# Kinetic Theory Modeling and Efficient Numerical Simulation of Gene Regulatory Networks Based on Qualitative Descriptions

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodologies

#### 2.1. Qualitative Modeling: Process Hitting

#### 2.2. Treating Qualitative Systems with Numerical Techniques

_{i}+ 1 possible levels (processes) s

_{i}∈ (0, 1, …, K

_{i}), i = 1, …, n, and a set of m reactions (hits) R

_{j}, j = 1, …, m, with propensity a

_{j}encoding the reaction rate:

_{j,i}controlling the appearance of sort S

_{i}in reaction j. For that purpose, χ is a Boolean variable, i.e., χ

_{j,i}= (0, 1).

**z**= (s

_{1}, …, s

_{n}). Thus, reaction j transforms the state $\widehat{\mathbf{z}}=({s}_{1}^{-},\dots ,{s}_{n}^{-})$ into

**z**, with $\mathbf{z}=\widehat{\mathbf{z}}+{\mathbf{v}}_{j}$, ${\mathbf{v}}_{j}=({s}_{1}^{+}-{s}_{1}^{-},\dots ,{s}_{n}^{+}-{s}_{n}^{-})$.

_{j}depending on the system state and time.

**z**, t), can be written as a sum of a product of separable functions of the interacting species, F

_{i}(s

_{i}), i = 1, … n, and time, F

_{t}(t):

_{1}× … × ω

_{n}, ω

_{i}= (0, 1, …, K

_{i}) and $t\in \mathrm{I}=(0,\mathcal{T}]$. In fact, the domains ω

_{i}in which the species levels are defined being discrete, the integral in ω

_{i}reduces to a discrete sum.

^{p}(

**z**, t) reads:

^{*}(

**z**, t) within a Galerkin framework writes:

^{p}(

**z**, t), the enrichment procedure continues for calculating using the same rationale ${\mathrm{\Psi}}^{p+1}(\mathbf{z},t)={\mathrm{\Psi}}^{p}(\mathbf{z},t)+{F}_{1}^{p+1}({s}_{1})\cdot \dots \cdot {F}_{n}^{p+1}({s}_{n})\cdot {F}_{t}^{p+1}(t)$. The enrichment stops as soon as ‖Ψ

^{Q}− Ψ

^{Q}

^{−1}‖ < ϵ. Alternative goal-oriented stopping criteria exist and were successfully applied in our former works [22].

_{i}= K, ∀i), solved at P time steps, the resulting complexity scales with Q · (n · K) (the complexity related to the solution of the time problem being negligible) instead of K

^{n}involved in an hypothetical mesh or grid that should be solved P times. The separated representation is sketched in Figure 3, which emphasizes the fact that a 3D problem can be solved as a sequence of one-dimensional problems. If, for a while, we consider K = 3 and 100 terms in the finite sum decomposition, Q = 100, the complexity of the PGD solver will scale as 300n, with n the number of species involved in the network, while standard mesh-based discretizations will scale as 3

^{n}. We can notice that for n ≈ 7, both complexities become equivalent, and for n ≈ 15, the one related to the PGD is three orders of magnitude lower than the one involved in standard mesh-based discretizations.

## 3. Application to a Biological Network

#### 3.1. Treating Qualitative Systems with Numerical Techniques

^{20}, or over one million possible states. The underlying probability distribution is a function of these species and time. Our goal is to approximate this solution by a summation of separable functions:

#### 3.2. Incorporation of Unknown Parameters

## 4. Evaluation and Analysis

## 5. Conclusions

## Appendix

## A. Qualitative Modeling: Process Hitting

#### A.1. Generalized Dynamics

#### A.2. Example: Generalized Dynamics of the Incoherent Feed-Forward Loop

_{1}, b

_{0}, c

_{0}) of the generalized dynamics. First b is activated by a. Then, as there is no knowledge of the cooperation between a and b and c, there cannot be any consensus on the value of c. As a result, the value of c oscillates due to the successive independent activations by b and inhibitions by a.

^{3}states, while the process hitting model is made of 3 × 2 actions. This is an important feature of process hitting, since it makes it possible to tackle very large systems in which the number of states grows exponentially with the number of components.

#### A.3. Refining Dynamics with Cooperativity

_{0}bounces to c

_{1}only if a

_{0}and b

_{1}are active. We call such a behavior a cooperativity between sorts a and b to act on c

_{0}. We show now how to interpret such cooperativities in process hitting.

#### A.4. Example (Continued): Refined Dynamics of the Incoherent Feed-Forward Loop

_{0}as a is active. Part of the transition graph is shown in Figure 8 when starting in the state (a

_{1}, b

_{0}, ab

_{00}, c

_{0}). It ends on the fixed point (a

_{1}, b

_{1}, ab

_{11}, c

_{0}). The initial process of the cooperative sort (named ab) has been intentionally chosen incoherent with the state of a and b.

## B. PGD Constructor

#### B.1. Calculation of Functions ${F}_{i}^{p}({s}_{i})$

#### B.2. Calculation of Function ${F}_{t}^{p}(t)$

_{1}×⋯ × ω

_{n}, it results:

## C. The ErbB Signaling Pathway

## Acknowledgments

## Author Contributions

## Conflicts of Interest

**PACS classifications:**02.60.-x; 02.50.Ga; 87.18.-h

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**Figure 1.**Creating a process hitting action. In gene regulation, we consider two kinds of interactions between species: activation and inhibition. If a is an activator of b, it is common to represent this by a signed, directed graph (left). These interactions have a characteristic form: unlike kinetic reactions, activation and inhibition usually depend on the regulator passing some threshold concentration in order to become effective (middle). Process hitting (right) represents these reactions via actions: a activates b becomes a

_{1}hits (solid arrow) b

_{0}to bounce (dashed arc) to b

_{1}. Generalized dynamics attempts to create the most permissive dynamics possible for the directed graph. Therefore, the absence of a effectively acts as an inhibitor, adding the action a

_{0}hits b

_{1}to bounce to b

_{0}. Every action can be associated with temporal and stochastic parameters; the reaction rate, for example [14].

**Figure 2.**Refinement of a model via cooperative sorts. Here, a is an activator of c, while b inhibits c. The generalized dynamics of the system have been constructed on the left. However, what should happen in the case that both and are present? According to the left-hand model, the system will oscillate. If we know more about how the system should function, however, we would like to be able to include this information in our model. With general dynamics, we are unable to express logical gates in which multiple species exhibit deterministic combined effects on a target, such as a ∧ −b, or the presence of the activator without the presence of the inhibitor. In order to add this combined interaction and eliminate the oscillatory behavior, we must refine the process hitting model with a cooperative sort, ab. This sort will handle the interactions of a and b on c, while leaving the original species to interact with other elements as before. In exchange, more actions must be added, such that a and b can effectively update ab, so that it truly reflects the current state of both elements. In our example, ab

_{1,1}will not interact with c

_{0}; thus, c remains inactive.

**Figure 3.**Decomposition of a state space. This illustration shows how a multidimensional space, for example a cubic space of three dimensions involving 3

^{3}degrees of freedom, can be decomposed into the product of the individual dimensions, 3 · 3. This mathematical property is exploited by PGD in that we search for the individual vectors, which are of a relatively small size, never touching the full state space. In such a way, we move from a complexity of K

^{n}to K · n.

**Figure 4.**Sample of the results from incorporating model parameters as dimensions in the PGD solution. Here, we have selected three potential values for the firing rate r: r/2 (left), 3r/2 (middle) and 2r (right), for any interaction involving the proteins p21 or p27. The resulting behaviors of five proteins along the chemical pathway are shown here. Since the system is binary, active expression is plotted in yellow and inactive in blue, following their probabilities on the y-axis. Notice that in all cases, behavior is equivalent until around 1.75 seconds, when the firing cascade reaches p21 and p27. Some signals are simply amplified, whereas others, such as p21 and pRB, develop more complicated behaviors as the firing rate increases, perhaps suggesting cyclical or dampening behaviors.

**Figure 6.**(

**a**) Generalized Boolean dynamics of the incoherent feed-forward loop in process hitting. (

**b**) Possible transitions from the state (a

_{1}, b

_{0}, c

_{0}).

**Figure 7.**Refined process hitting encoding the dynamics of the incoherent feed-forward loop where the activation of c needs both a inactive and b active. The process 01 of the sort ¬a ∧ b mirrors the (only) state where c

_{0}should be hit. Grayed processes indicate the initial state for the transition graph in Figure 8.

**Figure 8.**Transition graph of the process hitting in Figure 7 from the state represented by grayed processes.

**Figure 10.**The interaction graph for ErbB-mediated G1/S cell cycle transition. Here, elements directly related to the ErbB signaling portion of the network are represented by boxes, while the elements related to kinase activity are represented by circles. Activation interactions are shown in green arrows and inhibition in red blunted arrows. Since this is the initial, most basic network derived from the literature, no combined effects requiring Boolean logic gates are shown.

**Table 1.**Results for ErbB models using generalized dynamics and a refinement with cooperative sorts. Here, the two models were tested using three sanity checks related to our biological understanding of the system: the presence of fixed points, the lack of impossible behaviors and the presence of demonstrated behaviors. In order to be considered a functioning model, pRB should remain at rest when the system is universally inactive, including the absence of input protein EGF. However, in the presence of EGF, a signal should be able to propagate through the system, potentially activating pRB. We see that, while the generalized dynamics were able to propagate a signal from EGF to pRB (EGF present), it was not able to prevent sporadic activation of pRB in a system at rest (EGF absent), nor find any fixed points.

Model | Fixed points | EGF absent | EGF present |
---|---|---|---|

Gen. Dynam. | 0 | Fail | Pass |

Refined | 3 | Pass | Pass |

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**MDPI and ACS Style**

Chinesta, F.; Magnin, M.; Roux, O.; Ammar, A.; Cueto, E. Kinetic Theory Modeling and Efficient Numerical Simulation of Gene Regulatory Networks Based on Qualitative Descriptions. *Entropy* **2015**, *17*, 1896-1915.
https://doi.org/10.3390/e17041896

**AMA Style**

Chinesta F, Magnin M, Roux O, Ammar A, Cueto E. Kinetic Theory Modeling and Efficient Numerical Simulation of Gene Regulatory Networks Based on Qualitative Descriptions. *Entropy*. 2015; 17(4):1896-1915.
https://doi.org/10.3390/e17041896

**Chicago/Turabian Style**

Chinesta, Francisco, Morgan Magnin, Olivier Roux, Amine Ammar, and Elias Cueto. 2015. "Kinetic Theory Modeling and Efficient Numerical Simulation of Gene Regulatory Networks Based on Qualitative Descriptions" *Entropy* 17, no. 4: 1896-1915.
https://doi.org/10.3390/e17041896