# Feed Forward Loop

### From bio-physics-wiki

For a three node pattern there exitst 13 possible ways to create a directed (with arrows of definite direction) subgraph.

Similar calculations as for negative autoregulation (NAR) show, that one of this three node patterns occurs much more often in the *E. coli* network that expected at random. This pattern numbered with $5$ is called feed forward loop (FFL). *E. coli* contains 42 feed forward loops, while the expected number is less than one, thus the FFL is a network motif.

Depending on the sign of the arrows the feed forward loop can be classified into $8$ different types. Each arrow can be positive or negative (zwo possibilities) and there are three arrows this gives $2^3$ possibilities to make a FFL with different signs. This $8$ FFLs are eather **coherent** (when both incoming arrows of $Z$ have a positive sign) or **incoherent** (incoming arrows of $Z$ have different signs).

The input function of $Z$ is two-dimensional. In the logic approximation it can eather be an **AND** or an **OR** input funciton. It is common to use a symbolic notation like in electronic engineering and boolean logic.

The following four types of **Coherent Feed Forward Loops** [C#FFL] exist

The following four types of **Incoherent Feed Forward Loops** [I#FFL] exist

The most abundant FFLs in *E. coli* are C1FFL and I1FFL, $90\%$ of the *E. coli* network are of these types.

## C1FFL with AND input function

We already know the dynamics of simple regulation, which leads to a time course \begin{equation} Y(t)=Y_{st}(1-e^{-\alpha t}). \end{equation} We use this result to understand the dynamics of the feed forward loop with an AND input function. Let us now study the response to an ON-step. We assume that protein $X$ is present in its inactive form. Suddenly the signal $S_x$ appears and the protein is transformed in its active form $X \rightarrow X^*$. This leads to immediate production of protein $Y$ whose signal $S_y$ we assume present, so that $Y$ is active ($Y^*$).

Since, $Z$ has an AND input function, $Z$ needs both $X^*$ AND $Y^*$ to be present, before it starts producing $Z$, thus $Z$ is only expressed when $Y*$ exceeds a certain threshold $K_yz$ (threshold of Y to produce Z).

**delayed response**: the C1FFL-AND motif leads to a delayed production of $Z$ in response to the signal $S_x$

The time delay of production of protein $Z$ equals the time $Y$ needs to reach the activation threashold. \begin{equation} Y_{st}(1-e^{-\alpha \cdot t_K})=K_{yz}\\ e^{-\alpha \cdot t_K}=1-\frac{K_{yz}}{Y_{st}}\\ t_K=\frac{1}{\alpha} \ln \left(\frac{1}{1-\frac{K_{yz}}{Y_{st}}} \right) \end{equation}

If the signal $S_x$ is present for a time period smaller than $t_K$, $Y$ will not exceed the threshold $K_{yz}$ and $Z$ will not be produced.

The C1FFL-AND motif acts as filter for signals $S_x$ of a short time period $t < t_K$

An OFF-step occurs, when the signal $S_x$ suddenly disappears. Since, both $X^*$ AND $Y^*$ are needed to activate $Z$ production, the production stops immediately after $X^*$ disappears.

## C1FFL with OR input function

For an ON-step, where the signal $S_x$ appears, $Y^*$ starts beeing produced as well as $Z^*$, becaus it is enough that eather $X^*$ OR $Y^*$ is present.

The interessting part of this motif is is the OFF-step. When $S_x$ disappears $Y$ immediately stops beeing produced, but $Z$ is still produced until $Y$ reaches the threashold $K_yz$, then also $Z$ stops producing.

**delayed response**: the C1FFL-OR motif leads to a delay of production stop of $Z$ following an OFF-step of $S_x$

**Summary**

C1FFLs are **sign sensitive delay elements**. With a C1FFL-AND element brief fluctuations of ON-pulses can be filtered away, while with C1FFL-OR element brief OFF-pulses can be filtered away.

## I1FFL with AND input function

Let us now study the incoherent type one feed forward loop with an AND input function. Notice, that $Z$ is transcribed if transcription factor $X^*$ AND NOT $Y^*$ is present. For an ON-step $X^*$ is immediately active and produces $Y$. $Z$ is also produced, since $X^*$ AND NOT $Y^*$ is present. When $Y^*$ exceeds the threshold $K_{yz}$, $Z$ stops beeing produced and decays exponentially to reach the steady state $Z_{st}$.

As $S_x$ appears $Y$ is produced according to simple regulation and existis in its active form as we assume the signal $S_y$ present.
\begin{equation}
Y(t)=Y_{st}(1-e^{-\alpha_Y t}).
\end{equation}
$Z$ is also produced instantaneously and reaches a maximal level at $Z_m=\beta_Z/\alpha_Z$.
\begin{equation}
Z(t)=Z_m(1-e^{-\alpha_Z t}).
\end{equation}
The time span until $Z$ is repressed, equals the time $Y$ needs to reach the repression threashold.
\begin{equation}
Y_{st}(1-e^{-\alpha \cdot t_R})=K_{yz}\\
e^{-\alpha \cdot t_R}=1-\frac{K_{yz}}{Y_{st}}\\
t_R=\frac{1}{\alpha} \ln \left(\frac{1}{1-\frac{K_{yz}}{Y_{st}}} \right)
\end{equation}
After the repression thershold is reached the $Z$ expression level decays to a the steady state $Z_{st}=\beta_Z '/\alpha_Z$
\begin{equation}
Z(t)=Z_{st}+(Z_0-Z_{st})(1-e^{-\alpha t}).
\end{equation}
We now introduce the new concept of the **repression factor**
\begin{equation}
F=Z_m/Z{st}=\beta_Z/\beta '_Z
\end{equation}
For a hight repression factor $F$ the steady state $Z_{st}$ concentration is much lower than $Z_m$. For lower and lower $F$ the difference in concentration of $Z_m$ and $Z_{st}$ gets lower and lower. Hence, with high repression factors a pulse can be generated, while for low repression factors the response time can be shortened.

The I1FFL-AND motif can act as a pulse generator or speed the response time

Video Lecture:

- Uri Alon - Systems Biology Lecture 3

Further reading:

- Uri Alon - An Introduction to Systems Biology: Design Principles of Biological Circuits (Link)