# Negative feedback loop and oscillations

According to the textbook Alberts Molecular Biology of the Cell (5th ed., p. 902), negative feedback loops cause oscillations when they are long delayed. I just can't figure out why.

Except for that, in case of short delayed feedback, the inhibition doesn't seem to be full and to go down only mid-way, according to the attached plot. Why?

For that you would need to understand the dynamical systems theory behind the loop. The point at which the oscillation starts is called the Hopf-bifurcation. I shall explain this in simple intuitive terms. Lets assume that Protein-X activates the production of Protein-Y which in-turn causes inhibition of Protein-X production. This is a negative feedback loop.

X → Y
Y ⊣ X

Situation-1: No/little delay

You switch on the production of X. This leads to rapid production of Y which will in-turn quickly repress the formation of X. This would happen so quickly and all that would be observed is a reduction in the steady-state level of X (compared to an unregulated system). Mathematically, the eigenvalues of the system (jacobian matrix- denotes the slope of the function) evaluated at the steady state would be negative and real.

Situation-2: Some delay

In this case, it will take some time for Y to accumulate. This will cause X to shoot up above its final steady state level. This, in-turn, will cause higher production of Y. Because of higher production of Y, X will start getting repressed strongly when Y gets accumulated and starts acting. This will cause X to shoot down below the steady-state but the magnitude of this shoot-"down", will be less than the first shoot-"up". Finally, the system will settle to the steady state. This phenomenon is called damped oscillation. In this case, the eigenvalues of the jacobian are complex with negative real parts.

Situation 3: long delay

In this case the delay between production of Y and its action is so high that the system does not damp but sustains its oscillations. The eigenvalues of the jacobian are imaginary. These kind of oscillations are generally sensitive to different factors.

More robust forms of oscillations exist, called limit cycles. The network structures responsible for this are more complex than a delayed negative feedback loop. Typically it has a coupled positive and negative feedback. Understanding limit cycles is little bit more complicated and I shall not discuss them here.

Also, check out these reviews:

Tyson, John J., and Béla Novák. "Functional motifs in biochemical reaction networks." Annual Review of Physical Chemistry 61 (2010): 219.

Tyson, John J., Katherine C. Chen, and Béla Novák. "Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell." Current Opinion in Cell Biology 15.2 (2003): 221-231.

and this book:

Alon, Uri. An introduction to systems biology: design principles of biological circuits. CRC press, 2006. ISBN 9781584886426

This was our causal loop.

Fig-1 Our situation

Oscillation is simply an up and down of something, occurring in repeat with time. Now let's see why it causing an oscillation.

Fig 2. Analogy with toilet siphon. C = Cause. E = Effect. E(min)= minimum value of effect . E(max)= maximum value of effect (as much allowed by time delay).

I've compared here this process with a toilet siphon.

Here I've considered the constantly adding up water from tap; is cause. As in your case, Enzyme-E is always present at the left side.

In my case (siphon), the water-level is the effect (In your case it is phosphorylated enzyme (EP), that may present at right side, or may not, if inhibition is high.)

Step-1: starting point: cause is applied

Step-2 : Since inhibition take-place a while latter, your EP adds up (my water level goes up). The dotted inhibition sign on fig2 indicates it yet not reached to the target.

Step-3: When a certain time passed, no-more EP accumulates in your system-of interest. Because inhibition starts. In my case, siphon-effect starts, and water level decrease.

Step 4: Ongoing inhibition decrease the result (in your given-case it is phosphorylation product EP, in my case water siphon it is water level. The siphon start pump out some water.)

Step 5: At next stage, your product EP become so low-concentration, that feedback became feeble, and once stops. In my case, the water level come down below siphon input. so siphon-effect stops.

And Step 6 (Similar to step-1)the process repeats once again.

what will happen if the output from a not gate injected back to its own input

Ring Oscillator

Any Feedbacks are welcome.

One of the conditions for oscillations in a negative feedback system are delays. The question is how much of a delay and what do we even mean by a delay? Imagine a sine wave signal being transmitted along the pathway:

-> x1 -> x2 -> x3 -> x4 -> etc

As the signal moves from one species to the next it will get delayed because it takes time for species to accumulate and empty. Using a sine wave is useful because the delay will be related to shifts in the sine wave itself. It is possible to show that the maximum delay a sine wave can experience at each species is 90 degrees. See figure below. The blue wave is the wave we see at x1 and the red wave is approxiamtely the one we see at x2. Notice it has been pushed to the right 90 degrees.

For the linear pathway above, the delay at x2 will be up to 90 degrees, at x3 another 90 degrees (180 relative to x1) and another 90 degrees at x4 (total of 270 relative to x1). The actual amount of delay will depend on the frequency of the wave, the higher the frequency the more likely it will experience the maximum delay.

Now consider a system where x4 negatively feedbacks to the reaction x1 -> x2.

In a real system we won't be injecting a sine wave but there will be noise and that noise will contain many frequencies including one where the phase shift at x4 will be exactly 180. Note we can't get exactly 180 at x3 because we can only approach 180. Consider now that a noisy signal has been shifted exactly 180 degrees at x4, this signal is transmitted back via the negative feedback to the reaction x1->x2. Guess what the negative feedback does, it shifts the signal by another 180 degrees (eg a 0 turns into a 1 and a 1 turns into a 0, it inverts the signal), therefore the full phase shift is now 360 degrees. This means instead of the dampening down disturbances the negative fedback will actually start to amplify them. The negative feedback has turned into a positive feedback. At this point, the system becomes unstable. The instability doesn't cause things to go to infinity however because of physical contraints and nonlinearities in the system. This explains why it is impossible to get sustained oscillations in a negative fedback with only one step or two steps, oscillations only occurs when you have three steps inside the signal loop because that is the only time you can actually get a 180 phase shift.

The other thing to note is that one other thing needs to be present for oscillations to occur, even if you get a positive feedback. That is the instability has to be amplified, if the instbility is dampend as it goes round the loop nothing will happen. Therefore the other condition, the so-called loop gain, has to be at least one. This can be achieved is there is cooperativity of the signal molecule on the inhibition site.