Eigenvalue method

The eigenvalue method is similar to the method published by Vajda et al. [26]. The eigenvalue method proposed here differs from the approach proposed by Vajda and coworkers in that a manual inspection of eigenvalues and eigenvectors is not necessary here. As a result, an automatic analysis of large models becomes feasible. We first describe the eigenvalue method and subsequently describe differences to the method published by Vajda et al. [26] in more detail.

We consider the least squares parameter estimation problem that amounts to minimizing cost functions of the form

(9)

with respect to p, where the y _i (t _j ) are the measured or simulated data introduced in equation (2). In Gaussian approximation, the Hessian matrix H of equation (9) has the entries

(10)

for k, l ∈ {1,..., n _p }. As indicated in equation (10), H can be expressed in terms of the sensitivity matrix S introduced in equation (7). H is a symmetric matrix and therefore its eigenvalues are real. Furthermore, H is positive semi-definite.

In order to study the identifiability of the system in equation (1) for given x₀, u(t) and for nominal parameters we first solve equation (1) by numerical integration and subsequently calculate H as given by equation (10). We set p* to parameter values taken from the literature p^lit in all calculations. All identifiability tests carried out in the present paper therefore amount to asking whether the investigated signaling models are identifiable at the literature parameter values p^lit. As pointed out already in the Background section, identifiability tests of this type clarify whether a given model is reasonably detailed even before experiments are carried out. In this sense, the identifiability of a model at nominal or literature values for its parameters is considered to be a necessary condition for its practical identifiability with measured data.

Let λ ^j and u ^j denote the j th eigenvalue and the corresponding eigenvector of H, respectively. Assume the eigenvalues to be ordered such that , assume the eigenvectors u ^j to be normalized such that u ^jT u ^j = 1, and assume p* to minimize equation (9). Consider the change of ϕ when moving from p* in a direction αu ^j for some real α. Gaussian approximation yields

(11)

where the linear order is zero, since p* is is a minimum by assumption. Since Hu ^j = λ ^j u ^j and u ^jT u ^j = 1, the last equality in equation (11) holds. Equation (11) implies that ϕ does not change when moving from p* to p* + Δp if Δp = αu ^j for any eigenvector u ^j with λ ^j = 0 and any real α. In words, the directions u ^j corresponding to λ ^j = 0 are those directions in the parameter space along which the least squares cost function is constant. We call these directions degenerate for short. In the particular case of

(12)

where the entry 1 is in position k, the model is not identifiable with respect to the k th component . The approach proposed here will therefore remove this parameter from consecutive calculations by fixing to . In general however, u ^j will not be of the special form (12), but u ^j will be a vector with more than one non-zero entry, therefore the choice of k is not obvious. In this case, we select a k such that , and remove the k th component of p* from the parameter estimation problem by fixing it to its literature value. In the examples treated below this choice of k turns out to be appropriate in all cases. In general, this might not be the case, however. There might be entries of , l ≠ k, with an absolute value close or equal to the maximal value . We call such entries and the corresponding parameters co-dominant, since together with they dominate the degenerate direction. A simple example for a system with co-dominant parameters is given in Appendix A. We will discuss the issue of co-dominant parameters for the examples treated here in the Results section.

In practical applications the smallest eigenvalue will typically not be zero but close to zero. In this case an eigenvalue cut-off value ϵ ≈ 0, ϵ > 0 needs to be specified. An eigenvalue λ¹ <ϵ is considered to be small enough to be treated as zero. Note λ ^j ≥ 0 for all j since H is positive semi-definite.

The proposed algorithm splits the set of parameter indices {1,..., n _p } of a given model into two disjoint subsets which we denote by I and U. Upon termination, the sets I and U = {1,..., n _p } - I contain the indices to the identifiable and the unidentifiable parameters, respectively. We denote the current number of elements in I by n _I and the current elements in I by . The algorithm proceeds as follows.

1. 1.

   Choose Δt, t _f , ϵ. Set p* = p ^lit. Set I = {1,..., n _p }, n _I = n _p , and U = ∅.

    
2. 2.

   If I is empty, stop. The model is not identifiable with respect to any parameter in this case.

    
3. 3.

   Fix the parameters p _k , k ∈ U to their literature values and consider only the p _k , k ∈ I to be variable. Formally, this corresponds to setting p to and treating all remaining parameters p _k , k ∈ U as fixed numbers.

    
4. 4.

   Calculate ϕ(p*) according to equation (9). Calculate the Hessian matrix of ϕ with respect to the parameters p _k with k ∈ I and evaluate it at p = p*. Calculate the eigenvalues λ ^j and corresponding eigenvectors u ^j of the Hessian matrix. Assume the eigenvalues to be ordered such that . Assume the eigenvectors to be normalized.

    
5. 5.

   If λ ¹ ≥ ϵ ,stop. The model is identifiable with respect to the parameters p _k for all k ∈ I.

    
6. 6.

   If λ ¹ < ϵ ,select k such that . Remove k from the set I, add k to the set U, set n _I to the current number of elements in I, and return to step 2.

    

The order in which parameters are removed from I determines the ranking of parameters from least identifiable to most identifiable.

The method is similar to the approach introduced by Vajda et al. [26]. These authors also analyze the eigenvectors that correspond to small eigenvalues of the Hessian matrix, but they focus on the dependencies between eigenvectors that arise due to special parameter combinations of the form p₁/p₂ or p₁·p₂ in the model. In a two step procedure Vajda et al. first decide which parameters to lump together (for example, p _new = p₁·p₂) and subsequently recalculate the eigenvalues and eigenvectors for the system with the new lumped parameters. These two steps are repeated until the smallest eigenvalue is sufficiently large. The lumping step requires manual inspection of the eigenvalues and eigenvectors of the Hessian matrix. While the approach proposed by Vajda et al. worked very well for their example with 5 parameters, it becomes infeasible for models with considerably more parameters. The examples treated in the next section demonstrate that our approach, in contrast, can be applied to models with up to at least a hundred parameters. Moreover, our approach can be carried out automatically, while the approach suggested by Vajda et al. was never intended for this purpose.

Correlation method

The correlation method was first introduced by Jacquez and Greif [25] who compared identifiability results achieved with the correlation method to analytical results obtained with a transfer function method (see [14] for a review). Jacquez and Greif [25] consider only small linear compartmental models with up to 3 states and 5 parameters. In most of the examples the correlation and the transfer function method were in agreement.

In the systems biology literature the correlation method was applied by Zak et al. [9] to investigate identifiability of a large genetic regulatory network (nonlinear, 44 states and 97 parameters). More recently, Rodriguez-Fernandez et al. [10] embedded the correlation method into their framework for robust parameter estimation in order to exclude unidentifiable parameters from parameter estimation.

The central idea of the correlation method is to find unidentifiable parameters by investigating the linear dependence of the columns of S (for details see Appendix B). The correlation method approximates linear dependence by calculating the sample correlation [35] of two columns of S. The sample correlation is given by

(13)

where

(14)

(15)

(16)

In equations (13)–(16) corr(S. _i , S. _j ), cov(S. _i , S. _j ), σ (S. _i ), and denote the sample correlation between S. _i and S. _j , the sample covariance between S. _i and S. _j , the sample standard deviation of S. _i , and the mean of the entries of S. _i , respectively.

Two linearly dependent columns S. _i and S. _j of the sensitivity matrix give rise to an absolute value of the correlation corr(S. _i , S_.j) = 1. In the examples treated here, none of the parameters exhibit a correlation of exactly ± 1, however. In the numerical implementation of the method, two columns of S are considered correlated if the absolute value of their correlation is equal to or larger than 1 – ϵ _c , where ϵ _c ∈ [0, 1] is a parameter of the algorithm. We stress that while ϵ _c is a tuning parameter, the comparison of the identifiability methods is not affected by the choice of ϵ _c . As explained below, we systematically vary ϵ _c over its entire range 0 ≤ ϵ _c ≥ 1 in the comparison.

When applying the correlation method to the pathway model examples, two problems arise regularly that have not been discussed by Jacquez and Greif [25]. For one, the method detects pairs of correlated parameters, but there is no criterion which one of the parameters from a pair to consider unidentifiable. Moreover, if more than one pair of correlated parameters is detected, there is no criterion to choose among the pairs.

In order to mitigate these ambiguities we introduce the total correlation

(17)

where

(18)

and L denotes the set of indices of those parameters that have not been found to be unidentifiable in any previous iteration. We select the parameter with the highest total correlation for removal. Some cases remain, however, in which two parameters have the same total correlation. In these cases we pick one parameter at random.

In order to be able to compare results to those of the other methods, we use the correlation method to rank model parameters from least identifiable to most identifiable. More precisely, we initially set ϵ _c = 0 and L = {1... n _p } and calculate the total correlations (17) for all parameters in L. If parameters with nonzero total correlations exist we select the parameter with highest total correlation to be unidentifiable and remove its index from L. If no such parameter exists we increase ϵ _c until nonzero total correlations occur. The order in which parameters are removed from L creates a ranking of parameters from least to most identifiable. The algorithm is described in detail in Appendix B. Based on the ranking of parameters the method is compared to the other three approaches. The comparison is explained in detail in the subsection entitled Method comparison.

Principal component analysis (PCA) based method

Degenring et al. [27] introduce three criteria based on PCA that rank the influence of parameters on model output. Parameters that do not affect the model outputs are unidentifiable by definition and can be removed from the model equations. Degenring et al. successfully use their approach to simplify a complex metabolism model of Escherichia coli K12. A complete description how to get a full ranking using the three criteria can be found in [28]. A pseudo code description of the method can be found in Appendix C.

The PCA based method differs from the other methods described here in that it does not use the sensitivity matrix S as defined in equation (7) but a truncated matrix which we refer to by . This matrix is defined as

where i ∈ {1,..., n _y }. While the sensitivity matrix S as defined in equation (7) describes all responses in one matrix, is created for each response r _i separately.

Three PCA based criteria are applied to each of the n _y matrices . As a result 3n _y parameter rankings are created, which we represent by 3n _y lists

(19)

of integers , where j ∈ {1, 2, 3} and i ∈ {1,..., n _y }. The first ranking position and the last ranking position contain the index of the parameter with the lowest and the highest influence on the model output r _i , respectively. The lower index j ∈ {1, 2, 3} denotes the criterion by which the ranking is calculated. We will explain one of the criteria in more detail in order to give the reader an idea of the method. The remaining two criteria are summarized in Appendix C. All 3n _y rankings are integrated into one ranking by applying a strategy described in [28], which we briefly summarize below.

Let, and denote the j th eigenvalue and corresponding eigenvector of , respectively, and assume the eigenvalues to be ordered such that . For each i ∈ {1,..., n _p } evaluate the first criterion as follows.

1. 1.

   Set L ₁ = {1,..., n _y }.

    
2. 2.
   For q from 1 to n _p

   • Find r _l such that .

   • Set = r _l .

   • Set L_q+1= L _q - {r _l }.

   do:

    

The first criterion loops over the eigenvectors starting with the eigenvector that belongs to the smallest eigenvalue. It identifies the largest absolute entry of each eigenvector and selects the corresponding parameter for removal, if this parameter has not been selected yet. The index of the k th parameter selected for removal is ranked at position k, where k ∈ {1,..., n _p }.

The remaining two criteria of the PCA based method are similar but differ with respect to the process for chosing r _l . For brevity we omit details and refer the reader to Appendix C. A final ranking J that incorporates all 3n _y previously calculated rankings is obtained as follows. Set N _q = ∅, for all q = 1,..., n _p . Set J to an empty list.

For q from 1 to n _p do:

First note that an iteration q might exist, in which no N _q is appended to J. Therefore the final number n _J of elements in J is not necessarily equal to but might be smaller than n _p . Further note that entries of J do not necessarily correspond to indices of single parameters but to sets of parameter indices. Therefore, a ranking position J _i might contain several parameter indices an internal ranking of which cannot be obtained. This ambiguity hampers not only the interpretation of the ranking result but also the comparison with the rankings produced by the other methods. We describe a strategy to deal with the latter problem in section entitled Method comparison.

Orthogonal method

The orthogonal method was developed by Yao et al. [24] to analyze parameter identifiability of an Ethylene/Butene copolymerization model with 50 parameters. In the field of systems biology the method has been applied in a framework for model identification [36] and in the identifiability analysis of a NF-κ B model [37].

We summarize the concept of the method and refer the reader to Appendix D for details. The method iterates over the columns S. _k of the sensitivity matrix S, with k ∈ {1,..., n _p }. Those columns of S that correspond to identifiable parameters are collected in a matrix X ^q where q is the iteration counter. The algorithm starts by selecting the column of S with highest sum of squares in the first iteration. In iteration q + 1, q columns of S have been selected. In the order of their selection these columns form the matrix X ^q . The next step essentially amounts to selecting the column S. _k that exhibits the highest independence to the vector space V spanned by the columns of X ^q . More precisely, an orthogonal projection of S. _k onto V is calculated and is interpreted as the shortest connection from V to S. _k . The squared length of is used as measure of independence. If the length of is near zero, S. _k is nearly linearly dependent to the columns of X ^q . Conversely, a large value of indicates that parameter p _k is linearly independent to the columns of X ^q . In the orthogonal method the parameter p _l where is selected as the next identifiable parameter.

Essentially, the orthogonal method finds those columns of S that are as independent as possible. The quantity can be interpreted as a measure of independence that has been freed from dependencies on previously selected parameters. Therefore not only the influence of a parameter on the outputs, but also the dependencies between the parameters are accounted for by using as a measure. A visualization of the orthogonal method can be found in Appendix D.

Yao et al. [24] stop the selection process once the length of the maximal drops below a cut-off value ϵ _o ≈ 0, the choice of which is rather arbitrary. In the present paper we do not apply this cut-off criterion but merely rank all parameters from most to least identifiable where the terms most identifiable and least identifiable are used in the same sense as in the correlation method. A comparison to the results of the other methods is therefore not affected by the choice of ϵ _o . Our approach to comparing the methods is explained in the section entitled Method comparison.

Method comparison

The central idea behind our comparison of the methods is the equivalence between the local identifiability of a model and the local positive definiteness of the Hessian matrix (10) at a minimum of the least squares parameter cost function (9) for this model. Due to this theoretical result, which has been established by Grewal and Glover [38], we can test local identifiability at a point in the model parameter space by testing the Hessian matrix (10) for positive definiteness at this point. Since the Hessian matrix is positive definite if and only if all its eigenvalues are strictly positive, we consider a model to be identifiable if the eigenvalues are bounded below by a small strictly positive number ϵ. Note the eigenvalue method proposed here is based on the same idea in that it selects those parameters to be unidentifiable that cause eigenvalues smaller than ϵ.

By virtue of the relation between local at-a-point identifiability and the eigenvalues of the Hessian matrix we can compare all four identifiability testing methods to one another with the same criterion and an unique parameter value ϵ. For a given model we first create a ranking of the parameters from least to most identifiable with each method as described in the previous four sections. For each parameter ranking we then fix the parameter considered to be the least identifiable to its literature value, calculate the Hessian matrix (10) with respect to the remaining parameters, and determine the smallest eigenvalue. In all following steps we additionally fix the next least identifiable parameter and recalculate the Hessian and smallest eigenvalue. This process is carried out until the smallest eigenvalue exceeds ϵ. The parameters that need to be fixed in order for the smallest eigenvalue to exceed ϵ are considered to be the unidentifiable parameters. Clearly, the smaller the set of unidentifiable parameters, the larger the set of identifiable parameters or, equivalently, the larger the number of parameters for which the least squares parameter estimation problem can be solved. In this sense we consider the method to be the best one that results in the smallest number of unidentifiable parameters in our comparison.

Note in contrast to the other methods, the ranking created with the PCA based method does in general not contain one but several parameters. In such a case we fix all parameters ranked at the same position at once and reevaluate the smallest eigenvalue of the Hessian matrix. The last ranking position created by the PCA based method contains the indices of all parameters that have not been ranked previously. Fixing of parameters p _k , with k ∈ , would therefore not leave any parameter unfixed. For this reason we consider the PCA method to have failed, if fixing of all parameters with indices in does not lead to a value of the smallest eigenvalue larger than or equal to ϵ.

JAK-STAT pathway

The Janus Kinase (JAK) – signal transducer and activator of transcription (STAT) pathway is triggered by cytokines and growth factors. The pathway has an impact on the expression of genes that regulate diverse cellular processes, such as cellular proliferation, differentiation, and apoptosis. A detailed model of the Interferon-induced JAK-STAT pathway is given by Yamada et al. [2]. This model describes the following signal transducing steps. Upon ligand binding the receptor dimerizes thus triggering the activation of receptor-associated JAK. Activated JAK immediately phosphorylates the receptor complex, enabling the binding and subsequent phosphorylation of cytoplasmic STAT. Phosphorylated STAT can dimerize, and finally, dimerized STAT is imported into the nucleus. Here it activates target genes, one of which is the suppressor of cytokine signaling 1 (SOCS1), an important mediator of negative feedback.

EGF induced MAP kinase pathway

Mitogen-activated protein (MAP) kinases are involved in mitosis and the regulation of cellular proliferation. The Epidermal Growth Factor (EGF)-induced MAP kinase pathway is centered on a three kinase cascade that terminates with the dual phosphorylation of the third, so-called MAP kinase. EGF stimulation leads to dimerization of the EGF receptor. The dimerized EGF receptor triggers the phosphorylation of Raf, the first kinase of the cascade. Raf, in turn, phosphorylates the mitogen extracellular kinase, MEK, the second kinase in the cascade, which finally phosphorylates the MAP kinase (extracellular signal-related kinase, ERK). Phosphorylated ERK regulates several proteins and nuclear transcription factors and controls the expression of target genes. The model published by Schoeberl et al. [3] additionally includes the internalization of EGF receptor and two further modules: an SH2-domain containing protein (SHC) dependent module and an SHC independent one.

NF-κ B pathway

Nuclear Factor-κ B (NF-κ B) is a transcription factor, which regulates genes involved in inflammation, immune response, cell proliferation, and apoptosis. In the unstimulated cell NF-κ B is captured in the cytoplasm by the protein inhibitor of NF-κ B (Iκ Bα). Upon stimulation by pathway activating signals such as the Tumor Necrosis Factor (TNF), the Iκ B kinase (IKK) is activated. Activated IKK phosphorylates Iκ Bα thus inducing its degradation. As a consequence NF-κ B is released, translocates into the nucleus and regulates effector genes. NF-κ B induces its own downregulation by inducing the production of two proteins: 1) Iκ Bα that inhibits NF-κ B by relocating it to the cytoplasm and 2) the zink-finger protein A20 that represses IKK and consequently indirectly inhibits NF-κ B.

Theory

Let Δp = p - p* describe the deviation of a parameter vector p from the true parameter vector p*. For sufficiently small ||Δp||₂ the response function (4) can be approximated by its linearization at p*:

for all i ∈ {1,..., n _y } and j ∈ {1,..., n _t }. The sum of squared errors of the linearized response functions and the measured outputs y _i (t _j ) reads

(20)

The last equality holds, because the y _i (t _j ) are assumed to be noise free measurements and p* are the true parameter values by assumption. Therefore r _i (t _j , p*, x₀) - y_i(t_j) = 0 for all i ∈ {1,..., n _y } and j ∈ {1,..., n _t }.

Let denote the least squares estimates of Δp, i.e. the particular values of Δp that minimize equation (20). The necessary condition for optimality is equivalent to the so called normal equation S ^T S Δp = 0. Solving the normal equation for Δp therefore results in the least squares estimate .

If S ^T S has full rank, the normal equation only admits the solution Δp = 0. This implies Δp = 0, or equivalently = p*, is the unique minimizer of the least squares function R(Δp). Consequently, the model parameters p are locally identifiable by definition. Note only local identifiability can be inferred, since an approximation by linearization was introduced. If, on the other hand, S ^T S does not have full rank, there exists a non-trivial solution ≠ 0 of the normal equation. As a consequence, a parameter vector ≠ p* exists that results in the same model response as p*. Therefore the model parameters p are not identifiable by definition.

The matrix S ^T S has full rank if and only if the columns of S are linearly dependent [43]. The central idea of the correlation method is to detect linear dependence of two columns of S by calculating the correlation of these columns. Parameters corresponding to columns of S with a correlation of ± 1 are considered unidentifiable.

Algorithm for the correlation method

Let C denote the matrix that contains those absolute correlation values between columns of S that exceed the threshold 1 - ϵ _c . C is given by:

The list U contains the desired result as explained in step 4. Note ϵ _c is incremented in steps of Δϵ _c = 0.01. The choice of this increment is not critical as long as it is small enough. If Δϵ _c is chosen smaller than necessary the same ranking of parameters will be obtained but the algorithm may require more computation time.

Description of the second and third criterion of the PCA based method

Let and denote the j th eigenvalue and corresponding eigenvector of , respectively, and assume the eigenvalues to be ordered such that . For each i ∈ {1,..., n _p } evaluate the second and third criterion as follows.

Set L₁ = {1,..., n _y }.

The second criterion is based on the sum of squared eigenvector entries

(21)

Note in this sum, i and j are fixed, therefore the sum is not carried out over components of a particular eigenvector, but the l th entries of all eigenvectors are summed. The parameter with the largest value of (21) is selected.

Set L₁ = {1,..., n _y }.

The third criterion loops over the eigenvectors but, in contrast to the first and second criterion, starts with the eigenvector that belongs to the largest eigenvalue. As in the first criterion, parameters are selected based on the absolute value of the eigenvector component. The index of the parameter selected first will be placed last in ranking , the index of the parameter selected second will be placed at position n _p - 1, and so on.

Algorithm for the PCA based method

The final ranking that combines the PCA results of all and all three criteria is given by J, where the first position in J corresponds to the least and the last position in J corresponds to the most identifiable parameter.

Computational complexity of the orthogonal method

In compact notation the orthogonal projection in iteration i, where i ∈ {1,..., n _p } is arbitrary but fixed, can be written as:

(22)

The major runtime costs result from the computation 1) of Y and 2) of Y S' in equation (22). The number of operations necessary to calculate Y is:

(23)

(24)

where all iterations i = 1,..., n _p have been accounted for. The first and second term in the sum of equation (23) account for the multiplication of X ⁱ by (X ^iT X ⁱ )^-1 (n _y n _t × i matrix times i × i matrix) and the inversion of the matrix X ^iT X ⁱ (i × i matrix), respectively. The third term accounts for the cost of multiplying X ^iT by X ⁱ (i × n _y n _t matrix times n _y n _t × i). This term reduces from i²n _y n _t to in _y n _t , the cost or calculating the last row of X ^iT X ⁱ , since X^i-1TX^i-1is available from the previous iteration:

Here denotes the i th column of X ⁱ . The last row is equal to the last column, since X ^iT X ⁱ is symmetric. The cost for i vector multiplications with 1 ≤ k ≤ i amounts to in _y n _t , which gives rise to the last term in the sum in equation (23).

The number of operations necessary to calculate Y S' is:

(25)

The first term in the sum accounts for the multiplication of Y by S' (n _y n _t × i matrix times i × n _p matrix). The second term accounts for the calculation of S' from the product X ^iT S (i × n _y n _t matrix times n _y n _t × n _p matrix). The latter reduces to the cost n _y n _t n _p for calculating the last row of S', since X^i-1TS is vailable from the previous iteration:

where denotes the last column of X ⁱ and S. _i denotes the i th column of S.

The subtraction of S from Y S' (n _y n _t × n _p matrix minus n _y n _t × n _p matrix) in equation (22) requires O(n _y n _t n _p ) operations and therefore can be neglected.

In summary the dominant parts in equation (24) and (25) result in a computational complexity of order .

Computational complexity of the eigenvalue method

For ease of presentation the description of the algorithm in the Methods section slightly differs from the actual implementation. In the implemented version the Hessian matrix H = S ^T S has to be calculated only once at the beginning of the algorithm, which requires operations (multiplication of a n _p × n _y n _t matrix with a n _y n _t × n _p matrix). In each iteration i, the current Hessian H ⁱ can be obtained by removing the rows and columns from H that correspond to fixed parameters. In each iteration the eigenvalues of the current Hessian matrix have to be calculated. The number of operations necessary for the eigenvalue method is as follows.

(26)

(27)

The first term accounts for the initial calculation of the Hessian matrix. The second term accounts for the eigenvalue calculation.

Comparing computational complexity of the eigenvalue and the orthogonal method

The computational complexity of the eigenvalue method is of order (see equation (27)), whereas the computational complexity of the orthogonal method is of order (see equation (24)). For a comparison we have to distinguish between the following three cases. In the first case we assume n _y n _t ≤ n _p . In this case the complexity of both methods is dominated by the term and both method are equally efficient. In the examples treated here, this case does not occur. In the second case we assume . Here the computational complexity of the eigenvalue and the orthogonal method reduce to and , respectively. This case applies to the MAP-kinase example and the eigenvalue method is faster than the orthogonal method. The difference is the more pronounced the larger n _y n _t - n _p . In the last case we assume that . This is true in the JAK-STAT and the NF-κ B example. Here the complexity of the eigenvalue method and the orthogonal method reduce to and , respectively. In this case the eigenvalue method is an order of magnitude in the number of parameters faster than the orthogonal method.