Bioinformatics Advance Access originally published online on February 10, 2008
Bioinformatics 2008 24(6):848-854; doi:10.1093/bioinformatics/btn035
Complexity reduction of biochemical rate expressions
1Systems Biology and Bioinformatics Group, University of Rostock, Rostock, Germany, 2Department of Biomedical Sciences, University of Copenhagen, Copenhagen, Denmark and 3Department of Clinical and Experimental Medicine, Linköping University, Linköping, Sweden
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 APPROACH AND METHOD3 IMPLEMENTATION4 EXAMPLES5 DISCUSSIONAUTHOR CONTRIBUTIONSACKNOWLEDGEMENTSREFERENCES |
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Motivation: The current trend in dynamical modelling of biochemical systems is to construct more and more mechanistically detailed and thus complex models. The complexity is reflected in the number of dynamic state variables and parameters, as well as in the complexity of the kinetic rate expressions. However, a greater level of complexity, or level of detail, does not necessarily imply better models, or a better understanding of the underlying processes. Data often does not contain enough information to discriminate between different model hypotheses, and such overparameterization makes it hard to establish the validity of the various parts of the model. Consequently, there is an increasing demand for model reduction methods.
Results: We present a new reduction method that reduces complex rational rate expressions, such as those often used to describe enzymatic reactions. The method is a novel term-based identifiability analysis, which is easy to use and allows for user-specified reductions of individual rate expressions in complete models. The method is one of the first methods to meet the classical engineering objective of improved parameter identifiability without losing the systems biology demand of preserved biochemical interpretation.
Availability: The method has been implemented in the Systems Biology Toolbox 2 for MATLAB, which is freely available from http://www.sbtoolbox2.org. The Supplementary Material contains scripts that show how to use it by applying the method to the example models, discussed in this article.
Contact: henning.schmidt{at}uni-rostock.de
Supplementary information: Supplementary data are available at Bioinformatics online.
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 APPROACH AND METHOD3 IMPLEMENTATION4 EXAMPLES5 DISCUSSIONAUTHOR CONTRIBUTIONSACKNOWLEDGEMENTSREFERENCES |
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An important paradigm for dynamic modelling is that if a model is able to satisfactorily describe the measurement data used for the modelling task, and is able to make experimentally validated predictions, then this indicates that the mechanistic assumptions about the system might be correct and that the model might be accurate in some of its other (non-validated) predictions as well.
Most models of biochemical systems today are, however, highly overparameterized with respect to the available in vivo data. Overparameterization, or unidentifiability, means that there exist an infinite number of parameter combinations that produce an equally good agreement with the data (Isidori, 1995). Since these models might display a different behaviour in the non-measured parts of the system, one does not know which model predictions can be trusted, and which are more or less arbitrary (Cedersund, 2006). Additionally, large models are associated with problems regarding, e.g. numerics and model analysis (Okino and Mavrovouniotis, 1998). Nevertheless, a current trend in Systems Biology is to make models more and more complex, rendering identifiability analysis and model reduction as two very important aspects to address.
Chemical model reduction techniques have been reviewed by Okino and Mavrovouniotis (1998). Most of these fall into three classes: lumping methods, techniques based on sensitivity analysis and time-scale based techniques. Lumping is, probably, the most widely used technique. It returns reduced models with new variables corresponding to pools of the original variables. The reduced model structure is usually formed through biochemical intuition about very fast or very slow reactions [as in Wolf and Heinrich (2000)]. Pooling can, however, also be based on systematic analysis of, e.g. the correlation between the variables (Maertens et al., 2005). Sensitivity analysis-based methods use sensitivity analysis to identify those parts of a model that are (locally) unimportant for the property of interest, and these parts are then eliminated (Brown et al., 1997; Danø et al., 2006; Edelson, 1981; Gautier and Carr, 1985; Okino and Mavrovouniotis, 1998). Time-scale based methods are applicable if there are processes occurring on time-scales widely different from the interesting one. Very slow processes are neglected, and fast processes are projected onto the low-dimensional manifolds (Danø et al., 2005; Davis and Skodje, 1999; Lam and Goussis, 1994; Okino and Mavrovouniotis, 1998; Zobeley et al., 2005).
Other reduction methods are typically used in engineering sciences. One such method is balanced truncation (Glad and Ljung, 2000; Hahn and Edgar, 2002; Skogestad and Postlethwaite, 1996), which performs a coordinate transformation such that the new state variables are ordered according to their importance for the input–output relations of the model. The strength of such an input–output oriented approach is that it allows for a reduction of precisely those parts of the model that are not validated by available experimental data, i.e. the unidentifiable parts. The major drawback of balanced truncation is that the reduced model structure is generally without biochemical interpretation, but see Liebermeister et al. (2005) for its application in a special case.
In recent work on identifiability analysis (Hengl et al., 2007; Timmer et al., 2007) a method is proposed that, based on optimal transformations, helps in finding the non-linear relationships between the parameters of a model. Furthermore, Yue et al. (2006) proposed a method, based on parameter sensitivity analysis and the calculation of parameter correlation coefficients, that determines the identifiable parameters of a model. Both methods are data-based in the sense that experimental data has been used to perform parameter estimation once or several times prior to the identifiability analysis. As a result of both methods, the parameter estimation problem is reduced to only the identifiable parameters. This, however, does not reduce the complexity of the model itself and leaves the modeller with the important decision of choosing reasonable values for the unidentifiable parameters.
In this article, we propose a method for the reduction of complex kinetic rate expressions. Such expressions are often found in models of metabolic networks and are derived from detailed mechanistic understanding of the underlying enzyme kinetics and involved species. We avoid the problem regarding the biochemical interpretation by performing the model reduction in a reaction-wise manner, so that the complexity of the individual rate expressions is reduced, while the structure of the reaction network is unchanged. Our method takes advantage of unidentifiability to reduce the individual rate expressions to a simpler form. The rationale behind this is that an unidentifiable rate expressions has more than one set of parameters that can describe the data in question, and a simpler expression, which describes the data equally well, can be found.
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2 APPROACH AND METHOD |
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TOPABSTRACT1 INTRODUCTION2 APPROACH AND METHOD3 IMPLEMENTATION4 EXAMPLES5 DISCUSSIONAUTHOR CONTRIBUTIONSACKNOWLEDGEMENTSREFERENCES |
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2.1 Introductory example
Consider the following model of a simple biochemical system A
B
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| (1) |
Assume that [B] << KB in the interesting range of operation of the system, and that [A] is of the same order of magnitude as KA. Since, in this case, the term
will not be reflected in the rate r, it will not be possible to estimate KB from the data, and the more detailed and mechanistically correct rate expression (1) can be replaced by the simpler expression
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This little example shows that parameter identifiability depends strongly on the experimental protocols that are used to generate the measurement data, i.e. on how much and around which operating points the system is excited. Furthermore, it also depends on the measurement possibilities. Thus, in order to optimize the identifiability for a given model suitable experimental protocols and measurement techniques should be chosen. However, it is important to consider that the complexity of a model should be adapted to the phenomenon under consideration.
For our reduction method this means that the user should specify which behaviours of the system the reduced model should be able to reproduce. During the modelling of a system these behaviours will typically be defined by the experiments that have been performed to collect the necessary data for model fitting.
2.2 Method
Our model reduction method is applicable to all rate expressions that can be written in a rational form, i.e. as a fraction between two polynomials. The method thus reduces a model reaction by reaction. In addition to rate expressions, the method takes user-defined reference data as inputs. These reference data are determined from in silico simulations of the original models behaviours that the reduced model should be able to reproduce and consist of time-series or steady-state data of reaction rates and species concentrations. In the following, we will call this data reference data and the simulated experiments that have been performed on the model to obtain these data as reference experiments.
Step 1: Transformation of rational rate expressions
Consider the following general form:
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| (2) |
Multiplying Equation (2) with d(x,p) gives:
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| (3) |
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,
. In both cases i 
j. In order to account for that a new index k has been introduced, which is defined in the range [1,..., md – 1]. Furthermore, we also introduce corresponding variables for the right-hand side of the equation:
, and
.
Substituting these new variables into Equation (4) and rearranging it, the general rational rate expression in Equation (2) can be written as
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| (5) |
As a clarifying example consider rate expression (1). For this particular rate expression the vectors x and p are given by x = [ A], [B]] and p = [V, KA, KB]. By choosing j = 1, dpjdxj becomes equal to one and the vector description of the rate expression can be determined as follows:
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. Note that each of the pairwise elements in the vectors m and c correspond to a single term in either the numerator or the denominator of the original rate expression. Also, it is straightforward to transform the vector description in (5) back to the form of a kinetic rate expression, such as that given in Equation (1).
Step 2: Measurement matrix and reaction rate vector
A linear system of equations can now be constructed by introducing the simulated reference data (all species concentrations x and the reaction rate r):
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| (6) |
For the example reaction (1) and q sets of measurements, the linear system of equations (6) becomes
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If, on the other hand, M is singular, then the coefficients cannot be estimated, and the rate expression is unidentifiable. Hence, it can be simplified without loss of the ability to generate the simulated data. Thus, if the goal is expression reduction, j in (4) should be chosen such that the matrix M is as close to singular as possible. However, in practical applications, using several examples, the choice of j did not have an impact on the reduction results.
Step 3: Reduction based on linear dependencies
Equation (7) shows that it is possible to determine the elements of the c vector under the assumption that the measurement matrix M has full column rank. If M, however, is singular it means that linear combinations in the columns of M exist and, thus, the original rate expression can be reduced without loss of accuracy.
For the sake of explanation of this reduction step, it is assumed that M is exactly singular. Further, let Mi denote the i-th column in M
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| (8) |
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| (9) |
Solving (9) for the k-th element (
) gives Mk as a function of the other columns
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. However, for numerical reasons we make a specific choice that is discussed subsequently. Equation (8) can then be rewritten in reduced form as
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| (10) |
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| (11) |
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The reduced Equation (10) can now be transformed back to a rational rate expression that has one term less (in either nominator or denominator) than the original expression. The procedure can be repeated as long as the resulting measurement matrix is singular. This results in a reduced rate expression, in which the new terms have been determined by lumping of the original terms.
Apart from reducing exact linear dependencies between columns, the procedure in step 3 can also be used to reduce almost exact linear dependencies, which manifest themselves in singular values that are not exactly zero but much closer to zero than the other singular values. A threshold for the minimum singular value is used to decide if reduction based on linear dependencies or based on approximation (Step 4) is to be made. Furthermore, for numerical reasons it is advantageous to use in this step a scaled version of the matrix M, which we denote Ms. This matrix is scaled such that the element with the largest magnitude in each column has a magnitude of 1.
Step 4: Reduction based on approximation
In this step, we determine individual terms in the rate expression that can be removed because they only have a negligible effect on the reaction rate. Equation (6) can be written as a sum of the involved terms, see Equation (8).
Using the single summands we construct the following weighted measurement matrix Mc,
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Each column of Mc corresponds to a single term in the rate expression. The smaller the contribution of this term is to the reaction rate vector R, the more likely it is that this term can be neglected in a reduced expression.
Assume that the k-th column in Mc contains elements with a considerably smaller magnitude than the elements in the other columns, which might be due to a very small parameter ck, small elements in Mk or both. This column has then a considerably smaller contribution to R in Equation (8) and can thus be neglected if reduction by approximation is desired.
Applying a singular value decomposition to Mc, column k will show up as the dominant element of the input singular vector v* that corresponds to the smallest singular value. Thus, we determine the term that is to be cancelled in this step as the one that corresponds to the element with the largest magnitude in v*.
Since the columns of Mc correspond both to numerator and denominator terms of the expression, care has to be taken in the case that only a single nominator or denominator term is left in the expression. These single terms should then not be cancelled, even if their respective column in Mc shows up as the smallest.
This procedure is repeated until all unimportant terms are eliminated from the rate expression. Since the reduction in this step leads to an approximation of the original rate expression it is up to the modeller to decide when to stop the reduction. This can be done by comparing (eye inspection) the behaviour of the original expression with that of the reduced expression after each reduction step. Additionally, the plot of an error measure can be considered, e.g. the sum of squared errors, between the original and reduced reaction rates. In practical applications, a large jump in the plotted curve can be seen for reaction steps that lead to a behaviour of the reduced expression that is far away from the desired behaviour. For the examples below, such error plots are shown in the Supplementary Material.
In the case that information about the uncertainty (variance) of the parameters in the original model is available, it is possible to perform Monte Carlo simulations in order to determine the variance of the rate of the considered reaction. Having this information, goodness-of-fit statistical tests, such as the Akaike information criterion (Akaike, 1974) or chi-square test, can be applied for the selection of the reduced rate expression.
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3 IMPLEMENTATION |
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TOPABSTRACT1 INTRODUCTION2 APPROACH AND METHOD3 IMPLEMENTATION4 EXAMPLES5 DISCUSSIONAUTHOR CONTRIBUTIONSACKNOWLEDGEMENTSREFERENCES |
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An implementation of the reduction method is available in the Supplementary Material. Furthermore, the method has been added to the Systems Biology Toolbox 2 for MATLAB. Information about how to use the method and scripts allowing its application to the examples below can also be found in the Supplementary Material. In this section, we only mention some of the most important technical details of the implementation.
After each reduction step the parameters in the reduced rate expression are optimized in order to fit it to the reference data. A starting guess for the parameters is obtained from the elements of the c vector before the reduction step.
To decide between reduction based on linear dependencies or approximation, a tolerance threshold is used and can be changed by the user. If the smallest singular value of the scaled measurement matrix Ms is smaller than this threshold, reduction is performed based on linear dependencies.
After each term-cancellation, the program presents a comparison between the reaction rate of the original expression and that of the reduced expression, which is calculated using the original concentration time traces (this way, different reactions can be treated independently). Furthermore, the sum of squared errors between the original and reduced reaction rates is plotted. The modeller must then decide whether to perform a new reduction step, to end the reduction, or to go a step back. After the final reduction step a simulation is performed with the reduced model and compared to the simulation with the unreduced model, allowing the user to accept the reduction of the rate expression or to refine it.
As discussed above, the implementation of the method should take care that not all numerator or all denominator terms are cancelled. In our implementation of the method, it is possible to switch off the consideration of the numerator terms from the reduction. Apart from avoiding the named problem, this feature can also be used to conserve the reversibility of reactions, if desired.
When all reactions in a model have been reduced to a satisfying level of complexity, additional optimizations might be useful. For example, it might be important to ensure that the steady-state of the reduced model exactly matches the steady-state of the original model. This can be achieved by adjusting the velocity parameters of the reactions, using the direct method of optimization (Hynne et al., 2001; Danø et al., 2006).
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4 EXAMPLES |
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TOPABSTRACT1 INTRODUCTION2 APPROACH AND METHOD3 IMPLEMENTATION4 EXAMPLES5 DISCUSSIONAUTHOR CONTRIBUTIONSACKNOWLEDGEMENTSREFERENCES |
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We have applied the method to two complex rate expressions that appear in published models of yeast glycolysis (Hynne et al., 2001; Teusink et al., 2000), and to all reactions of the model presented in Hynne et al. (2001).
All information and scripts needed to reproduce the figures in this section are available in the Supplementary Material.
4.1 Aldolase reaction in the model by Teusink et al.
The first example is the aldolase reaction in the model presented in Teusink et al. (2000). The original rate expression is based on in vitro investigations and it is given by:
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For the sake of demonstration of our method we have chosen two very simple reference experiments for the reduction. In the first experiment the initial concentration of external glucose in the model is set to 100 mM, and in the second experiment it is set to 80 mM. Both models are then simulated and the reference data are collected each minute for a time of 40 min.
The first two reduction steps are based on linear dependencies, followed by three reduction steps based on approximation. The resulting reduced rate expression is an irreversible, one-substrate Michaelis–Menten expression
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The behaviour of the reduced expression matches the original perfectly, as can be seen in the upper part of Figure 1. The first 40 min correspond to the first experiment and the last 40 min to the second experiment. However, further reduction of the rate expression is not possible without loosing the original behaviour. This is seen in the lower part of Figure 1, which shows the comparison between the original reaction rate and the one step further reduced expression
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4.2 Glucose transport in the model by Hynne et al.
As a second example, the glucose transport reaction in the model by Hynne et al. is considered:
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The parameters of the model have been fitted in order to be in agreement with the experimentally observed behaviour of the true system.
The reference experiment simulates the behaviour of the model in a set of states consistent with the actual experimental characterization:
- A stable stationary state close to the Hopf-bifurcation at a mixed flow glucose concentration, [Glcx]0, of 18.0 mM and other parameters as indicated in the Supplementary Material.
- The neutrally stable Hopf point ( [Glcx]0 = 18.5 mM).
- Points on the stable limit cycle at [Glcx]0 = 20.0, 22.0, 24.0, 25.5, 27.0, 29.0, 30.5, 32.0, 33.5, and 35.0 mM.
- The trajectory after an acetaldehyde quenching experiment at [Glcx]0 = 35.0 mM, a specific reactor flowrate, k0, of 0.0479 min–1, and a mixed flow cyanide concentration, [CN–x]0, of 5.37 mM.
Using our reduction method it is possible to perform five reduction steps based on approximation, without noticing a larger difference between the original and reduced model. After the sixth step, however, the reduced reaction rate is not any more able to catch the desired behaviour. This is illustrated in Figure 2. The final reduced rate expression is given by
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An important experiment that is observable is timetraces of NADH autofluorescence (Hynne et al., 2001). Keeping this is mind, we inspected the [NADH] trajectory after the sixth reduction step of the glucose transporter. As shown in Figure 3, the removal of the G6P inhibition term K2 [G6P] [Glcx] has no effect on the experimentally observable NADH time trace.
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4.3 Reduction of the entire model by Hynne et al.
We now apply the method to all reactions in the model by Hynne et al. using the same reference experiment as above. As mentioned in the implementation section, the individual reactions can be treated independently since the reduction process is only based on the concentration time traces of the unreduced model.
Reactions described by mass action kinetics were kept unconsidered in the reduction process. Of the remaining reactions, we found that the phosphofructokinase-1 (PFK) and pyruvate decarboxylase (PDC) reactions could not be reduced. This is consistent with the findings by Madsen et al. (2005) that PFK is central to the formation of instability in the system. The PDC rate expression cannot be further reduced because it already has a simple Michaelis–Menten rate expression.
The full model was reduced from 59 to 46 parameters. Out of originally 61 terms in the reactions that have been reduced, 24 could be removed. Figure 4 compares the original model with the fully reduced version.
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The reduction slightly altered the location of the Hopf-bifurcation from [Glcx]0,bif = 18.5 mM to 17.6 mM. To test the behaviour of the reduced model, we calculated the unscaled stability-, frequency- and flux-control coefficients and polar phase plane plots at the Hopf-point as described in Danø et al. (2006) (figures shown in the Supplementary Material). All plots are very similar to those of the full model, with only minor redistributions of control between the glucose transporter and the flow rate of the reactor.
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5 DISCUSSION |
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TOPABSTRACT1 INTRODUCTION2 APPROACH AND METHOD3 IMPLEMENTATION4 EXAMPLES5 DISCUSSIONAUTHOR CONTRIBUTIONSACKNOWLEDGEMENTSREFERENCES |
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The main feature of the presented model reduction method is that it reduces the algebraic complexity of individual rate expressions, but retains the structure and interpretation of the reaction network. The reduction of the rate expressions is done in a term-wise manner, where terms that are unimportant for the ability of the model to describe the behaviour of interest are removed. The behaviour of interest is specified by a set of reference experiments that are used to generate in silico reference data. The method is well suited for applications where it is desirable to reduce the number of kinetic parameters, and thus serves as a complement to lumping-oriented reduction techniques, which instead reduce the number of states in the model.
The proposed method eliminates those terms in the rate expressions that are unimportant for the model's behaviour during the reference experiments. One way to view this is that the different terms represent the different functionalities of the enzyme; by eliminating the unimportant terms, the important functionalities of the enzyme are pinpointed. In this sense, the method is well suited for analysis of model behaviour. It is, however, important to realize the importance of the reference experiments: a different choice of reference experiments might bring other enzymatic functionalities into play, and thus result in a different reduced model.
A limitation of the method is that it cannot handle parameters that appear as exponents in Hill-type rate expressions. The only way around this problem is to exchange these parameters against their numerical values prior to using the method.
The examples given above illustrate that the model reduction performs well in the sense that a substantial number of terms and kinetic parameters can be removed from the rate expressions with only very limited effect on the overall model behaviour. In particular, the characterization of the fully reduced version of the model by Hynne et al. (2001), which is found in the Supplementary Information, can be compared with those given in Danø et al. (2006).
Anguelova et al. (2007) presented a related method that reduces structural unidentifiability in rate expressions, including unidentifiability due to moiety conservations. Such cases of unidentifiability will not always be captured by our method, since they might appear as affine instead of linear relations in the M matrix. However, this apparent weakness is solved by eliminating the redundant states due to the conservation laws prior to the application of our method.
In principle, the reduction method is also applicable directly to real measurement data. For a given reaction expression it would mean that time-series or steady-state data for all involved species concentrations and the corresponding reaction rates need to be available. Since real measurement data is always noisy, this noise will have an effect on the resulting reduced expression. However, the discussion of these effects is outside the scope of this article.
Finally, the method that we are proposing could be performed in two different ways. Either, the reaction rates of the reduced reactions can be calculated from the concentration time traces of the original model, or they can be calculated by integration of the reduced model. Calculation of reaction rates from the original concentration time traces has the advantage that the individual rate expressions can be reduced in a truly independent manner. The disadvantage, on the other hand, is that the integrated effects of the model reduction will not become apparent before the reference experiments are simulated using the fully reduced model. In practice, however, this will not be a major problem, since one will typically stop the iterative model reduction at a point where there is still good agreement between the behaviour of the original and the reduced model.
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AUTHOR CONTRIBUTIONS |
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TOPABSTRACT1 INTRODUCTION2 APPROACH AND METHOD3 IMPLEMENTATION4 EXAMPLES5 DISCUSSIONAUTHOR CONTRIBUTIONSACKNOWLEDGEMENTSREFERENCES |
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S.D., M.F.M., and H.S. conceived the idea. G.C. and H.S. sorted out the linear algebra and H.S. developed and implemented the method. S.D. and M.F.M. provided the examples and their biochemical interpretations. All authors discussed the research and took part in the writing of the manuscript.
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ACKNOWLEDGEMENTS |
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TOPABSTRACT1 INTRODUCTION2 APPROACH AND METHOD3 IMPLEMENTATION4 EXAMPLES5 DISCUSSIONAUTHOR CONTRIBUTIONSACKNOWLEDGEMENTSREFERENCES |
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This work has been supported by the Swedish Foundation for Strategic Research and the European Science Foundation through the func
tyn project, which are gratefully acknowledged. It is also part of the EU NoE BIOSIM project, contract no. LSHB-CT-2004-005137. Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Jonathan Wren
Present address: Topsoe Fuel Cell, Nymøllevej 66, DK-2800 Lyngby, Denmark. 
Received on September 28, 2007; revised on December 17, 2007; accepted on January 22, 2008
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