensitivity equations are more stiff than model equations. In the algorithm, the modified collocation method is used to transform model and sensitivity equations into algebraic equations, and the approximated solution is solved by an iteration method. This algorithm can be extended easily to solve problems of mixed differential and algebraic equations (DAEs) by combing algebraic equations with that transferred from differential equations. In our algorithm for computing dynamic sensitivities of an ODE system, the model equations and sensitivity equations are solved alternatively in two stages. First, the model equations are advanced from t_{ i }to t_{ i }+ η using the iteration method, where η is the step size decided by model equations based on the fixedpoint theorem. Second, the solution of model equations at t_{ i }+ η and the same step size are used to propagate the sensitivity equations from t_{ i }to t_{ i }+ η. For dynamic systems with continuous timedependent admissible input, the dynamic log gains are computed based on the parameterization techniques. The PM is used to approximate the original infinitedimensional problem by a finite dimensional one with piecewise constant input. The dynamic log gain for this approximation problem is defined as the percentage change of a dependent variable in response to an infinitesimal percentage change for each piecewise constant input.
To show this algorithm can perform dynamic sensitivity analysis on very stiff ODE systems with moderate accuracy, it is implemented and applied to two sets of chemical reactions: pyrolysis of ethane and oxidation of formaldehyde. The accuracy of this algorithm is demonstrated by comparing the dynamic parameter sensitivities from this new method and that from the direct method with Rosenbrock stiff integrator based on the indirect method. The same dynamic sensitivity analysis is performed on an ethanol fedbatch fermentation system with a timevarying feed rate to evaluate the applicability of the algorithm to realistic models with timedependent admissible input.
Results and discussion
To illustrate the accuracy of our algorithm, it is implemented and applied to stiff chemical mechanisms for the pyrolysis of ethane as well as the oxidation of formaldehyde. These systems have been shown to be unstable using both the DM and the Green's function method [15]. The same dynamic sensitivity analysis is performed on an ethanol fedbatch fermentation system with a timevarying feed rate to evaluate the applicability of the algorithm to realistic models with timedependent admissible input.
Pyrolysis of ethane
The mechanism for ethane pyrolysis.
Reaction  Rate constants 

C_{2}H_{6} → CH_{3} + CH_{3}  1.14 × 10^{2} 
CH_{3} + C_{2}H_{6} → CH_{4} + C_{2}H_{5}  1.19 × 10^{6} 
C_{2}H_{5} → C_{2}H_{4} + H  1.57 × 10^{3} 
H + C_{2}H_{6} → H_{2} + C_{2}H_{5}  9.72 × 10^{8} 
H + H → H_{2}  6.99 × 10^{13} 
Sensitivity coefficients for ethane pyrolysis.
Species x_{ i }  ∂ ln [x_{ i }]/∂ k_{1} at 1 s  ∂ ln [x_{ i }]/∂k_{1} at 20 s  

IDM  R/DM  our algorithm  IDM  R/DM  our algorithm  
CH _{3}  1.000  1.000  0.99986  1.000  1.000  1.00000 
CH _{4}  0.976  0.976  0.97625  0.644  0.644  0.64350 
C _{2} H _{4}  0.680  0.680  0.68039  0.324  0.323  0.32348 
C _{2} H _{5}  0.662  0.662  0.66149  0.209  0.210  0.20950 
C _{2} H _{6}  0.044  0.044  0.04425  0.819  0.819  0.81896 
H  0.478  0.478  0.47783  0.091  0.091  0.09053 
H _{2}  0.602  0.602  0.60214  0.221  0.221  0.22098 
Oxidation of formaldehyde
The mechanism for formaldehyde oxidation.
Reaction  Rate constants 

HCO + O_{2} → HO_{2} + CO  6.02 × 10^{10} 
HO_{2} + CH_{2}O → H_{2}O_{2} + HCO  3.43 × 10^{10} 
H_{2}O_{2} + M → 2OH + M  4.01 × 10^{6} 
OH + CH_{2}O → H_{2}O + HCO  9.64 × 10^{13} 
OH + H_{2}O_{2} → H_{2}O + HO_{2}  3.07 × 10^{12} 
H_{2}O_{2} → H_{2}O_{2}(wall)  1.05 × 10^{2} 
HO_{2} → HO_{2}(wall)  1.05 × 10^{1} 
HO_{2} + HO_{2} → H_{2}O_{2} + O_{2}  1.81 × 10^{12} 
OH + CO → CO_{2} + H  1.99 × 10^{11} 
HO_{2} + CO → CO_{2} + OH  7.23 × 10^{8} 
H + CH_{2}O → H_{2} + HCO  1.63 × 10^{12} 
H + O_{2} → OH + O  3.32 × 10^{10} 
H + O_{2} + M → HO_{2} + M  3.63 × 10^{15} 
HO_{2} + M → H + O_{2} + M  2.83 × 10^{5} 
O + H_{2} → OH + H  1.82 × 10^{11} 
O + CH_{2}O → OH + HCO  6.02 × 10^{13} 
H + H_{2}O_{2} → HO_{2} + H_{2}  7.83 × 10^{11} 
H + H_{2}O_{2} → H_{2}O + OH  3.55 × 10^{12} 
O + H_{2}O_{2} → OH + HO_{2}  6.02 × 10^{10} 
HCO → H + CO  4.60 × 10^{12} 
OH + H_{2} → H_{2}O + H  6.02 × 10^{12} 
CH_{2}O + O_{2} → HCO + HO_{2}  1.75 × 10^{4} 
H + HO_{2} → 2OH  3.01 × 10^{12} 
H + HO_{2} → H_{2}O + O  3.01 × 10^{13} 
H + HO_{2} → H_{2} + O_{2}  2.71 × 10^{13} 
Sensitivity coefficients for formaldehyde oxidation.
Rate constant  ∂ ln [HO_{2}]/∂k_{ i }  ∂ ln [O]/∂k_{ i }  

IDM  R/DM  our algorithm  IDM  R/DM  our algorithm  
k _{2}  0.683  0.683  0.68255  0.827  0.827  0.82719 
k _{3}  0.700  0.700  0.69986  0.835  0.835  0.83486 
k _{4}  0.210  0.209  0.20917  1.160  1.156  1.15579 
k _{8}  0.306  0.306  0.30569  0.296  0.296  0.29599 
k _{9}  0.210  0.210  0.20962  1.156  1.156  1.15628 
k _{10}  0.164  0.164  0.16373  1.031  1.031  1.03065 
k _{11}  0.121  0.121  0.12087  0.660  0.659  0.65906 
k _{12}  0.188  0.188  0.18848  0.979  0.979  0.97926 
k _{13}  0.327  0.327  0.32713  
k _{16}  1.002  1.000  0.99990  
k _{22}  0.686  0.685  0.68536  0.742  0.742  0.74169 
Ethanol fedbatch fermentation
Conclusion
To deeply study the dynamic behavior of a biological system, one of the methods is to model it as a mathematical model. The most used mathematical model for simulating biological systems is the ODE model. The essential task for modeling and simulating a biological system is to find the solution of an ODE model efficiently and accurately. We present an algorithm with an adaptive step size control that can be used for computing the solution and dynamic sensitivities of an ODE system simultaneously. Instead of using error control to decide the step size in solving the model equations, our algorithm computes the step size based on the fixedpoint theorem and the same step size can be used in solving the sensitivity equations.
Dynamic sensitivity analysis is a useful tool to investigate the behavior of dynamic systems. In the direct methods for solving the dynamic sensitivities, sensitivity equations and model equations are coupled and solved together at the expense of more computation time. In contrast, sensitivity equations and model equations are solved separately in the decouple direct methods. The DDMs are more efficient than the DMs due to the dimension of ODEs. The chief disadvantage of DDMs is the requirement of error control on both model equations and sensitivity equations. Our algorithm with an efficient step control approach based on the fixedpoint theorem is used to address the disadvantage of DDMs. Analogous to the DMs, the same step size obtained by model equations is used on both model and sensitivity equations. It has been implemented and applied to wellknown stiff problems with the same accuracy compared to the direct method with Rosenbrock stiff integrator (R/DM). As our algorithm is one of the DDMs, it has the efficiency of the DDMs and the same accuracy of the DMs as presented in the section describing the results. By combining the efficiency and accuracy, our algorithm is an excellent method for computing dynamic parameter sensitivities in stiff problems.
We extend the scope of classical dynamic sensitivity analysis to the investigation of dynamic log gains of models with timedependent admissible input. The parameterization method is used to approximate the infinitedimensional computation problem for dynamic log gains in models with timedependent admissible input by a classical finitedimensional computation problem of dynamic log gains. Then, all dynamic log gains and parameter sensitivities can be obtained simultaneously from our algorithm. Appropriate parameterization allows one to obtain a more efficient way to compute the dynamic log gains with respect to a continuous timedependent input than that by finite difference approximation. Finally, the new proposed algorithm is applied to the ethanol fedbatch fermentation system, a real dynamic biological system which never reaches a steady state, with a timevarying feed rate for elucidating the applicability to realistic models with timedependent admissible input. Through the dynamic sensitivity analysis of the ethanol fedbatch fermentation model, we conclude that to get a higher ethanol production rate by increasing the ethanol yield factor is a good choice.
Methods
where the function f is assumed to be continuous and differentiable in all its arguments x, y,and θ. This assumption on f is satisfied for both of equations (3) and (4).
we let x_{n+1}= t and dx_{n+1}/dt = 1, equation (6) can be rewritten as equation (5) with x(t) ∈ ℝ^{n+1}. This is an (n + 1)dimensional autonomous dynamic system. Similarly, an ndimensional time dependent equation is a special case of an (n+ 1)dimensional autonomous dynamic system. Using this trick, we can always remove any time dependence by adding an extra dimension to the system. Thus, without loss of generality, we will consider the autonomous dynamic systems expressed by equation (5) unless stated otherwise.
ODE solver
Instead of solving the ODEs in equation (5) directly, we find the solution of algebraic equations in equation (8) stepbystep for each time interval [t_{i1}, t_{ i }], i = 1,..., k. The solution obtained from equation (8) is a good approximation solution of ODEs in equation (5) when the step size η_{ j }is small enough. How to decide the step size is an important problem for all ODE solvers. A larger step size can cause the solution to be inaccurate and divergent, and a smaller step size is inefficient for the computations. Wang [6] uses the NewtonRaphson method with step length restriction to solve equation (8). The restricted step size is fixed and computed by trial and error. To overcome this drawback, we propose an adaptive step size control approach based on the Banach fixed point theorem for the modified collocation method in this paper. We will show that this approach can determine the step size automatically and efficiently when computing the solutions and dynamic sensitivities of equation (5) simultaneously.
The Banach fixed point theorem and some terminologies for describing the theorem are defined below.
Definition 1. Metric space [32]
A metric space (X, d) is a set X where a notion of distance d (called a metric) between elements of the set is defined.
Definition 2. Cauchy sequence [32]
A sequence (x_{ n }) in a metric space (X, d) is said to be Cauchy if for every ε > 0 there is a positive integer N such that for all natural numbers m, n >N, the distance d(x_{ m }, x_{ n }) is less than ε.
Definition 3. Complete metric space [32]
A metric space (X, d) in which every Cauchy sequence has a limit in X is called complete.
Theorem 4. Banach fixed point theorem [32]
Let (X, d) be a nonempty complete metric space. A mapping ψ: X → X is called a contraction on X if there is a nonnegative real number q < 1 such that for all a, b in X
d(ψ(a), ψ(b)) ≤ q·d(a, b). (9)
The contraction ψ on X admits one and only one fixed point x* ∈ X such that ω (x*) = x*.
where c = x(t_{j1})+0.5η_{ j }f(x(t_{j1}), y(t_{j1}); θ) is a constant vector. When the values of independent variable y and θ are given, the solution of x = g(x) can be found by an iteration process. A sequence of values of x(t_{ j }) is obtained using the iterative rule. If x^{ i }(t_{ j }) tends to a limit x*(t_{ j }) when i → ∞, it is the answer of x = g(x) and called a fixed point of the function g(x). Let X be the set of x^{ i }(t_{ j }), i = 1,..., ∞ and d(a, b) = a  b_{ p }where a and b are arbitrary x^{ i }(t_{ j }) and ·_{ p }is the pnorm. Then (X, d) forms a nonempty complete metric space. If g is a contraction on X, the Banach fixed point theorem guarantees the existence of a fixed point and the convergence of the iteration process to that fixed point. By the equation (9) and the definition of distance function d, for a, b ∈ X we obtain
g(a)  g(b)_{ p }≤ qa  b_{ p }, q < 1. (11)
where N is the stoichiometric matrix, v_{ i }is the i^{ th }element of v ∈ ℝ^{ q }and g_{ ij }is the kinetic order for each x_{ j }∈ x in v_{ i }. For efficiency, we approximate the value of 2norm of the Jacobian matrix ∂f/∂x by ∂f/∂x_{Δ}, where n is the dimension of x and ∂f/∂x_{Δ} is the maximum absolute value of the element of the Jacobian matrix. The proposed algorithm AMCM, Adaptive Modified Collocation Method, is shown as follows
Algorithm AMCM
 1.
A set of n ordinary differential equations $\dot{x}$ = f(x, y) with n dependent variables x_{ i }, i = 1,..., n and m independent variables y_{ i }, i = 1,..., m.
 2.
Two order sets x_{0} = {x_{ i }(t_{0})i = 1,..., n} of initial values of x and y_{0} = {y_{ i }(t_{0})i = 1,..., m} of initial values of y.
 3.
An order set T = {t_{1},..., t_{ k }} of sampling points, t_{ i }, 1 ≤ i ≤ k is the sampling time of the solution of each ODE, k is the number of sampling points.
 4.
A tolerance ε
Output: The set of solutions of dependent variables at each sampling time.

For each sampling time t_{ i }in T.
 1.
η_{ j }← t_{ i } t_{i1}, d_{ t }← 0, x^{ c }← x(t_{i1}), y^{ c }← y(t_{i1})
 2.
Repeat the following steps until d_{ t }= t_{ i } t_{i1}.
 (a)
x^{ p }← x^{ c }, y^{ p }← y^{ c }
 (b)
Evaluate the Jacobin matrix $A\leftarrow \frac{\partial f({x}^{p},{y}^{p})}{\partial x}$
 (c)
Compute the upper bound μ of the value of A_{2} by $\sqrt{n(m+n)}\left\rightA{}_{\Delta},\leftA\right{}_{\Delta}\equiv \underset{i,j}{\mathrm{max}}\left{a}_{ij}\right$
 (d)
If μ * ε ≥ 1, it means the ODEs are stiff, then exit this algorithm.
 (e)
If μ * η_{ j }> 1, then update η_{ j }with 0.9/μ
 (f)
Call iteration algorithm to compute the value of x^{ c }stepped forward η_{ j }from x^{ p }.
 (g)
If the iteration algorithm succeeds in computing x^{ c }, then d_{ t }← d_{ t }+ η_{ j }and η_{ j }← t_{ i } t_{i1} d_{ t }, otherwise exist this algorithm.
 3.
x(t_{ i }) ← x^{ c }, y(t_{ i }) ← y^{ c }.

return x(t_{ i }), i = 1,..., k.
End of Algorithm AMCM
Algorithm Iteration
 1.
A set of n ordinary differential equations $\dot{x}$ = f(x, y).
 2.
x(t), y(t), η_{ j }and the iteration limitation.
 1.
Evaluate the value of f(x(t), y(t)).
 2.
x(t + η_{ j }) ← x(t) + f(x(t), y(t)) * η_{ j }.
 3.
y(t + η_{ j }) ← y(t) + f(x(t), y(t)) * η_{ j }.
 4.
Repeat the following steps until the iteration limitation is reached or the value of x(t + η_{ t }) converges.
 (a)
Evaluate the value of f(x(t + η_{ j }), y(t + η_{ j })).
 (b)
x(t + η_{ j }) ← x(t) + 0.5 * η_{ j }* (f(x(t), y(t)) + f(x(t + η_{ j }), y(t + η_{ j })))
 5.
If the iteration limitation is reached, then exit this algorithm; otherwise, return x(t + η_{ j }).
End of Algorithm Iteration
Dynamic sensitivity Solver
Once the local sensitivity is known, the calculation of the relative sensitivity is straightforward. So, for briefing, we limit our explanation on the absolute sensitivity only below.
where j = 1,..., N_{ u }.
The value of w(t_{ j }) can be evaluated if the values of x(t_{ j }) have been computed when solving equation (8) by the ODE solver. The value of the M(t_{ j }) in equation (22) must be calculated before the solution of equation (21) can be obtained by an iteration method. When the value of the Jacobian matrix ∂f/∂x in time t_{ j }is known, the value of the M(t_{ j }) can be obtained straightforwardly. There is no requirement for computing the value of the Jacobian matrix here due to it has been computed for step size control when solving equation (8) by the ODE solver.
Equation (23) is the same as equation (14), so the same criterion is used to determine the step size η_{ j }for computing the time course and sensitivity profile with the guarantee of convergence and existence of a fixed point. The dynamic sensitivities can be obtained directly by solving equation (21) after the time profile of x has been computed and the step size η_{ j }has been decided by the ODE solver.
Declarations
Acknowledgements
The financial support from the National Science Council, Taiwan, ROC (Grant NSC972221E194010MY3 and NSC972627B194001), is highly appreciated.
This article has been published as part of BMC Bioinformatics Volume 9 Supplement 12, 2008: Asia Pacific Bioinformatics Network (APBioNet) Seventh International Conference on Bioinformatics (InCoB2008). The full contents of the supplement are available online at http://www.biomedcentral.com/14712105/9?issue=S12.
Authors’ Affiliations
References
 Albrecht P: A New Theoretical Approach to RungeKutta Methods. SIAM Journal on Numerical Analysis. 1987, 24: 391406. 10.1137/0724030.View ArticleGoogle Scholar
 Albrecht P: The RungeKutta Theory in a Nutshell. SIAM Journal on Numerical Analysis. 1996, 33: 17121735. 10.1137/S0036142994260872.View ArticleGoogle Scholar
 Inc M, Bildik N, Bulut H: A comparison of numerical ODE solvers based on Euler methods. Math Comput Appl. 1998, 3: 153159.Google Scholar
 Gerald CF, Wheatley PO: Applied numerical analysis. 1994, AddisonWesleyGoogle Scholar
 Villadsen JV, Stewart WE: Solution of boundaryvalue problems by orthogonal collocation. Chem Eng Sci. 1967, 22: 14831501. 10.1016/00092509(67)800745.View ArticleGoogle Scholar
 Wang FS: A modified collocation method for solving differentialalgebraic equations. Applied Mathematics and Computation. 2000, 116: 257278. 10.1016/S00963003(99)001381.View ArticleGoogle Scholar
 Savageau MA: Parameter sensitivity as a criterion for evaluating and comparing the performance of biochemical systems. Nature. 1971, 229: 542544. 10.1038/229542a0.View ArticlePubMedGoogle Scholar
 Savageau MA: The behavior of intact biochemical control systems. Current Topics in Cellular Regulation. 1972, 6: 63129.View ArticleGoogle Scholar
 Voit EO: Computational analysis of biochemical systems: a practical guide for biochemists and molecular biologists. 2000, Cambridge, UK: Cambridge University PressGoogle Scholar
 Savageau M: Biochemical Systems Analysis. A Study of Function and Design in Molecular Biology. 1976, Reading, MA: AddisonWesleyGoogle Scholar
 Kacser H, Burns JA: The control of flux. Symp Soc Exp Biol. 1973, 27: 65104.PubMedGoogle Scholar
 Fell DA: Metabolic control analysis: a survey of its theoretical and experimental development. Biochem J. 1992, 286: 313330.PubMed CentralView ArticlePubMedGoogle Scholar
 Heinrich R, Schuster S: The Regulation of Cellular Systems. 1996, New York: Chapman & HallView ArticleGoogle Scholar
 Caracotsios M, Stewart WE: Sensitivity Analysis of Initial Value Problems with Mixed Ode's and Algebraic Equations. Comput Chem Eng. 1985, 9: 359365. 10.1016/00981354(85)850146.View ArticleGoogle Scholar
 Dunker AM: The decoupled direct method for calculating sensitivities coefficients in chemical kinetics. J Chem Phys. 1984, 81: 23852393. 10.1063/1.447938.View ArticleGoogle Scholar
 Dougherty EP, Hwang JT, Rabitz H: Further developments and applications of the Green's function method of sensitivity analysis in chemical kinetics. J Chem Phys. 1979, 71: 17941808. 10.1063/1.438530.View ArticleGoogle Scholar
 Dickinson RP, Gelinas RJ: Sensitivity analysis of ordinary differential equation systems. A direct method. J Comput Phys. 1976, 21: 123143. 10.1016/00219991(76)900073.View ArticleGoogle Scholar
 Leis JR, Kramer MA: The simultaneous solution and sensitivity analysis of systems described by ordinary differential equations. ACM Trans Math Softw. 1988, 14: 4560. 10.1145/42288.46156.View ArticleGoogle Scholar
 Mauch K, Arnold S, Reuss M: Dynamic sensitivity analysis for metabolic systems. Chemical Engineering Science. 1997, 52: 25892598. 10.1016/S00092509(97)000754.View ArticleGoogle Scholar
 Ghosh A, Miller D, Zou R, Pais H, Sokhansanj B, Kriete A: Integrated Spatiotemporal Model of Cell Signaling. FOSBE. 2005Google Scholar
 Hwang JT, Dougherty EP, Rabitz S, Rabitz H: The Green's function method of sensitivity analysis in chemical kinetics. J Chem Phys. 1978, 69: 51805191. 10.1063/1.436465.View ArticleGoogle Scholar
 Ingalls BP, Sauro HM: Sensitivity analysis of stoichiometric networks: an extension of metabolic control analysis to nonsteady state trajectories. Journal of Theoretical Biology. 2003, 222: 2336. 10.1016/S00225193(03)000110.View ArticlePubMedGoogle Scholar
 Shen J: A direct method of calculating sensitivity coefficients of chemical kinetics. J Chem Phys. 1999, 111: 72097214. 10.1063/1.480049.View ArticleGoogle Scholar
 Zou R, Ghosh A: Automated sensitivity analysis of stiff biochemical systems using a fourthorder adaptive step size Rosenbrok integration method. AIEE Proc Sys Biol. 2006, 153: 7990. 10.1049/ipsyb:20050058.Google Scholar
 Shiraishi F, Takeuchi H, Hasegawa T, Nagasue H: A Taylorseries solution in Cartesian space to GMAsystem equations and its application to initialvalue problems. Applied Mathematics and Computation. 2002, 127: 103123. 10.1016/S00963003(01)000078.View ArticleGoogle Scholar
 Shiraishi F, Hatoh Y, Irie T: An efficient method for calculation of dynamic logrithmic gains in biochemical systems theory. Journal of Theoretical Biology. 2005, 234: 7985. 10.1016/j.jtbi.2004.11.015.View ArticlePubMedGoogle Scholar
 Gorbunov VK: The parameterization method for optimal control problems. Comput Math Math Phys. 1979, 19: 212224.View ArticleGoogle Scholar
 Gorbunov VK, Lutoshkin IV: The parameterization method in optimal control problems and differentialalgebraic equations. Journal of Computational and Applied Mathematics. 2006, 185: 377390. 10.1016/j.cam.2005.03.017.View ArticleGoogle Scholar
 Li R, Teo KL, Wong KH, Duan GR: Control parameterization enhancing transform for optimal control of switched systems. Mathematical and Computer Modelling. 2006, 43: 13931403. 10.1016/j.mcm.2005.08.012.View ArticleGoogle Scholar
 Teo KL, Jennings LS, Lee HW, Rehbock V: The control parameterization enhancing transform for constrained optimal control problems. J Austral Math Soc Ser B. 1999, 40: 314335.View ArticleGoogle Scholar
 Wang FS, Su TL, Jang HJ: Hybrid differential evolution for problems of kinetic parameter estimation and dynamic optimization of an ethanol fermentation process. Ind Eng Chem Res. 2001, 40: 28762885. 10.1021/ie000544+.View ArticleGoogle Scholar
 Kreyszig E: Introductory Functional Analysis with Applications. 1989, WileyGoogle Scholar
 Varma A, Morbidelli M, Wu H: Parameter sensitivity in chemical systems. 1999, Cambridge: Cambridge University PressView ArticleGoogle Scholar
 Tomovic R, Vukobratovic M: General Sensitivity Theory. 1972, New York: American ElsevierGoogle Scholar
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