PHYSICAL REVIEW E 91, 032704 (2015)

Quantum decoherence time scales for ionic superposition states in ion channels V. Salari,1,2’* N. Moradi,1 M. Sajadi.3 F. Fazileh,1 and F. Shahbazi1^ 1Department o f Physics, Isfahan University o f Technology, Isfahan 84156-83111. Iran 2Foundations o f Physics Group, School o f Physics, Institute fo r Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran 3Department o f Physics, University o f Shahrekord, Shahrekord 88176-53849, Iran (Received 1 November 2014; published 9 March 2015) There are many controversial and challenging discussions about quantum effects in microscopic structures in neurons of the brain and their role in cognitive processing. In this paper, we focus on a small, nanoscale part of ion channels which is called the “selectivity filter” and plays a key role in the operation of an ion channel. Our results for superposition states of potassium ions indicate that decoherence times are of the order of picoseconds. This decoherence time is not long enough for cognitive processing in the brain, however, it may be adequate for quantum superposition states of ions in the filter to leave their quantum traces on the selectivity filter and action potentials. PACS number(s): 87.16.Vy

DOl: 10.1103/PhysRevE.91.032704

I. INTRODUCTION

Mainstream cognitive neuroscience has so far largely ignored the role of quantum physical effects in the neuronal processes underlying cognition and consciousness. Classical physics is viewed by most scientists today as an approximation to the more accurate quantum theory, and therefore due to the nature of this classical approximation the causal effects of our conscious activity on the material substrate may appear to be eliminated. Flowever, there are many controversial and challenging discussions about this issue. The challenge is mainly because of the quick decoherence of quantum states due to the hot, wet, and noisy environment of the brain, which does not allow long-lived coherence for brain processing. Despite these critical discussions, there are only a few published papers on the numerical aspects of decoherence in neurons. Perhaps the most important issue is offered by Max Tegmark [ 1], who has calculated decoherence times for the systems of “ions” and “microtubules” in neurons of the brain. Tegmark has shown that decoherence times for superposition states of ions in neurons are of the order of 10“ 19 to 10-20 s, and those for the superposition states of microtubules are of the order of 10-13 s, which is not long enough for quantum states to survive for quantum processing in the brain. Hagan et al. [2] and Rosa and Faber [3] have corrected the decoherence times of Tegmark for quantum states of microtubules in neurons, and some of their results indicate that there are still possibilities for quantum coherent states to be effective in brain processing. The big picture of comparisons between the above approaches is given in Table I. where the main reason for decoherence is because of the interactions between the system and environmental particles due to scattering. Regarding ionic superposition states, Tegmark has used a simple model and did does not consider the real structure of ion channels, which are responsible for ion displacement through the membrane and are the building blocks of electrical membrane signals in neurons. Ion channels are proteins in the membranes of excitable cells (e.g., neurons) that cooperate

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in the onset and propagation of electrical signals across membranes by providing a highly selective conduction of charges bound to ions through a channel-like structure. The question here is whether quantum superposition states in ion channels can affect signal propagation in neurons of the brain. Tegmark has calculated decoherence times for quantum superposition of ions crossing the entire membrane based on a simple ion-pore diffusion model [1]. The calculated decoherence times for crossing ions in his work are derived from the scattering of environmental particles based on the Coulomb interaction between ions and particles. He has assumed that ions are in a superposition state “inside” and “outside” of the cell and are separated by a distance of 10 nm as the thickness of the membrane. In the view of atomic scaled resolution maps and recent molecular dynamics (MD) studies of the filter region in this protein, this type of interaction is oversimplified and the pore-diffusion scattering model is not applicable for description of ion-protein interactions. Here, we reinvestigate decoherence times for ionic superposition states in the real structure of an ion channel by using the data obtained via MD simulations, which have not been studied before. II. THE SELECTIVITY FILTER STRUCTURE IN ION CHANNELS

The selectivity filter is a part of the protein forming a narrow tunnel inside the ion channel which is responsible for the selection process and fast conduction of ions across the membrane. The determination of the atomic resolution structure of the ion channel and selectivity filter by MacKinnon led to his award of the Nobel prize for chemistry in 2003 [4], The 3.4-nm-long KcsA channel is comprised of a 1.2nm-long selectivity filter that is composed of four P-loop strands. Each P loop is composed of five amino acids—T (threonine, Thr75), V (valine, Val76), G (glycine, Gly77), Y (tyrosine, Tyr78), G (glycine, Gly79)—linked by peptide units (H -N -C = 0), where N -C = 0 is an amide group and C = 0 is a carbonyl group. Carbonyls are responsible for trapping and displacement of the ions in the filter (see Fig. 1). At the atomic scale the filter region exposes negative charges owing to the lone electron pairs from oxygen ions bound to 20

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©2015 American Physical Society

SALARI, MORADI, SAJADI, FAZILEH, AND SHAHBAZI

PHYSICAL REVIEW E 91, 032704 (2015)

TABLE I. The history of calculations for decoherence times in neurons.

System

Approach

Soliton-ion Ion-ion Ion-water Ion-distant ions Soliton-ion MT-ion MT-ion MT-dipole

Tegmark Tegmark Tegmark Tegmark Hagan et al. Hagan et al. Rosa and Faber Rosa and Faber

Year [Ref. No.] 2000 2000 2000 2000 2002 2002 2004 2004

C/3

1

III. CALCULATION ON THE BASIS OF DECOHERENCE TIM ES

10~13 s IQ-20 s O

carbonyl groups arranged into five rings o f the lining P-loop peptide. All together this structure provides a highly ordered atom ic coordination pattern am ong oxygens and ions. If a positive charged ion, such as sodium or potassium , enters the selectivity filter, the ion can be transiently trapped by C oulom b interactions, w ith the negative charges provided by the oxygen ions. The selectivity filter accounts for roughly one-sixth o f the total transm em brane length o f the channel protein. Congruent w ith the crystallographic K+ electron density profiles, adjacent K+ locations are separated by 0.3 nm in their S 1-S 3 and S 2 -S 4 configurations [5], From a quantum m echanical point o f view, these separations can be seen as quantum superposition states o f ions w ith a 0.3-nm distance in the selectivity filter. It was recently suggested that quantum effects in the selectivity filter could be the reason for highly efficient functioning o f ion channels [6,7],

Order of decoherence time

O

Microtubule (MT) Neuron (ions) Neuron (ions) Neuron (ions) MT MT MT MT

Mechanism (collision/interaction)

10_i9 s 1 0 7—10~6 s 1 0 2-10 1 s 10-9 s 10“ 16 s

[1] [1] [1] [1] [2] [2] [3] [3]

w here p is the density m atrix in the position basis and P refers to the Fourier transform . Then the tem poral distribution o f the scattering event can be m odeled by a Poisson distribution with intensity A = er, where a is the cross section and 0 is the flux. By this assum ption, the evolution o f the density m atrix can be w ritten as [8]

p(x,x\t + dt) = p(x,x')P(x - x \ t ) A d t +

p ( x , x ’)(l - A d t ) ;

v-----------------------------------------------------------------------'

v----------------------- "V ------------------------- '

effect of intracted particles

effect of nonintracted particles

(2)

arranging this equation leads to

d p ( x, x', t )

------—----- = - A(1 - P ( x - x , t ) ) p ( x , x ) . at '------------- v------------- >

(3)

F ( x - x ')

The above equation has a sim ple solution. We can define the decoherence tim e as the tim e in w hich p reaches e ~ 1 tim es its initial value, so 1

T egm ark’s calculations to obtain decoherence tim es [1] are based on the scattering o f environm ental particles from the system . In fact, scattering changes the density m atrix o f the system m athem atically via m ultiplication to a function w hich is the Fourier transform o f the probability function o f the transform ed m om entum to the system,

P f ( x , x ' ) = P i( x, x' )P (x - x'),

(1)

in order to calculate P we can w rite its cum ulant expansion as In P(r) = ~ ( Q j ) cf j - ~ ( Q i Q j ) crirj + 0 ( x 3),

(5)

w here ( Q,)c and (Q, Q j ) c are, respectively, the first and second cum ulants o f the probability distribution P. If we consider the

Z (nm)

X (nm)

FIG. 1. (Color online) Left: Representation of a KcsA ion channel. Right: Two P-loop monomers in the selectivity filter, composed of sequences of TVGYG amino acids [T (threonine, Thr75), V (valine, Val76), G (glycine, Gly77), Y (tyrosine, Tyr78), G (glycine, Gly79)] linked by peptide units H -N -C =Q .

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PHYSICAL REVIEW E 91, 032704 (2015)

QUANTUM DECOHERENCE TIME SCALES FOR IONIC . ..

(a)

(b)

FIG. 2. (Color online) Schematic difference between Tegmark’s model [1] and the real model, in which Tegmark has not considered the ion channel structure, (a) Tegmark’s model [1], in which two ions are in a superposition state of “inside and “outside” of the membrane. The superposition distance is the thickness of the membrane. Tegmark has assumed 10 nm for the thickness and therefore the superposition distance, (b) The real model, in which quantum superposition states occur in the selectivity filter of the ion channel. The superposition distance here is 0.3 nm. The 3.4-nm-long KcsA channel is comprised of a 1,2-nm-long selectivity filter that is composed of four P-loop monomers. Each P loop is composed of five amino acids linked by peptide units (H -N -C = 0 ), where N -C = 0 is an amide group and C = 0 is a carbonyl group. If a positive charged ion, such as sodium or potassium, enters the selectivity filter, the ion can be transiently trapped by Coulomb interactions, with the negative charges provided by the oxygen ions.

isotropic distribution, we have simply ( Q) c = (Q) = 0 and the covariance matrix is proportional to the identity matrix (Qi Qj ) c = s2Sij, and thus we obtain / ( s2l * - * 'l2\ P(x - x ) = exp ( ------- — -----J ,

(6)

and consequently the decoherence time is determined as follows: 1 ^dec

(7)

A ( l - e x p (-£ !£ £ ))■

We can now define a characteristic length by using the de Broglie relation as Aeff = h/s [8]. In the scattering approach, the decoherence time can be expressed in two limits, long and short wavelengths [9]: A’

Vice —

V A |A * |2 ’

X «( | Ax | ; X » | Ax | .

anticipate that it will take a large number of scattering events to induce a significant degree of spatial localization of the object [9]. Considering Tegmark’s calculations for the system of a neuron [ 1], he has assumed a quantum superposition of resting and firing states for an order of a million ions being in a spatial superposition inside and outside the axon membrane, separated by a distance of about S ~ 10 nm (see Fig. 2). At room temperature, the de Broglie wavelength of Na+ is about A 0.03 nm and the spatial separation is 8 ~ 10 nm (which is assumed to be the superposition distance in the coherent state). So, based on Tegmark’s assumptions, the calculations should be done in the short-wavelength limit X