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Key physical concepts, relevant for the cross-fertilization between condensed matter physics and the physics of life seen as a collective phenomenon in a system out-of-equilibrium, are discussed. The onset of life can be driven by: (a) the critical fluctuations at the

Today we know nearly all molecular components of a minimal cell and their interactions using novel microscopy and spectroscopy methods taking advantage of synchrotron radiation, lasers, magnetic resonance using photons from x-ray to radiofrequency, neutrons and quantum effects such as in atomic force microscopy and the Förster resonance energy transfer (FRET). The extraordinary discoveries of molecular biology have elucidated the structure of many organism genomes, revealing complex systems made of thousands of interacting genes. We have reached a fantastic knowledge on the complexity of the living cell thanks to advanced computers for storage and data analysis unveiling the networks of thousands of different bio-molecules in the cells. The collection of all these data, however, does not give us substantial clues on the overall cell functioning; therefore a number of scientists have started to think that it is becoming feasible to search for new physical laws of the living matter for the foundations of a new physics of life.

Founding the physics of the living cell means to find the general laws governing the cooperative network of networks of thousands of different selected biomolecules spanning a large time scale from picoseconds to years and space scales from nanometers to meters. The genomic data allows us to look for new ideas of evolution, which can be considered as a collective phenomenon far from equilibrium [

Percolation is a critical behavior proposed by Giorgio Careri in [

From the point of view of a theoretical physicist, percolation is interesting mainly as a model of critical behavior, but here we want to stress its relevance in biological processes. In 1986 G. Careri and collaborators [_{c}_{c}

In a subsequent paper, in 1989, Rupley

In conclusion, there may be a statistical component in the theoretical explanation of fundamental biological processes such as proton and ion transport across membranes: topological disorder and a statistical process may be needed in order to construct models with fundamental biological functions.

A major lesson from the statistical physics of critical phenomena is that the details of the structural chemistry of the system are not all equally important for the onset of collective phenomena [

In recent years, research on biological and social systems has seen the unprecedented availability of experimental data on the simultaneous state of a large number of interacting units, like for example the electrical recordings of neuronal activity from large retinal patches [

In [

In this section we present briefly what percolation theory is about and its main concepts. We refer to the excellent book of Stauffer [

Let us imagine a large square lattice. Each square site can be in two mutually exclusive states: occupied by a big dot or empty (see _{c}

Let us call _{s}_{s}^{−} ^{cs}_{s}

The functional form of ^{z}^{z}

From

where

An important lesson from percolation is that near the transition most quantities obey scaling laws that are almost universal in that they do not depend on the details of the lattice structure (be it triangular or squared) and other microscopic parameters. The dimensionality of the problem, on the contrary is a truly relevant parameter. This is a common feature of phase transitions and its explanation requires sophisticated techniques of renormalization group analysis that we cannot address here, but we observe that universality in critical phenomena can be clearly addressed mathematically and conceptually within the framework of limit theorems of probability theory.

We will see in the next section that randomness is a crucial component also in other biological processes where stochastic fluctuations are exploited in order to gain efficiency in information processing.

Noise in human-made devices has been considered for long time as something to avoid or to reduce as much as possible. In electronic devices for example noise can affect the reliable communication between the electrical components and it can be a source of failures. In biological systems the situation can be profoundly different. In this context the noise can have different origins. First of all living organisms operates at non-zero temperature, and indeed thermal noise is present everywhere in cellular processes like the diffusion of proteins in chemoreception [

Ranging from the whole organism to single cells and their microscopic constituents, biological systems exploit the noise as a resource with different astonishing mechanisms. Important examples are molecular motors,

Let us consider a heavily damped particle of mass

The particle is subject to a stochastic force that is due to the collisions with the molecules of the fluid at temperature

where ζ_{t} is a normalized white noise (

where Δ_{0}cos(Ω

An asymptotic approximate formula (which is valid for long times and small amplitudes) for the system response is

where

Different mechanisms have been proposed but a common feature emerges: the system experiences an increased sensitivity to small perturbations when the noise level is tuned to an optimal value [

In spatially extended systems, in which units interact through a certain interaction network, the noise is not the only thing that has to be tuned in order to have an optimal resonance: also the topology of the network is important. The optimal topology is a compromise between a scale-invariant network that enhances long-range connections and the presence of hubs, and a short-range connection topology [

Stochastic resonance is a well-accepted paradigm in biological sciences, especially in neurobiology where information processing is the essential function. If we consider the case of the crayfish [

The living state of the cell is determined by the coherence in time and space of the biochemical reactions as proposed by G. Careri in the eighties [

It has been proposed that a specific feature of the quantum world, the “shape resonance” called also “Fano resonance” or “Feshbach resonance” (since it was introduced by Fano and developed by Feshbach) could play a key role in the transition from the non-living matter to the dynamical order in the living matter [

Here we describe the evolution of the scientific idea of shape resonance to understand its relevance. At the time of the foundations of quantum mechanics in 1929 O. K. Rice in California noticed that “when the discrete vibration rotation absorption bands connected with transitions to a certain final electronic state of a molecule overlap the continuous region for the transitions to another final electronic state, some of the discrete bands may be diffuse,

Around the same time also Ettore Majorana developed the theoretical idea of G. Wentzel for the radiationless quantum jumps in the Auger decay introducing the idea of configuration interaction between a discrete and a continuum set of states. In this new quantum scenario the quasi-stationary bound states interfere with the continuum states as shown in

In the eighties the association and dissociation problems found renewed interest in atomic collisions with a focus on molecular reactions like _{2} →_{3} →

The interference at the resonances occurs in the dissociation of van der Waals complexes and is particularly important in order to understand the fragmentation where overlapping resonances have been observed. The physical possibility of such a state is what could allow proteins to synchronize their interaction after a time t accordingly to the equilibrium constant describing the chemistry of the reaction. It is worth to mention that such behavior is only possible in a quantum mechanical formulation of the association-dissociation process among the proteins interactions.

Processes of association and dissociation have been later discovered in ultracold gases designing different experiments depending on the diatomic energy spectra. The ultracold molecules can be dissociated by ramping the magnetic field in the opposite direction through the Feshbach resonance. Once above the Feshbach resonance, the molecules dissociate spontaneously with a finite rate. The association of ultracold molecules with a Feshbach resonance can be used to obtain full quantum control of this chemical reaction [

Recently the shape resonance has been found in multigap superconductors [_{c}. The complexity and shape resonances appear to be the key for keeping quantum coherence at high temperature. The time evolution of this complexity is now of high interest since its control will allow the development of advanced material with novel functionalities [

The quantum theory of configuration interactions between closed and open channels was applied to explain the lineshape of absorption spectra above the ionization limit of atomic spectra due to two electron excitations in He: ^{2})+^{0} 2^{1}^{1} ) with mixing between quasi boud stationary state with continuum states ^{0} 2^{1}^{1} )↔^{1} )+

Using a phenomenological parameter _{0})/Γ where _{0} is the resonant energy and Γ is the width of the autoionized state, this equation predicts a maximum and a minimum in the Fano line-shape. The Fano formula is a superposition of the Lorentzian line shape of the discrete level
_{0} and the maximum moves well above the quasi-bound state. In the extreme case of strong interference, proportional to 1_{0} and becomes very large while the maximum moves far away from _{0} and its intensity vanishes. The result is that for very high interference the shape resonance gives essentially an anti-resonance driven by the negative interference effects.

The shape resonance can be understood through a classical mechanical analogue. Let us consider a pair of classical oscillators ω_{1} and ω_{2} connected by a spring. If some energy is injected into one of the oscillators at _{1} −_{2} → 0. In quantum physics the analogue case is the configuration interaction between two bound states confined in the same atom or in the nucleus and it depends on the energy difference between the levels.

There is not such a resonance in classical physics between the localized mode and the free wave. The lower panel of

On the contrary, according to Fano, the quantum world shows the shape resonance between the free wave and the localized mode. The difference between the energy of the free wave and the energy of the local oscillator is the tuning parameter.

A second classical case where the anisotropic line-shape appears is the case where an external periodic force of variable frequency ω is applied to the pairs of oscillators with different frequencies shown in the upper panel of

where v_{12} describes the strength of the coupling between the two oscillators. Solving this set of equations and calculating the amplitudes, one finds that the resonant behavior of the amplitude as a function of the frequency of the external force has two peaks (symmetric and asymmetric line-shape) near the two “eigen” frequencies. The coupled oscillator responds with two resonances 1) a symmetric resonant profile at _{1} and an asymmetric profile at _{2}. The first is described by a Lorentzian function (known as a Breit-Wigner resonance) and the second is characterized by the asymmetric Fano line-shape as an effect of the phase interference of the two oscillators when driving frequency passes through the resonance.

A solvable example of a resonance between a chain of oscillators and a local mode is the Fano-Anderson Hamiltonian:

This model describes the interaction of two subsystems [_{n}_{k}_{b}. The third term the coupling among the chain and the oscillator given by the coupling coefficient _{0} of the discrete chain. A propagating wave may directly pass through the chain or instead visit the Fano state, return back, and continue with propagation. The main resulting action of the Fano state is that the strength of the effective scattering potential ^{2}/(ω_{k}_{b})resonantly depends on the frequency of the incoming wave ω_{k}_{b} lies inside the propagation band of the linear chain |_{b}| < 2_{b}_{b}, completely blocking propagation as shown by the strong anti-resonance with a dip in

The asymmetric Fano lineshape of the shape resonance, where both the anti-resonant suppression and the resonant enhancement of the wave transmission are located close to each other is obtained by an extension of the basic Anderson model. In fact, each degree of freedom introduced in the single oscillator site provides an additional local path for the scattering wave to propagate, which may lead to a variety of interference phenomena.

This approach is said to derive from the Fano-Anderson Hamiltonian in solid-state physics introduced in 1961 to describe localized magnetic states in metals [

The Hamiltonian (_{c}^{+}_{κ}^{+}^{+}_{k}_{k}^{+}

However the key feature of the “shape resonance” is the mixing of different many body wave-functions introduced by the quantum mechanics formalism of Fano and Feshbach.

The clearest configuration interaction between many body functions is the case of shape resonance in the superconducting gaps [

The first two terms give the kinetic energy ɛ_{kb}_{kc}_{cc}_{bb}

Here each macroscopic condensate is characterized by an order parameter, the superconducting gap, that can be measured and the interband term acts as the link between the wave-functions of the two or more condensates giving a single critical temperature. J. Kondo in 1961 proposed [_{k,k}_{′} ^{*}_{k}_{↑} ^{*} _{−}_{k}_{↓} _{−}_{k}_{′↓}_{k}_{′↑} +

where

The concepts of shape resonance may play a role in biomolecules association and dissociation. Using the previous simple oscillator examples, we can develop some concrete examples in biology.

A zinc finger protein is a good candidate for creating a quasi-stationary state while it is transiently in contact with DNA. The oscillator chains, described in the previous section, would be the idealized case of the zinc finger protein leaving far away in the cell from DNA, while the peculiar state where the zinc finger protein is in contact with DNA would represent the single oscillator coupled to the chain. The more are the degrees of freedom linked to this state such as transcription factors and other biomolecules, the more would be the complex interference pattern possible. Instead, if no shape resonating mechanism is working, the zinc finger would leave DNA without interacting with it anymore, since the energy flow would be dissipated off immediately after the reaction. On the other hand, if the shape resonance mechanism is possible, we could expect the zinc finger proteins behaving as a wave leaving to the infinite, and returning after a finely tuned time back to the DNA in a synchronized fashion.

In metabolic networks nodes are small molecules (substrates) and links are chemical reactions controlled by specific enzymes [

It is known that in protein-protein networks, two proteins may interact and form a new complex depending on whether they are linked by a link. Links in networks therefore suggest a transient presence of an associated state between two proteins. If a node A has more than a link, for example two, this means that the same protein (node A) can associate and dissociate at subsequent times with two different proteins B and C. Each link is weighted [

In this simple example therefore we have two different quasi-bound states AC and BA for the same protein A and each of these states is characterized by an association-dissociation equilibrium constant.

It is an empirical fact that a protein after some time leaves the protein with which it has interacted. It is therefore possible to imagine the leaving protein as a wave that scatters at infinity. In other words, the protein spends some time near the protein to which is linked, but eventually ends up far from it.

The links attached to the node can belong to the same network or be in a modular region between overlapping networks, each of them performing a variety of tasks. However, when a first node protein is transiently associated with a second linked protein, the third protein becomes dissociated and free in solution.

However, how can we try to explain the dynamical process which forbids the free proteins to bind with the many other highly concentrated proteins in a typical cell milieu? We propose that the protein-protein interaction has to be tuned (changing physical parameters such as density, temperature, membrane architectures and pH) near a shape resonating state. In this resonating state it is not essential how far the protein is when it is dissociated because it will return to associate transiently again.

A condensation of a biochemical reactions network develops if the same protein associate and dissociate transiently in time with a great number of proteins. In other word, this means that the protein is a highly linked node of the network. The link condensation can be controlled by weighting the links [

In chemistry and biochemistry, a dissociation constant is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions. The dissociation constant is usually denoted _{d}

in which a complex _{x}_{y}

where [_{x}_{y}_{x}_{y}

A careful engineering of the dissociation constants within network of protein interactions could be a way to achieve a shape resonance in biological matter, establishing finally a robust and functional system against the decoherence attacks of temperature [

Let us assume that there exists a metastable state in which the two proteins are linked for some time. After a certain amount of time the protein interaction terminates and the two proteins are again in free motion. In order to estimate the transition from the quasi-stationary state to the continuous state, we can describe the wave function of the two protein systems in the metastable state (quasi-stationary state)

while in the continuum we have a many-body wave function of the following form:

The transition amplitude is given by the exchange interaction (energy transfer: monopole)

that describes the oscillations between the quasi-stationary state and the free state of the proteins and in some energy range the shape resonances are expected to determine the coherence of the protein dynamics.

In this review, we have discussed three ingredients that could be of relevance in a physical theory of life. The first is the granular percolation of multiple phases far from equilibrium. Indeed, near a critical state, it has been shown that a fractal granular percolation plays a relevant role in the robustness of a coherent quantum state in high temperature superconductors [

_{2}

_{2}

^{2+}and Fe

^{3+}Hexacyanide Complexes

_{2}HF + H Reaction

_{C}superconductors by producing metal heterostructures as in the cuprate perovskites

_{C}

_{C}superconductivity in a superlattice of quantum stripes

_{C}Superconductors

_{2}CuO

_{4+y}

(_{c} = 0.5927) on a square lattice of size ^{3}. There is a cluster (the dark blue one) spanning the whole system from the bottom to the top of the square.

The mechanism of stochastic resonance. The double well potential _{(x)} represents a physical system with two stable states. In absence of noise a particle, which at one time is within one of the two wells, relaxes towards the intrawell minimum and then stops. When stochastic perturbation is added to this system the particle can hop between the two wells with a rate predicted by reaction-rate theory. If a weak periodic forcing is present the overall potential minimum changes periodically from one side to the other. Stochastic resonance happens when the typical escape time from the lower barrier and the periodic forcing are synchronized.

The typical shape of a stochastic resonance peak. The behavior of the amplitude of the system response <x> is plotted as a function of the noise level given by the diffusion coefficient

The dissociation process of a quasi-bound molecule in an excited predissociation state (called a quasi stationary state) in a potential valley degenerate with continuum states of the dissociated particles in the plain region of the potential as described by Rice [

The “shape resonance” is determined by the quantum configuration interaction between the quasi-bound states forming the bound molecules in the potential

The evolution of the Fano line-shape for the shape resonance, as a function of the strength of the interference term proportional to 1

(Upper panel): A pair of oscillators ω_{1} and ω_{2} is weakly linked. The energy injected into one oscillator flows to the other one and