Quantum physics determines the cooperatively of various many-body systems like solids and superconductors suggesting that quantum physics may also play a cooperative role in biology. The appeal of this idea is that quantum coherence emulates an attribute of living cells whereby many degrees of freedom are reduced to a collective mode enabling long range coordination and order—a property clearly analogous to the coordination of individual processes in living cells. Fröhlich initiated this concept and advocated that all living cells maintain coherent vibrations generated by metabolism [1] . The present note extends this view and proposes that a cooperative process exists in water in the cell. As water is the majority substance of the cell, quantum cooperative behavior in water could have a pervasive influence.

Schrödinger posited that quantum effects might distinguish living from non-living matter revealing “new” physical behavior [2]. Until recently, this seemed unlikely because quantum effects readily decohere in warm wet environments like the cell. Investigators have since confirmed that quantum effects exist in photosynthesis [3] and other biological phenomena [4]. The model cooperative process proposed here originates from waves of energy propagating through water in cells. For our present purposes it is not necessary to identify a metabolic origin for this energy. Water has long been known to transfer energy at an ultrafast rate following Woutersen and Bakker [5]. The agents of these energy transmissions are water dimer oscillations that spread by successive excitations of dipole-dipole interactions (a Förster transfer mechanism) [5,6]. According to the matter-wave duality of quantum physics, the resulting energy waves have corresponding particle quanta here termed chaions. We address the decoherence problem and find the waves are sufficiently short and fast to evade decoherence. The model is then shown to reproduce several empirical results and make testable predictions of wave speed and wavelength.

Quanta of the ultrafast energy transfers do not convey mass so chaion particles are analysed using a statistical treatment similar to that for a massless photon gas. This treatment is detailed in [7] and here we summarize results needed for our applications.

Integrating the occupation number over phase space gives the average number of energy-carrying particles, ánñ,

where v is the particle velocity, h is Planck’s constant, k_B is Boltzmann’s constant, T is temperature, and V is the volume of water. Similarly, integrating chaion energies occupied over all of phase space gives the average energy vested in the quanta with the result that the average energy per chaion particle is 2.701 k_BT. Hereafter we abbreviate the average energy per particle as 2.7 k_BT while using the more precise value in computations. Frequency f is given as usual by f = 2.7 k_BT/h and the energy density ρ is uniform throughout the volume.

We now determine chaion velocity and wavelength directly from the model. Although relativity requires that massless particles in vacuo must propagate at the speed of light, c, water transmits energy at a nonrelativistic speed, v. This is easily reconciled if excited water dimers transmit energy successively with probability p such that the effective speed is a fraction of light speed p = v/c (we neglect the transit time between dimers).

Consider an assembly of N water molecules containing one excited dimer able to emit its energy of 2.7 k_BT with probability v/c. Let V_w represent the volume of a spherical water molecule with effective diameter 0.32 nm [8] . Eq.1 becomes

Alternatively, the probability of a single emission is also expressed by the canonical distribution for the dimer energy attributed to one molecule.

.

The latter two equations are solved for N and v at standard temperature giving N = 129 and v = 156 km/s. (This speed coincides with that of an electron excited with chaion energy 2.7 k_BT.) The wave relation λf = v now determines the characteristic wavelength is λ = 9.3 nm.

Although a warm wet cell seems an unlikely place for coherent quantum processes, the waves considered here are sufficiently fast and short enough to avoid decoherence. The characteristic times considered in the chaion theory are about while biochemical reaction times are larger by orders of magnitude. The measured equilibration time for pure water is 0.55 ps [9] exceeding chaion transit time almost tenfold. Energy waves therefore “see” their environment as being relatively timeindependent.

The chaion model presupposes that quantum coherence extends over wavelength λ in the warm wet cell. Coherent quantum behavior requires that the time for decoherence t exceeds the characteristic chaion transfer time. An approximation for massless particles with relatively long wavelengths and high temperature is given by Schlosshauer [10] ,

where t is the characteristic time to decohere in a distance by a factor of e and L is a scattering constant,

parameterized by wave speed, temperature, and water molecule size. Substituting the chaion quantities v and T = 310 K with in Eq.2 gives, two orders of magnitude longer than the chaion transfer time.

The plausibility of quantum coherent states existing in biological systems has been addressed elsewhere [11] from fundamental considerations. In contrast, our approach here was to accept the existence of coherence at the outset and then explore the consequences of the assumption. We then see after the fact that the proposed mechanism does avoid decoherence.

The model is now tested against experimental data. We find accurate numerical agreements for the dipole moment of water dimers, microwave radiation on yeast, and the Kleiber law of metabolic rates. The computations use only the model quantities and standard physical values with a minimum of mathematics.

This note posits that ultrafast energy transfers in water are a basis for cooperative properties in living cells. Direct applications of the model accurately determined disparate findings for microwaves on yeast, the Kleiber law, and the dipole moment of water dimers. It is noteworthy that no free parameters were introduced in these successful applications.

The model also predicts wave speed and wavelength as well as the energy of the associated particles. These are subject to experiment and await confirmations.

The theory implies that chaion waves tend to be selfperpetuating. Harmonic perturbations induce transitions in the neighborhood of a resonant frequency. As a familiar example, photons matching the resonant frequency of target molecules cause cascades of coherent light by stimulated emission. Similarly, chaions match the resonant frequency of excited water dimers and induce further chaion emissions. Such emissions are more probable than competing reactions and the quantum system favors this state over a multitude of possibilities. The system can then select the best choices for its perpetuation perhaps including enzyme-substrate matches or metabolic reactions that can support chaion production.

In this view the living cell favors chemistry that sustains chaion production. Proto-life forms can be imagined to select primitive enzymes and even coding systems without having to assemble the separate parts serially. Patel [23] argues that the universal coding system of 4 nucleotide bases and 20 amino acids results from an optimum quantum database search according to the Grover algorithm [24] . This may be plausible in a quantum environment on the order of ~9 nm. A sphere enclosing a single chaion is readily found from Eq.1 to have a diameter ~20 nm. If the quantum effect is required for a living state, this is a lower bound for cellular dimensions in this theory. Nanobes this small are observed [25], but are not universally accepted to be living.

Although these projected consequences are intriguing, the model must be supported first by quantitative results. We’ve seen several results that are consistent with the premise that cellular life is characterized by waves of rapid energy transfers in water. This process may assist in the chemical organization and coherence of the cell and we conclude that further study of the model is warranted.