# Metabolic Pathway Analysis in the Presence of Biological Constraints

## Abstract

**:**

## 1. Introduction

#### 1.1. Metabolic Networks

#### 1.2. Steady-State Behavior and Flux Subspace

#### 1.3. Irreversible Reactions, Flux Cones and Polyhedral Cones

#### 1.4. Extreme Vectors and Generating Sets

#### 1.5. Elementary Vectors and Conformal Generating Sets

#### 1.6. Elementary Modes

#### 1.7. Inhomogeneous Linear Constraints and Polyhedra

**0**, and its underlying structure of vector space and we will identify a point in the affine space with the corresponding vector in the vector space). Its dimension is defined as the dimension of its affine span. In this way, a flux polyhedron $FP$ corresponds to a particular matrix $\mathbf{A}\in {\mathbb{R}}^{(2m+{r}_{I}+l)\times r}$ and vector $\mathbf{b}\in {\mathbb{R}}^{2m+{r}_{I}+l}$ given by

#### 1.7.1. Extreme Points and Vectors and Generating Sets

#### 1.7.2. Elementary Points and Vectors and Conformal Generating Sets

#### 1.7.3. Elementary Modes

#### 1.8. Complexity Results

## 2. Metabolic Pathways in the Presence of Biological Constraints

#### 2.1. Biological Constraints

#### 2.1.1. Thermodynamic Constraints

**Lemma**

**1.**

#### 2.1.2. Kinetic Constraints

#### 2.1.3. Regulatory Constraints

**Lemma**

**2.**

#### 2.2. Characterizing the Solution Space

#### 2.2.1. Application to Thermodynamics

**Proposition**

**1.**

**Lemma**

**3.**

**M**and ${O}_{{\mathit{ts}}_{\underline{\mathit{M}}}}$.

**M**and ${\mathbf{ts}}_{\underline{\mathbf{M}}}$, where all unions are finite.

**Proposition**

**2.**

**M**and ${\mathit{ts}}_{\underline{\mathit{M}}}$.

**Gale’s theorem**(a form of Farkas’ lemma). For any $\mathbf{A}\in {\mathbb{R}}^{p\times q}$ and $\mathbf{b}\in {\mathbb{R}}^{p}$, exactly one of the following statements holds:

- (a)
- there exists $\mathbf{y}\in {\mathbb{R}}^{q}$ such that $\mathbf{A}\mathbf{y}<\mathbf{b}$;
- (b)
- there exists $\mathbf{z}\in {\mathbb{R}}^{p}\backslash \{\mathbf{0}\}$ such that $\mathbf{z}\ge \mathbf{0}$, ${\mathbf{z}}^{T}\mathbf{A}=\mathbf{0}$ and ${\mathbf{z}}^{T}\mathbf{b}\le 0$.

**Proposition**

**3.**

**FP**satisfying

**TC**is a finite union of flux polyhedra, obtained as intersections of the flux cones above with the polyhedron defined by $\mathit{G}\mathit{v}\ge \mathit{h}$. All these results hold for constraint

**TC**by just replacing the ${\widehat{K}}_{eq}^{i}$’s by the ${K}_{eq}^{i}$’s.

**Proposition**

**4.**

**FP**satisfying

**TC**

^{b}is a finite union of flux polyhedra, obtained as intersections of the flux cones above with the polyhedron defined by $\mathit{G}\mathit{v}\ge h$.

#### 2.2.2. Application to Kinetics

**Proposition**

**5.**

**h**<

**0**).

**Proposition**

**6.**

#### 2.2.3. Application to Regulatory Constraints

_{&η}for FC(resp., FP

_{&η}for FP), ${CFC}_{{\mathbf{R}\mathbf{C}}_{Bc}\le \eta}$ (resp., ${CFP}_{{\mathbf{R}\mathbf{C}}_{Bc}\le \eta}$) is a disjoint union of open faces of FC

_{&η}(resp., FP

_{&η}). As we grouped together open orthants into semi-open orthants in Lemma 2, we can also group together with such an open face $F$ all those other open faces ${\stackrel{\xb0}{F}}^{\prime}$ in question where ${F}^{\prime}$ is a face of F to obtain thus a (minimal) disjoint union of semi-open polyhedral cones (resp., semi-open polyhedra). Here, we call semi-open polyhedral cone ${C}^{\xb0}$ (resp., semi-open polyhedron ${P}^{\xb0}$ a polyhedral cone C (resp., polyhedron P) without certain (between zero and all) of its faces of lesser dimension, that can be thus expressed as a disjoint union of certain (between all and only $\stackrel{\xb0}{C}$, resp.,$\stackrel{\xb0}{P}$) of the open faces of C (resp., P). We have: $\stackrel{\xb0}{C}\subseteq {C}^{\xb0}\subseteq C$, resp., $\stackrel{\xb0}{P}\subseteq {P}^{\xb0}\subseteq P$).

**Proposition**

**7.**

**Example**

**1.**

**Example**

**2.**

**Proposition**

**8.**

**Example**

**3.**

**Example**

**4.**

**Proposition**

**9.**

**v**i = 0}, in order to determine ${\mathrm{EFMS}}_{{\mathbf{RC}}_{D}}$\${\mathrm{EFMS}}^{{\mathbf{RC}}_{D}}$. Moreover, in practice,

**ILC**is generally only used to bound (below and/or above) fluxes, in which case each extreme ray ${\mathbb{R}}_{+}\mathbf{e}$ of ${FC}_{\le \mathsf{\eta}}$ is still partly present in ${FP}_{\le \mathsf{\eta}}$ as for example an edge [α

^{−},α

^{+}]

**e**, defined by the two vertices α

^{−}

**e**and α

^{+}

**e**. The results above can then be transposed by using convex-conformal decomposition into vertices and conformal decomposition into elementary vectors to characterize ${\mathrm{EFMS}}_{{\mathbf{RC}}_{D}}$\${\mathrm{EFMS}}^{{\mathbf{RC}}_{D}}$.

**Example**

**5.**

- Let $R=\{m\in M\mid {\mathbf{e}}^{m}\in {F}^{\circ}\}=\{m\in M\mid \forall i\in I\phantom{\rule{4.pt}{0ex}}{\mathbf{e}}_{i}^{m}\ne 0\}$. Then EFMs ${}^{{\mathbf{RC}}_{D}}={\{{\mathbf{e}}^{m}\}}_{m\in R}$. Note that R can vary from ∅ to M, thus EFMs ${}^{{\mathbf{RC}}_{D}}$ from ∅ to EFMs $\left(F\right)$. If $R=M$, then EFMs ${}^{{\mathbf{RC}}_{D}}=$ EFMs ${}_{{\mathbf{RC}}_{D}}=$ swNSDFVs ${}_{{\mathbf{RC}}_{D}}=$ EFMs $\left(F\right)$ and the analysis is done (if $\mathsf{\eta}$ has been chosen maximal in $sign\left(CF{C}_{{\mathbf{RC}}_{D}}\right)$, this case corresponds to $I=\varnothing $). We consider the case $R\subset M$ here below.
- Let us consider successively all faces ${F}^{\prime}$ of F of dimension at least two, such that $\stackrel{\u02da}{{F}^{\prime}}\subseteq {F}^{\circ}$, i.e., ${F}^{\prime}$ is not included in any hyperplane $\{{\mathbf{v}}_{i}=0\}$ with $i\in I$ (the lattice of faces of F can be explored for example in a way such that a sub-face is visited before a super-face; once such an ${F}^{\prime}$ has been found, all faces of F containing it are also suitable). Let ${\{{\mathbf{e}}^{k}\}}_{k\in K}$, with $K\subseteq M$, be representatives of the extreme vectors of ${F}^{\prime}$. Thus, $\stackrel{\u02da}{{F}^{\prime}}=con{e}_{\oplus}^{+}\left({\{{\mathbf{e}}^{k}\}}_{k\in K}\right)$ and $I\subseteq {\bigcup}_{k\in K}supp\left({\mathbf{e}}^{k}\right)$. For each of these ${F}^{\prime}$, three exclusive cases can now be distinguished.
- If no facet of ${F}^{\prime}$ has its interior included in ${F}^{\circ}$, i.e., any facet of ${F}^{\prime}$ is included in a certain hyperplane $\{{\mathbf{v}}_{i}=0\}$ with $i\in I$ (a necessary but insufficient condition is $K\subseteq M\backslash R$), then $\stackrel{\u02da}{{F}^{\prime}}\subseteq $ EFMs ${}_{{\mathbf{RC}}_{D}}\backslash $ EFMs ${}^{{\mathbf{RC}}_{D}}$ and ${\bigcup}_{k\in K}supp\left({\mathbf{e}}^{k}\right)$ is a minimal support for vectors in ${F}^{\circ}$.
- If exactly one facet of ${F}^{\prime}$ has its interior included in ${F}^{\circ}$, i.e., it is the only facet not included in any hyperplane $\{{\mathbf{v}}_{i}=0\}$ with $i\in I$, then $\stackrel{\u02da}{{F}^{\prime}}\subseteq $ swNSDFVs ${}_{{\mathbf{RC}}_{D}}\backslash $ EFMs ${}_{{\mathbf{RC}}_{D}}$ and ${\bigcup}_{k\in K}supp\left({\mathbf{e}}^{k}\right)$ is a non-strictly-decomposable non-minimal support for vectors in ${F}^{\circ}$ (non-strictly-decomposable support means that any vector which is a conical sum of the ${\mathbf{e}}^{k}$’s having this support is support-wise non-strictly-decomposable, independently of the choice of the non-negative coefficients fixing the contribution of each ${\mathbf{e}}^{k}$ in the distribution of the fluxes). This result follows immediately from the facts that one facet is not enough to decompose a certain vector in $\stackrel{\u02da}{{F}^{\prime}}$ strictly in terms of supports and that the support of the vectors in the interior of the facet in question is strictly included in the support of the vectors in $\stackrel{\u02da}{{F}^{\prime}}$.
- If at least two facets of ${F}^{\prime}$ have their interior included in ${F}^{\circ}$, i.e., these facets are not included in any hyperplane $\{{\mathbf{v}}_{i}=0\}$ with $i\in I$, then let ${\{{\mathbf{e}}^{l}\}}_{l\in L}$, with $L\subseteq K$, be representatives of the extreme vectors of all these facets (note that we have then necessarily $K\cap R\subseteq L$, i.e., $K\backslash L\subseteq K\backslash R$). Thus, the strict conical sum of the interiors of these facets, which is equal to $con{e}_{\oplus}^{+}\left({\{{\mathbf{e}}^{l}\}}_{l\in L}\right)$, is not empty in $\stackrel{\u02da}{{F}^{\prime}}$ (as there are at least two such facets) and is made up of the support-wise strictly-decomposable vectors of $\stackrel{\u02da}{{F}^{\prime}}$ (by construction): $con{e}_{\oplus}^{+}\left({\{{\mathbf{e}}^{l}\}}_{l\in L}\right)=\stackrel{\u02da}{{F}^{\prime}}\backslash $ swNSDFVs ${}_{{\mathbf{RC}}_{D}}$. Two subcases must therefore be distinguished.
- If $L=K$, i.e., the strict conical sum of the interiors of these facets is equal to $\stackrel{\u02da}{{F}^{\prime}}$, then $\stackrel{\u02da}{{F}^{\prime}}\subseteq {F}^{\circ}\backslash $ swNSDFVs ${}_{{\mathbf{RC}}_{D}}$ and ${\bigcup}_{k\in K}supp\left({\mathbf{e}}^{k}\right)$ is a strictly-decomposable support for vectors in ${F}^{\circ}$ (which means that any vector which is a conical sum of the ${\mathbf{e}}^{k}$’s having this support is support-wise strictly-decomposable, independently of the choice of the non-negative coefficients fixing the contribution of each ${\mathbf{e}}^{k}$ in the distribution of the fluxes).
- If $L\subset K$, then $\stackrel{\u02da}{{F}^{\prime}}$ is split into two nonempty subsets: $con{e}_{\oplus}^{+}\left({\{{\mathbf{e}}^{l}\}}_{l\in L}\right)\subseteq {F}^{\circ}\backslash $ swNSDFVs ${}_{{\mathbf{RC}}_{D}}$ and $\stackrel{\u02da}{{F}^{\prime}}\backslash con{e}_{\oplus}^{+}\left({\{{\mathbf{e}}^{l}\}}_{l\in L}\right)\subseteq $ swNSDFV ${}_{{\mathbf{RC}}_{D}}\backslash $ EFMs ${}_{{\mathbf{RC}}_{D}}$ (note that $\stackrel{\u02da}{{F}^{\prime}}\backslash con{e}_{\oplus}^{+}$$\left({\{{\mathbf{e}}^{l}\}}_{l\in L}\right)$ is made up of the vectors of $con{e}_{\oplus}^{+}\left({\{{\mathbf{e}}^{k}\}}_{k\in K}\right)$ the conical decomposition of which on the ${\mathbf{e}}^{k}$’s requires at least one ${\mathbf{e}}^{k}$ with $k\in K\backslash L$). This means that part of the vectors of $\stackrel{\u02da}{{F}^{\prime}}$ are support-wise non-strictly-decomposable and part are support-wise strictly-decomposable, while having the same non-minimal support ${\bigcup}_{k\in K}$