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We review in this paper the use of the theory of scale relativity and fractal space-time as a tool particularly well adapted to the possible development of a future genuine systems theoretical biology. We emphasize in particular the concept of quantum-type potentials, since, in many situations, the effect of the fractality of space—or of the underlying medium—can be reduced to the addition of such a potential energy to the classical equations of motion. Various equivalent representations—geodesic, quantum-like, fluid mechanical, stochastic—of these equations are given, as well as several forms of generalized quantum potentials. Examples of their possible intervention in high critical temperature superconductivity and in turbulence are also described, since some biological processes may be similar in some aspects to these physical phenomena. These potential extra energy contributions could have emerged in biology from the very fractal nature of the medium, or from an evolutive advantage, since they involve spontaneous properties of self-organization, morphogenesis, structuration and multi-scale integration. Finally, some examples of applications of the theory to actual biological-like processes and functions are also provided.

The theory of scale relativity and fractal space-time accounts for a possibly nondifferentiable geometry of the space-time continuum, based on an extension of the principle of relativity to scale transformations of the reference system. Its framework was revealed to be particularly well adapted to a new theoretical approach of systems biology [

This theory was initially built with the goal of re-founding quantum mechanics on prime principles [

This new “macroquantum” mechanics (or “mesoquantum” at, e.g., the cell scale) no longer rests on the microscopic Planck constant ℏ. The parameter, which replaces ℏ is specific to the system under consideration, emerges from self-organization of this system and can now be macroscopic or mesoscopic. This theory is specifically adapted to the description of multi-scale systems capable of spontaneous self-organization and structuration. Two privileged domains of applications are, therefore, astrophysics [

In this contribution dedicated to applications in biology, after a short reminder of the theory and of its methods and mathematical tools, we develop some aspects which may be relevant to its explicit use for effective biophysical problems. A special emphasis is placed on the concept of macroquantum potential energy. Scale relativity methods are relevant because they provide new mathematical tools to deal with scale-dependent fractal systems, like equations in scale space and scale-dependent derivatives in physical space. This approach is also very appropriate for the study of biological systems because its links micro-scale fractal structures with organized form at the level of an organism.

For more information the interested reader may consult the two detailed papers [

The theory of scale relativity consists of introducing, in an explicit way, the scale of measurement (or of observation)

The coordinates can now be explicit functions of these variables,

The description of such an explicitly scale dependent system needs three levels instead of two. Usually, one makes a transformation of coordinates

However, in the new situation, since the coordinates are now scale dependent, one should first state the laws of scale transformation,

The motion equations in scale relativity are therefore obtained in the framework of a double partial differential calculus acting both in space-time (positions and instants) and in scale space (resolutions), basing oneself on the constraints imposed by the double principle of relativity, of motion and of scale.

The simplest possible scale differential equation which determines the length of a fractal curve (_{F}_{F}_{0}(^{τF}

This result indicates that, in a general way, fractal functions are the sum of a differentiable part and of a non-differentiable (fractal) part, and that a spontaneous transition is expected to occur between these two behaviors.

On the basis of this elementary solution, generalized scale laws can be naturally obtained by now considering second order differential equations in scale space. This is reminiscent of the jump from the law of inertial motion, ^{2}^{2} =

Many of these generalizations may be relevant in biology, in particular:

log-periodic corrections to power laws:

law of “scale dynamics” involving a constant “scale acceleration”:

law of “scale dynamics” involving a scale harmonic oscillator:
_{0} it gives the standard scale invariant case
_{0}(λ_{0}/^{τ0}, _{F}_{0}. But its intermediate-scale behavior is particularly interesting, since, owing to the form of the mathematical solution, resolutions larger than a scale λ_{1} are no longer possible. This new kind of transition therefore separates small scales from large scales, _{1}) from an “exterior” (scales larger than λ_{1}). It is characterized by an effective fractal dimension that becomes formally infinite. This behavior may prove to be particularly interesting for applications to biology, as we shall see in Section 6.

laws of special scale relativity [_{F}_{F}_{H}_{0}. We have identified this invariant scale to the Planck length

Many other scale laws can be constructed as expressions of Euler-Lagrange equations in scale space, which give the general form expected for these laws [

The laws of motion in scale relativity are obtained by writing the fundamental equation of dynamics (which is equivalent to a geodesic equation in the absence of an exterior field) in a fractal space. The non-differentiability and the fractality of coordinates implies at least three consequences [

The number of possible paths is infinite. The description therefore naturally becomes non-deterministic and probabilistic. These virtual paths are identified with the geodesics of the fractal space. The ensemble of these paths constitutes a fluid of geodesics, which is therefore characterized by a velocity field.

Each of these paths is itself fractal. The velocity field is therefore a fractal function, explicitly dependent on resolutions and divergent when the scale interval tends to zero (this divergence is the manifestation of non-differentiability).

Moreover, the non-differentiability also implies a two-valuedness of this fractal function, (_{+}, _{−}). Indeed, two definitions of the velocity field now exist, which are no longer invariant under a transformation |

These three properties of motion in a fractal space lead to describing the geodesic velocity field in terms of a complex fractal function
_{+} + _{−})/2 − _{+} − _{−})/2. The (+) and (−) velocity fields can themselves be decomposed in terms of a differentiable part _{±} and of a fractal (divergent) fluctuation of zero mean _{±}, _{±} = _{±} + _{±} and therefore the same is true of the full complex velocity field,

Jumping to elementary displacements along these geodesics, this reads _{±} = _{±}_{±}, with (in the case of a critical fractal dimension _{F}_{±} represents a dimensionless stochastic variable such that <ζ_{±}> = 0 and

These various effects can be combined under the construction of a total derivative operator [

The next step consists of making a change of variable in which one connects the velocity field
_{0} is a constant for the system considered (it identifies to the Planck constant ℏ in standard quantum mechanics). Thanks to this change of variable, the equation of motion can be integrated under the form of a Schrödinger equation [_{0} and
_{0} = ℏ, so that

By setting finally
^{2} gives the number density of virtual geodesics. This function becomes naturally a density of probability, or a density of matter or radiation, according to the various conditions of an actual experiment (one particle, many particles or a radiation flow). The function

Reversely, the density

After this brief summary of the theory (see more details in [

The first representation, which can be considered as the root representation, is the geodesic one. The two-valuedness of the velocity field is expressed in this case in terms of the complex velocity field

We have recalled in the previous section how a wave function

It is also possible, as we shall now see, to go directly from the geodesic representation to the fluid representation without writing the Schrödinger equation.

To this purpose, let us express the complex velocity field in terms of the classical (real) velocity field _{N}

The additional “fractal” potential is obtained here as a mere manifestation of the fractal geometry of space, in analogy with Newton's potential emerging as a manifestation of the curved geometry of space-time in Einstein's relativistic theory of gravitation. We have suggested ([

Another equivalent possible representation consists of separating the real and imaginary parts of the complex velocity field,

The fundamental two-valuedness which is a consequence of non-differentiability has been initially described in terms of two mean velocity fields _{+} and _{−}, which transform one into the other by the reflexion |_{+}, as for a classical stochastic process:
_{−} does not correspond to any classical process:

However, one may remark that the previous representation is not fully coherent, since it involves three quantities _{+} and _{−} instead of two expected from the velocity doubling. Therefore it should be possible to obtain a system of equations involving only the probability density _{+}. To this purpose, one remarks that _{−} is given in terms of these two quantities by the relation:
_{+}.

This derivation is once again reversible. This means that a classical diffusive system described by a standard Fokker-Planck equation which would be subjected to such a generalized quantum-type potential would be spontaneously transformed into a quantum-like system described by a Schrödinger _{+} as
_{+} = 2
_{+}.

Such a system, although it is initially diffusive, would therefore acquire some quantum-type properties, but evidently not all of them: the behaviors of coherence, inseparability, indistinguishability or entanglement are specific of a combination of quantum laws and elementarity [

This is nevertheless a remarkable result, which means that a partial reversal of diffusion and a transformation of a classical diffusive system into a quantum-type self-organized one should be possible by applying a quantum-like force to this system. This is possible in an actual experiment consisting of a retro-active loop involving continuous measurements, not only of the density [_{+}, followed by a real time application on the system of a classical force _{Q}_{+} = −∇_{+} simulating the new macroquantum force [

One may also wonder whether living systems, which already work in terms of such a feedback loop (involving sensors, then cognitive processes, then actuators) could have naturally included such kinds of quantum-like potentials in their operation through the selection/evolution process, simply because it provides an enormous evolutionary advantage due to its self-organization and morphogenesis negentropic capabilities [

One of the recently obtained results which may be particularly relevant to the understanding of living systems concerns the reversal of the quantum-type potential. What happens when the potential energy keeps exactly the same form, as given by

We have shown [

Indeed, let us start from a Fokker-Planck equation

We have suggested that this behavior may be relevant for the understanding of cancer [

The phenomenon of superconductivity is one of the most fascinating of physics. It lies at the heart of a large part of modern physics. Indeed, besides its proper interest for the understanding of condensed matter, it has been used as model for the construction of the electroweak theory through the Higgs field and of other theories in particle physics and in other sciences.

Moreover, superconductivity (SC) has led physicists to deep insights about the nature of matter. It has shown that the ancient view of matter as something “solid”, in other words “material”, was incorrect. The question: “is it possible to walk through walls” is now asked in a different way. Nowadays we know that it is not a property of matter by itself which provides it qualities such as solidity or ability to be crossed, but its interactions.

A first relation of SC with the scale relativity approach can be found in its phenomenological Ginzburg-Landau equation. Indeed, one can recover such a non-linear Schrödinger equation simply by adding a quantum-like potential energy to a standard fluid including a pressure term [

Consider indeed an Euler equation with a pressure term and a quantum potential term:
_{s}_{0} ≪ _{0}, one may use the additional approximation
^{2}, so that ^{2}, and one still obtains a non-linear Schrodinger equation of the same kind [

The intervention of pressure is highly probable in living systems, so that such an equation is expected to be relevant in theoretical systems biology Laboratory experiments aiming at implementing this transformation of a classical fluid into a macroscopic quantum-type fluid are presently under development [

Another important question concerning SC is that of the microscopic theory which gives rise to such a macroscopic phenomenological behavior.

In superconducting materials, the bounding of electrons in Cooper pairs transforms the electronic gas from a fermionic to a bosonic quantum fluid. The interaction of this fluid with the atoms of the SC material becomes so small that the conducting electrons do not “see” any longer the material. The SC electrons become almost free, all resistance is abolished and one passes from simple conduction to superconduction.

In normal superconductors, the pairing of electrons is a result of their interaction with phonons (see, e.g., [

Therefore the problem of HTS can be traced back to that of identifying the force that links the electrons. We suggest that this force actually derives from a quantum potential.

Most HTS are copper oxide compounds in which superconductivity arises when they are doped either by extra charges but more often by ‘holes’ (positive charge carrier). Moreover, a systematic electronic inhomogeneity has been reported at the microscopic level, in particular in compounds like Bi_{2}Sr_{2}CaCu_{2}O_{8+}_{x}

Basing ourselves on these observations, we have suggested that, at least in this type of compound, the electrons can be trapped in the quantum potential well created by these electronic modulations.

Let us give here a summary of this new proposal. We denote by _{n}_{s}

We set _{n}_{s}_{d}_{d}

The doping induced charges constitutes a quantum fluid which is expected to be the solution of a Schrödinger equation (here of standard QM,

Let us separate the two contributions _{s}_{d}_{s}_{s}_{d}_{d}_{d}_{d}_{s}_{d}_{d}

Even if in its details this rough model is probably incomplete, we hope this proposal, according to which the quantum potential created by the dopants provides the attractive force needed to link electrons into Cooper pairs, to be globally correct, at least for some of the existing HT superconductors.

Many (up to now) poorly understood features of cuprate HTS can be explained by this model. For example, the quantum potential well involves bound states in which two electrons can be trapped with zero total spin and momentum. One can show that the optimal configuration for obtaining bound states is with 4 dopant defects (oxygen atoms), which bring 8 additional charges. One therefore expects a ratio _{s}_{n}

The characteristic size of LDOS wells of ∼30 Angstroms is also easily recovered in this context: the optimal doping being ^{2} in units of _{CuO} = 3.9 Angstroms,

In this context, the high critical temperature superconductivity would be a geometric multiscale effect. In normal SC, the various elements which permit the superconductivity, Cooper pairing of electrons, formation of a quantum bosonic fluid and coherence of this fluid are simultaneous. In HTS, under the quantum potential hypothesis, these elements would be partly disconnected and related to different structures at different scales (in relation to the connectivity of the potential wells), achieving a multi-scale fractal structure [

If confirmed, this would be a nice application of the concept of quantum potentials [

Living systems are well known to exhibit fractal structures from very small scales up to the organism size and even to the size of the collective entities (e.g., a forest made of trees). Therefore it is relevant to assess and quantify these properties with sophisticated models.

Some advanced fractal and multifractal models have been developed in the field of turbulence because fractals are the basic fundamental feature of chaotic fluid dynamics [

There is therefore a strong analogy with living systems. An investigation of turbulence

Dissipation: both turbulent flows and living systems are dissipative.

Non-isolated: existence of source and sink of energy.

Out of equilibrium.

Chaotic.

Existence of stationary structures. Individual “particles” enter and go out in a very complex way, while the overall structure grows (growth of living systems, development of turbulence) then remains stable on a long time scale.

Fundamentally multi-scale and multi-fractal structuring.

Injection of energy at an extreme scale with dissipation at the other (the direction of the multiplicative cascade is reversed in living systems compared to laboratory turbulence).

In a recent work, L. de Montera has suggested an original application of the scale relativity theory to the yet unsolved problem of turbulence in fluid mechanics [

The difference is that coordinates remain differentiable, while in this new context velocity becomes non-differentiable, so that accelerations ^{−1/2} become scale-divergent. Although this power law divergence is clearly limited by the dissipative Kolmogorov small scale, it is nevertheless fairly supported by experimental data, since acceleration of up to 1500 times the acceleration of gravity have been measured in turbulent flows [

De Montera's suggestion therefore amounts to apply the scale relativity method after an additional order of differentiation of the equations. The need for such a shift has already been remarked in the framework of stochastic models of turbulence [

Let us consider here some possible implications of this new, very interesting, proposal.

The necessary conditions which underlie the construction of the scale relativity covariant derivative are very clearly fulfilled for turbulence (now in velocity space):

The chaotic motion of fluid particles implies an infinity of possible paths.

Each of the paths (realizations of which are achieved by test particles of size <100 μm in a Lagrangian approach, [_{F}

The two-valuedness of acceleration is manifested in turbulence data. As remarked by Falkovich _{a}^{2} vs _{da}^{2}. This fundamental result fully supports the acceleration two-valuedness on an experimental basis.

The dynamics is Newtonian: the equation of dynamics in velocity space is the time derivative of the Navier-Stokes equation,

The range of scales is large enough for a K41 regime to be established: in von Karman laboratory fully developed turbulence experiments, the ratio between the small dissipative scale and the large (energy injection) scale is larger than 1000 and a K41 regime is actually observed [

The application of the scale relativity method is therefore fully supported experimentally in this case. Velocity increments ^{2} >= 1. One recognizes here the K41 scaling in ^{1/2}. One introduces a complex acceleration field

By coming back to a fluid representation—but now in terms of the fluid of potential paths—using as variables ^{2} and _{υ}_{0} is Kolmogorov's numerical constant (whose estimations vary from 4 to 9). Concerning the two small scale (dissipative) and large scale (energy injection) transitions, one could include them in a scale varying
_{υ}_{υ}

The intervention of such a missing term in developed turbulence is quite possible and is even supported by experimental data. Indeed, precise experimental measurements of one of the numerical constants which characterize the universal scaling of turbulent flows,
_{0} = 6 in the developed turbulence domain _{λ} ≥ 500 [_{0} = 4, smaller by a factor 2/3.

Let us derive the scale relativity prediction of this constant. We indeed expect an additional contribution to the DNS, since they use the standard NS equations and do not include the new quantum potential.

The considered experiments are van Karman-type flows. The turbulence is generated in a flow of water between counter-rotating disks (with the same opposite rotational velocity) in a cylindrical container [_{υ}_{0}_{υ}_{L}_{x}_{a}_{cl}_{a}_{a}_{L}

For example, in one of the experiments with _{λ} = 690, Voth _{a}^{2} and _{L}_{0})_{DNS} = 4.5 while (_{0})_{exp} = 6.2, so that (1 − (_{0})_{DNS}/(_{0})_{exp})^{1/2} = 0.52, very close to the theoretical expectation from the scale relativity correction (0.54).

Although this is not yet a definitive proof of a quantum-like regime in velocity space for developed turbulence (which we shall search in a finer analysis of turbulence data), this adequation is nevertheless a very encouraging result in favor of de Montera's proposal [

Let us now give some explicit examples of applications of the scale relativity theory in life sciences, with special emphasis to cases where the cell scale is directly or indirectly concerned.

Actually this theory, through its generalized scale laws and its motions laws that take the form of macroscopic quantum-type laws, allows one to naturally obtain from purely theoretical arguments some functions, characteristics and fundamental processes which are generally considered as specific of living systems. We shall briefly consider the following ones (in a non-exhaustive way): confinement, morphogenesis, spontaneous organization, link to the environment, “quantization”, duplication, branching, (log-periodic) evolution, multi-scale integration (see [

Living systems are often characterized by properties of “quantization” and discretization at a very fundamental level. We mean here that they are organized in terms of specific structures having characteristic sizes that are defined in a limited range of scales. The example of cells, which can be considered a kind of “biological quantum”, is the most clear, but this is also true of the cell nucleus, of organs and of organisms themselves for a given species.

This kind of property is naturally expected from the scale relativity approach. Indeed,, the three conditions under which the fundamental equation of dynamics is transformed in a Schroödinger equation (infinity or very large number of potential paths, fractality of these paths and infinitesimal irreversibility) could reasonably be achieved, at least as approximations, in many biological systems.

Such a Schrödinger equation yields stationary and stable solutiöns only for some discretized values of the parameters (energy, momentum, angular momentum,

Example of quantized structures: solutions of a Schrödinger equation for an harmonic oscillator potential.

These structures and their type of quantization appear in dependence of the various limit conditions (in time and space) and environment conditions (presence of forces and fields). This is a very appealing result for biology, since it is clear that not all possible shapes are achieved in nature, but only those corresponding to definite organization bauplans, and that these bauplans appear in relation with the environmental conditions. This is manifest in particular in terms of species punctuated selection and evolution. Some examples will be given in what follows.

As we have seen in the theoretical part of this paper (Section 2.1), one can obtain usual fractal or multifractal scale laws as solutions of first order differential equations acting in the scale space. But these laws can also be generalized to a “scale dynamics” involving second order differential equations. As we have already remarked, this is similar to the passage from inertial laws to Newton's laws of dynamics as concerns motion. Pushing further the analogy, the deviation from a constant fractal dimension (corresponding to scale invariance) can be attributed to the action of a “scale force”.

A particularly interesting application to biology is the case when this force is given by an harmonic oscillator. Indeed, harmonic oscillators appear in a very common way, since they describe the way a system evolves after having been removed from its equilibrium position. But here, the “position” is a scale, which means that, in the case of an attractive oscillator, the system will change its scale in a periodic way. This may yield model of breath/lung dilation and contraction. An interesting feature of such models is that the scale variable is logarihmic, so that the dilation/contraction remains symmetrical only for small deviations from equilibrium, while it becomes disymmetrical for larger ones, as observed in actual situations.

In the case of a repulsive oscillator, one obtains a three-domain system, characterized by a inner and an outer fractal dimension which may be different, separated by a zone at intermediate scales where the fractal dimension diverges (see

A model of cell wall. The figure gives the value of the fractal dimension which is solution of a second order scale differential equation involving a repulsive harmonic oscillator in scale space. One finds a constant fractal dimension in the inner region, a diverging dimension in an intermediate region which may represent a “wall”, then another constant dimension (possibly non-fractal) in the outer region.

Moreover, the zone where the fractal dimension rapidly increases (up to divergence in the mathematical model) corresponds to an increased ‘thickness’ of the material and it can therefore be interpreted as the description of a ‘membrane’. It is indeed the very nature of biological systems to have not only a well-defined size and a well-defined separation between interior and exterior, but also systematically an interface between them, such as membranes or walls. This is already true of the simplest prokaryote living cells. Therefore this result suggests that there could be a connection between the existence of a scale field (for example a pressure), the confinement of the cellular material and the appearance of a limiting membrane or wall [

This is reminiscent of eukaryotic cellular division which involves both a dissolution of the nucleus membrane and a deconfinement of the nucleus material, transforming, before the division, an eukaryote into a prokaryote-like cell. This could be a key toward a better understanding of the first major evolutionary leap after the appearance of cells, namely the emergence of eukaryotes.

The Schrödinger equation, which is the form taken by the equation of dynamics after integration in scale relativity, can be viewed as a fundamental equation of morphogenesis. It has not been yet considered as such, because its unique domain of application was, up to now, the microscopic domain concerned with molecules, atoms and elementary particles, in which the available information was mainly about energy and momentum.

However, scale relativity extends the potential domain of application of Schrödinger-like equations to every systems in which the three conditions (1) infinite or very large number of trajectories; (2) fractal dimension of individual trajectories; (3) local irreversibility, are fulfilled. Macroscopic Schrödinger equations can be constructed, which are not based on Planck's constant ℏ, but on constants that are specific of each system (and may emerge from their self-organization). In addition, systems which can be described by hydrodynamics equations including a quantum-like potential also come under the generalized macroscopic Schrödinger approach.

The three above conditions seem to be particularly well adapted to the description of living systems. Let us give a simple example of such an application.

In living systems, morphologies are acquired through growth processes. One can attempt to describe such a growth in terms of an infinite family of virtual, fractal and locally irreversible, fluid-like trajectories. Their equation can therefore be written under the form of a fractal geodesic equation, then it can be integrated as a Schrödinger equation or, equivalently, in terms of hydrodynamics-type energy and continuity equations including a quantum-like potential. This last description therefore shares some common points with recent very encouraging works in embryogenesis which describe the embryo growth by visco-elastic fluid mechanics equations [

Let us take a more detailed example of morphogenesis. If one looks for solutions describing a growth from a center, one finds that this problem is formally identical to the problem of the formation of planetary nebulae, and, from the quantum point of view, to the problem of particle scattering, e.g., on an atom. The solutions correspond to the case of the outgoing spherical probability wave.

Depending on the potential, on the boundary conditions and on the symmetry conditions, a large family of solutions can be obtained. Considering here only the simplest ones, ^{2} (measured by the quantum number _{z}

Finally the ‘most probable’ morphology is obtained by ‘sending’ matter along angles of maximal probability. The biological constraints leads one to skip to cylindrical symmetry. This yields in the simplest case a periodic quantization of the angle

Morphogenesis of a ‘flower’-like structure, solution of a Schrödinger equation that describes a growth process from a center (

Another very interesting feature of quantum-type systems (in the present context of their possible application to biology) is their behavior under a change of energy. Indeed, while the fundamental level solution of a stationary Schrödinger equation describes a single structure, the first excited solution is usually double.

Therefore, the passage from the fundamental (‘vacuum’) level to the first excited level provides us with a (rough) model of duplication/cellular division (see

Steps in the “opening” of the flower-like structure of

Model of duplication. The stationary solutions of the Schrödinger equation in a 3D box can take only discretized morphologies in correspondence with quantized values of the energy. An increase of energy results in a jump from a single structure to a binary structure. No stable solution can exist between the two structures.

Steps of duplication. Stationary solutions of the Schrödinger equation can take only discretized morphologies in correspondence with quantized values of the energy. The successive figures (from top left to bottom right) give different steps of the division process, obtained as solutions of the time-dependent Schödinger equation in an harmonic oscillator potential, which jump from the fundamental level (top left) to the first excited level (bottom right). These extreme solutions are stable (stationary solution of the time-independent Schrödinger equation), while the intermediate solutions are transitory. Therefore it is seen that the system spontaneously jumps from the one structure to the two-structure morphology.

Model of branching and bifurcation. Successive solutions of the time-dependent 2D Schrödinger equation in an harmonic oscillator potential are plotted as isodensities. The energy varies from the fundamental level (

It is clear that, at this stage, such a model is extremely far from describing the complexity of a true cellular division, which it did not intend to do. Its interest is to be a generic and general model for a spontaneous duplication process of quantized structures, linked to energy jumps. Indeed, the jump from one to two probability peaks when going from the fundamental level to the first excited level is found in many different situations of which the harmonic oscillator and the 3D box cases are only examples. Moreover, this property of spontaneous duplication is expected to be conserved under more elaborated versions of the description provided the asymptotic small scale behavior remains of constant fractal dimension _{F}

Such a model can also be applied to a first rough description of a branching process (

Such a model is still clearly too rough to claim that it truly describes biological systems. It is just intended to describe a general, spontaneous functionality. But note that it may be improved and complexified by combining with it and integrating various other functions and processes generated by the scale relativity approach. For example, one may apply the duplication or branching process to a system whose underlying scale laws include (i) a model of membrane—or cell wall—through a fractal dimension that becomes variable with the distance to a center; (ii) a model of multiple hierarchical levels of organization depending on ‘complexergy’ (see below).

A fundamentally new feature of the scale relativity approach as concerns the question of the origin of life is that the Schrödinger form taken by the geodesic equation can be interpreted as a general tendency for systems to which it applies to make structures,

Such an approach could allow one to ask the question of the origin of life in a renewed way. The emergence of life may be seen as an analog of the ‘vacuum’ (lowest energy) solutions in a quantum-type description,

The problem of the origin of life, although clearly far more difficult and complex, shows common features with this question of structure formation in cosmology. In both cases one needs to understand the apparition of new structures, functions, properties,

We have therefore tentatively suggested a new way to tackle the question of the origin of life (and in parallel, of the present functioning of the intracellular medium) [^{4}−10^{5}, see ([

Schematic illustration of a model of hierarchical organization based on a Schrödinger equation acting in scale space. The fundamental mode corresponds to only one level of hierarchy, while the first and second excited modes describe respectively two, then three embedded hierarchical structures.

The spontaneous transformation of a classical, possibly diffusive mechanics, into a quantum-like mechanics, with the diffusion coefficient becoming the quantum self-organization parameter

In such a framework, the fundamental equation would be the equation of molecular fractal geodesics, which could be transformed into a Schrödinger equation for wave functions

Finally, the Schrödinger equation may in its turn be transformed into a continuity and Euler hydrodynamic-like system (for the classical velocity _{+} (see Section 3). It is therefore possible to generalize the standard classical approach of biochemistry which often makes use of fluid equations, with or without diffusion terms (see, e.g., [

Under the point of view of this third representation, the spontaneous transformation of a classical system into a quantum-like system through the action of fractality and irreversibility on small time scales manifests itself by the appearance of a quantum-type potential energy in addition to the standard classical energy balance. We have therefore suggested to search whether biological systems are characterized by such an additional potential energy [

However, we have also shown that the opposite of a quantum potential is a diffusion potential (Section 3.7). Therefore, in case of simple reversal of the sign of this potential energy, the self-organization properties of this quantum-like behavior would be immediately turned, not only into a weakly organized classical system, but even into an increasing entropy diffusing and disorganized system. We have tentatively suggested [

Another application of the scale relativity theory consists of applying it in the scale space itself. In this case, one obtains a Schrödinger equation acting in this space, and thus yielding peaks of probability for the scale values themselves. This yields a rough but already predictive model of the emergence of the cell structure and of the value of its typical scales.

Indeed, the three first events of species evolution are the appearance of prokaryot cells (about 3.5 Gyrs in the past), then of eukaryot cells (about 1.7 Gyr), then of the first multicellulars (about 1 Gyr). These three events correspond to three successive steps of organizational hierarchy.

Indeed, at the fundamental (‘vacuum’) level, one can expect the formation of a structure characterized by one length-scale (_{m}_{M}

What are the minimal and maximal possible scales? From a universal viewpoint, the extremal scales in nature are the Planck-length _{P}^{−1/2} given by the cosmological constant Λ in the macroscopic domain [_{ℙ} = 5.3 × 10^{60}, so that the mid scale of the universe is at 2.3 × 10^{30} _{ℙ} ≈ 40 μm.

Now, in a purely biological context, one would rather choose the minimal and maximal scales characterizing living systems. These are the atomic scale toward small scales (0.5 Angströms) and the scale of the largest animals like whales (about 10–30 m).It is remarkable that these values yield the same result for the peak of probability of the first structure of life, λ = 40 μm. This value is indeed a typical scale of living cells, in particular of the first ‘prokaryot’ cells appeared more than three Gyrs ago on Earth. Moreover, these prokaryotic first cells are, as described in this simple model, characterized by having only one hierarchical level of organization (monocellulars and no nucleus, see

The second level describes a system with two levels of organization, in agreement with the second step of evolution leading to eukaryots about 1.7 Gyrs ago (second event in

The following expected major evolutionary leap is a three organization level system, in agreement with the apparition of multicellular forms (animals, plants and fungi) about 1 Gyr ago (third event in

The following major leaps correspond to more complicated structures, then possibly to more complex functions (supporting structures such as exoskeletons, tetrapody, homeothermy, viviparity), but they are still characterized by fundamental changes in the number of organization levels. We also recall that a log-periodic acceleration has been found for the dates of these events [

The first steps in the above model are based on spherical symmetry, but this symmetry is naturally broken at scales larger than 40 μm, since this is also the scale beyond which the gravitational force becomes larger than the van der Waals force. One therefore expects the evolutionary leaps that follow the apparition of multicellular systems to lead to more complicated structures (such as those of the Precambrian-Cambrian radiation), than can no longer be described by a single scale variable. This increase of complexity will be dealt with by extending this model to more general symmetries, boundary conditions and constraints.

We hope the scale relativity tools and methods to be also useful in the development of a systems biology framework [

Now, one of the challenges of systems biology is the problem of multiscale integration [

The first step consists in defining the elements of description, which represents the smallest scale considered at the studied level. For example, at the cell level, these elementary structures could be intracellular “organelles”.

The second steps amounts to writing for these elementary “objects” an equation of dynamics which account for the fractality and irreversibility of their motion. As we have seen, such a motion equation written in fractal space can be integrated under the form of a macroscopic Schrödinger-type equation. This equation would no longer be based on the microscopic Planck constant, but on a macroscopic constant specific of the system under consideration (this constant can be related, e.g., to a diffusion coefficient). Its solutions are wave functions whose modulus squared gives the probability density of distribution of the initial “points” or elements.

Actually, the solutions of such a Schrödinger-like equation are naturally multiscaled. It describes, in terms of peaks of probability density, a structuring of the “elementary” objects from which we started (e.g., organelle-like objects structuring at the larger scale cell-like level). As we have previously seen, while the vacuum state (lowest energy) usually describes one object (a single “cell”), excited states describe multiple objects (“tissue-like” level), each of which being often separated by zones of null density—therefore corresponding to infinite quantum potentials—which may represent “walls” (

A simple two-complementary-fluid model (describing, e.g., hydrophile/hydrophobe behavior) can easily be obtained, one of them showing probability peaks in the “cells” (and zero probability in the “wall”) while the other fluid peaks in the “walls” and have vanishing probability in the “cells”. This three-level multi-scale structure results from a general theorem of quantum mechanics (which remains true for the macroscopic Schrödinger regime considered here), according to which, for a one-dimensional discrete spectrum, the wave function corresponding to the (

Multiscale integration in scale relativity. Elementary objects—at a given level of description (left figure)—are organized in terms of a finite structure described by a probability density distribution (second figure from the left). By increasing the energy, this structure spontaneously duplicates (third figure). New increases of energy lead to new duplications (fourth figure), then to a “tissue”-like organization (fifth figure—the scale of the figures is not conserved).

Moreover, this scale relativity model involves not only the resulting structures themselves but also the way the system may jump from a two-level to a three-level hierarchical organization. Indeed, the solution of the time-dependent Schrödinger equation describes a spontaneous duplication when the energy of the system jumps from its fundamental state to the first excited state (see Section 6.4 and

One may even obtain solutions of the same equation organized on more than three levels, since it is known that fractal solutions of the Schrödinger equation do exist [

Fractal multiscale solutions of the Schrödinger equation. Left figure: one-dimensional solution in a box, in terms of position ^{2} is drawn for

Note that the resulting structures are not only qualitative, but also quantitative, since the relative sizes of the various embedded levels can be derived from the theoretical description. Finally, such a “tissue” of individual “cells” can be inserted in a growth equation which will itself take a Schrödinger form. Its solutions yield a new, larger level of organization, such as the flower-like structure of

The theory of scale relativity, thanks to it accounting for the fractal geometry of a system at a profound level, is particularly adapted to the construction and development of a theoretical biology. In its framework, the description of living systems is no longer strictly deterministic. It supports the use of statistical and probabilistic tools in biology, for example as concerns the expression of genes [

However, it also suggests to go beyond ordinary probabilities, since the description tool becomes a quantum-like (macroscopic) wave function, which is the solution of a generalized Schrödinger equation. This involves a probability density such that ^{2}, but also phases which are built from the velocity field of potential trajectories and yield possible interferences.

Such a Schrödinger (or non-linear Schrödinger) form of motion equations can be obtained in at least two ways. One way is through the fractality of the biological medium, which is now validated at several scales of living systems, for example in cell walls [

In this framework, one therefore expects a fundamentally wave-like, and often quantized, character of numerous processes implemented in living systems. In the present contribution, we have concentrated on the theoretical aspect of the scale relativity approach, then we have given some examples of applications to some biological processes and functions.

Several properties that are considered to be specific to biological systems, such as self-organization, morphogenesis, ability of duplicating, reproducing and branching, confinement, multi-scale structuration and integration are naturally obtained in such an approach [

The author gratefully thanks Philip Turner for his careful reading of the manuscript and for his useful comments.

The author declares no conflict of interest.