Quantum theory is one of the greatest discoveries of the twentieth century. In recent years, quantum effects in biological systems have been discovered in several areas, including olfaction,^[1,2] avian magnetoreception,^[3,4] photosynthesis,^[5–9] quantum entanglement in living bacteria,^[10] and so on.^[11] The theoretical study of quantum effect in bio-systems and its possible relevance in explaining the functional properties of these systems has also drawn considerable attention, such as consciousness in the brain.

How to explain consciousness? Classical or quantum? Consciousness is very mysterious and over the years researchers have proposed many models.^[12–24] Some studies have suggested that the quantum effect might play an important role in the functioning of the brain.^[16–24] Penrose and Hameroff proposed the orchestrated objective reduction (Orch OR) model, which suggests that microtubules (MTs) in neurons act as a quantum computer,^[18–20,24] Fisher proposed that quantum entanglement may exist between two neurons.^[23]

Decoherence is an important phenomenon in quantum information. The “warm, wet, and noisy” environment might destroy the quantum state,^[26–29] thus the decoherence time scale τ is an important parameter to the quantum model.

To study the decoherence process in MTs, researchers determined this parameter according to different mechanisms of decoherence, including quantum gravity,^[18] cavity quantum electrodynamics (QED) model,^[25,26] and single ion-MT interactions. These results are listed in Table 1, which shows that the value of decoherence time varies greatly.

Table 1.

Table 1.

Table 1. Decoherence time scales and their mechanisms. . Mechanisms Researchers Timescale Quantum gravity Penrose (∼10 ms Cavity QED Mavromatos 0.1 ns ∼0.1 ms Ion-dipole Tegmark ∼0.1 ps Ion-dipole Hagan et al.

Table 1. Decoherence time scales and their mechanisms. .

Decoherence mainly derives from the interaction between quantum systems and the environment. Four basic interactions that have been discovered in nature: in the range of molecule interactions, the main interaction between environment and tubulin dimers is electromagnetic interaction. In this paper, a model based on the electromagnetic interaction Hamiltonian between microtubules and plasmon in the neurons is proposed. Previous studies considered the effect of a single ion on the decoherence process in MTs; however, cells are known to contain different kinds of ions that have different charges and masses, i.e., some ions have positive charge, whereas others have negative charge. Over a long time scale, cells can be considered to be electrically neutral; however, this is not true over very short time scales. Therefore, the decoherence rates cannot be calculated only considering the effect of a single ion since decoherence is a result of the interaction between tubulin dimers and cellular fluid environment. In this paper, we construct the interaction Hamiltonian by using the second quantization method, and the decoherence time is estimated according to the interaction Hamiltonian.

This article is organized as follows. Section 2 includes the introduction for decoherence mechanisms in our model, and also the total Hamiltonian of tubulin dimers and cell fluid environment. The decoherence timescale τ are computed and how τ changes with environment parameters will also be discussed. In Section 3, other mechanisms of decoherence will be discussed. Finally, some important formulas and their derivations are given in the appendix.

In this section, the decohenrence mechanisms in MTs will be discussed. A MT is a hollow cylinder with an outer diameter of 24 nm and an inner diameter of 15 nm. The basic unit of MT is tubulin dimer, which has two subunits (denoted by α and β). All of the tubulin dimers form MT crystal lattice by helical encircle. The tubulin dimers have different kinds of conformational states, which are regarded as quantum bit in Orch model, and MTs can store information thanks to different combinations of these conformational states. Electron transition in each tubulin dimer could change the conformational states. The MT is a polar molecule and it has an intrinsic electric dipole moment (fig. 1).^[25]

Fig. 1.

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	Fig. 1. The structure diagram of MTs.

2.1. Mechanisms of decoherence

Cellular fluid is considered to have both positive and negative charges (similar to plasma). Consequently, two basic and very important parameters can be used to describe it, namely: Debye length λ_D and plasma frequency ω_p. These two parameters will be discussed compendiously and their range will be given later in this paper.

The Debye length λ_D represents the space scale when the plasma is kept as a neutral state and is determined by

where is the average density of the k-th kind of ion, q_k is the quantity of charge, is the dielectric constant of water, k_B is the Boltzmann constant, and T is the temperature of the cellular fluid. For physiological Ringer solution, λ_D ∼0.7 nm,^[29] and in the following calculation the value of λ_D is set to be around 0.7 nm.

The surface of tubulin dimers have net charge,^[30] so a counterion layer will be formed because of the Debye shielding. The thickness of the counterions is approximately λ_D, as shown in Fig. 2. The counterions could shield the interaction between MTs and the environment, as shown in Appendix C, although the coupling coefficient is decreased if the shielding effect is considered.

Fig. 2.

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	Fig. 2. Schematic diagram of counterion layer with a thickness of λD.

The second parameter is called plasma frequency, which describes the collective oscillations of ions and is determined by

For typical parameters in a cell,^[29] , , . Therefore, in the following calculation, the value of ω_p is set to be around 0.6 THz.

When the plasmon is in an excited state, the electric neutrality is destroyed, and some net charges appear. The net charges can interact with the dipole in the tubulin dimmers, as shown in Fig. 3. As shown in Appendix A, the local ion density fluctuation could excite ion density waves. There are different ion density waves but the only one called plasma oscillation could be coupled with MTs. The dispersion relation of plasma oscillation is

where β is the average value of ion thermal velocity, which has the same order of magnitude as the thermal velocity.

Fig. 3.

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	Fig. 3. Schematic diagram of the coupling between tubulin dimers and the cellular fluid environment.

The total Hamiltonian of the MT-environmental systems can be derived as follows:

where is the Hamiltonian of the excited systems in the MTs, is the Hamiltonian of the plasmons in the cellular fluid environment, and represents the interaction between the MTs and cellular aqueous environment caused by the interactions between the dipole and net charges. is the reason for decoherence; if , then the decoherence time is .

As shown in Fig. 3, the interaction Hamiltonian between a single tubulin dimer with the electric dipole moment and the cellular fluid environment can be shown as follows:

2.2. Computation method for a decoherence timescale

Now, the total Hamiltonian equation (4) will be derived; some basic assumption or approximation is listed below, and will be discussed in Section 3 and in an appendix.

(i) Water is treated as an medium with a dielectric constant , and detailed interaction of ion-water molecules and MT-water molecules is ignored.

(ii) Due to the Debye shielding, plasma oscillations could only be excited above the Debye length, that is to say, the wave numberkhas an upper limit of in our calculation, we consider k will decay rapidly as a small quantity for the short wavelength modes.

(iii) Random phase approximation (RPA) for many-particles system, In equilibrium state or near equilibrium state, as the position of particles is random, unless k = 0.

(iv) The tubulin dimers are seen as a mass point with electric dipole moment .

As introduced in Subsection 2.1, the tubulin dimers have different conformational states, denoted by , and let be the creation operator and annihilation operator of the quantum state , so the Hamiltonian of tubulin dimers can be expressed by

where is the eigenvalue of quantum state the numerical estimation of can be seen in Ref. [21].

The detailed calculation of , will be given in Appendix A and Appendix B, and the total Hamiltonian of the MTs and cellular environment can be expressed as follows:

where is the dispersion relation of plasma oscillation, and are the creation operator and annihilation operator of plasma oscillations, respectively, and the coupling coefficient is given by

Equation (8) is given in Appendix C, and is the electric dipole moment in state .

Next, Tolkunov’s model is used,^[31,32] which describes the interaction between the spin system and Boson thermal reservoir. In 2-level approximation, the Hamiltonian equation (7) of our model is the same with that of Tolkunov’s in form, so the non-diagonal elements of density matrix will also change with time in the same way

Here, , and the integral region is . The plasma-like oscillations will be excited only when the wavelength is larger than the Debye length.^[33] Therefore, it could only be used to refer to the excited state, and equation (9) will becomewhere

Obviously, . In the quantum information theory, the decoherence process is reflected in the damping of the nondiagonal element of the density matrix, so we define decoherence time τ as the timescale when decays into , namely

Equations (11) and (12) could be used to compute decoherence time.

2.3. Typical order of magnitude of the decoherence timescale

In this section, the typical value of decoherence time scale will be estimated by Eqs. (11) and (12). The parameters in Eq. (10) are chosen as follows:

is the electric dipole moment of tubulin dimer.^[25]

is the dielectric constant of water.^[29]

, are two basic plasma parameters, which have been discussed in Subsection 2.1.

T = 310 K is the environment temperature.

since it has the same order of magnitude as the thermal velocity.

is the Boltzmann’s constant.

is the Planck’s constant.

The function G(t) can be computed in a numerical method (fig. 4). Set and the decoherence time could be easily obtained

The decoherence timescale is about 10 fs.

Fig. 4.

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	Fig. 4. Schematic diagram of how to compute the decoherence time by the exponential factor G(t).

2.4. The dependence of decoherence time with other parameters

Decoherence time may change with other parameters; how these parameters affect the decoherence time will be studied in this section. As discussed in Subsection 2.3, the typical time scale for decoherence is T₀=10 fs. So we set T₀=10 fs as the time unit,. The six dimensionless physical quantities are shown below

Then, equations (11) and (12) become The typical values of these parameters are given in Subsection 2.3; in this section, their values are given in a wide range as follows:Here, some values may never be reached, such as T = 900 K, , and so on; however, the purpose in this model is to analyze how the decoherence time changes with physical parameter, so the parameter distribution is in a very wide range.

Case 1: Decoherence time changes with plasma frequency ω_p

As shown in Fig. 5, decoherence remains almost unchanged when the plasma frequency changes.

Fig. 5.

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	Fig. 5. Decoherence time changes with plasma frequency ωp when other parameters are consistent with those in Subsection 2.3.

Case 2: Decoherence time changes with average thermal velocity β

As shown in Fig. 6, decoherence remains almost unchanged when the average thermal velocity β changes, similar to Case 1.

Fig. 6.

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	Fig. 6. Decoherence time changes with average thermal velocity β when other parameters are consistent with those in Subsection 2.3.

Case 3: Decoherence time changes with Debye length λ_D

In Fig. 7, the decoherence time increases with Debye length; since the plasma oscillation modes could only be excited when , a larger Debye length means that fewer modes will be excited, so the number of the modes interacting with MTs will decrease, and the decoherence time will increase.

Fig. 7.

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	Fig. 7. Decoherence time changes with Debye length λD when other parameters are consistent with those in Subsection 2.3.

Use to fit the curve in Fig. 7 (or equivalently , the power exponent s and linearly dependent coefficient for are

Doing the same work to other parameters and we find that could fit the relationship between τ, , so we can approximately consider that

Case 4: Decoherence time changes with dielectric constant of water

In Fig. 8, the decoherence time increases with dielectric constant of water, and the reason is obvious. According to Eq. (5), a larger dielectric constant means the weaker interaction between MTs and environment.

Fig. 8.

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	Fig. 8. Decoherence time changes with dielectric constant of water when other parameters are consistent with those in Subsection 2.3.

Doing the same work as Case 3, we find that

Case 5: Decoherence time changes with dipole moment of tubulin dimer p_n

In Fig. 9, we show the decoherence time decreases as the dipole moment of tubulin dimer increases. According to Eq. (5), the increase of the dipole moment will enhance the interaction between MTs and environment, and then the decoherence time will decrease.

Fig. 9.

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	Fig. 9. Decoherence time changes with dipole moment of tubulin dimer pn when other parameters are consistent with those in Subsection 2.3.

Doing the same work as Case 3, we find that

Case 6: Decoherence time changes with environment temperature T

In Fig. 10, the decoherence time decreases as the environment temperature increases, and it is also easy to understand. The higher temperature means that more oscillation modes will be excited, and this will have a greater impact on the MTs, so the decoherence time decreases.

Fig. 10.

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	Fig. 10. Decoherence time changes with environment temperature T when other parameters are consistent with those in Subsection 2.3.

Doing the same work as Case 3, we find that

According to Eqs. (16)–(19), the decoherence time could be approximately expressed asSince the decoherence time relies less on ω_p, β, equation (20) will be changed into:In fact, equation (21) could be proven because the plasma frequency , the decoherence time –100 fs, and the temperature . Therefore,Under the condition of Eq. (22), equation (12) could be approximately expressed as

Then, the decoherence time satisfies

Equation (24) could only be used to calculate the decoherence time under the condition of Eq. (22). However, equation (24) is useful for various actual parameters.

If the Orch OR model can be verified both in theory and experiment, then the influence will be inestimable; however, the conformational state is affected by the “warm and wet” cellular environment, and the decoherence time is a very important parameter.

In this paper, the decoherence time scale is even smaller than 0.1 ps. This timescale is so short that the quantum state will soon be destroyed by the cell solution environment. This model only considers the coupling between the tubulin dimers and ions in the cellular fluid system, treating the water as a medium and overlooking the interactions of MTs-water molecules. Water molecules may shield some interactions of ion-MTs, and the interaction of water-ions and water-MTs may have influence on the decoherence process.^[32,33] According to Eq. (24), if the interaction strength attenuates to ε , then the decoherence will increase to than before. Enough decoherence requires and the strength of shielding by water molecules needs to be measured by experiment.

Another mechanism for decoherence that is not considered is the coherent pumping of the system via the environment.^[21] According to Fröhlich’s theory, if a system is strongly coupled to its environment via some degrees of freedom, and a coherent pumping source exists in environment, then it might inhibit other degrees of freedom known as coherent oscillations.^[35,36] These oscillations might increase the decoherence time. Guanosine triphosphate (GTP) hydrolyzation in the cells might act as a pumping source. However, this mechanism was not considered in this paper.

Decoherence is an important phenomenon in quantum information. Decoherence mainly comes from the interaction of quantum systems with the environment. In the range of molecule interactions, the main interaction between environment and tubulin dimers is the electromagnetic interaction. The electromagnetic field comes from ions and thermal radiation of the environment. However, in this model, the thermal radiation is ignored and, in the range of room temperature, the thermal frequency spectrum mainly concentrates in the range of the THz band. The water molecules in the cell environment could strongly absorb the THz photon, while the model only takes into account the electromagnetic field from ions. Besides, if the thermal radiation is considered, then the decoherence time would be smaller than the result given before and this will not change the conclusion.

This model needs to be verified both experimentally and theoretically. This model may offer a helpful theoretical framework to compute the decoherence time in quantum bio-systems, even though the environment of biological system is different. However, the electromagnetic interaction is essential in the scale of molecules, so this model could be used for reference when dealing with the interaction between the ions in cell environment and dipoles of bio-molecules. A direct experiment to verify this model is hard to be carried out at this time but with the development of ultrafast biophysics, quantum information, quantum optics, and imaging technology,^[37–41] the experiment could be carried out in the future.