Could quantum mechanics be necessary to analyze some biology scenarios? 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 ?
 A: 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 $M=\frac{4\pi\rho R^3}{3}$, GPE $E_G=\frac{3GM^2}{5R}=\frac{16\pi^2 G\rho^2 R^5}{15}$ and OR timescale $\frac{15\hbar}{16\pi^2 G\rho^2 R^5}$. For $\rho =10^3\text{kg}\,\text{m}^{-3},\,R=10^{-5}\text{m}$ (if you'll pardon such approximations of a neuron) this is $1.5\mu\text{s}$. Take any such number with a pinch of salt, though, because neurons vary in size.
A: I can't pull a simple quote from this Physics World article, but it has a pretty decent history of the discoveries and analyses which may or may not demonstrate quantum effects in the photosynthesis to energy storage process in plants.   
My take is that it hasn't definitely been disproved or proved just yet.  
A: Physics of RNA and protein folding is based in quantum mechanical description, as it deals with the nature of different chemical bonds. The celebrated Nussinov algorithm and its more modern incarnations are essentially - a more complicated version of DNA zipping problem, routinely studied in statistical physics. It is however fair to say, that it involves a lot more statistical physics and electrostatics than QM.
Less direct from the physical point of view, but more involved mathematically is the application of matrix field theory to classifying RNA structures by Henri Orland.
An unrelated example is studying the proton transport in biochemical reaction chains.
A: 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
A: 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.
Note that the actual physical size does not matter at all to QM: 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.
When you combine those two together you get a result that once a system is immersed in constant interactions with an environment, quantum mechanics only has two sorts of effects:


*

*the system carves out a space inside of it which is isolated from the environment, and arbitrary quantum stuff happens in that space, or

*the system displays some big features of a bunch of little quantum "nudges" to the classical picture -- something doesn't happen in quite the way that you would have expected for example.


So for example the pigments that plants use to convert light into chemical energy only absorb certain wavelengths of light, and this is a little quantum nudge (quantum systems frequently have discrete energy transitions and preferentially absorb photons that have an energy between the two states), and there is a quantum "stickiness" that molecules have towards each other called the van der Waals interaction that is crucial for understanding lots of different chemistry.
Biological structures that would display deeply quantum features would therefore generally 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 quantum computation in cells and so they are very interested in 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,” he wants to use the bulk features that come from a lot of little nudges rather than make the appeal Penrose is making that somehow the brain is a quantum computer because its cells are quantum computers.
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 one might expect synapses for example to probably have a very good classical approximation with maybe a couple quantum nudges, because those synapses are coupled strongly with all of the warm, wet, noisy things around it. 
