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As an example, we could talk about a neuron cell in brain, with a size of $1 mu m$ (=$10^{-6}$ m), being the distance between one neuron and next one in a synaptic connection of around 40 nm (=$40 cdot 10^{-9}$ m) (reference).

According to my information, atomic radius are around 100 pm ($10^{-10}$m), not very far of the synapse size (factor of 400).

Thus, my question is, could quantum mechanics be necessary to analyze biological scenarios, such as neuron cell interaction, etc ?

I've read several examples about why take into account quantum effects, as Heisenberg's uncertainty principle, is "useless" at the scale of usual objects (say a rocket), but what about at the scale of cells ?

It should be said that a few years ago (around 2007 I believe) there has been some fuzz in the physics community after some researchers found (some) evidence of quantum behavior in biological systems. Most notably some bacteria. In one of these experiments quantum effects (at ambient temperature!) were observed in the FMO complex and involved say, coherent assisted transport of excitations.
I don't think the results are disputed but I believe the consensus nowadays is that, in a way, those measurements are so precise that after all is not that surprising if a (tiny) effect becomes observables.

There were other biological systems where quantum effects were predicted or observed (avian compass is another one, and even a model for sensing odor) but these were more controversial.

I will add some references if you are interested. Googling FMO complex or quantum-biology should give you plenty of hits.

Added Edit

In fact there is even a Wikipedia page which is quite explanatory

https://en.m.wikipedia.org/wiki/Quantum_biology

So the short answer is that we don't 100% know but most physicists do not think so.

The reason that they do not think so comes down to two things: Ehrenfest’s theorem and decoherence.

Ehrenfest’s theorem is a bound on how weird quantum mechanics can be. It says that on average quantum mechanics is not weird: particular measurement outcomes get correlated in weird ways but the average picture looks always like classical mechanics would say it looks.

Decoherence says that quantum things start to average out as soon as they get entangled with some broader outside world. So for example a protein folding in water is constantly entangling with those water molecules which constantly entangle with each other, and so the interesting correlations cannot be measured on the protein itself anymore but we would have to involve all of the water molecules too.

Biological structures that would display quantum features would therefore have to create a safe, non-interacting space for a quantum state to be preserved. This is why the slightly cooky among us like Penrose start from examples like cytoskeleton tubules: they are looking for the tiny little spaces that are walled off from the rest of the world. It is also why smart non-physicists like Searle are very careful to say something like “look I just want to import the bulk features of our quantum realm like nondeterminism but then explain things as classical physics+nondeterminism rather than getting super cooky for quantum mechanics,” it's not that it's wrong to say that it's a quantum system because undoubtedly it is, everything is—it's just that it probably has a very good classical approximation because it couples strongly with all of the noisy things around it.

Quantum does not really mean “small” and we have created tests of QM spanning kilometers. It just requires “isolated” things, and small nanoscale systems and single atoms happen to be isolated from their surroundings more often than big things like baseballs flying through the many air atoms knocking them all out of the way.

I'll discuss two controversial "quantum mechanics explains it" issues in biophysics.

A biophysical explanation of olfaction remains incomplete. It mostly centres on two models, neither of which can explain all data, but it's possible olfaction uses a combination of both effects (and possibly also something else). One model, the docking theory, is preferred; it relies on how molecules interact through shape and chemistry. The other, the vibrational, theory, depends on quantum tunnelling.

Orchestrated objective reduction posits that consciousness relies on quantum effects in neurons. This is at odds with the usual view that connections between neurons are responsible. However, physicists as eminent as Roger Penrose have worked on and championed Orch OR, which is why I'm risking it being mainstream enough for inclusion in an answer here despite our policies. Penrose conjectures that superpositions form spacetime "blisters" that undergo OR in a time $hbar/E_G$, with $E_G$ the blister's gravitational self-energy. A radius-$R$ density-$rho$ neuron has mass $M=frac{4pirho R^3}{3}$, GPE $E_G=frac{3GM^2}{5R}=frac{16pi^2 Grho^2 R^5}{15}$ and OR timescale $frac{15hbar}{16pi^2 Grho^2 R^5}$. For $rho =10^3text{kg},text{m}^{-3},,R=10^{-5}text{m}$ (if you'll pardon such approximations of a neuron) this is $1.5mutext{s}$. Take any such number with a pinch of salt, though, because neurons vary in size.