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Quantum biology, water and living cells Eugen A. Preoteasa HH-NIPNE, LEPD (DFVM)

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I.Introduction and background – Biology from classical to quantum II.New models of collective dynamics for liquid water and living cell – Ionic plasma in water – The cell dimensions problem – Free water coherent domains’ Bose condensation: The minimum volume of the cell – Water coherent domains in an impenetrable spherical well: The maximum cell volume of small prokaryotic cells – Plausible interaction potential between coherence domains – Two coupled water coherent domains as a harmonic oscillator and the maximum cell volume – Isotropic oscillator in a potential gap and the spherical cells: larger prokaryotes and small eukaryotes – Cylindrical potential gap and disc-like cells: the erythrocyte – Cylindrical potential gap and rod-like cells: typical bacilli – The semipenetrable spherical well: The toxic effect of heavy water in eukaryotic cells III.Conclusions

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Introduction and background Biology from classical to quantum

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Life is a phenomenon strikingly different of the non-living systems. Some distinctive traits Metabolism Homeostasis Replication Stability of descendents Spontaneous, low-rate random mutations Diversity by evolution: ~ 8.000.000 species Adaptation (e.g., bacteria eating vanadium, bacteria living in nuclear reactor water, life in desert and permafrost) Damage repair (e.g., wound healing) Integrality / indivisibility

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The phenomenology and evolution of the living world are described by classical biology. Classsical biology started with the optical microscope and developed in XVII-XIX centuries (by people like Leeuwenhoek, Maupertuis, Linne, Lamarck, Cuvier, Haeckel, Virchow, Darwin, Wallace, Mendel, Pasteur, Cl. Bernard, etc.). The main ideas of biology were influenced by classical physics (Newton, Pascal, Bernoulli, Carnot, Clausius, Bolzmann, Gibbs, Helmholtz, Maxwell, Faraday, Ostwald, Perrin, …) and chemistry (Lavoisier, Berzelius, Woehler, Berthelot, …). Biological phenomenology and evolution Tree of life

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Molecular biology – a new reductionism Recently, phenotypic plasticity and self-organization re- vealed limits of “the central dogma” of molecular biology: DNA RNA Enzymes Genome (DNA from the ovocite of a species’ individual) Phenotype (particular individual organism of a species) “DNA (or RNA) encodes all genetic infor-mation” (Crick & Watson 1950) devastating effect on biology. Two images since 1967: integrative (Jacob); vs. reductionist, (Monod):

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The “central dogma” raises questions, e.g.: Is all information contained in DNA, RNA? Are mutations purely random? Is the environment only selecting mutations? No feed-back? The main ideas of molecular biology : All biological phenomena reduced to information stored in some (privileged) molecules. Only short-range specific interactions. Classical (Bolzmann-Gibs), equilibrium statistics. Water – mainly a passive solvent. The cell – a bag filled with a solution of molecules. This picture – rooted in XIX century thinking – is disputable. It fails to seize complexity, integrality of living organisms.

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Cells, complexity, integrality The cell – basic unit of life / at the origin of any organism. Cells – an unparalleled complexity, a singular, unique type of order. Integrality – cells are killed by splitting. Biological complexity – order (almost) without repetition – different of the physical complexity (= nonintegrable, 3 bodies). A bacterial cell – 4.10 10 molecules H 2 O, and 5.10 8 various organic molecules. An eukaryotic cell ~ x10 5 more molecules. Huge complexity of metabolic network. Shown above only ~5%.

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Limits of molecular biology Complexity, integrality – pointing to nonlinear, optimal, self- organized systems, to long-range correlations. Molecular biology “sticks and balls” picture – isolated classical particles, short-range interactions. Success of molecular biology – at the roots of its limits. Origin of life unexplained – probability of first cell ~10 -40,000, of man ~10 -24,000,000 in 4.10 9 yr. –“Chance is not enough” (Jacob 1967). Metabolic co-ordination: How a huge number of specific chemical reactions occur in a cell at the right place / time? Information content in the cell much larger than in DNA (a read- only memory) – where the rest comes from? Unexplained: brain activity, biological chirality, etc.

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Collective dynamics of many freedom degrees. Life – a metastable state. Various types of local and global order. Structural and dynamic hierarchy, successive levels. Biological complexity – order without repetition. Short- and long-range correlations and interactions. Living organisms are open, irreversible, disipative systems. They are self-organized, optimal systems (->homeostasis), with cooperative interactions. Nonlinear interactions, highly integrated dynamics. Such features – to some degree in various complex non-living systems – but only organisms join them altogether. Features of life unsolved by molecular biology

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Molecular biology, biophysics, quantum mechanics A) Physical methods for “special materials” studies. B) Molecular structure and properties – quantum chemistry – integrated in the “balls and sticks” picture of molecular biology. Though A), B) based on QM – ancillary / “trivial” role for QM. Could QM yield insight on the essence of life? What is the usual place of biophysics and QM in molecular biology?

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Correlations, functions and soft matter Organisms evolve by functions – space-time correlations between freedom degrees. Functions are controlled by specific messages. Messages express biological complexity. Both imply order without repetition convey information. Cells – soft matter facilitate functions by (re)aggrega- tions and conformational changes. Flexible geometic structure, conservative topological correlations of freedom deg.s. Dynamical organization. Cells – condensed matter – facilitate long-range correlations and information transfer. Either correlations and information admit both classical and quantum support.

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Classical and quantum correlations – long range interactions between (quasi)particles Long range correlations – self-correlation functions – in biological, chemical and physical systems – formally similar for: a classical observable z(r): G( ) = a wavefunction (r) : G( ) = The self-correlation or coherence function is connected to interference of waves associated with a (quasi)particule: I( ) ~ | 1 (r) + 2 (r+ )| 2 ~ 1 + |G( )|cos k Necessary condition – long range interactions between particles or quasiparticles.

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Biological order and information Biological order – order without repetition. Such order - aperiodic and specific (Orgel 1973) conveys information. Periodic nonspecific order – minimal information : AAAAAAAAAAAA… Periodic specific order – useful information overwhelmed in redundance: CRYSTAL CRYSTAL CRYSTAL … Complexity: aperiodic nonspecifica order – maximal total information, minimal useful information: AGDCBFE GBCAFED ACEDFBG … Complexity: aperiodic and specific order : THIS IS A MESSAGE. Well-defined sequence = message, precise code, maximum useful information, comands an unique function. Biological systems – informational syst. – adressable both C/Q.

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Quantity of information ( Shannon, Weaver 1949 ): H = – Σ p i log 2 p i ; p = |ψ| 2 ; Ex. H(Xe) = 136 bit. Information gain between 2 probability distrib.s P, W: I (P|W) = Σ p i log 2 (p i / w i ) Information gain in a quantum transition |m> → |l> ( Majernik 1967 ): I ( φ m | φ l ) = ∫ φ m φ m * log 2 ( φ m φ m * / φ l φ l *) dv Ex.: Potential gap, I(u 2 |u 1 ) = 3,8 bit. Hydrogen atom, I(u 2 |u 1 ) = 83,1 bit. Hypothesis : In biological systems, certain wave- functions may play a role in transmission, storage, processing, and control of information. Information and quantum mechanics

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Alternatives to molecular biology Postulate: Living organisms contain both classical and quantum (sub)systems. Alternatives to describe biological complexity and integral properties of organisms: 1.Far from equilibrium dynamics, dissipative structures (classical or quantum); 2.Models of periodic phenomena based on equations with eigenfunctions and eigenvalues (classical or quantum); 3.Quantum biology.

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Irreversibility, far from equilibrium dynamics, dissipative structures ( Prigogine, Nicolis, Balescu ) Limit cycle (strange attractor): All trajectories, whatever their initial state, lead finally to the cycle. Makes the origin of life from non-living much more probable. Spontaneous synchronization of oscillations in glycolysis (glucose consumption) in yeast cells (Bier) Belousov-Zhabotinsky reaction: Heterogeneous (order) out of homogeneous (disorder).

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Integral properties without molecular biology. I. The fur of mammals by partial derivative equations Diffusion-reaction of melanin: Results:

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Integral properties of cells without molecular biology. II. Flickering modes of erythrocyte membrane by Fourier / correlation analysis

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Quantum biology Bohr, Heisenberg, Schrodinger, John von Neumann, C. von Weizsacker, W. Elsasser, V. Weisskopf, E. Wigner, F. Dyson, A. Kastler, and others – QM essential for understanding life. Quantum biology (QB): “speculative interdisciplinary field that links quantum physics and the life sciences” (Wikipedia) – runs the first phase, inductive synthesis, of every science. Some directions : quantum physicslife sciences – Quantum-like phenomenology – QM without H and/or h. – Non-relativistic QM. – “Biophoton” (ultraweak emission) statistics. – Solitons (Davydov), phonons, conformons, plasmons, etc. – Decoherence, entanglement, quantum computation. – Long-range coherent excitations – Frohlich. – QED coherence in cellular water – Preparata, Del Giudice.

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Decoherence, entanglement, quantum computation Origin of life – Davies; Al-Khalili & McFadden Photosynthesis – Castro et al; coherence found experimentally. Decoherence in proteins, tunelling in enzymes – Bothema et al Protein biosynthesis and molecular evolution – Goel Cytoskeleton, decoherence, memory – Nanopoulos; Hameroff Genetic code, self-replication – Pati; Bashford & Jarvis; Patel Quantum cellular automata – Flitney & Abbot Evolutionary stability – Iqbal & Cheon Quantum-like phenomenology Consciousness, Psyche – Orlov; Piotrowski & Sladkowski Embriogenesis – Goodwin Non-relativistic QM Protein folding – Bohr et al. Scaling laws and the size of organisms – Demetrius

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Embriogenesis by variational principle (Goodwin) Introduce a field function u ( , ) – i.e., a morphogenetic field; Its nodal lines – lines of least resistance; Define the surface energy density: The cleavage planes given by the minima of the integral: Eigenfunctions – spherical harmonics Y lm ( , ): Biological constraint / selection rule – the number of cells = 2 p : 2 48 1632

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Consciousness by spinor algebra (Orlov) Yuri Orlov (Soviet physicist and disident). Consciousness states cannot be reduced to the QM states of brain molecules. Consciousness is a system that observes itself, being aware of doing so. – No physical analogue exists. – Partly true for life (?) Consciousness state – described by a spinor. Let a proposition: Every elementary logical proposition can be represented by the 3 rd component of Pauli spin: Hamlet’s dilemma: and

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Protein topology and folding by quanta of torsion (Bohr, Bohr, Brunak) Heat consumed both for disorder-order and order- disorder transitions. Spin-glass type Hamiltonian: Topology – White theorem: writhings + twists = const. Quantified long-range excitations of the chain, wringons. Explain heat consumption both in disorder-order and order-disorder transitions of some proteins in aqueous solution.

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Herbert Fröhlich postulated a dynamical order based on correlations in momentum space, the single coherently excited polar mode, as the basic living vs. non-living difference. Assumptions: (1) pumping of metabolic energy above a critical threshold; (2) presence of thermal noise due to physiologic temperature; (3) a non-linear interaction between the freedom degrees. Physical image and biological implications: A single collective dynamic mode excited far from equilibrium. Collective excitations have features of a Bose-type condensate. Coherent oscillations of 10 11 -10 12 Hz of electric dipoles arise. Intense electric fields allow long-range Coulomb interactions. The living system reaches a metastable minimum of energy. This is a terminal state for all initial conditions (e.g. Duffield 1985); thus the genesis of life may be much more probable. Fröhlich’s long-range coherence in living systems

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Applications – theoretical models: Biomembranes, biopolymers, enzymatic reactions, metabo-lism (stability far from equilibrium), cell division, inter-cellular signaling, contact inhibition, cerebral waves. Examples of experimental confirmations: Cell-cycle dependent Raman spectra in E. coli (Webb); Micro-waves accelerated growth of yeast (Grundler); Cell-cycle effects on dielectric grains dielectrophoresis (Pohl); Optical effects at ~5 m in yeast (Mircea Bercu); Erythrocyte rouleaux formation – 5 m forces (Rowlands). Other models consistent to Fröhlich’s theory: 1) Water dynamical structure – coherence domains (Preparata, Del Giudice), 2) cell models based on water coherence domains (Preoteasa,Apostol), 3) ionic plasma water (Apostol,Preoteasa). Aims and evidences of Fröhlich’s theory

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Liquid and cellular water Water – an unique liquid with remarkable anomalies (density, compresibily, viscosity, dielectric constant, etc.). Water remarkable properties: The dipole moment d = 1,84 D – would yield a dielectric constant r ~10, while experimental value r = 78,5. Dissociation, H 2 O…HOH H 3 O + + OH – H 3 O(H 2 O) 3 + + OH –. O-H…O hydrogen bond, H 2 O…HOH, L(O-H…O) = 2,76 Å, E(O-H…O) = 20 kJ/mol > E(Van der Waals) = 0.4 – 4 kJ/mol ~ k B T ~ 2.6 kJ/mol. Angle 104,5 o between O-H bonds in H 2 O Tetrahedral structure formation. Intuitive explanation: two- phase phenomenological model (Röntgen, Pauling).

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Two-phase model of water – H-bond flickering “ice-like” clusters in dynamical equilibrium with a dense gas-type fluid with unbound molecules. Near polar interfaces and intracellular surfaces – altered – long-range interactions. Interfacial water – bound w. (< 5 nm), vicinal w. 15-50 nm (Drost-Hansen), gel w. ~ 1-10 m (Pollack). The non-repeating structure of proteins / nucleic acids and short-range forces may not explain a concerted collective dynamics in the cell. Water – possible vehicle for long-range specific interactions. Hypothesis: water converts position-space correlations to momentum-space correlations, – emergence of cellular order. Water physical state changes in cell cycle.

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QED theory of water coherence domains in living cell (of the Milano group) New models – based on the concept of coherence domains (CD) of water from the QED theory of Preparata, DelGiudice. Water forms polarization coherence domains (CDs) where the water dipoles oscillate coherently, in-phase. The water CDs are elementary excitations with a low effective mass (excitation energy) m eff ~12.7-13.6 eV (m e = 511000 eV). CDs are bosons (S = 0), obey Bose-Einstein statistics below a critical temperature T c. Due to low effective mass, much longer de Broglie wavelength = h/m eff enhaced wavelike properties high T c. The coherence domains are shaped as filaments, R~15 - 100 nm, L~100 - 500 nm. In cells some water filaments are located around chain-like proteins and some are free. Around water filaments appear specific, non-linear forces.

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Experimental proofs of water QED model QED model predicts water anomal properties. QED model predicts expelling of H + ions CDs external electric field + dialysis pH between compartments. Biological proof: Ionic Cyclotron Resonance & Zhadin effect. Density anomaly Specific heat at 4 o C at constant pressure

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New models of collective dynamics for liquid water and the living cell

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A model for liquid water – by plasmon-like excitations. The dynamics of water has a component consisting of O –2z anions and H +z cations, where z is a (small) effective charge. Due to this small charge transfer, the H and O atoms interact by long- range Coulomb potentials in addition to short-range potentials. This leads to a H +z – O –2z two-species ionic stable plasma. As a result, two branches of eigenfrequencies appear, one corresponding to plasmonic oscillations and another to sound-like waves. Density oscillations in water and other similar liquids (M. Apostol and E. Preoteasa Phys Chem Liquids 46:6,653 — 668, http://arXiv.org/abs/0803.2949v1 20 March 2008) http://arXiv.org/abs/0803.2949v1

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Calculating the spectrum given by the eq. of motion without neglecting terms in q 2 gives: For vanishing Coulomb coupling, z -> 0, this asymptotic frequency looks like an anomalous sound with velocity:

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Hydrodynamic sound velocity vo ~ 1500 m/s. ‘Anomalous’ sound velocity v s : Hence we get the short-range interaction The plasma oscillations can be quantized in a model for the local, collective vibrations of particles in liquids with a two- dimensional boson statistics. The energy levels of the elementary excitations: This allowed an estimate of the correlation energy per particle and cohesion energy (vaporization heat) of water: corr ~ 10 2 K at room temperature. Similar results – for OH - – H + or OH - – H 3 O + dissociation forms.

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In the living cell, the ionic plasma oscillations of water and their fields may interact with various electric fields associated to biomembranes, biopolymers and water polarization coherence domains – may play a certain role in intra- and intercellular communications. The water ionic plasmons should have a very low excitation energy (effective mass), of ~200z [meV], and are almost dispersionless the associated de Broglie wavelength may be very large entanglement of their wavefunctions is possible support for intercellular correlations at very long distance, of major interest for phenomena such as embrio-, angio-, and morphogenesis, malign proliferation, contact inhibition, tissue repair, etc. The model is consistent to the general Fröhlich theory. Ionic plasma model of brain activity postulated (Zon 2005).

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The cell size problem Cells are objects of dimensions of typically ~ 1 – 100 µm specific dynamical scale. Smaller biological objects are not alive. Biological explanations: Lower limit – min. ~5.10 2 – 5.10 3 different types of enzymes necessary for life. Upper limit – due to metabolism efficiency (prokaryotes), surface / volume ratio (animal eukaryotic cells), and large vacuoles (plant eukaryotic cells). The explanation relies on empirical bio- chemical / biological data – it only displaces the problem. “Systems biology” – starting not from isolated genes but from particular whole genome network (Bonneau 2007, Feist 2009) – classical dynamics, is it sufficient?

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Physical explanations: Schrödinger (1944) – a minimum volume cooperation of a sufficient number of molecules against thermal agitation. Dissipative structures (Prigogine) – cell as a giant density fluctuation cell size must exceed the Brownian diffusion during the lifetime. Empirical allometric relationship P = W ; P metabolism, W size – both in uni- / multicellular organisms. Mechanistic / fractal models fail for unicellular organisms. Quantum model (Demetrius) electron/proton oscillations in cell respiration and oxidative phosphorilation – applies Planck’s quantization rule and statistics deduces P = W for both uni- and multicellular organisms. Demetrius QM model depends on metabolism a “purely physical” basis for cell size is possible? We propose a new quantum model for the cell size and shape based on coherence domains of water, without explicit reference to metabolism.

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Bose-type condensation of water coherent domains’: the minimum cell volume At a critical density and temperature, the wavefunctions of CDs overlap and collapse common wavefunction, single phase. Water CDs’ low effective mass temperature T c of Bose-type condensation of CDs – where a ‘coherent state’ arise – might exceed the usual temperature of organisms (~310 K). A Bose-type condensate of CDs in whole cells at ~310 K. The assemble of water CDs in cell - a boson ideal gas in a spherical cavity. The wavefunctions of the water CDs boson gas reflect totally on the membrane. The cell – a resonant cavity of volume V limited by membrane containing N CDs.

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For T < T c, a coherent state of CDs in the whole cell emerges. The dynamical states of all CDs – correlated – supercoherence (Del Giudice). The collective wavefunction of CDs – an unified system for transmission, storage and processing of information, maximizing correlation of molecular dynamics in the cell. High order, CD-correlated, coherent dynamics – supercoherence new macroscopic dynamical properties – essential for life. Postulates enhancing the role of water CDs 1.The living state is defined in the essence by metabolism, and not by replication (Dyson’s “metabolism first, replication after” hypothesis). 2.The metabolism is dinamically co-ordinated by interactions between enzymes and water CDs (Del Giudice’s hypothesis). 3.The maximum dynamical order in cell – life – reached when a Bose-type condensation of the water CDs free in the cytoplasm occurs – supercoherence (D.G.).

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T c = [ (N/V) / (3/2) ] 2/3 2 ħ 2 / m eff k B For T c = 310 K, m eff = 13.6 eV = 2.4 10 -35 kg, imposing N > 2 (N c = 2 – the smallest possible number of condensing CDs), V > V min = 1.02 m 3 Correct as magnitude order – or better ! The smallest cell known, Mycoplasma, V = 0.35 m 3 Typical prokaryotic cells – e.g. E. coli, V = 1.57 m 3 ; Eukaryotic cells – RBC, V = 85 m 3 ; Typical volumes for eukaryotic cells – 10 3 –10 4 m 3. For a critical density of CDs wavefunctions overlap and collapse in a common wavefunction a “coherent state” arises. The temperature T c where the ‘coherent state’ arise – given by the Bose-Einstein equation of a boson gas condensation:

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Basic postulates for models giving cell’s maximum volume and shape In the following models – new basic postulates: Water CDs in the cell – bound quantum systems. Quantized dynamics of water CDs (translation in potential gaps, harmonic oscillations). Biological constraints certain levels / certain transitions between the quantized energy levels forbidden for biological stability thermally inaccessible energy levels / forbidden transitions. Cell size and shape selected in evolution – fit the QM potentials and wavefunctions of CDs.

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Water coherent domains in a spherical potential well: maximum volume of typical prokaryotic cells In addition to coherent internal oscillations, a CD may have translation, rotation, deformation, etc. freedom degrees. The cell – a spherical well of radius a with impenetrable walls (infinite potential barrier, U o . The orbital movement is neglected (l = 0). The translation energy of the CD inside the spherical well is quantized on an infinite number of discrete levels E 1, E 2, E 3, … E n = 2 ħ 2 /2m eff a 2 n 2 = 9.87 u n 2 (n = 1, 2,...) Notation: u = ħ 2 /2m eff a 2 A water CD – a quasi-particle of m eff ~13.6 eV in a potential well.

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For a spherical well with semipenetrable walls, i.e. finite potential barrier, e.g. U o = 4 u = 4 ħ 2 /2m eff a 2, E n = 1,155 ħ 2 /2m B a 2 n 2 = 1,155 u n 2 (n = 1, 2,...) For a spherical cell of 2 m diameter, a = 1 m, the energy/frequency of the first level, in these two cases, is: - impenetrable wall: E 1 ~ 3.5 10 12 Hz, - semipenetrable wall: E 1 ~ 4.0 10 11 Hz, in agreement as order of magnitude to the frequencies of coherent oscillations predicted by Fröhlich. To estimate the maximum volume of a cell, we postulate: The metastable living state requires that the second level E 2 to be thermally inaccessible from the first level E 1. Thus the energy difference E 2 – E 1 should exceed thermal energy at physiological T, 37 o C = 310 K. Hence for the spherical well with impenetrable walls: 2 ħ 2 /2m B a 2 (2 2 – 1 2 ) > 3kT/2 Staphylo- coccus

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The maximum radius of the spherical impenetrable cell – defining also a basic biological length a o (T-dependent): a(T) < a max (T) = a o = ħ / (m B kT) 1/2 = 1.02 m for T = 310 K The cell maximum volume V max = 4.45 m 3. Together with the minimum volume estimated by Bose-type condensation, we have the limits of the cell volume: 1.02 m 3 = V min < V cell < V max = 4.45 m 3 Satisfactorily confirmed for typical prokaryotic cells, e.g. E. coli 1,57 m 3, Eubacteria, Myxobacteria 1-5 m 3. Seemingly not confirmed to eukaryotic cells, ~10 2 –10 4 m 3. But: Eukaryotic cells - highly compartmenta- lized, organelles divide cell in small spaces. These spaces obey the above volume limits. This sustains the evolutionary internalization of organelles as small foreign cells. The dimensions of the first protocells may have been similar to the prokaryotic cells.

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The previous models do not assume interactions involving CDs and neglects their nature and structure. Water CDs form by interaction between H 2 O dipoles and radiation – by self-focusing, self-trapping of dipoles, filamenta-tion ( Preparata, Del Giudice ) – nonlinear optics phenomena disco-vered by G. Askaryan (Soviet-Armenian physicist, 1928 - 1997). Therefore CDs are supposed to have filament shape. Around water filaments strong electric field gradients appear, developing frequency-dependent, specific, long-range, non-linear forces to dipolar biomolecules (“Askaryan forces”): F ~ { (ω ο 2 − ω 2 ) / [ (ω ο 2 − ω 2 ) 2 + Γ 2 ] } Ε 2 They have the same form as the dielectrophoresis forces of an oscillating e.m. gradient field on a dielectric body (Pohl). Interactions between water CDs: the possibility of a harmonic potential

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Depending on the ω to ω o ratio, they can be attractive or repulsive. Askaryan force is higher when ω is close to ω o in a narrow frequency band resonant and selective character. They can bring non-diffusively into contact dipolar specific biomolecules, controlling thus cell metabolism ( Del Giudice ). The Askaryan force derives from a “Fröhlich potential” U A (r): F A = - U A / r The potential depends on distance ( central component) and on relative orientation ( non-central component) of dipolar molecule vs. CD. Neglect the explicit dependence of the non-central part: U A (r ) = U A (r ) | |, A – geometric factor 47

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Central part of Fröhlich potential – 2 terms (Tuszinsky): U(r ) = – F/r 6 – E/r 3 –F/r 6 – Van der Waals; –E/r 3 – Fröhlich potential water CD – dipole molecule. At resonance long-range (~1-10 m) potential between a CD and a dipolar molecule. At sufficient long distance U ~ r -3. P1: The potential between two water CDs is similar to the potential between a CD and a permanent dipole molecule. P2: At sufficiently short distance, the potential will have always a repulsive term at least. Repulsive forces in water : 1.“Pauli forces”, +A/r 12 – repulsion between electron clouds of H 2 O in the two CDs (~3.10 -11 erg), 2.Forces due to tetrahedral structure of water (~10 -13 erg); 3.Quadrupolar interactions (2.10 -12 erg);

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3.Interactions due to the CD’s surface electric field polarization of the cavity created in the dielectric medium following the displacement of solvent water by the CD – Polarization pushes cavity toward lower field – Spheres, potential ~ r - 4. 4.Solvent cosphere free energy potential - repulsive or attractive, depending on the relative volumes of solute and solvent species. 5.Lewis acid-base interactions – attractive or repulsive (v.Oss). Qualitative account of potential: 1. repulsion due to the cavity created in the dielectric ( + r – 4 ); Fröhlich attraction (–r –3 ): U(r ) = + G/r 4 – E/r 3 Neglect Pauli repulsion (+r -12 ), Van der Waals attraction (-r -6 ). The potential U(r) minimum/gap equilibrium distance r e between the two CDs – a ‚diatomic molecule’ of 2 water CDs. Expand U(r) to 2 nd degree approx. harmonic potential: U(r) U(r e ) + U’(r e ) (r-r e ) + ½ U”(r e ) (r-r e ) 2 +... /2 (r-r e ) 2 + U(r e ), = U”(r e )

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The interaction potential between two CDs approx. around r e as a harmonic potential, the two CDs form a harmonic oscillator, with eigenfrequency: = ( / ) 1/2 – effective mass of the oscillator. Gap depth U(r e ) exceed thermal energy, avoid dissociation: |U(r e )| > 3/2 k B T At pysiol. T, 37 o C = 310 K 3/2 k B T = 6.45 10 -14 erg. Assume: water CD oscillator remains in ground state during cell lifecycle, define a minimum eigen-frequency: T = 310 K, min = 3k B T/2ħ, min = 0.97 × 10 12 Hz ~ 10 13 Hz – very close to the Fröhlich band upper limit.

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Min. frequency min in the harmonic potential ½ (r-r e ) 2 : min = min 2 = 4.7 10 -5 dyn/cm from U = ½ (r-r e ) 2 G/r 4 – E/r 3 must satisfy > min. An example – a possible potential of a CD of 15 nm radius: U = +0.021 / (R-15) 4 – 5 10 -5 / (R-15) 3 (0.021, 5 10 -5 – param.s) R e = 582 nm ~ 0.6 m ok, comparable to cell size; = 2.7 10 -4 dyn/cm > 4.7 10 -5 dyn/cm = min ok ; |U(R e )| = 7.1 10 -14 erg > 6.45 10 -14 erg = 3/2 K B T ok, not thermally dissociated; = 2.4 10 13 s -1 > 10 13 s -1 = min ok, slightly above Froehlich band; ħ = 1.6 10 –13 erg > 6.45 10 –14 erg = 3/2 K B T ok, oscillator excitation produces dissociation forbidden. Postulated potential – realistic.

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Two water coherent domains coupled in a spherical harmonic oscillator: maximum cell volume of small prokaryotic cells Two CDs – a spherical harmonic oscillator, in the center of mass coordinate system, distance d, reduced mass m: Harmonic potential: In the ground state, n r = 0 (n = 1), l = 0 (no orbital motion), m = 0, Gaussian wavefunction, of halfwidth d o : d 0 = d = ( 2 – ) 1/2

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The diameter 2a of a spherical cell equals the sum of equilibrium distance r e between CDs and a length proportional to halfwidth d 0 : c > 1; c = 4 for 4 ; probab. > 99.99 % for oscillator inside cell. In the ground state we take r e, for instance: r e ~ 1/2 = [3 ħ / m eff ]½ Cell radius a as a function of eigenfrequency : a = (3 ½ / 2 + 2. 2 ½ ) [ ħ / m eff ] ½ Postulate: In the living cell, the oscillator is in the ground state of energy E 000 = 3ħ /2. For stability, the thermal energy must be lower than the energy quantum ħ = E 100 – E 000 to first excited level:

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Maximum radius of a spherical cell: a < 0,987 µm, maximum volume V < 4,03 µm 3. Comparison of harmonic oscillator and spherical gap: 4,03 µm 3 harmonic spherical 0,42 µm 3 = V min < V cell < V max = oscillator 4,45 µm 3 impenetrable sphe- rical potential gap Concordance of radius better than 3 % the two models are consistent with, and sustain, each other. Experimental confirmation typical prokariotes Eubacteria, Myxobacteria 1 -5 µm 3, E. Coli 0.39 – 1.57 µm 3, small Cyanobacteria. Confirmation sustains a harmonic potential between CDs.

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The isotropic oscillator in a spherical potential well: maximum volume of larger prokaryotes and small eukaryotes Excellent agreement of a by spherical well and isotropic oscillator models both realistic no discrimination make a combined model isotropic harmonic oscillator enclosed in a spherical box with impenetrable walls larger than that required to accommodate only the oscillator. Centre of mass of the oscillator independent translation system with two freedom degrees. Cell – spherical well of radius a one particle of mass 2m eff translate in a smaller well of radius b + oscillator of reduced mass m eff / 2 in virtual sphere of radius r e +cd 0 : a = b + r e + cd 0 Perturbation treatment: Unperturbed energy levels in box : E n = 2 ħ 2 n 2 / 4 m eff b 2

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Energy difference between first two unperturbed levels : E 21 (0) = E 2 (0) – E 1 (0) = (3/4) 2 ħ 2 /4m eff b 2 E n levels of unperturbed Hamiltonian of the potential well. Wave functions: n (r) = (2/b) 1/2 sin (n r / b) The harmonic potential V(r) - centred at the half b/2 of radius V(r) = /2 (r-b/2) 2 Harmonic potential V – a small perturbation on the unperturbed functions. The shifts of the first two unperturbed energy levels, b V’ 11 = /b ∫ (r – b/2) sin 2 b/r dr = b 2 /4 (1/6 – 1/ 2 ) 0 b V’ 22 = /b ∫ (r – b/2) sin 2 2 b/r dr = b 2 /4 (1/6 – 1/4 2 ) 0 Their difference: V’ 22 - V’ 11 = 3/16 2 b 2 adds to the difference E 21 (0) between the unperturbed levels of the spherical gap.

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Difference between the perturbed first two levels E 21 (1), assumed higher than thermal energy: E 21 (1) = (3/4) 2 ħ 2 /4m eff b 2 + 3/16 2 b 2 > 3/2 k B T For the minimum oscillator frequency = min = 3k B T/2ħ → min : min = (3/2 k B T/ħ) 2 m eff /2 Obtained → 4 th degree equation in b (b ≠ 0) : 9m eff 2 k B 2 T 2 b 4 – 64 2 ħ 2 m eff k B Tb 2 + 32 4 ħ 4 = 0 with one real positive solution: b = ħ/(m eff k B T) 1/2 [2/3 (4 + 46 1/2 ) 1/2 ] = [2/3 (4 + 46 1/2 ) 1/2 ] a 0 = 2,1891 a 0 = 2,23 m Total maximum radius of the spherical cell obtained: a = [2/3 (4 + 46 1/2 ) 1/2 ] a 0 + 1/ [(2/3) 1/2 (3 1/2 /2 + 2 2 1/2 )] a 0 = = 3,1493 a 0 = 3.21 μm where a 0 = a 0 (T) = ħ/(m eff k B T) 1/2 = 1.02 m for T = 310 K. Maximum cell volume = 138.6 μm 3.

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V max = 138.6 μm 3 experimental confirmation – biological data: –Larger prokariotes Taxa Myxobacteria including extremes (V = 0.5 – 20 μm 3 ) Sphaerotilus natans (V = 6 – 240 μm 3 ) Bacillus megaterium (V = 7 – 38 μm 3 ). –The smallest eukayotic cells: Beaker’s yeast Saccharomyces cerevisiae (V = 14 – 34 μm 3, a = 1,5 – 2 μm), Unicellular fungi and algae (V = 20 – 50 μm 3 ), Erythrocyte, enucleated eukaryotic cell (V = 85 μm 3 ), Close to the lymphocyte (V = 270 μm 3 ). Yeast

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Correction of minimum cell volume/radius estimated on the basis of the Bose condensation, due to m eff (single free CD) 2m eff (two CDs in harmonic oscillator): V min decrease by a factor of 2 –3/2 = 0,3536 to 0.15 µm 3, a min decrease by 2 –1/2 = 0.7071, from 0.46 to 0.33 µm. Biological implication: included the smallest known cells, blue-green alga Prochlorococcus of Cyanobacteria genre (V = 0.1–0.3 μm 3 ), Mycoplasma (V = 0.35 m 3 ).

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The cylindrical potential well and the shape and size of discoidal cells: the erythrocyte A disc-like cell – a cylindrical well, of finite thickness a, radius r o. Along the rotational axis the problem reduces to a linear gap with impenetrable walls and the length a energy levels E n. In the circular section of the disk polar co-ordinates solution of the form (r, ) = f(r) g( ) radial part : Bessel functions of the first degree and integer index, f(r) = J l (r). Probability density vanish on the walls of the cylinder, J l ( r o ) = 0, radius given by the roots x lm of the function J l ( r), with energy eigenvalues E lm.

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No immediate restriction to the values of l, m (radial movement) with respect to n (axial movement). Total energy - sum of the two energies: E nlm = E n + E lm The only restriction for l and m – due to the obvious rule: E n < E n + E lm < E n+1. Total energy of an arbitrary quantified level: E nlm = ħ 2 /2m eff ( 2 n 2 /a 2 + x lm 2 /r o 2 ) Choose E 110 as the ground level, E 221 higher level. Impose E 221 – E 110 as a thermally inaccessible transition : E 221 – E 110 = ħ 2 /2m eff (4 2 /a 2 + x 21 2 /r o 2 - 2 /a 2 – x 10 2 /r o 2 ) ≥ 3/2 k B T x 10 = 0, x 21 ≈ 5.32 – first roots of J 1 (r) and J 2 (r) Bessel functions. We are lead to a second degree inequality, with the solution: r o ≤ x 21 a o a / (a 2 – a o 2 ) 1/2, for a ≠ 0, a > a o, where a o = ħ / (m eff k B T) 1/2 = 1,02 µm.

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Biological implications: The model neglects nucleus / the erythrocyte is an eukaryotic enucleated, non-replicating cell. The experimental confirmation of predicted shape and size - sustains water CDs dynamics in erythrocyte. According to our basic assumption that water CDs dynamics is essential for living state – the enucleated, non-replicating, but metabolically active erythrocyte is a living cell indeed. This sustains the general hypothesis of the „metabolism first, replication after” origin of life (Dyson). Radius r o of discoidal cell – monotonously decreases with thickness a. Thickness a ↔ radius r o. The ratio r o /a determines the cell shape. For a = 1.15 µm, r o ≤ 3.8 µm. Red blood cell: 2 µm thickness, 3.75 µm radius. The model describes a non-spherical cell, neglecting biconcave shape, rounded margins. Erythrocyte

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The cylindrical potential well and the shape and size of rod-like cells: typical bacilli Other radial levels E l’m’ – thermally inccessible biologically forbidden transitions between such levels. Model of cylindrical gap with impenetrable walls rod-like bacilli of typical size. Advantage used – liberty in choosing the l and m values of x lm roots of the Bessel functions J l (r). Approximate roots of Bessel functions for l + m > 2: x lm ≈ ¾ + l /2 + m Specific postulate – in the rod-like cell biologically relevant transitions leave unchanged the axial translation energy E n, n = 0 Some radial levels E lm fall between the E n levels – close of each other the lowest – thermally occupied. E. coli

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For n = 1 and n = 0, a thermally inaccessible state |1l’m’> defines a biologically forbidden transition |1lm> ↔ |1l’m’>. Thus: E 1l’m’ – E 1lm = ħ 2 /2m eff (x l’m’ 2 – x lm 2 )/r o 2 > 3/2 k B T r o < 1/ [(x l’m’ 2 – x lm 2 )/3] 1/2 ħ/(m eff k B T) 1/2 r o < 1/ [(x l’m’ 2 – x lm 2 )/3] 1/2 a o, with a o = 1,02 µm. Postulate: ground state |102>, „life-forbidden” transition |102> ↔ |121>. Substitute x 02 = 5.52 and x 21 = 5.32 roots of the J 0 and J 2 Bessel functions. radius r o < 0.28 µm or diameter 2r o < 0.55 µm; axial length a o = 1,02 µm; form ratio 2r o / a o = 0.54. Species2r o (µm)a o (µm)2r o /a o Calculated0.551,020.54 Brucella melitensis0.5-0.70.6-1.50.5-0.8 Francisella tularensis0.20.3-0.7 Yersinia pestis0.5-1.01.0-2.0~ 0.5 Escherichia coli0.5-1.02.0-2.50.25-0.4

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Other biologically forbidden couple of states: |103> ↔ |122>, 2r o = 0.41 µm, a o = 1,02 µm. Similar results with the pairs of states |113> ↔ |104>, |124> ↔ |105>, |125> ↔ |106>,.... Some of these levels may be unoccupied at 310 K. Empirical „selection rule” emerges for „biologically forbidden” transitions in relatively small, typical bacilli, with diameters close to half of a 1.02 µm length. : (l + m) 0, +1 The model neglects rounded ends of rod-like bacteria – and possible influence of inhomogeneous distribution of DNA inside. *** The model size and shape of axially symmetric cells – there are no intermediate cell shape between erythrocyte and bacilli. Some of the above assumptions still need sufficient rationales – they are postulates, justified so far only by results. Further studies needed – to describe larger bacilli.

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The toxic effect of heavy water and water coherent domains in a spherical well D 2 O and H 2 O chemical properties - almost identical; most physical properties difer by ~5 – 10 %, However, D 2 O induces severe, even mortal biological effects. Complete substitution with isotopes 13 C, 15 N, 18 O well tolerated. Effects - irreversible and much worse to eukaryotes than procaryotes. Looking for an explanation: 1)in the cell; 2)in the physical properties of D 2 O vs. H 2 O. 1) Eukaryotes – divided by organelles, prokaryotes – not. 2) D 2 O vs. H 2 O substantial physical differences: H + ion mobility (- 28.5%), OH - ion mobility (-39.8%), Ionization constant, Ionic product (-84,0%), Inertia moment (+100%).

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The unique twofold different physical property of D 2 O vs. H 2 O - inertia momentum of water molecule (m D 2 m H ): I(D 2 O) = m D d 2 2 m H d 2 = 2 I(H 2 O) Doubling of inertia momentum implies radically different physical properties of CDs in D 2 O and H 2 O, as evidenced in QED theory (Del Giudice et al 1986, 1988). Rotation frequecy w o of water molecule: Size d of a water CD: 67

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Effective mass m eff of CDs: Consequence: Substitution of H 2 O by D 2 O reduction to a half of water CD effective mass: The eukaryotic cell – approximated as an aggregate of small water- filled spheres of radius a closed by membranes. CDs confined in spherical wells with finite potential walls. 68 Postulate: The CDs’ potential barrier heigth admitted the same in H 2 O- and D 2 O-filled cells: U 0 = 4 ħ 2 /2m eff a 2 = 4 u = const. (4u – arbitrary)

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Constant U o – by compensation of opposed D 2 O effects due to lower ionization constant, ionic product, D + and OD - ions mobility, and of higher CD mobility due to lower m eff. For the spherical well of finite height there is a minimal heigth U min for the occurrence of the first quantified energy level: With m eff = m eff (H 2 O) and m eff (D 2 O) m eff (H 2 O)/2 the minimal height of well is double for D 2 O vs. H 2 O. The relation of U min vs. U o is thus fundamentally changed: U min (H 2 O) ~ 2.5 u < 4 u = U o U min (D 2 O) ~ 5 u > 4 u = U o

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U min (H 2 O) < U o U min (D 2 O) > U o In D 2 O-filled cells the first energy level is higher than the height of potential well – in contrast to the H 2 O-filled cells. Therefore the D 2 O coherence domains will not be in a bound state in the cell compartments – the CDs will move freely in the whole volume of D 2 O-filled eukaryotic cells. Contrarywise, CDs are bound in H 2 O-filled compartments of eukaryotic cells. This qualitative difference a totally perturbed dynamics of heavy water may explain D 2 O toxicity in eukaryotes. Eukaryotes internal membranes high D 2 O toxicity. Prokaryotes no internal membranes no qualitative CD dynamics difference of D 2 O vs. H 2 O low D 2 O toxicity. 70

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A last hour finding in rod-like bacteria: a possible proof of long-range interactions inside living cells A new mechanism in bacteria support CDs long-range interactions. Some proteins navigate in the cell sensing the membrane’s curvature. Proteins recognize geometric shape rather than specific chemical groups. Bacillus subtillis: DivIVA protein – convex; SpoVM – concave curvatures, i.e. poles of rod-like bacteria (Ramamurthi, Losick 2009). Protein adsorption model – explanation limited to highly concave membrane: curvatures of protein and cell are very different a single protein could not sense the curved surface cooperative adsorption of small clusters of proteins once a protein located on the curved membrane, may attract others.

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Long-range hypothesis for rod-like cells effect Limits of cooperative adsorption model: How is directed the first protein? Difficulty: proteins which recognize convex surface. Alternative explanation: Proteins are carried by long-range forces derived from strong potential gradients – as expected from our cylindrical well model and oscillating electromagnetic fields generated by CDs (Del Giudice). Attraction to the cell extermities superimposes a deterministic dielectrophoretic (Askaryan) force on Brownian motion. Probability of transport to curved cell ends much enhanced. Because the Askaryan dielectrophoretic forces can be attractive / repulsive specific proteins attracted by negatively / positively curved surfaces. Suggested test: different electrical characteristics of SpoVM (concave), DivIVA (convex), and of proteins not attracted. The effect first evidence of protein-cell long-range forces.

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Conclusions and final remarks

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Quantum biology is one among several approaches aiming of coming close to the collective, non-linear, “holistic” phenomena of the living cell, beyond the reductionist view of life given by molecular biology. A large variety of models – based on different assumptions – already succeeded to deal with biological facts unexplained by molecular biology. Long-range coherence and Bose-type condensation postulated in Fröhlich’s theory as essential features of living systems, explain many biological phenomena. Long-range interactions in cells - experimentally proved. Coherence – proved in photosynthesis. Models of water consistent to Fröhlich’s theory explain its remarkable properties and its key role in living cells.

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A ionic plasma model explains the ‘second sound’ and more usual properties of water (Apostol & Preoteasa). The QED model of water CDs explains water anomalies, dynamical order in cell, cell activity effects, Zhadin effect and ICR, etc. (Preparata, Del Giudice). The cell size (~1-100 m) – between classical and quantum – a spatial scale for a specific dynamics. A quantum model: size vs. metabolic rate (Demetrius). We propose new, metabolism-independent, quantum models for cell size, based on CDs’ low mass (12-13.6 eV) dynamics (Preoteasa and Apostol). The models suggest that cell size and shape selected in evolution, fit the size and shape of potentials and QM wavefunctions describing water CDs dynamics.

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Bose-type condensation may explain lower size limit. Impenetrable spherical well, isotropic oscillator, isotropic oscillator in spherical well, explain upper size limits of cocci, yeast, algae, fungi. Axially-symmetric wells (disk-like, rod-like) explain size / shape of erythrocyte and typical bacilli. Cell shape sensing by proteins in bacilli backs model. A model of spherical well with semipenetrable walls explains the toxic effects of D 2 O, much stronger in eukaryotic than in prokaryotic cells. Explanation of D 2 O toxicity sustains water-based QM models! The same model connects D 2 O toxicity and cell size/shape – two very different phenomena. QM water dynamics models still provide a vast potential for further explaining other cellular facts.

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