# Ensemble Kinetic Modeling of Metabolic Networks from Dynamic Metabolic Profiles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Ensemble Kinetic Modeling

**p**is the parameter vector,

**X**(t,

**p**) is the vector of m metabolite concentrations,

**v**(

**X**,

**p**) denotes the vector of n enzymatic reactions/fluxes, and

**S**is the m×n stoichiometric matrix. The metabolic fluxes are further specified as the function of

**X**, for example using a power-law dependence:

_{j}is the rate constant of the j-th flux and f

_{ji}is the kinetic order, reflecting the influence of metabolite X

_{i}on the j-th flux (positive: substrate or activation, negative: inhibition). Aside from the power-law function, Michaelis-Menten and Hill equations have also been commonly used to describe the kinetics of enzymatic reactions

**v**(

**X**,

**p**).

**X**

_{M}(t) is decomposed into a few steps, involving (1) the computation of slopes of time-series data

**Ẋ**

_{M}(t), (2) the calculation of dynamic flux profiles

**v**(t) from

**Ẋ**

_{M}(t), and finally (3) the regression of parameters, which can be done one flux at a time. In the original formulation of the incremental identification and DFE, the number of measured species is assumed to be larger than the number of reactions, such that the second step possesses a unique solution. However, since metabolic pathways typically involve more fluxes than metabolites, there now exist (infinitely) many dynamic flux values, each of which is a mathematically valid solution. This is the premise of the new ensemble modeling method. Specifically, models in the ensemble represent a subset of the dynamic flux solutions to

**Ẋ**

_{M}=

**Sv**, with additional criteria that the kinetic parameters produce statistically equivalent and biologically relevant model predictions. The construction of the model ensemble is detailed in the Method section.

#### 2.1. A Generic Branched Pathway

_{1}(t

_{0}) X

_{2}(t

_{0}) X

_{3}(t

_{0}) X

_{4}(t

_{0})] = [4 1 3 4] and [0.2 0.3 4.2 0.01], respectively. The in silico noisy data were smoothened using a 6-th order polynomial, which gave the best polynomial fit to the data according to adjusted R

^{2}[34] and Akaike Information Criterion (AIC) [35] (see Figure 1b). Subsequently, a central finite difference approximation was applied to compute the time-slopes of the smoothened data.

**Figure 1.**A generic branched pathway. (

**a**) Metabolic pathway map. Metabolic fluxes: double-line arrows, regulatory interactions: dashed arrows with signs; (

**b**) The smoothened data (red line) versus the noisy data (

**×**).

_{1}and v

_{6}were chosen as the independent fluxes, since this selection led to an invertible

**S**

_{D}and comprised the least number of independent parameters. The involved independent parameters

**p**

_{I}included the rate constants {γ

_{1}, γ

_{6}} and the kinetic orders {f

_{13}, f

_{64}}, which were constrained to within [0, 100] and [0, 5], respectively. The bounds for dependent parameters were set to be the same, i.e. {γ

_{2}, γ

_{3}, γ

_{4}, γ

_{5}} ∈ [0,100], {f

_{21}, f

_{33}, f

_{43}, f

_{44}, f

_{51}} ∈ [0,5]. In addition, the upper bound for allowable metabolic fluxes in this artificial network was set as 5×10

^{5}mM/min.

_{R}(minimum = 0.130), and the upper 95% confidence bound of the error function value was determined using Monte Carlo approach (viable < 0.347). Table 1 summarizes the outcome of the ensemble modeling. The multiple ellipsoid-based sampling (MEBS) algorithm produces a model ensemble with 59,928 members within the viable parameter subspace. The corresponding volume of the viable subspace represented only 0.284% of the original parameter space (i.e. the space defined by the upper and lower parameter bounds). Figure 2 shows the projections of the viable regions onto the two-dimensional parameter axes of each independent flux. The true parameter values are contained in the viable subspace, and thus belong to the ensemble (red dot in Figure 2). The member models of the ensemble were able to predict the concentration and slope profiles reasonably well (see Table 1), even when the ensemble was constructed using a different error function. The comparison of data and model predictions in Figure 3 demonstrates the equivalence among five randomly selected models in the ensemble. Finally, Figure 4 shows the comparison of model simulations from the same five models and independent (simulated) experimental datasets, indicating that these models could predict the systems dynamics under different initial conditions reasonably well.

CPU time (sec) ^{a} | 1664 |

Calculated volume of initial parameter space (V_{ci}) ^{b} | 2.5 × 10^{5} |

Estimated volume of viable parameter space (V_{ev}) ^{c} | 710.1 ± 5.1 |

Ratio of
V_{ev} to V_{ci} | (284.0 ± 2.0) × 10^{−3}% |

Range of slope errors | [1.370 × 10^{−1}, 5.081 × 10^{−1}] |

Range of concentration errors | [3.554 × 10^{−2}, 2.150 × 10^{−1}] |

_{ci}was calculated by simple multiplications of the independent parameter ranges.c. V

_{ev}was calculated by integrating the volumes of an ensemble of ellipsoids that cover the viable parameter space [25].d. The range of slope error was computed using Equation (14) for all models in the ensemble.e. The range of concentration error was computed by Equation (15) for all models in the ensemble.

**Figure 2.**Two-dimensional projections of the viable parameter space onto the parameter axes of each independent flux (v

_{1}: left, v

_{6}: right). The true parameters are marked in red.

_{R}minimization, the proposed kinetic ensemble modeling approach can also use other error functions. The viable parameter space using the slope error Φ

_{S}, for example, closely resembles that shown in Figure 2 (see Supplementary Material), demonstrating the robustness of the procedure in capturing the model uncertainty.

#### 2.2. The Trehalose Pathway in Saccharomyces cerevisiae

_{1}, intracellular glucose (inGlc) – X

_{2}, glucose 6-phosphate (G6P) – X

_{3}, trehalose (Tre) - X

_{4}, fructose 1, 6-biphosphate (FBP) – X

_{5}, extracellular end-products (ethanol, glycerol and acetate) – X

_{6}, pentose phosphate pathway (PPP) – X

_{7}and other pathways (Leakage) – X

_{8}. The variables V

_{ex}and V

_{in}denote the extracellular (5.00×10

^{−2}L) and intracellular (7.17×10

^{−3}L) volumes of the bioreactor and the cell population, respectively. The time-course concentration data have been obtained using in vivo NMR, but only X

_{1}, X

_{3}, X

_{4}, X

_{5}and X

_{6}were measured [27]. In the following, we used the dataset from normally grown cells at 30 °C that were fed with a pulse of glucose. The raw experimental data were smoothened using a piecewise cubic spline, the fitting of which was validated by adjusted R

^{2}[34] and AIC [35] (see Figure 5b). Like before, a central difference approximation was applied to obtain the time-slopes of concentration data.

**Figure 3.**Concentration simulations of five randomly selected models from the ensemble (solid blue, brown, green, red and purple lines) versus the noisy data (

**×**).

**Figure 4.**Concentration simulations of the same five models as in Figure 3 (solid blue, brown, green, red and purple lines) versus independent datasets (

**×**), with initial concentrations of [4 1 3 4] (

**a**) and [0.2 0.3 4.2 0.01] (

**b**).

_{7}and X

_{8}are removed, as their concentrations do not affect the other metabolites (i.e. they are sinks in the system). While the intracellular glucose X

_{2}was not measured, its rate of change can be obtained from the measured metabolites by performing an overall mass balance around the pathway, resulting in the following relationship:

_{4}, v

_{7}and v

_{8}were chosen as the independent fluxes, by the same rationale as before. Correspondingly, the independent parameters

**p**

_{I}comprised the rate constants {γ

_{4}, γ

_{7}, γ

_{8}} and the kinetic orders {f

_{44}, f

_{73}, f

_{85}}, which were constrained within [0, 100] and [0, 5], respectively. Note that the glucose transport flux (v

_{1}) was modeled using Michaelis-Menten (MM) kinetics instead of the power law, as this was found to be a better fit to the time profile of X

_{1}(a constant decrease at high X

_{1}and an exponential-like time profile at low X

_{1}). The regression of the MM kinetic parameters can also be casted as a linear regression problem as follows:

_{1}∙ v

_{1}] is the vector of element-wise multiplication of X

_{1}and v

_{1}. Finally, the upper bound for flux values was set as 5 × 10

^{5}mM/min, according to the maximal flux value reported in a similar glycolytic pathway [36].

_{R}(minimum = 7.64 × 10

^{−2}) and the upper 95% confidence bound was found using a Monte Carlo approach (viable < 0.186). Table 2 gives the summary of the model ensemble for the trehalose model. The model ensemble was represented by 3423 member models, and the volume of the corresponding viable subspace constitutes 2.59 × 10

^{−3}% of the original constrained parameter space. The slope errors were acceptable, but the concentration errors had a high upper bound. Upon a closer inspection, only a minority of the model (3 out of 3423) had concentration errors larger than 10

^{2}, and removing these, the upper bound for the concentration error reduces to 35.92. This issue is not unexpected as the model ensemble was created based on the flux error function and not the concentration error. In particular, there is no guarantee that parameter values with a small flux error will also provide a low concentration error. However, we note that the divergence between the flux error and concentration error functions occurred only rarely (< 0.1%). Figure 6 shows the projections of the viable parameter subspace onto the two-dimensional parameter axes of each independent flux. Finally, Figure 7 shows a comparison between the concentration predictions of five randomly chosen models from the ensemble and the measured metabolite time profiles, again demonstrating that models in the ensemble can reproduce the data equally well.

**Figure 5.**The trehalose pathway in Saccharomyces cerevisiae. (

**a**) Metabolic pathway map. Metabolic fluxes: double-line arrows; (

**b**) The smoothened data (red line) versus the noisy data (

**×**).

**Figure 6.**Two-dimensional projections of the viable parameter space onto the parameter axes of each independent flux (v

_{4}: left, v

_{7}: middle, v

_{8}: right).

**Figure 7.**Concentration simulations of five randomly selected models from the ensemble (solid blue, brown, green, red and purple lines) versus the experimental data (×).

CPU time (sec) | 6489 |

Calculated volume of initial parameter space (V_{ci}) | 1.25 × 10^{8} |

Estimated volume of viable parameter space (V_{ev}) | 3237 ± 125 |

Ratio of
V_{ev} to V_{ci} | (25.90 ± 1.00) × 10^{−4}% |

Range of slope errors | [5.825, 46.42] |

Range of concentration errors | [1.125, 3.880 × 10^{2}] |

## 3. Discussion

**Ẋ**(t

_{k}) =

**Sv**(t

_{k}), to restrict the parameter subspace within which the model ensemble is created. Since this DOF is associated with the stoichiometric matrix

**S,**the same ambiguity also exists, albeit implicitly, when the original ODE model:

**Ẋ**(t) =

**Sv**(t) is integrated during the parameter estimation. In corollary, there can exist more than one

**v**(t) that agree with the same

**X**(t

_{k}). However, in this case, the calculation of

**v**

_{D}(t) will involve an infinite dimensional vector space (function space). Furthermore, we note that the ambiguity mentioned above is different from the parametric uncertainty that is represented by the ensemble modeling. In particular, the equivalency of models in the ensemble is judged by the error function Φ and different error functions can produce dissimilar model ensembles. As shown in the second case study, a few models of the ensemble created by Φ

_{R}produced large concentration errors Φ

_{C}. This discrepancy is perhaps not surprising as Φ

_{R}is based on the algebraic model

**Ẋ**(t

_{k}) =

**Sv**(

**X**(t

_{k}),

**p**), while the calculation of Φ

_{C}involves the integration of the ODE model

**Ẋ**(t) =

**Sv**(

**X**(t),

**p**)..

**X**(t). For the error function used in the case studies above, this computational cost should increase linearly with the number of dependent fluxes, assuming that the number of unknown parameters in each dependent flux stays about the same.

**p**

_{D}from

**p**

_{I}. For GMA models, this assumption requires that (1) the number of time points exceed the number of parameters

**p**

_{D}from each flux (not the total number) and (2) the logarithm of the metabolite concentration time profiles appearing in each flux are linearly independent. The first requirement is usually satisfied as the number of parameters involved in every flux ranges only between 2 and 5. The second requirement depends on the experimental conditions, but is again usually fulfilled since each flux depends only on a handful of metabolites and data are contaminated with random noise. If this assumption becomes invalid for one or more dependent fluxes, then these fluxes can be included into the set of independent fluxes, at the cost of increasing the dimensionality and computational time of the parameter exploration step. In such a case, the calculation of dependent fluxes from the independent flux values will require taking a pseudo-inverse of

**S**

_{D}(see Method).

_{1}(see Figure 2). Meanwhile, parameter constraints affect the second case study more than the first, where the lower and upper constraints of all rate constants and the lower bounds of all kinetic orders limited the viable parameter subspace (see Figure 6). Furthermore, in both case studies, the requirement for positivity of the flux values (i.e. lower bounds of the fluxes) was an important constraint, as this was frequently violated during the parameter exploration (data not shown).

## 4. Method

#### 4.1. Problem Formulation

**X**

_{M}(t

_{k}), k = 1, K, the estimation procedure starts with the computation of time-slopes. Data smoothing is usually applied to improve the numerical estimation of

**Ẋ**

_{M}(t

_{k}). The slopes can be estimated using a finite difference approximation of the smoothened data or by differentiating the smoothened curve function, if available. Subsequently, the values of dynamic reaction fluxes are approximated from the mass balance

**Ẋ**

_{M}(t

_{k}) =

**Sv**(t

_{k}). Finally, the kinetic parameters are determined from dynamic flux values using a least square regression

**v**(t

_{k}) =

**v**(

**X**

_{M}(t

_{k}),

**p**) which can now be done for each flux individually. By decomposing the identification problem into smaller easy-to-do subproblems, the step-wise identification can offer a significant reduction in the computational cost of performing the estimation. Furthermore, for power-law flux functions, the third step involve only simple (log-)linear regressions. However, in the original formulation of incremental identification and DFE, one assumes that the subproblems have a unique solution, which is often invalid for a metabolic pathway.

**v**(t

_{k}) that can satisfy the mass balance equation

**Ẋ**

_{M}(t

_{k}) =

**Sv**(t

_{k}), each of which represents a valid mathematical solution to the parameter estimation problem. The dimensionality of the dynamic flux solutions is equal to the degree of freedom (DOF) in the mass balance, defined as the difference between the number of fluxes and the number of metabolites: n

_{DOF}= n−m > 0. Thus, only a subset of n

_{DOF}fluxes (called independent fluxes) need to be specified at each time point t

_{k}, while the remaining (dependent) fluxes can be computed from the mass balance equation.

**v**(t

_{k}) = [

**v**

_{I}(t

_{k})

^{T}

**v**

_{D}(t

_{k})

^{T}]

^{T}, where the subscripts I and D denote the independent and dependent subsets, respectively. Similarly,

**S**and

**p**are restructured as

**S**= [

**S**

_{I}

**S**

_{D}] and

**p**= [

**p**

_{I}

**p**

_{D}]. As mentioned above, given the values of

**v**

_{I}(t

_{k}), one can compute the corresponding values of

**v**

_{D}(t

_{k}), according to:

**S**has a full row rank, one can choose n

_{DOF}independent fluxes such that

**S**

_{D}is invertible. For numerical efficiency, the independent fluxes are chosen by considering the following: (i) the

**S**

_{D}is invertible, (ii) the number of the independent parameters

**p**

_{I}is small, and/or (iii)

**p**

_{I}values are known a priori within a small range. Similar numerical considerations for selecting flux functions have also been discussed elsewhere [38]. Subsequently, by replacing

**v**

_{I}(t

_{k}) with the flux function

**v**

_{I}(

**X**

_{M}(t

_{k}),

**p**

_{I}) and assuming that the dependent parameters

**p**

_{D}can be uniquely determined from

**v**

_{D}(t

_{k}), then the model parameters can be completely defined by assigning the values of the independent parameters

**p**

_{I}. For power-law models, the uniqueness of

**p**

_{D}is a weak assumption, requiring the least square regression problem

**v**

_{D}(t

_{k}) =

**v**

_{D}(

**X**

_{M}(t

_{k}),

**p**

_{D}) to be fully or over-determined (see Discussion section).

**S**

_{I,M}and

**S**

_{D,M}are submatrices of

**S**

_{M}, such that

**S**

_{M}= [

**S**

_{I,M}

**S**

_{D,M}] following the decomposition of

**v**(t

_{k}) = [

**v**

_{I}

^{T}

**v**

_{D}

^{T}]

^{T}. As expected, the degree of freedoms will increase (n

_{DOF}= n−m*, where m* is the number of measured metabolites), and so will the number of independent fluxes. The independent fluxes should be selected such that

**S**

_{D,M}is invertible and should also include fluxes that appear in

**Ẋ**

_{U}. The same practical considerations for choosing

**v**

_{I}, e.g. considering the number of and the prior information on

**p**

_{I}, are also applicable. Finally, like before, given the values of

**p**

_{I}, the dependent parameters can be obtained by least square regression of

**v**

_{D}(t

_{k}). However, since

**v**

_{I}(t

_{k}) can also depend on

**X**

_{U}, i.e.

**v**

_{I}(

**X**

_{M}(t

_{k}),

**X**

_{U}(t

_{k}),

**p**

_{I}), we will need to simulate

**Ẋ**

_{U}=

**S**

_{U}

**v**(

**X**

_{M},

**X**

_{U},

**p**), using the smoothened

**X**

_{M}(t) as input variables.

**p**

_{I}and

**p**are obtained from the relationship

_{D}**Ẋ**

_{M}(t

_{k}) =

**Sv**(t

_{k}), they may not give the same goodness-of-fit to the concentration measurements

**X**

_{M}(t

_{k}). Briefly, the difference in the quality of data fitting is due to the fact that the mathematical equivalence above is established based on the slopes of the (smoothened) concentration data, not on the concentrations themselves, and also due to noise in data. Here, the ensemble modeling is performed by exploring the parameter space

**p**

_{I}and demarcating the viable subset of parameters that satisfy both

**Ẋ**

_{M}(t

_{k}) =

**Sv**(t

_{k}) and two additional criteria: (1) all kinetic parameter values and fluxes are within biologically relevant bounds and (2) the model prediction error is within acceptable statistical bounds. Details of the parameter exploration algorithm and parameter viability criteria are given below.

#### 4.2. HYPERSPACE Toolbox

_{MC}containing coarse-grained viable parameter points. Figure 8 illustrates the procedure of this algorithm.

**Figure 8.**Flowchart of the out-of-equilibrium adaptive Metropolis Monte Carlo (OEAMC) algorithm. On the right, the red closed curves represent hypothetical contour plots of the viable parameter space. The viable points are marked in blue and the nonviable points are marked in red. Finally, the grey areas illustrate the minimum volume enclosing ellipsoids. This figure is adapted from the original publication [25].

_{MC}set, the MEBS searches for viable parameter points near the boundary of the viable region. A Minimum Volume Enclosing Ellipsoid (MVEE, dashed ellipsoids in Figure 9) is then created to cover the local viable region. Subsequently, the MVEE is scaled up by a multiplier g

_{i}(solid curves in Figure 9), and a uniform sample of points is generated inside this scaled ellipsoid. Among these random points, the nonviable points (red points) are discarded, and another iteration of MVEE and another uniform sampling using a new multiplier g

_{i+1}are done using the remaining viable ones (blue points). The performance of the algorithm depends strongly on the multiplier g

_{i}, and here we have used the recommended scaling parameters in the original publication [25]. The iteration is repeated until the scaling multiplier tends to one or a fixed number of iterations is reached. Finally, the whole procedure above is repeated for another viable parameter point from V

_{MC}until all parameter points in this set are exhausted. The output of the MEBS is a comprehensive set of viable parameter points. Figure 9 summarizes the procedure of the MEBS algorithm.

**Figure 9.**Flowchart of the multiple ellipsoid-based sampling (MEBS) algorithm. In the right part of the figure, the red closed curves represent hypothetical contour plots of the viable parameter space defined by some criteria. The viable points are marked in blue and the nonviable points are marked in red. Finally, the grey areas illustrate the minimum volume enclosing ellipsoids. This figure is adapted from the original publication [25].

#### 4.3. Model Viability Criteria

**p**

_{I}, the corresponding dependent fluxes and parameters may not necessarily be biologically relevant, for example the dependent fluxes may become negative or the parameters may assume unrealistic values. Thus, in the ensemble modeling procedure, these cases are excluded by enforcing constraints on the values of fluxes and parameters, as follow:

**L**and

**U**denote the lower and upper bounds for the parameters, and

**U**is the maximum value of metabolic fluxes. The second viability criterion is meant to establish equivalence among the member models in terms of their goodness of fit to data. If one makes the assumption that data noise comes from a Gaussian distribution, then the confidence bound of error function Φ can usually be estimated using standard statistical analyses and model sensitivities [42]. When data noise is not Gaussian, the confidence bounds can be estimated using a Monte Carlo approach [43].

_{v}#### 4.4. Ensemble Modeling Procedure

**p**

_{D}are regressed from the dynamic flux estimates

**v**

_{D}(t

_{k}). Note that the calculation of this error function was actually done one flux at a time, as the least square regression of

**p**

_{D}from

**v**

_{D}(t

_{k}) was performed for each flux function separately. For power-laws, this regression can be performed very efficiently, as the logarithm of the flux function depends linearly on the parameters (leading to a linear least square regression). In this case,

**v**

_{D}(t

_{k}) was calculated from

**v**

_{I}(t

_{k}) according to Equation (8), while

**v**

_{I}(t

_{k}) was computed from the time series data and

**p**

_{I}using the flux function

**v**

_{I}(

**X**

_{M}(t

_{k}),

**p**

_{I}). In other words, the error function depends only on the independent parameters

**p**. The initial parameter point

_{I}**p**

_{I}for the OEAMC algorithm was obtained from the following optimization:

**X**(t

_{k},

**p**) is the concentration simulation.

**p**

_{I}that represents the ensemble of models. Note that while this work concerns with power-law fluxes, the ensemble generation procedure has general applicability to any kinetic models that can be written as

**Ẋ**(t,

**p**) =

**Sv**(

**X**,

**p**).

## 5. Conclusions

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Jia, G.; Stephanopoulos, G.; Gunawan, R.
Ensemble Kinetic Modeling of Metabolic Networks from Dynamic Metabolic Profiles. *Metabolites* **2012**, *2*, 891-912.
https://doi.org/10.3390/metabo2040891

**AMA Style**

Jia G, Stephanopoulos G, Gunawan R.
Ensemble Kinetic Modeling of Metabolic Networks from Dynamic Metabolic Profiles. *Metabolites*. 2012; 2(4):891-912.
https://doi.org/10.3390/metabo2040891

**Chicago/Turabian Style**

Jia, Gengjie, Gregory Stephanopoulos, and Rudiyanto Gunawan.
2012. "Ensemble Kinetic Modeling of Metabolic Networks from Dynamic Metabolic Profiles" *Metabolites* 2, no. 4: 891-912.
https://doi.org/10.3390/metabo2040891