QUANTUM CORRELATIONS IN
COOPERATIVE TUNNELLING
PROCESSES
FOR A MODEL OF BIOLOGICAL ION
CHANNELS
David Goodwin
david.goodwin@liverpool.ac.uk
12:00, Tuesday 24th August 2010
Outline
1
Biological Ion Channels
Creating a Model
2
Quantum ion channels
Master Equation and Liouvillian
Multi-level wells and themalisation
3
Simple quantum systems - electrons
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Motivation for Research
• Ion channels dictate charge transfer through cell membranes.
• Channels contain a small number of ions at any one time.
• Selecivity to particular ions.
• Diameter of channel is comparable to the size of an ion.
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Selectivity Filter
(a) Plan view
(b) Cross-sectional view
(c) Selectivity filter
• Four binding sites within selectivity filter 1 .
• Model as square potential wells of equal depth at V = 0.
1
Images of a potassium ion channel from PRB:1K4C
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Model of selectivity filter
Energy, eV
V = 0.0
V = −0.1
V = 0.1
0.16
0.16
0.16
0.14
0.14
0.14
0.12
0.12
0.12
0.1
0.1
0.1
0.08
0.08
0.08
0.06
0.06
0.06
0.04
0.04
0.04
0.02
0.02
0.02
0
0
0.5
1
0
0
0.5
1
0
0
Distance from outer membrane barrier, nm
0.5
1
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
1
Coherent evolution between adjacent binding sites.
2
Incoherent jumps into, and out of particle reservoirs.
3
Radiative decay to lower discrete energy levels within potential
well.
4
Non-radiative decay (mainly thermal dissipation) to an
energetic equilibrium at finite temperature.
5
Dephasing
6
Thermal hopping between sites
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Unconditional master equation
dρ
=
dt
i
X
1 h
Ĥ, ρ +
ΓD[σ̂r ]ρ + ΓD[σ̂r† ]ρ +
i~
r =L,R
X
†
(n̄ + 1)γD[âw ]ρ + n̄γD[âw
]ρ
w
1
n̄ =
exp
~ω
KB T
−1
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Stochastic evolution
• Uncondition master equation is useful to find analytic solution
for small systems.
• Use stochastic evolution for larger systems.
• Point processes characterise quantum jumps.
• Hamiltonian evolution for coherent character.
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Single well test, for ions
1
0.9
0.8
V = 0.0
V = −0.1
V = 0.1
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0.6
0.5
0.4
Energy, eV
Probability to be in state
0.7
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
time, seconds
0.6
0.7
0.8
0.9
1
−3
x 10
0
0
0.1
0.2
0.3
0
0
0.1
0.2
0.3
0
0
0.1
0.2
0.3
Distance from outer membrane barrier, nm
(d) Analytic solution to master equation of in- (e) Geometry and potential landcoherent jumps. Colour scale, red to yellow, scape of system
lower to higher energy levels, black is vaccuum
state. Solid line V=0, dotted line V=-0.1,
dashed line V=+0.1.
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Single well, with two energy levels
1
0.9
0.8
Probability to be in state
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
time, seconds
0.6
0.7
0.8
0.9
1
−13
x 10
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Two symmetric wells, each with two levels
1
0.9
0.8
Probability to be in state
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
time, seconds
0.6
0.7
0.8
0.9
1
−13
x 10
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Three symmetric wells, each with two levels
1
0.9
0.8
Probability to be in state
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
time, seconds
0.6
0.7
0.8
0.9
1
−13
x 10
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Four symmetric wells, each with two levels
1
0.9
0.8
Probability to be in state
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
time, seconds
0.6
0.7
0.8
0.9
1
−13
x 10
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Four symmetric wells with an additional asymmetric well
1
0.9
0.8
Probability to be in state
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
time, seconds
0.6
0.7
0.8
0.9
1
−13
x 10
Biological Ion Channels
Quantum ion channels
Simple quantum systems - electrons
Working to find quantum character for ion channels
1
Stochastically evolve ion channel system for selectivity filter.
2
Add cavity to selectivity filter.
3
Model a conical, extra-cellular region in additon to reservoir.
4
Allow 2 to 3 partices in the system at once.
5
Take account particle indistinguishability.
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