Projection to latent pathways (PLP): a constrained projection to latent variables (PLS) method for elementary flux modes discrimination
- Ana R Ferreira†^{1, 2},
- João ML Dias†^{1},
- Ana P Teixeira†^{2, 3},
- Nuno Carinhas†^{2, 3},
- Rui MC Portela†^{1},
- Inês A Isidro†^{1},
- Moritz von Stosch†^{4} and
- Rui Oliveira†^{1, 2}Email author
DOI: 10.1186/1752-0509-5-181
© Ferreira et al; licensee BioMed Central Ltd. 2011
Received: 9 May 2011
Accepted: 1 November 2011
Published: 1 November 2011
Abstract
Background
Elementary flux modes (EFM) are unique and non-decomposable sets of metabolic reactions able to operate coherently in steady-state. A metabolic network has in general a very high number of EFM reflecting the typical functional redundancy of biological systems. However, most of these EFM are either thermodynamically unfeasible or inactive at pre-set environmental conditions.
Results
Here we present a new algorithm that discriminates the "active" set of EFM on the basis of dynamic envirome data. The algorithm merges together two well-known methods: projection to latent structures (PLS) and EFM analysis, and is therefore termed projection to latent pathways (PLP). PLP has two concomitant goals: (1) maximisation of correlation between EFM weighting factors and measured envirome data and (2) minimisation of redundancy by eliminating EFM with low correlation with the envirome.
Conclusions
Overall, our results demonstrate that PLP slightly outperforms PLS in terms of predictive power. But more importantly, PLP is able to discriminate the subset of EFM with highest correlation with the envirome, thus providing in-depth knowledge of how the environment controls core cellular functions. This offers a significant advantage over PLS since its abstract structure cannot be associated with the underlying biological structure.
Background
An elementary flux mode (EFM) can be defined as a minimal set of enzymes able to operate at steady state, with the enzymes weighted by the relative flux they need to carry for the mode to function [1]. The universe of EFM of a given metabolic network define the full set of non-decomposable steady-state flux distributions that the network can support. Any particular steady-state flux distribution can be expressed as a non-negative linear combination of EFM. Motivated by these unique properties, EFM analysis has become a widespread technique for systems level metabolic pathways analysis [1–8].
Classification of methods for EFM reduction
Principle | Method | Data required | References |
---|---|---|---|
Network connectivity and stoichiometry | K-shortest EFM: Enumerates the EFM in increasing order of number of reactions. Yield Analysis: Excludes EFM with negligible contribution to convex hull in yield space. | Parameter free | [11] [12] |
Thermodynamics | Fractional contributions of EFM: Estimates the EFM Coefficients based on calculated EFM thermodynamic properties. Maximum Entropy Principle: Calculates the EFM Coefficient by maximizing Shannon's entropy, which is an indirect measure of system complexity. | Thermodynamic data | [13] [14] |
(Non)linear programming | α-spectrum: Uses linear optimization to maximize and minimize the weightings of each metabolic pathway that produces steady state flux distributions. Flux regulation coefficients: Estimates the EFM coefficients that optimize a given performance function (e.g. minimum error in flux or yield prediction). Quadratic program: Calculates the weights for a large set of EFM by using quadratic program to reconstruct flux distributions from subsets of EFM. | '-omics' data can be used to shrink the α-spectrum. Fluxomics and possibly other omic datasets | [18] [17] |
Enzyme kinetics | Quantitative elementary mode analysis of metabolic pathways: Combines structural and kinetic modelling to assess the effect of changes in enzyme kinetics on the usage of EFM. | Enzyme kinetic parameters | [19] |
Some of the proposed methods reduce EFM based solely on structural information of the metabolic network. de Figueiredo et al. [11] presented a method to enumerate the EFM in increasing order of number of reactions. This approach enabled to identify the K-shortest EFM in Escherichia coli and Corynebacterium glutamicum metabolic networks, which are in principle energetically more efficient. Song and Ramkrishna [12] proposed a reduction algorithm based on the effect of EFM on the convex hull volume. This allowed the a priori reduction, without any experimental data, from the initial 369 to 35 EFM for a yeast metabolic network fermenting both glucose and xylose.
EFM can also be discriminated on the basis of reaction thermodynamics. Wlaschin et al. [13] demonstrated with experimentally determined intracellular fluxes that EFM weights are inversely correlated with the entropy generated by the involved metabolic reactions. This suggests that evolution induced cellular regulatory patterns to favour efficient pathways with low entropy generation. Zhao et al. [14] proposed a method for correlating enzyme activity and flux distribution which uses the Shannon's maximum entropy principle, a measure of system complexity, as an objective function to estimate the enzyme control flux.
Several methods have been proposed that merge linear programming and experimental data. Palsson and co-authors [15, 16] suggested linear optimization to determine how extreme pathways (the systemically independent subset of EFM) contribute to a given (measured) steady-state flux distribution. There is a range of possible nonnegative weighting values associated to extreme pathways that produce a given steady-state flux distribution. This range was calculated by maximizing and minimizing the extreme pathway weighting factors, resulting in the so called α-spectrum. Wang et al. [17] presented a method to calculate the EFM coefficients for a large set of EFM by devising a quadratic program to explore the possibility and performance of using a subset of the EFM to reconstruct flux distributions. Alternatively, a framework based on EFM analysis and the convex properties of EFM was developed to calculate EFM flux regulation coefficients (FRC) corresponding to an appropriate fractional operation of this mode within the complete set of EFM [18].
Schwartz and Kanehisa [19] showed that a combination of structural and kinetic modelling in yeast glycolysis significantly constraints the range of possible behaviours of a metabolic system. All EFM are not equal contributors to physiological cellular states, and this approach may open a direction towards a broader identification of physiologically relevant EFM among the very large number of stoichiometrically possible modes.
In a previous paper [20], we have delineated a conceptual approach to map envirome factors to cellular functions based on the correlation of EFM weighting factors and measured envirome variables. Here we study in detail the computational algorithm to reduce EFM based on the degree of correlation of EFM weighting factors with measured envirome factors, which we call projection to latent pathways (PLP). The underlying principles are: (i) only a moderate number of EFM are active at given environmental conditions, (ii) the envirome plays a critical role in their regulation, and (iii) active EFM deliver a characteristic environmental footprint that can be used for their identification. In what follows we present all mathematical details underlying PLP and compare it with PLS in relation to a case study.
Results
Projection to Latent Pathways (PLP) Algorithm
Problem statement
with em_{ i } a q × 1 vector of reaction weighting factors that defines EFM i and λ_{ i } a scalar variable defining the partial contribution of em_{ i } to the overall flux phenotype, r, and n_{ em } the number of EFM.
- 1.
Maximisation of explained variance of flux data sets, R = {r(t)}
- 2.
Maximisation of correlation of λ_{ i } against envirome data, X = {x(t)}
- 3.
Minimisation of the number of active EFM
with EM = {em_{ i }} a nr × nem matrix of nem EFM, em_{ i } (dim(em_{i}) = q), Λ = {λ(t)} a np × nem matrix of weight vectors λ(t) of EFM (dim(λ) = nem) and C a nem × nx matrix of regression coefficients.
Projection to Latent structures (PLS)
PLS is a multivariate linear regression technique between an input (predictor) matrix, X, and an output response matrix, Y. It differs from traditional multivariate linear regression in that it decomposes both the predictor and the response matrices into reduced sets of uncorrelated latent variables, which are then linearly regressed against each other.
- 1.Set the initial ny × 1 Y-loading vector, q, equal to an arbitrarily chosen nonzero row of Y, y_{ t }$q=\frac{{y}_{t}^{\mathsf{\text{T}}}}{\u2225{y}_{t}\u2225}$(4)
- 2.Compute the np × 1 Y-score vector, u$u=Y\cdot q$(5)
- 3.Compute the nx × 1 weight vector, w$w=\frac{{X}^{\mathsf{\text{T}}}\cdot u}{\u2225{X}^{\mathsf{\text{T}}}\cdot u\u2225},$(6)
- 4.Compute the np × 1 X-score vector, t$t=X\cdot w$(7)
- 5.Recalculate the Y-loading vector, q$q=\frac{{Y}^{\mathsf{\text{T}}}\cdot t}{\u2225{Y}^{\mathsf{\text{T}}}\cdot t\u2225}$(8)
- 6.
Repeat steps 1-5 until the convergence criterion ||t-t_{ old }|| < eps is obeyed with, for instance, eps = 1 × 10^{-8}. In case of univariate PLS, Eq. 8 yields q = 1 hence no iterations are performed.
- 7.Compute the X data block loadings, p, and rescale accordingly:$p=\frac{{X}^{\mathsf{\text{T}}}\cdot t}{\u2225{t}^{\mathsf{\text{T}}}\cdot t\u2225}$(9)${p}_{new}=\frac{p}{\u2225p\u2225}$(10)$t=t\cdot \u2225p\u2225$(11)$w=w\cdot \u2225p\u2225$(12)
- 8.Compute the regression coefficient of the inner linear model$b=\frac{{u}^{\mathsf{\text{T}}}\cdot t}{{t}^{\mathsf{\text{T}}}\cdot t}$(13)
- 9.Compute the X and Y residuals${E}_{X}=X-t\cdot {p}^{\mathsf{\text{T}}}$(14)${E}_{Y}=Y-b\cdot t\cdot {p}^{\mathsf{\text{T}}}$(15)
- 10.Then go back to step 1 and repeat the procedure for the next latent variable after making$X={E}_{X}$(16)$Y={E}_{Y}$(17)
For more details about PLS and NIPALS see Geladi and Kowalski [23].
Projection to latent pathways (PLP)
- 1.For each EFM k, set the loadings equal to em_{ k } and compute the respective score vector, λ_{ k }:${q}_{k}=e{m}_{k}$(23)${\lambda}_{k}=R\cdot {q}_{k}\left(\equiv {u}_{k}\right)$(24)
- 2.Perform a univariate PLS (with q = 1) with input X and target Y = λ_{ k } for Fac latent variables as described in the previous section and compute the predicted λ_{ k }${\widehat{\lambda}}_{k}:\phantom{\rule{0.3em}{0ex}}predicted\phantom{\rule{0.3em}{0ex}}{\lambda}_{k}\phantom{\rule{0.3em}{0ex}}from\mathsf{\text{}}univariate\mathsf{\text{}}PLS$(25)
- 3.Compute the predicted R by the k EFM and the respective explained variance${\widehat{R}}_{k}={\widehat{\lambda}}_{k}\cdot {q}_{k}^{\mathsf{\text{T}}}$(26)$va{r}_{k}\left(\mathsf{\text{\%}}\right)=\mathsf{\text{100}}\cdot \left(\mathsf{\text{1-}}\frac{\sum _{\mathsf{\text{i}}}{\left(R-{\widehat{R}}_{k}\right)}^{\mathsf{\text{T}}}\cdot \left(R-{\widehat{R}}_{k}\right)}{\sum _{\mathsf{\text{i}}}{R}^{T}\cdot R}\right)$(27)
- 4.Repeat steps 1-3 for every EFM k = 1,..., nem and choose the best, kopt, as the one that exhibits the highest variance value given by Eq. 27.$kopt:\mathsf{\text{EFMwithhighest}}\phantom{\rule{0.3em}{0ex}}va{r}_{k}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{value}}$(28)
- 5.Remove kopt from the list of EFM and make$R=R-{\widehat{R}}_{kopt}$(29)
- 6.
Go back to step 1 and repeat the procedure for a maximum number of EFM or until the explained variance of R does not increase any further.
- 1.
The number of latent variables in PLS is analogous to the number of active EFM in PLP. Thus the subset of EFM that explain most of the variance of R are interpreted as the set of metabolic pathways activated by environmental factors.
- 2.
The regression coefficients vector, RC_{ kept }, of the inner univariate PLS, being directly associated with EFM, show the contribution of each environmental factor to the up- or down-regulation of EFM.
The PLS and PLP algorithms were coded in Matlab™(Mathworks, Inc). The code is freely available for academic use under a free academic license and can be downloaded at http://www.dq.fct.unl.pt/sbegroup.
In what follows we compare both algorithms in relation to a case study.
Case study: recombinant BHK cell line
Data of a recombinant baby hamster kidney (BHK) cell line expressing a fusion glycoprotein IgG1-IL2 was used to compare PLS and PLP. The data set comprises 134 observations acquired from 7 independent bioreactor experiments operated in batch and fed-batch modes. The predictor matrix, X (dim(X) = 134 × 26), includes measured data of 26 environmental factors (pH, osmolarity and concentrations of viable cells, glucose, lactate, ammonia, IgG1-IL2 and 19 amino acids) while the target matrix, R (dim(R) = 134 × 24), comprises 24 production or consumption fluxes of extracellular compounds. Further details about the data can be found elsewhere ([20]).
A relatively small BHK metabolic network comprising 35 metabolites and 57 metabolic reactions was constructed. Its EFM were computed using Metatool 5.0 [24] resulting in 251 EFM. Details can be found as Additional Files 1 and 2. These 251 EFM were used as constraints to PLP decomposition.
Comparing PLP and PLS decomposition results
PLS decomposition results in terms of % of explained variance (Var) over number of latent variables (LV).
# Lv | Var X (%) | Var R (%) |
---|---|---|
1 | 48.9 | 32.4 |
2 | 59.6 | 51.8 |
3 | 79 | 58.0 |
4 | 84.6 | 64.3 |
5 | 89.8 | 67.4 |
6 | 92.2 | 70.9 |
7 | 94.5 | 74.1 |
8 | 96.2 | 76.4 |
9 | 97.7 | 78.6 |
10 | 98.3 | 82.1 |
11 | 98.9 | 83.0 |
12 | 99.4 | 84.1 |
13 | 99.6 | 85.8 |
14 | 99.8 | 86.7 |
15 | 99.9 | 87.9 |
16 | 99.9 | 89.0 |
17 | 99.9 | 89.6 |
18 | 100 | 90.1 |
PLP decomposition results showing the subset of EFM with highest correlation with the envirome (as denoted by the r^{2} and p-value).
EFM | # LV | r^{2} | p-value | Var(λ) | Var(R) |
---|---|---|---|---|---|
179 | 4 | 0.95 | 1.14E-32 | 88.90 | 52.60 |
1 | 4 | 0.89 | 5.16E-23 | 79.90 | 57.30 |
210 | 4 | 0.87 | 1.56E-20 | 65.90 | 57.80 |
173 | 4 | 0.82 | 8.78E-17 | 62.40 | 58.30 |
116 | 4 | 0.82 | 2.49E-16 | 58.40 | 58.70 |
139 | 4 | 0.86 | 1.34E-19 | 52.50 | 58.90 |
206 | 4 | 0.92 | 2.14E-27 | 73.90 | 60.20 |
143 | 4 | 0.86 | 9.71E-20 | 66.70 | 60.60 |
69 | 4 | 0.82 | 1.04E-16 | 57.30 | 61.00 |
72 | 4 | 0.84 | 3.52E-18 | 57.80 | 61.30 |
4 | 4 | 0.92 | 1.96E-27 | 81.40 | 64.10 |
68 | 4 | 0.81 | 3.96E-16 | 60.20 | 64.80 |
11 | 4 | 0.91 | 4.16E-25 | 76.80 | 79.20 |
6 | 4 | 0.94 | 1.99E-30 | 84.60 | 80.90 |
7 | 4 | 0.82 | 1.72E-16 | 59.10 | 81.60 |
12 | 4 | 0.83 | 1.52E-17 | 58.30 | 82.10 |
2 | 4 | 0.85 | 7.26E-19 | 71.00 | 82.50 |
Assessment of EFM reduction consistency
Metabolic interpretability
It can be seen that the first PLS loadings vector q_{ 1 } calculated by Eq. 8 does show a residual correlation with the first selected EFM 179 structure (r^{2} = 0,26). However, the second loadings vector q_{2} shows no correlation at all with second select EFM 1. Despite the fact that both the calculation of the output loadings q and the selection of EFM obey to the same criterion of maximization of the correlation between X and Y, it is clear that the data structure identified by PLS cannot be easily associated with the underlying biological structure.
It is beyond the scope of this paper to present a detailed metabolic interpretation of the discriminated EFM by PLP (for a detailed analysis see [20]). Here we just highlight a few illustrative examples, the most frequently selected EFM for biomass synthesis is EFM 179 followed by EFM173. The product formation EFM (EFM 1) is also frequently selected. The anaerobic conversion of glucose into lactate was also frequently selected (EFM 11). Serine transamination into glycine (EFM 6) was also among the most frequently selected EFM. EFM 4 corresponds to the glutaminolysis pathway, well known as a major carbon source for energy production in mammalian cells. In general, these are important pathways known to be active in mammalian cells.
Regression coefficients
Predictive power
In order to screen out the possibility of a casual better performance of PLP in relation to PLS due to the particular data partitioning employed, the same variance analysis was performed for the z = 200 PLP and PLS runs performed with randomly selected calibration and validation data points according to the bootstrapping technique previously described. The results show that the explained variance of the validation data set varied between 78.8-85.6% for PLP and 50.4-82.7% for PLS (see Additional File 3). In 194 out of 200 runs the PLP outperformed the PLS, thus confirming that while PLS is consistently more accurate in describing the calibration data than PLP, the latter is consistently more accurate at predicting the validation data than PLS.
Discussion
The key PLS feature is identifying independent X and Y-scores so that the relationship between successive pairs of scores is as strong as possible. PLS may be thus viewed as a robust form of redundancy analysis, seeking directions in the factor space that are associated with high variation in the response Y but biasing them toward directions that are more accurately predicted. Due to its advantages in handling highly redundant data sets, PLS has become a widely used regression analysis technique in systems biology. It has been applied as an inference tool for predicting metabolic fluxes using isotopomer flux data [29], analysing genomic and proteomic data [25], identifying signalling networks by inducing cellular response to different stimuli [30–32] and network structure using metabolomic data [33]. Moreover, PLS has also been applied for the identification of active cellular pathways as a function of the environment using metabolic and gene expression profiles [34], detection of gene-gene interactions from microarrays data [35, 36] and culture media optimization using nutritional profiling data [26, 27].
The main disadvantage of PLS lies in its empirical data-driven nature with limited added-value in terms of mechanistic knowledge generation. Although carrying some internal structure, this structure is not inspired by any a priori mechanistic knowledge of the system. PLP may be viewed as a constrained version of PLS, attuned to the structure of the biological system under study. While in PLS the loadings and score are abstract variables, in PLP loadings and scores refer to well defined metabolic structures. Specifically, PLP explores EFM as "principle components" of a metabolic network. Indeed, EFM obey to the principle of non-decomposability, meaning that any particular flux distribution can be expressed as a nonnegative weighted sum of EFM. Thus the ranking obtained in PLP refers to active pathways as inferred by their level of correlation with the environmental state. In terms of data requirement, PLS belongs to the class of multivariate regression techniques particularly suitable to handle highly dimensional data sets even if the number of observations is limited [25]. PLS is typically used to model spectral data such as near infrared or 2D-fluorescence maps [37]. A basic requirement is that the number of latent variables must be lower than the number of observations in the calibration data set. This means that reliable linear models can be identified from a moderate number of observations of highly dimensional datasets. The same properties apply to PLP. A basic constraint is that the number of discriminated EFM cannot be higher than the number of observations in the calibration data set. However the method offers no restriction in terms of the dimensionality of the input data set.
Finally it should be commented on the computational power requirements, which scales linearly with the number of EFM. In the present study with 251 EFM, computation requirements are in the order of seconds in a common PC. For a genome scale network with several million of EFM, computation power might easily rise to the scale of days in a common PC.
Conclusions
In this work we have developed an algorithm for the discrimination of active EFM on the basis of dynamical envirome data called projection to latent pathways (PLP). The algorithm is designed to maximise the covariance between envirome data and observed flux data under the constraint of universe of genes translated into a plausible set of EFM. In general lines, the algorithm discriminates a minimal set of envirome correlated EFM that maximise the variance of measured flux data. Thus the algorithm may be viewed as a reverse, envirome-to-function metabolic reconstruction methodology as opposed to the generally accepted gene-to-function reconstruction approach. Although presented here as a method to analyse envirome data sets, PLP has broader scope. It is rather a general methodology for statistical elimination of redundant metabolic structures that, in a broader sense, has the potential to bring together all layers of 'omic' information under a common computational framework.
Notes
Declarations
Acknowledgements
Financial support for this work was provided by the Portuguese Fundação para a Ciência e Tecnologia through projects PTDC/EBB-EBI/103761/2008 and MIT/Pt7BS-BB/0082/2008, PhD grant SFRH/BD/36285/2007 (ARF), SFRH/BD/36676/2007 (NC), SFRH/BD/51577/2011 (RP), SFRH/BD/70768/2010 (IAI) and SFRH/BD/36990/2007 (MvS) and Post-Doc grant SFRH/BPD/46277/2008 (JD).
Authors’ Affiliations
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