Bioelectric phenomena
2014/2015, lecturer: Martin Zápotocký
1. Electric quantities and their measurement,
components of electric circuits
2. Electrode potential, Nernst equation, galvanic and
electrolytic cell
3. Electric phenomena at the cell membrane, ion channels,
Donnan equilibrium, resting membrane potential
4. Action potential, effects of electric current on the organism
Warning: figures and equations
drawn on the blackboard are not
included in these slides
Electric charge
• Electric phenomena are due to the interaction and motion of electric
charges
• The net electric charge Q of a physical body is given by the total
contained number of electrons 𝑁𝑒 and number of protons 𝑁𝑝 :
𝑄 = 𝑁𝑒 −𝑒 + 𝑁𝑝 𝑒
•
•
•
•
where 𝑒 = 1.6 × 10−19 𝐶 is the elementary quantum of electric charge
Example: calcium ion Ca2+ has net charge 18 −𝑒 + 20𝑒 = +2𝑒
Unit of electric charge: 1 Coulomb (C)
Total charge of one mole of univalent cations: the Faraday constant
𝐹 = 𝑒𝑁𝐴 = 96.5 𝑘𝐶/𝑚𝑜𝑙
The net electric charge of an isolated body is conserved
Electrostatic force, electric field
• Coulomb law: the electrostatic force between charges q and Q
separated by distance r has magnitude
1 𝑞𝑄
𝐹=
4𝜋𝜖 𝑟 2
and points in the direction connecting the charges
(repulsive for like charges, attractive for opposite charges)
• The permittivity constant 𝜖 is characteristic of the given material
(medium); it is lowest in the vacuum, 𝜖0 = 8.85 × 10−12 C V −1 m−1
• Two charges of +1 C, one meter apart, will repel each other with force
of 9 × 109 N! Macroscopic objects are nearly neutral.
• The interaction is mediated by the electric field. The charge Q creates
the electric field intensity 𝐸 = 𝐹 /𝑞. The electric field lines of a
positive point charge point radially outwards.
Electric dipole, polarization, dielectric screening
• Inside a molecule with net zero electric charge, positive and negative
charges can spatially separate, giving the molecule a dipole moment
• A dipole placed in an external electric field will rotate to align with the
field lines (parallel to intensity 𝐸 )
• A material composed of molecular dipoles will get polarized and
reduce (screen) the effect of the external electric field
• The force between two charges placed inside a dielectric material is
1 𝑞𝑄
reduced compared to their interaction in vacuum: 𝐹 =
2 , 𝜖 > 𝜖0
4𝜋𝜖 𝑟
• Relative permittivity 𝜖/𝜖0 for some materials:
air: 1.0006, glass: 5-10, lipid membrane: 8, water: 78.5
Solvation, hydration
• Polar solvents (water, ethanol, ammonia,…) efficiently
solvate ions
• The dielectric screening helps to dissociate ionic bonds
(e.g. Na+Cl- dissolved in water)
• Hydration number: the number of molecules of water
with which an ion forms a complex. Effective radius of
ion: the size of ion together with its hydration shell.
Smaller ions can fit more water molecules. Cations
polarize water stronger than anions.
Ion
Ionic
Radius
(Å)
Approx.
hydrated
radius
Approx.
hydration
number
Li+
0.90
3.40
25
Na+
1.16
2.76
17
K+
1.52
2.32
11
Electric potential
• Electric potential U at a given location 𝑟: defined as the amount of
work needed to bring a unit electric charge from the reference location
(with zero potential) to the given location
• In the field of a point charge Q located at 𝑟 = 0 :
𝑄
𝑈 𝑟 =
4𝜋𝜖𝑟
•
•
•
•
•
In this case U=0 at 𝑟 = ∞.
Note that 𝑊 = 𝑞𝑈 is the potential energy of charge q in the electric
field of charge Q
Unit of electric potential: 1 Volt (V); note V=J/C
Equipotential lines for the field of a point charge: concentric circles
Only the potential difference (voltage) is measurable
In a reactive medium, the notion of electrochemical potential is
necessary
Electrochemical potential
• Consider the boundary of two phases (e.g. solid electrode / electrolyte
solution). We define the following potentials:
• External (Volt) potential: work needed to bring unit charge from ∞ to 10-8 m
within the phase boundary. (This is the usual electrostatic potential)
• Internal (Galvani) potential Φ: the external potential + work needed to bring
unit charge across the boundary surface
• Electrochemical potential (of specific component): work required to
transport 1 mole of molecules into the given phase
𝜇𝑖 = 𝜇𝑖0 + 𝑅𝑇 𝑙n 𝑎𝑖 + 𝑧𝐹𝜙
where F=Faraday’s constant, z = net number of elementary charges on
each molecule
Electric current, Ohm’s law
• A positive electric charge which is mobile will move in the direction of
the electric field, i.e. to locations with lower electric potential.
• Conductors: materials in which (some) electric charges are mobile.
Examples: metals, ionic solutions. Note: dielectrics are insulators;
water is a conductor as it contains a fraction of OH- and H+
• Electric current through a conductor: defined as amount of charge
passing per unit time. Units: 1 Ampere (A) = C/sec.
• Ohm’s law: relates the magnitude I of (direct) electric current passing
through a conductor with the applied electric potential difference U
𝑈 = 𝑅𝐼
where R is the resistance of the conductor; units: 1 Ohm (Ω) = V/A
Resistivity, molar conductivity
• Resistance of conductor of length L and cross-section A:
𝑅 = 𝜌 𝐴𝐿
where ρ is the resistivity of the material (units: Ω m)
• Typical resistivity for metal conductors: 10-8 Ω m, for insulators: wood
1010 Ω m, teflon 1014 Ω m. Resistivity depends on temperature.
• For ionic solutions, the specific conductivity is defined as
1
𝜅=
𝜌
units: Ω-1 m-1 = S m-1 , S=Siemens=Ω-1
• Molar conductivity: Λ = 𝜅/𝑐 , where c is the molar ionic concentration.
Typical values: 10-3 S m2 mol-2
• In dilute solutions, molar conductivities add:
Λ = Λ𝑖 (Law of independent migration of ions)
Direct / alternating current, effective value
• Direct current (DC): electrons move only in one direction. Example:
constant current.
• Alternating current (AC): direction of motion alternates. Example:
periodic harmonic current,
𝐼 = 𝐼0 sin 𝜔𝑡
where 𝐼0 is the amplitude of the current, 𝜔 = 2𝜋𝑓 is the angular frequency
• In a (linear) electric circuit, both current and voltage are harmonic with
identical frequency (but can be mutually phase-shifted)
• Effective value of current (or voltage): 𝐼eff =
𝐼0
2
Meaning: energy dissipated in resistor through which a DC current of
magnitude 𝐼eff flows = energy dissipated in resistor through which an
AC current of amplitude 𝐼0 flows
Components of electric circuits, impedance
• Electric circuits are combinations of the 3 ideal components: resistors,
capacitors, inductors. Circuit is driven by a voltage source, e.g. a
battery (device using chemical energy to separate positive and negative
charges – see next lecture).
• In general, applying an alternating voltage 𝑈0 sin 𝜔𝑡 results in an
alternating current with same frequency 𝜔 and with amplitude 𝐼0 ,
related to 𝑈0 by electric impedance Z
𝑈0 = 𝑍 𝐼0
Capacitor
•
A capacitor consists of two conductors separated
by an insulator (dielectric).
• The capacitor stores separated electric charges. The
capacitance C relates the potential difference U
across capacitor and stored charge Q:
𝑄 = 𝐶𝑈
• Units: 1 Farad (F) = Coulomb / Volt
• For capacitor made of two plates of area A
separated by distance d, dielectric of permittivity ε:
𝜖𝑆
𝐶=
𝑑
• When alternating voltage 𝑈0 sin 𝜔𝑡 is applied,
current with amplitude 𝐼0 = 𝜔𝐶𝐼0 will flow
• Impedance 𝑍 =
1
𝜔𝐶
= capacitive reactance
The oscilloscope
• Used to visualize the time course
of electric quantities
• Most common type: based on the
cathod ray tube (electron gun)
• Electrons emitted by heated wire
filament (cathode), accelerated
through anode (high voltage),
collimated into narrow beam
• Beam passes through horizontal
pair of plates (deflection in
vertical direction, signal) and
vertical pair of plates (deflection
in horizontal direction, time basis)
• Used e.g. in heart and breathing
activity monitors
Electric phenomena in biological cells
• Every living cell maintains an electric potential difference across its
plasma membrane. Usually, inside is negative compared to outside, by
40 to 90 mV.
• The cell membrane acts as capacitor and as highly variable resistor
• A living cell functions as a battery, i.e. uses chemical energy (ATP) to
recharge the membrane capacitor
• Signaling (information transmission) in the neural system is based on
electric excitations traveling along the membrane
• Muscle activity, heart rhythm also controlled by electric excitations
The plasma membrane is a selectively permeable
interface of distinct ionic environments
• Lipid bilayer gives capacitance 10 nF/mm2
• Charge neutrality in bulk, + and – charge on
membrane. It is enough to move 1/105 of total
K+ ions to set up V=-60 mV.
• Resistance: highly variable, depends on
number and state of ion channels. Typical
value 1 MΩ/mm2
• Ion concentrations for typical neuron:
Outside Inside
[mol/m3] [mol/m3]
Na+
150
15
K+
5.5
150
Cl-
125
9
Ion channels
• Conformational transitions between open state and closed state.
Control of transition rates: ligand-gated vs. voltage-gated channels.
• Most ion channels are selective mainly for one type of ion.
• A single channel behaves like a resistor, but with modified Ohm’s law:
𝐼channel = 𝑔(𝑉 − 𝐸)
g is the single-channel conductance, 𝑔 = 0 or 𝑔 = 𝑔open ≃ 10 pS
V = transmembrane voltage, E = reversal potential
• Conductance for given ion type: 𝐺 = 𝑛 𝑝open 𝑔open
n = No. of channels / unit area, 𝑝open = open probability
Nernst equation for reversal potential
• For the given ion type, no net flow through channel when the two sides
are in electrochemical equilibrium, 𝜇𝑖𝑛 = 𝜇𝑜𝑢𝑡 :
𝜇0 + 𝑅𝑇 𝑙n 𝑐𝑖𝑛 + 𝑧𝐹𝜙𝑖𝑛 = 𝜇0 + 𝑅𝑇 𝑙n 𝑐𝑜𝑢𝑡 + 𝑧𝐹𝜙𝑜𝑢𝑡
• Nernst – Donnan potential:
𝑅𝑇 𝑐𝑜𝑢𝑡
𝐸 = 𝜙𝑖𝑛 − 𝜙𝑜𝑢𝑡 =
𝑙n
𝑧𝐹
𝑐𝑖𝑛
• When 𝑉 = 𝐸, gradient of concentration is balanced by electrical
potential gradient. Net electrochemical driving force is zero.
• Note:
𝑅𝑇
𝐹
= 27 mV. Typical values:
𝐸𝐾 = −80𝑚𝑉, 𝐸𝑁𝑎 = +50 𝑚𝑉, 𝐸𝐶𝑙 = −60𝑚𝑉
Donnan equilibrium
• Can we achieve equilibrium with two ion types?
• Consider K+ and Cl-: reversal potentials identical if
(Donnan equilibrium rule)
𝐾+
𝑖𝑛
𝐶𝑙 −
𝑖𝑛
= 𝐾+
𝑜𝑢𝑡
𝐶𝑙 −
𝑜𝑢𝑡
• Will the in and out concentrations equilibrate? No,
due to impermeant anions (proteins) inside the cell
(the Donnan effect):
𝐾 + 𝑖𝑛 > 𝐾 + 𝑜𝑢𝑡
𝐶𝑙 − 𝑖𝑛 < 𝐶𝑙 − 𝑜𝑢𝑡
• Neurons do not satisfy the Donan equilibrium rule!
Exchangers and pumps (ATPases) maintain system
out of equilibrium.
Example of Donnan equilibrium
• Consider these concentrations of K+, Cl-, A- [μM]:
• The Donnan equilibrium condition is satisfied:
80 × 20 = 40 × 40
• The reversal potentials are:
𝐸𝐾 =
𝑅𝑇
40
𝑙n
𝐹
80
= −17mV,
𝐸𝐶𝑙 =
𝑅𝑇
40
𝑙n
(−1)𝐹
20
= −17mV
• If transmembrane voltage is 𝑉 = −17mV, no K+ current and
no Cl- current flows through the open channels.
Deviation from Donnan equilibrium
• Donnan equilibrium rule for K+, Cl- is well
satisfied in some muscles.
• Donnan equilibrium is broken in neurons.
Resting potential is in between the distinct
reversal potentials.
• This is the result of Donnan equilibrium,
+ influence of other ions (Na+, Ca2+)
+ action of pumps and echangers (ATP used,
deviation from thermodynamic equilibrium).
• In general, the “resting” membrane is
maintained in a non-equilibrium steady state.
Concentrations have to be maintained by
pumps and exchangers.
• Na-K pump: 3 Na+ out, 2 K+ in (to the side
with higher concentration) – chemical energy
is used to restore the concentration gradient
Resting potential, hyperpolarization, depolarization
• By convention, the potential outside the cell is 0.
• The living cell can indefinitely maintain a “resting state”, with the
cell interior at the resting potential, usually -40 mV to -90 mV
• Perturbations (channel gating or injection of current by electrode)
lead to hyperpolarization or depolarization
• Current injection (of positive ions) leads to depolarization, i.e. less
negative potential inside the cell
• Outflux of positive ions or influx of negative ions leads to
hyperpolarization, i.e. more negative potential (below the resting
potential).
Resting membrane potential
• When K, Na, Cl ions have different reversal potentials, their
thermodynamic equilibria cannot be achieved simultaneously.
• At the resting membrane potential, currents will flow across channels,
but total current = 0. The Hodgkin-Horowitz equation expresses the
net balance of currents:
0 = 𝐺𝐾 𝐸rest − 𝐸𝐾 + 𝐺𝑁𝑎 𝐸rest − 𝐸𝑁𝑎 + 𝐺𝐶𝑙 (𝐸rest − 𝐸𝐶𝑙 )
therefore
𝐺𝐾 𝐸𝐾 + 𝐺𝑁𝑎 𝐸𝑁𝑎 + 𝐺𝐶𝑙 𝐸𝐶𝑙
𝐸𝑟𝑒𝑠𝑡 =
𝐺𝐾 + 𝐺𝑁𝑎 + 𝐺𝐶𝑙
• Goldman equation for rest membrane potential (reduces to Nernst eq.
if only one ion type present) :
𝐸𝑟𝑒𝑠𝑡
𝑅𝑇 𝑃𝐾 𝐾 + out + 𝑃𝑁𝑎 𝑁𝑎+ out + 𝑃𝐶𝐿 𝐶𝑙 + out
=
𝑙n
𝐹
𝑃𝐾 𝐾 + 𝑖𝑛 + 𝑃𝑁𝑎 𝑁𝑎+ 𝑖𝑛 + 𝑃𝐶𝐿 𝐶𝑙 + 𝑖𝑛
P = permeability = ion diffusion const. across membrane
divided by membrane thickness
The neuron
• Morphologically and functionally
distinct parts: dendritic tree, soma, axon
• Dendrite receives input from sensory
stimulus or other neurons
• Soma: integration of input, generation
of action potential (at axon hillock)
• Axon: conducts action potentials.
Myelinated vs. non-myelinated. Up to
1 m in length.
• Membrane in dendrite, soma, axon
differs by distinct types of ion channels
Neuron
• Hlavní části neuronu:
– tělo (integrace)
– dendrity (vstup)
– axon (výstup, až 1m)
• Komunikace mezi neurony: skrz kontaktní body, synapse
Electric activity in pyramidal neuron of brain cortex
Action potentials
Passive spread of depolarization
• Used for signal transmission in the dendrites and within myelinated axon
segments. Depolarization decays exponentially with distance from point of
steady current injection:
𝑉(𝑑) = 𝑉0 𝑒 −𝑑/𝜆
where the electrotonic length λ = 0.1 to 5 mm (smaller in narrow neurites).
• Equivalent circuit:
The action potential (“neural spike”)
• Short propagating pulse, generated when
depolarization exceeds a threshold
• The pulse is always the same (“all or
nothing”), does not depend on details of
stimulus (if big enough)
• Basis for all neural communication,
coding by spike frequency
• Only in excitable membranes
• Mechanism explained by A. L. Hodgkin
and A. Huxley (1952), Nobel Prize 1963.
Electrode through giant axon of the squid
(1 mm diameter). Mathematical model.
Properties of ion channels predicted,
confirmed only in 1970.
Mechanism of excitability
• In the resting state, for squid axon:
𝑉 = 𝐸𝑟𝑒𝑠𝑡 = −60𝑚𝑉, 𝐸𝐾 = −75𝑚𝑉,
𝐸𝐶𝑙 = −40𝑚𝑉, 𝐸𝑁𝑎 = +55𝑚𝑉
• In the resting state, G(K) >> G(Na)
• Voltage-gated ion channels:
conductivity G depends on V
• Depolarization stage:
if 𝑉 > 𝑉𝑡ℎ𝑟𝑒𝑠ℎ = −45𝑚𝑉,
Na channels open (gated by voltage)
→ further depolarization
→ further opening → ... (positive
feedback loop)
• After a few msec: inactivation of Na
channels, delayed activation of K
channels. Leads to repolarization.
• Refractory period: time before next
action potential can be initiated
(several msec). Determines maximum
spike frequency (several hundred Hz).
Propagation of action potential
• Spatial shape of the action potential:
width of 1-2 mm
• Travels on the axon with v = 1m/sec
to 100 m/sec, faster in bigger axons
• Mechanism: enough voltage spreads
passively from center of spike to
neighboring region to exceed
threshold there
• Refractory period ensures that spike
keeps traveling in one direction
(usually away from soma, starting at
axon hillock). Backpropagation of
spikes also possible.
Myelinated vs. non-myelinated axons
• Long axons are usually myelinated
• Myelin sheath increases transmembrane
resistance and reduces capacitance →
better passive conduction along axon
• It is enough to generate action potential
only at the gaps, Nodes of Ranvier,
separated by ~ 1mm. This is saltatory
conduction.
• More efficient transmission of spikes:
smaller diameter without loss of velocity,
smaller energetical requirements
Electric control of muscle activity
• Neuromuscular junction: motor neuron
terminates on muscle fiber
• Separate action potentials on motor neuron and
muscle. In the muscle, the action potential
leads to release of Ca2+ from intracellular
stores, activation of motor proteins (myosin).
• For skeletal muscles, tension is controlled by
a) frequency of action potentials sent from
motor neuron to each fiber, b) recruitment of
additional fibers.
• For cardiac muscle, the action potential lasts
~200 msec. Extended plateau, which is due to
long-lasting opened Ca2+ channels. Pacemaker
cells spontaneously generate action potentials.
Cardiac action potentials
Action potential in ventricular myocyte
Antiarrhythmic agents
Effect of electric current on the organism
• DC current: electrolytic effects, result in change in tissue excitability.
• Stimulatory effect following a change in DC current, such as the pulse
stimulus. Rheobase: minimum amplitude of pulse which will
eventually result in an excitation (action potential, muscle twitch).
Chronaxie: pulse duration needed to induce excitation when pulse
amplitude = 2 rheobase. Typically msec. Rheobase depends strongly
on location of stimulation.
• AC current at intermediate frequency (10-500 Hz): strong stimulatory
effect. Above 100 Hz, stimulatory threshold increases as f 2, i.e. effect
is weaker. Current densities used in electrotherapy: 1-10 mA/cm2.
Review Lab No. 4b!
• AC current at high frequency: weak stimulatory effect up to 100 kHz,
strong thermal effect. Can be used to generate heat deep in tissue.