# Design Principles as a Guide for Constraint Based and Dynamic Modeling: Towards an Integrative Workflow

^{*}

## Abstract

**:**

## 1. Introduction

#### The Chess Metaphor

**Figure 1.**A sequential workflow from metabolic reconstructions to dynamic models. Each of the three techniques featured here incorporates different kinds of information into the process, this information can be of two kinds: different types of data, shown in the upper part of the figure, and evolutionary considerations at different levels, shown in the lower part of the figure. Furthermore, the results of each step are fed to the next. The flux distributions provided by FBA (Flux Balance Analysis) [4,5] can be used as an input to TFA (Thermodynamic Feasibility Analysis) [6] and the results of both approaches can be translated to parameters for a dynamic model, which can be formulated and analyzed accoding to BST (Biochemical Systems Theory) [7,8,9] and MCC (mathematically controlled comparison) [47].

## 2. Established Methods

#### 2.1. Flux Centric Approaches: Constraining the Flux Space

**Figure 2.**Results of applying the two optimization programs discussed in the text on a genome scale model of E. coli. Solving Equation (2) for increasing upper limits for glucose uptake (${V}_{in}$) yields solutions ascending along the blue curve. Solving Equation (3) for decreasing lower limits on the growth rate (${V}_{bio}$) results in solutions descending along the same curve. All optimizations were performed using COBRA for python version 0.3.2 and the E. coli model iJO1366 included with the software. All fluxes were optimized with their default limits except EX_o2_e, for which a lower bound of −10.0 was set.

#### 2.2. Thermodynamics: The Bridge to Metabolites

**Figure 3.**The lines represent the $\Delta G$ of each reaction. The area below both curves (shaded) covers all possible values for B. The maximal possible value of B is obviously the intersection between the two lines.

#### 2.3. Catalytic Efficiency of Enzymes

#### 2.3.1. Theoretical Limits and Some Reference Values

**Figure 4.**When $\Delta G$ is similar enough to $R\phantom{\rule{0.166667em}{0ex}}T$ (typically assumed to be 2.48 kJ/mol for biochemical reactions), the thermodynamic factor, $1-\theta $, decreases rapidly, since $\theta =exp\left(\frac{\Delta G}{R\phantom{\rule{0.166667em}{0ex}}T}\right)$. $\Delta {G}_{95\%}$ and $\Delta {G}_{99\%}$ are the affinities at which the enzyme can operate at 95% and 99% of its ${V}_{max}$ respectively.

**Figure 5.**(

**a**) Distribution of enzyme abundance in E. coli in copies/cell [40]; (

**b**) Values of ${k}_{cat}$ for different enzymes. Bimodal distribution with modes 2.5 and 230 ${s}^{-1}$; (

**c**) Values of ${k}_{cat}$ vs. copies per cell for different enzymes. In logarithmic coordinates, the product ${k}_{cat}\phantom{\rule{0.166667em}{0ex}}{E}_{T}$, is a straight line, so a red line has been added as a reference. All enzymes falling in a line parallel to it have the same ${V}_{max}$; (

**d**) Distribution of ${V}_{max}$. Bimodal distribution with modes approx 1 and 75 μM/s. Some histograms are complemented by Gaussian kernel density estimations performed using Python’s package Scipy.

#### 2.4. Adding Regulation to Obtain a Dynamic Model

**Figure 6.**Thermodynamic contribution, $\frac{\theta}{1-\theta}$, to the kinetic order for different values of $\Delta G$. The thermodynamic contribution is plotted in logarithmic coordinates. To simplify reading the curve, the area, where the thermodynamic contribution is between 0.1 and 1 is grayed. In this area, the thermodynamic contribution is comparable to the kinetic contribution, ${g}_{i,j}^{k}$ in Equation (12) for hyperbolic kinetics like MM, which varies between 0 and 1. Allosteric enzymes have kinetic contributions of magnitudes between 0 and 4, so an extra area for thermodynamic contributions between 1 and 4 has been marked.

#### 2.5. Mathematically Controlled Comparison (MCC)

**Internal equivalence**The reference and alternative system belong to different classes due to differences in one or more reactions/processes—e.g., the first reaction of the reference system is inhibited by the end product of the pathway, while the alternative system does not have this feedback loop. Since those distinctive reactions have been modified from one system to another, their parameters may differ. We will keep the notation, where, if p is a parameter or a property of the reference system, then $\widehat{p}$ is the same parameter or property in the alternative system. For all the parameters not involved in distinctive reactions, internal equivalence is satisfied by making the parameters of every other reaction equal in both systems $p=\widehat{p}$.

**External equivalence**Two systems satisfying the internal equivalence condition will have identical values for most parameters except for the handful involved in processes distinctive of their class. These degrees of freedom are further reduced by ensuring that both systems are perceived by their environments as being as similar as possible. That eliminates differences that are not inherent of the class but characteristic of some particular cases. Let’s imagine a pathway that has an optimal flux to provide a certain precursor for biomass. Introducing a feedback inhibition will decrease the flux through the whole pathway, making it suboptimal. Does it mean that the feedback is deletereous for the cell? Obviously not. The system with feedback is perfectly capable to supply the precursor in the same amounts, but that will require an increase in the activity of the inhibited enzyme. Only after setting the fluxes equal in both systems, can the comparison between the two alternative architectures be considered fair. An additional boon of this approach is that the degrees of freedom for the choices of parameter values is reduced, so the free parameters of the alternative system become a function of those of the reference system, and comparisons are made practically on a one to one basis. Typical requirements for external equivalence would be that both systems carry the same fluxes, have the same concentrations of initial and end product in the steady state, or that they have as many identical logarithmic gains as possible.

## 3. Results

**Thermodynamic shortening**As has been shown above, as a reaction approaches equilibrium, its rate becomes hypersensitive to perturbations of substrates and products, their kinetic orders tending to infinite magnitude in the sense that counters the perturbation, positive for substrate and negative for product. Also, by definition, the mass action quotient will tend to the value of the equilibrium constant. This enables to reformulate the model in a reduced form that will, nevertheless, be consistent with the behavior of the full metabolic network, as has been established using perturbation theory [51].

#### 3.1. Case Study 1: Ammonia Assimilation

_{3}to α-ketoglutaric acid by Glutamate Synthase (GOGAT). The sum of these two reactions are equivalent to the action of GDH plus the hydrolysis of one ATP. Viewing this small network (see Figure 7) from a stoichiometric perspective, would show two Elementary Flux Modes, one for GDH and another for GS/GOGAT. Optimizing a bigger network using FBA will often result in GDH as the optimal path to carry the flux, since that would free ATP for other pathways. It is however well known that GS/GOGAT is the preferred pathway for nitrogen assimilation with the GS/GOGAT system being very strictly controlled at many levels. In fact, the whole set of regulatory processes around this simple set of reactions is extremely complex, see [52] for a thorough review. However, focusing our attention in the biochemistry can help us understand much about the constraints that have conditioned the evolution of the system. Complementing FBA with additional analysis presented above and interpreting the result from the point of view of “design principles” will already clarify things significantly.

_{3}available, as we’ll see below.

**Figure 8.**The lines represent the $\Delta G$ of each reaction at different concentrations of glutamine for a fixed NH

_{3}= 1 mM. Concentrations of other metabolites taken from the literature (see supplementary information), except α-ketoglutaric acid that was set as high as the default boundary permitted. Red GDH, green GS and blue GOGAT.

_{3}, while all other variables were kept equal to those in Figure 8. Figure 9 shows the resulting $\Delta G$ for each reaction. For low concentrations of nitrogen, GDH is thermodynamically unable to carry any flux and its driving force remains below that of GS and GOGAT. The area marked in grey shows scenarios, where the enzyme can operate but, due to its proximity to equilibrium, will need more enzyme to carry the flux, than each of the other reactions. Even if all enzymes had the same ${V}_{max}$, a GDH operating under $\Delta {G}_{50\%}$—dark grey area—would have to be present in a concentration higher than the sum of the other two enzymes to carry the same flux.

**Figure 9.**Comparison of $\Delta G$ of GDH and GS/GOGAT applying the Max-min Driving Force principle so that both GS and GOGAT have the same $\Delta G$. The greyed areas mark values, where $\Delta G$ is similar enough to $R\phantom{\rule{0.166667em}{0ex}}T$ that the thermodynamic factor $1-\theta $ decreases rapidly. The darker area marks $\Delta {G}_{50\%}$, and the lighter $\Delta {G}_{95\%}$.

#### 3.2. Case Study 2: Thermodynamic Shortening of an Unbranched Pathway

**Reactions near equilibrium**Thermodynamic analysis has established that reactions operating close to equilibrium originate a higher cost in terms of enzyme. Now, the question remains: can they also provide any advantage?

**Equivalence conditions**Due to internal equivalence, only the parameters involved in the second reaction can change with θ, which leaves four degrees of freedom: θ, ${g}_{1,2}^{k}$, ${g}_{1,2}^{k}$ and ${\alpha}_{2}$.

**Systemic properties**Once the equivalence conditions have been set, the behavior of the pathway, when a reaction approaches equilibrium, can be checked analytically by taking limits of the systemic properties, when $\theta \to {1}^{-}$. As a reaction approaches equilibrium, the kinetic orders for both substrate and product tend to plus and minus infinity respectively, attesting for an extremely fast response to any deviation. This hypersensitivity of the reaction comes together with a desensitization of the pathways to the reaction (Logarithmic gains/Control Coefficients tend to zero [24]),

**Thermodynamic shortening**It has been shown that pathways with feedback inhibition tend to have a narrower margin of stability—right hand side of Equation (13)—as they grow longer [3], see also supplementary information. Short pathways can have really strong feedback loops, while longer ones must keep the strength of the inhibition signal weaker or risk becoming unstable. The stability of a long pathway is improved, when the kinetic parameters of its enzymes are distributed along a wide interval, such that each enzyme has very different kinetics. This phenomenon, called kinetic shortening, has been observed in long aminoacid synthesis pathways and could be considered a design principle in its own right. Now we will show how bringing some reactions close to equilibrium can have a similar effect that, by analogy, we will call thermodynamic shortening.

#### 3.3. Case Study 3: Two Alternative Designs for an Unbranched Pathway

**Figure 10.**The circles in (

**a**,

**c**) schematically show the reduction of enzyme activity due to different causes (graphical depiction of the η terms in Equation (6)). In white, the fraction of the enzyme that is actually catalyzing the reaction forward. In grey the fraction that is inactive due to allosteric inhibition (or more exactly the fraction in which the total activity is reduced). In black, the fraction of activity that is lost due to the flux of the reverse reaction or due to insaturation of the enzyme; (

**b**) Shows the two alternative pathways in performance space and the trade-off between them; (

**c**) A more detailed depiction of the two modes of operation: The economic variant carries a high flux and all its enzymes are operating close to their ${V}_{max}$ due to weak inhibition and distance to equilibrium. In the responsive variant, the first enzyme is inhibited and two of the reactions are close to equilibrium. The efficiency of the enzymes is much lower but so is the flux they have to carry.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

## A. Supplementary Information: The Unbranched Pathway

**Effect of reversible kinetics in the unbranched pathway**Alves and Savageau [47] analyzed the effect of reversible vs. fully irreversible kinetics in unbranched pathway models like Equation (25). Let’s define our reference system as a a fully irreversible version of Equation (25). In such case, ${g}_{2,2}={g}_{3,3}=0$. The alternative system will differ in the second reaction, which, still being far away from equilibrium, is affected by its own product, $P\simeq {K}_{P}\Rightarrow {\widehat{g}}_{2,2}\ne 0$. Since the characteristic difference between both systems lies in the second reaction, internal equivalence establishes that the parameters of all other reactions are equal: ${\alpha}_{i}={\widehat{\alpha}}_{i}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}i\ne 2$ and ${g}_{i,j}={\widehat{g}}_{i,j}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}i\ne 2$. External equivalence, on the other hand, helps us find values for the three parameters that are different in the alternative model: ${\widehat{g}}_{2,1}$, ${\widehat{g}}_{2,2}$ and ${\widehat{\alpha}}_{2}$. By imposing that both pathways have the same concentrations of ${x}_{1}$, ${x}_{2}$, and ${x}_{3}$ in the steady state and that both have the same response to an increase in supply, it follows that

## B. Supplementary Information: Ammonia Assimilation

**metabolites**glutamate 100 mM α-ketoglutaric acid 1 mM

**cofactors**[6] ATP = 7.9E-3 ADP = 1.04E-3 Pi = 7.9

_{3}as an independent variable ${X}_{0}$ and assume that the ratios of cofactors are fixed. The dependent variables will be α - ketoglutaric acid ${X}_{1}$, glutamine ${X}_{2}$ , glutamate ${X}_{3}$. We will also use the classical ${y}_{i}=ln{X}_{i}$

**Glutamate dehydrogenase (GDH)**

_{3 <}= > NADP

^{+}+ glutamate + H

_{2}O

**Glutamine synthase (GS)**

_{3 <}= > ADP + glutamine + P

_{i}

**Glutamate synthase (GOGAT)**

_{<}= > NADP

^{+}+ 2 glutamate

**ATP hydrolisis**

_{2}O

_{<}= > ADP + P

_{i}

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**MDPI and ACS Style**

Sehr, C.; Kremling, A.; Marin-Sanguino, A.
Design Principles as a Guide for Constraint Based and Dynamic Modeling: Towards an Integrative Workflow. *Metabolites* **2015**, *5*, 601-635.
https://doi.org/10.3390/metabo5040601

**AMA Style**

Sehr C, Kremling A, Marin-Sanguino A.
Design Principles as a Guide for Constraint Based and Dynamic Modeling: Towards an Integrative Workflow. *Metabolites*. 2015; 5(4):601-635.
https://doi.org/10.3390/metabo5040601

**Chicago/Turabian Style**

Sehr, Christiana, Andreas Kremling, and Alberto Marin-Sanguino.
2015. "Design Principles as a Guide for Constraint Based and Dynamic Modeling: Towards an Integrative Workflow" *Metabolites* 5, no. 4: 601-635.
https://doi.org/10.3390/metabo5040601