<figure><img src="image_1.png" /></figure>

**Context:** Bacterial metabolism can be mathematically represented as a large network of chemical reactions. In such networks, propagating perturbations can carry information about environmental perturbations. To quantify this information, we need to study the responses of bacterial components in the presence of noise. Here we study the system response to perturbations following probability distributions.

**Aim:** Mutual information between model variables is used to quantify their dependencies as a form of information transfer. We expect that work to help us understand the signal-processing capacities of bacteria, taking into account internal and environmental noise and uncertainties.

## 1. Metabolic network modeling

We consider a simple linear metabolic network with two internal metabolites and two external metabolites serving as a boundary condition.

$$X_{\text{inp}} \xrightarrow{v_1} A \xrightarrow{v_2} B \xrightarrow{v_3} X_{\text{out}}$$

$$N = \begin{array}{c} \begin{bmatrix} -1 & 0 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix} \\ m \qquad \vec{c} = \begin{bmatrix} [X_{\text{inp}}] \\ [A] \\ [B] \\ [X_{\text{out}}] \end{bmatrix} \\ r \qquad \vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} \\ \vec{p} = \begin{bmatrix} e_1 \\ e_2 \\ e_3 \end{bmatrix} \end{array}$$

The metabolic network structure can be described by a stoichiometric matrix **N**. Each variables of concentration, flux and parameters are grouped into respective vectors **c**, **v** and **p**.

We suppose the system is at a steady state

$$\Leftrightarrow \quad \frac{\partial \vec{c}(t, p)}{\partial t} = N \cdot \vec{v}(\vec{c}, \vec{p}) = \vec{0}$$

The local dependency is represented by the elasticities $\epsilon$, which are the derivatives of a reaction's rate law with respect to a concentration or a parameter.

Normalized elasticities assuming half-saturated enzyme:

$$\bar{\epsilon} = \begin{vmatrix} \bar{\epsilon}_{1,1} & \dots & \bar{\epsilon}_{1,m} \\ \vdots & \ddots & \vdots \\ \bar{\epsilon}_{r,1} & \dots & \bar{\epsilon}_{r,m} \end{vmatrix} = -\frac{1}{2} N^T \quad \text{with} \quad \bar{\epsilon}_{i,j} = \frac{\partial \bar{v}_i}{\partial \bar{c}_j}$$

## 2. Response to parameter noise

In the context of a system at steady state, we use the Metabolic Control Analysis (MCA) framework for a first-order approximation of the global response of the system to small perturbations. The consequences of small perturbations can be described by a linear response matrix **R**.

$$R_p^c = -J^{-1} N \cdot \epsilon_p \\ R_p^v = \epsilon_s \cdot R_p^c + \epsilon_p$$

With the Jacobian **J**

$$\begin{vmatrix} \Delta y_1 \\ \vdots \\ \Delta y_n \end{vmatrix} = \begin{vmatrix} R_{1,1} & \dots & R_{1,p} \\ \vdots & \ddots & \vdots \\ R_{n,1} & \dots & R_{n,p} \end{vmatrix} \cdot \begin{vmatrix} \Delta p_1 \\ \vdots \\ \Delta p_p \end{vmatrix}$$

Then the response of variables to Gaussian-distributed noisy parameters will also be Gaussian-distributed with different standard deviations.

$$\Delta \vec{p} = \begin{vmatrix} \vdots \\ \vdots \\ \vdots \end{vmatrix} \xrightarrow{X_{\text{inp}} \xrightarrow{v_1} A \xrightarrow{v_2} B \xrightarrow{v_3} X_{\text{out}}} \begin{vmatrix} \vdots \\ \vdots \\ \vdots \end{vmatrix} = \Delta \vec{y}$$

## 3. Mutual information of distributions

To characterize the dependency between the distribution of parametric perturbation and the induced variation of the system's variables, we use mutual information (MI) and Shannon entropy to tell us how much knowing a distribution helps us predict the other variable.

<figure><img src="image_2.png" /></figure>

$$I(X;Y) \stackrel{\text{def}}{=} H(X) + H(Y) - H(X;Y)$$

* MI can capture non-linear relationships between variables

* It can help reveal the underlying structures in models by quantifying the dependency between different groups.

* Easy to compute with Gaussian distribution :

$$H(x, y) = n_x \cdot \log_b(\sqrt{2\pi e}) + \log_b\left(\prod_{i}^{n_x} \lambda_i\right)$$

* With continuous distributions, the MI is difficult to interpret and can only be used as a means of comparison

<figure><img src="image_3.png" /></figure>

## 4. Study of mutual information in a linear system with inhibitory regulation

Linear system with an inhibition

$$X_{\text{in}} \xrightarrow{v_1} A \xrightarrow{v_2} B \xrightarrow{v_3} X_{\text{out}}$$

The regulation arrow between [B] and $v_1$ can be described by changing the elasticities of the system.

<figure><img src="image_4.png" /><figcaption>(b) correlation without regulation</figcaption></figure>

<figure><img src="image_5.png" /></figure>

In the case of linear dependence between Gaussian distributions, the correlation can be used to compute the mutual information.

$$I(x_i, y_j) = -\frac{1}{2} \ln (1 - \rho(x_i, y_j)^2)$$

<figure><img src="image_6.png" /><figcaption>(d) correlation with huge inhibition</figcaption></figure>

<figure><img src="image_7.png" /></figure>

### Conclusion and perspectives

We use tools from information theory to study the dependency between parameters and variables. The next step will be applying this approach to a bigger model and implementing other model details like temperature variation, periodic perturbation, and correlated gene expression.

Finally, optimization could be used to find the variables that share the largest mutual information with a biologically relevant flux.