#
Effective Estimation of Dynamic Metabolic Fluxes Using ^{13}C Labeling and Piecewise Affine Approximation: From Theory to Practical Applicability

^{*}

## Abstract

**:**

^{13}C labeling to identify piecewise-affine (PWA) flux functions has been described recently. Obtaining flux estimates nevertheless still required frequent manual reinitalization to obtain a good reproduction of the experimental data and, moreover, did not optimize on all observables simultaneously (metabolites and isotopomer concentrations). In our contribution we focus on measures to achieve faster and robust dynamic flux estimation which leads to a high dimensional parameter estimation problem. Specifically, we address the following challenges within the PWA problem formulation: (1) Fast selection of sufficient domains for the PWA flux functions, (2) Control of over-fitting in the concentration space using shape-prescriptive modeling and (3) robust and efficient implementation of the parameter estimation using the hybrid implicit filtering algorithm. With the improvements we significantly speed up the convergence by efficiently exploiting that the optimization problem is partly linear. This allows application to larger-scale metabolic networks and demonstrates that the proposed approach is not purely theoretical, but also applicable in practice.

## 1. Introduction

- (1)
- Genetic adaption by mutagenesis and selection with a timescale of several generation times [3],
- (2)
- Adaption of gene expression levels with a timescale of minutes [4],
- (3)
- Post-translational modifications with a time constant on the order of seconds and
- (4)
- Kinetic response which is considered an inherent property of the enzymes in the metabolic network and therefore persistent.

**c**

_{i}) and kinetic parameters (θ) and are typically non-linear. The latter can be either derived from mechanistic assumptions e.g., Michaelis-Menten kinetics, or be non-mechanistic e.g., power law leading to parameters with no physical meaning [7].

- (1)
- (2)
- The kinetic functions have to be chosen a priori, which means that the mechanism of every enzyme and all interactions between metabolites and enzymes involved in the regarded metabolic network have to be selected before parameter optimization. Especially, the kinetic formats of each reaction in larger metabolic networks are yet unknown, unclear, or shall be deduced from the captured observables.

**Figure 1.**“Classical” kinetic modeling requires a priori defined kinetic mechanisms. Using the hybrid modeling approach, only the metabolic network structure (incl. atom transitions) is required a priori. As a result, the flux profile in time can be identified rather than kinetic parameters. With the flux functions on hand, the kinetics of the metabolic network can be investigated decoupled from the overall network.

^{13}C labelling [19]. Therewith the resulting parameter estimation problem becomes much more challenging, as the balances for the tracer atoms introduce non-linearity into the so far linear system, which moreover has to be solved by numerical integration instead of analytically. The metabolic tracers add a second set of observables, leading to a multi-objective parameter estimation problem that requires definition of a trade-off between the different types of observables. Moreover parameter correlations, and especially high dimensionality, need to be tackled. Classical non-linear optimization approaches were not found practical for the arising ill-posed inverse problems at a larger scale as they usually lead to very slow convergence and are often not robust enough for practical application in larger-scale metabolic networks.

#### Experimental Requirements for in Vivo Dynamic Flux Estimation

- •
- A stimulus-response experiment should lead to perturbation(s) strong enough to cover a significant (metabolite) concentration space for good identification of the kinetic parameters [22].
- •
- The metabolomics should preferably have a complete coverage of the regarded intracellular metabolic response, e.g., intracellular concentrations, which moreover have to be quantitative.
- •

^{13}C labeled internal standards [24], as they require non-labeled intracellular metabolite pools for quantification. The putative experimental setups therefore either require repetitive (cyclic) conditions or multiple experimental runs to capture the metabolite and

^{13}C enrichment information.

- (1)
- They generate repetitive concentration patterns in time, allowing for dense sampling from multiple cycles as well as application of
^{13}C labeling from a single experiment. - (2)
- The feast/famine perturbation includes both: the transient from limitation to excess as well as a return to limitation in a short timeframe of minutes.
- (3)
- In this setup, the starting metabolite concentrations of each cycle are the same as the endpoint, which means there is no net metabolite accumulation during one cycle (material is washed out with the biomass).

## 2. Results and Discussion

#### 2.1. Computation of an Initial Set of Breakpoints Using the Concentration Measurements

^{13}C labeling aside. If the reproduction of enrichment measurement is not satisfactory, additional domains can be introduced at any time in the optimization workflow. Moreover, a trade-off between the number of parameters to be estimated and the convergence that can be achieved with the PWA flux functions has to be found. Especially, a too high number of breakpoints can significantly increase the tendency of the system towards overfitting.

- (1)
- The number of breakpoints and
- (2)
- The placement of breakpoints.

^{2}and adjusted ${\overline{R}}^{2}$ with reference to the scenario without any free breakpoint (thus only the fixed points t

_{0}= 0 s and t

_{end}= 140 s). Further, the Akaike information criterion (AIC) is calculated and summarized in Table 1. All criteria equally suggest a not yet sufficient reproduction (R

^{2}= 0.77).

**Figure 2.**Approximation of the concentration measurements using piecewise affine derivatives with 0 (blue), 1 (orange), 2 (yellow and magenta) and 3 (green and light blue) free breakpoints. The respective residual sum of squared errors can be found in Table 1.

**Table 1.**Comparison of goodness of fit for different number of breakpoints and placement. In all cases 98 observations (measurements) are present, the number of parameters corresponds to n × p (n = number of fluxes, p = number of breakpoints).

Break Points (s) | #p | RSS | R^{2} | ${\overline{R}}^{2}$ | AIC |
---|---|---|---|---|---|

- | 8 | 579.7 | - | - | 91.7 |

126 | 16 | 134.6 | 0.77 | 0.72 | 45.5 |

25, 131 | 24 | 15.1 | 0.97 | 0.97 | −31.6 |

2, 58 | 24 | 22.2 | 0.96 | 0.95 | −15.2 |

17, 79, 133 | 32 | 2.3 | 1.00 | 0.99 | −95.7 |

1, 14, 83 | 32 | 4.7 | 0.99 | 0.99 | −65.5 |

_{2}= 25 s, t

_{3}= 131 s), a sum of squares of 15.1 (R

^{2}= 0.97) is reached and the concentration measurements can be reproduced within the expected error range of 10%. Addition of another, third free breakpoint further reduces the RSS to 2.3 (R

^{2}= 0.99). Next to these global optima, local minima are observed.

_{2}= 23 s and t

_{3}= 132 s (RSS = 16.9). The territory of approximation error (RSS) has several local minima (data not shown). The second best combination of breakpoints was obtained with t

_{2}= 2 s and t

_{3}= 30 s (RSS 25.3).

^{2}= 0.996) is found for a combination with

**t**= (17 s, 79 s, 133 s), the second cluster has a minimum of 4.6 at t

_{2}= 2 s, t

_{3}= 14 s, t

_{4}= 83 s (Table 1. Comparison of goodness of fit for different number of breakpoints and placement. In all cases 98 observations (measurements) are present, the number of parameters corresponds to n × p (n = number of fluxes, p = number of breakpoints). These local minima differ in sum of squares about two-fold; nevertheless, based on expected experimental noise both should be taken into account and compared when labeling data is incorporated (isotopomer simulation).

_{c}, but likely never the optimal one, subject to the complete set of observables. Still, the domain selection approach often gives a good sequence of breakpoints in practice. As seen, the implementation of the domain selection in the flux estimation leads to a highly non-convex optimization landscape, and this makes incorporation of the domain selection into the main optimization (using the enrichment information) laborious and computationally hardly feasible on standard computer hardware. From a practical standpoint, it is desirable to start with a simple model, as extension of a model is usually more straightforward than a model reduction, moreover, a smaller number of parameters can often be better identified from the observables [20].

^{2}or AIC, can help in finding a minimal set of breakpoints. However, also using those criteria does not necessarily guarantee a sufficient representation of the concentrations, i.e., occurrence of overfitting and no negative concentrations. Having in mind those practical limitations of the domain selection, we implemented constraints in the approach that allow computing flux functions, even with a suboptimal choice of breakpoints and shape-prescriptive constraints.

#### 2.2. Introduction of Shape-Prescriptive Constraints

**t**= (10 s, 79 s, 133 s) is discussed. Two major flaws can be observed (Figure 3): (1) Negative concentrations have been computed for metabolites A

_{ex}and A, and(2) Metabolite A shows overfitting in the last domain. Negative concentrations are undoubtedly not feasible and cannot be accepted; moreover they compromise the numerical integration of the isotopomer balances being detrimental to the numerical robustness of the forward simulation. To enforce non-negativity quadratic inequality, constraints are introduced. This measure prevented all metabolites to reach negative concentrations (Figure 3, green approximation), but does not improve the overfitting behavior observed for metabolite A. Therefore, a shape constraint was introduced enforcing the concentration in the last domain to be monotonous decreasing. The constraint eliminates the overfitting, but leads to higher deviations from the observables in the first and second domain, whereas the fit on A

_{ex}is also improved in the last domain with respect to overfitting. Thus, the example illustrates that the introduction of additional constraints is essential as it ensures a feasible concentration profile for the subsequent

^{13}C-based estimation of fluxes (see Figure 3). Moreover shape constraints are a very efficient measure to prevent overfitting, and this can also help to smoothen noise in the optimization landscape which leads to a better-posed optimization problem. Moreover they can help to achieve a higher convergence of the flux functions towards the real fluxes, as overfitting in the concentration space can be efficiently eliminated. Considering that the flux estimation problem is usually high-dimensional, the constraints also reduce the parameter space that has to be searched, which can lead to an increase in convergence speed. This example was kept simple for demonstration, an extensively constrained network using experimental data can be found in the next subchapter (also refer to Supporting Material for more details on the introduced constraints).

**Figure 3.**Estimated concentration profile using three free breakpoints

**t**= (0s 10s 79s 133s 140s) (

**red**), non-negativity constraints (

**green**) and shape constraints for metabolite A (

**blue**), the corresponding R

_{c}are 4.06 (

**red**) 4.60 (

**green**) and 72.79 (

**blue**).

#### 2.3. Estimating Flux Functions Using the Implicit Filtering Algorithm

**t**= (0 s 18 s 36 s 90 s 185 s 230.5 s 360 s). Because of the dependency of the last and first domain (feast/famine setup), 28 × 6 = 168 flux values need to be estimated.

**Figure 4.**Objective function as a function of executed integrations. Using only central finite differences (cfd), resembling the search directions of a quasi-Newton solver or cfd together with shape constraints (shape) leads to a slow decrease in the residual sum of squares. The use of null-space-based sampling (null) significantly increases the convergence speed. Combining the null-space-based sampling with shape constraints leads to a slight improvement. Even better results are obtained when thresholds for improvement within the current stencil are introduced. Using a threshold of minimally 1% improvement in objective function (acc) to keep the stencil reduces inefficient iterations (green line).

## 3. Materials and Methods

#### 3.1. Used Models and Data

^{−1}. The feed contained minimal medium with a glucose concentration of 15 g/L. After several cycles, repetitive offgas and DO measurements were obtained, a biomass concentration of 5.7 g/L was obtained (average over the cycle). Samples for intracellular metabolites were withdrawn using a rapid-sampling device [27], quenched, extracted and analyzed using the ID-MS protocol [17,28]. After sampling for intracellular concentrations, the feed was switched to a medium with the same composition, but containing fully

^{13}C-labeled glucose. The enrichment was monitored during three consecutive cycles using rapid sampling and MS analysis.

#### 3.2. Mathematical Modeling of Dynamic ^{13}C Labeling Experiments Using PWA Flux Functions

**N**.

#### 3.3. Flux Functions and Nomenclature in the PWA Flux Framework

_{j}valid for all fluxes in the considered metabolic network, partitioning the flux function v

_{i}into number of breakpoints minus one domains in time:

_{i}at the breakpoint t

_{j}and defined strictly positive (so a reversible reaction will have two fluxes). Flux values between breakpoints are calculated by linear interpolation on its two adjacent breakpoints. Therefore a metabolic network with i fluxes and j breakpoints has i × j parameters to be estimated. As will be discussed later, this number can decrease for specific experimental setups that introduce additional constraints on the concentrations or flux functions. This definition is also valid for higher order piecewise-defined flux functions as long as they are to be defined unique on the breakpoints, e.g., smooth quadratic splines, but will not be discussed in the scope of this paper.

#### 3.4. Balancing of Metabolites and Solution of the Metabolite Mass Balances

_{b}and unknown v

_{n}fluxes [30].

_{n}is square and nondegenerate, the matrix can be inverted and a unique solution of unknown fluxes can be identified from the concentration transients and the known fluxes.

**v**, thus $\frac{d\mathbf{c}}{dt}$ also results in a PWA function. Therewith the rhs can be described by a linear function

**u**on the same breakpoints as

**v**:

**c**

_{0}:

_{i,j}; with available concentration measurements, a linear regression can be performed to obtain the best estimate of

**u**. The measurements

**c**

_{M}at timepoints

**t**

_{M}can be obtained by generation of a matrix

**Y**:

**W**

_{M}) then reads:

#### 3.5. Sufficient Breakpoint Selection

^{13}C tracer.) with the inner loop solving Equation (12). and the outer loop iterating on the breakpoints minimizing the weighted residual sum of squares on the concentration observables R

_{c}using the PSwarm optimization algorithm [31] (Figure 5).

**Figure 5.**Overall workflow of the method with breakpoint selection as the first step followed by the main optimization and used optimizers in parentheses. The model inputs are successively expanded as described on the left columns. The first two steps only use the concentration observables, whereas the final step additionally incorporates the enrichment observables from the

^{13}C tracer.

#### 3.6. Introducing Constraints

#### 3.6.1. Specific Constraints for Feast-Famine Conditions

- (1)
- In a stable feast-famine regime the metabolite concentration at the end of one cycle has the same concentration as in the beginning (of the next cycle).
- (2)
- Similarly, the flux at the end of the feast famine cycle has to be the same as in the beginning. Otherwise no stable, repetitive cycles were obtained.

**u**(and c

_{0}). A column vector

**C**is generated, representing a column of matrix

**Y**for the timepoint t

_{end}:

_{1}= v

_{j}. The vector

**C**(and

**b**) are extended by an additional row:

#### 3.6.2. Non-Negativity Constraints

#### 3.6.3. General Shape-Prescriptive Constraints

#### 3.6.4. Monotonicity and Convexity Constraints

**u**at the breakpoints adjacent to the respective domain.

#### 3.6.5. Equality Shape Constraints

#### 3.7. ^{13}C DMFA

^{13}C enrichment observables. The weighted residual sum of squares (R) was chosen, whereas a constant a was introduced as scaling factor to weight the two R

_{i}, which will lead to different points on the respective pareto frontier; it has been set to 1.317 to normalize for the different number of observations in either dataset. This leads to a multi-objective optimization problem and the L

_{2}global criterion was used to define the overall objective function on previously computed aspiration values (best fit on either dataset) for the two residual sum of squares (R

_{c}and R

_{x}).

#### 3.7.1. The Implicit Filtering (Imfil) Optimization Algorithm

#### 3.7.2. Implementation of the Constraints in Implicit Filtering (Imfil)

_{c}has been implemented, and this was typically set to two times the initial R

_{c}and prevents that a numerical integration of flux functions is performed for parameter vectors with a very bad objective function value on the concentrations. It is also useful in the initial phase of the optimization, when R

_{x}dominates the objective function, in order to constrain the redistribution of error into R

_{c}.

#### 3.7.3. Null-Space-Based Sampling

^{13}C labeling measurements to be identified. This parameter space reduction also eliminates the putative trade-off between concentration and enrichment observable residuals in the multi-objective function (i.e., the concentration residual remains equal). In order to find the best overall optimum on all observables the central finite differences for all parameters are sampled together with the null space basis vectors.

**Figure 8.**Sampling space for a network with two fluxes and one measured metabolite (glucose). When using central finite differences the blue points are sampled. Using the null space sampling additionally the two green points are sampled that have the same concentration residuals as the current best iterate and therewith also always fulfill the constraints.

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Symbols and Abbreviations

Subscript | |

0 | Initial condition |

asp | Aspiration value |

e | Metabolite identifier |

end | End of feast famine cycle |

i | index of flux |

j | index of breakpoint |

k | index of domain |

m | Index of measurement |

M | All measurements |

Superscript | |

inp | Enrichment of used substrate for labelling |

b | Known flux |

n | Unknown flux |

f | Number of unknown fluxes |

General | |

a | Scaling factor |

e | Enzyme activity |

R, RSS | (weighted) residual sum of squares |

R^{2} | Coefficient of determination |

θ | Flux function parameters |

α | Kinetic parameters |

c | (metabolite) concentration |

x | (C-molar) enrichment |

v | Flux |

u | Right-hand-side (dc/dt) |

W | Weight matrix |

^ | Optimal estimate |

t | Time |

λ | Lagrange multiplier |

w | Translation vector of the null space |

N | Stoichiometry matrix |

σ | Standard deviation |

imfil | Implicit filtering algorithm |

PWA | Piecewise affine |

acc | Accelerator for imfil |

DMFA | Dynamic metabolic flux analysis |

AIC | Aikaike information criterion |

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**MDPI and ACS Style**

Schumacher, R.; Wahl, S.A.
Effective Estimation of Dynamic Metabolic Fluxes Using ^{13}C Labeling and Piecewise Affine Approximation: From Theory to Practical Applicability. *Metabolites* **2015**, *5*, 697-719.
https://doi.org/10.3390/metabo5040697

**AMA Style**

Schumacher R, Wahl SA.
Effective Estimation of Dynamic Metabolic Fluxes Using ^{13}C Labeling and Piecewise Affine Approximation: From Theory to Practical Applicability. *Metabolites*. 2015; 5(4):697-719.
https://doi.org/10.3390/metabo5040697

**Chicago/Turabian Style**

Schumacher, Robin, and S. Aljoscha Wahl.
2015. "Effective Estimation of Dynamic Metabolic Fluxes Using ^{13}C Labeling and Piecewise Affine Approximation: From Theory to Practical Applicability" *Metabolites* 5, no. 4: 697-719.
https://doi.org/10.3390/metabo5040697