The Biology of Ion Channels
Ben Gustafson
Katherine Gurdziel
Introduction
Ion channels are protein molecules in cell membranes which permit the flow of ions
from one side of the membrane to the other. They are among a group of membrane
transport proteins which span the cell membrane and allow the selective passage of
substances which otherwise would not be able to pass through the cell membrane.
Alan Hodgkin and Andrew Huxley first described the flow of potassium and sodium ions
through the membrane of the nerve cell in 1952. They surmised from the very close
relationship between the flow of ions and changes in membrane potential that charged
particles were moving through the membrane. They could not detect the charge
movement at the time, but the current produced by the ion movement could be easily
measured. They deduced that for each moveable membrane charge there must be many
ions moving across the cell membrane.1
In 1976, Erwin Neher and Bert Sackman developed the patch clamp technique for
measuring ion flow through a single channel. Their technique uses a glass microelectrode
with a polished tip whose diameter is small enough to encompass a small patch of cell
membrane containing a few ion channels. Using a feedback amplifier, they held the
voltage across the patch steady, or “clamped,” thus allowing them to measure the
currents flowing through the individual ion channels. They found that the channels
opened and closed at apparently random intervals, thus producing a pulse of current. As
the amperage of the current remained constant during the bursts, this suggested that an
ion channel is either fully opened or fully closed.2 Fig. 1 below shows the patch clamp
technique being used to monitor ion channel activity.
1
Aidley, David J. and Peter R. Stanfield, Ion Channels: Molecules in Action, Cambridge:
Cambridge University Press, 1996, p. 4
2
Ibid., p. 5
Fig. 1 Patch Clamp Recording of Single Ion Channels. (Source: Alberts et al., Essential Cell Biology,
Second Edition, 2004, p. 406)
The use of molecular biology to study the structure of ion channels has advanced our
knowledge of their structure. Teams led by Shosaku Numa, Jean-Pierre Changeux and
others determined the amino acid structure of the nicotinic acetylcholine receptor in
1982, followed by the voltage-gated sodium channel two years later.3
Physical Structure
The cell is enclosed in a membrane composed of two layers of lipid molecules. These
lipids are composed of a hydrophilic head and a hydrophobic tail structure. In the
watery environment of the cell, these molecules organize themselves into the most
energetically favorable conformation, which is with the hydrophilic head facing the
3
Ibid., p. 6
watery cytosol and extracellular environments, and the tails of each layer oriented to
the inside of the lipid bilayer.
The normal functioning of cells requires the movement of molecules from one side of
the membrane enclosing it to the other. The hydrophobic interior of the cell wall’s
lipid bilayer blocks transport across the membrane of large uncharged polar
molecules such as amino acids and sugars, and all ions, due to their charge and
strong attraction to water molecules, and must rely on other means of transport
across the cell wall.
A membrane transport protein is a type of protein molecule, spanning the lipid
bilayer membrane of a cell, and is responsible for moving molecules between the
cytosol of the cell and the extracellular space.
The basic structure of an ion channel is a few protein molecules arranged around a
central aqueous pore. This pore may be opened or closed by conformational changes
caused by the channel’s surrounding environment. Ion channels differ in their
gating, or the factors which make them open and close at a more rapid rate.
Environmental factors causing conformational changes include combinations of
chemicals such as messenger molecules inside the cell or neurotransmitters outside
the cell.
Simple ion channels exist in an open or closed conformation, or state. The stimulus
responsible for gating the ion channel can vary. Some are voltage gated, opening and
closing with changes in cell membrane potential. Ligand-gated ion channels open
when a specific type of molecule, or ligand, binds to a site on the channel, causing its
conformation to change and the channel to open. A mechanically-gated channel
opens when physical stimulus pushes against lever-like structures on the channel.
The irregular patterns of the opening and closing of ion channels makes them stochastic
events, able to be analyzed using statistical tools. By observing the behavior of a single
ion channel given different environmental factors, we can project to a larger population
the statistical probability that they will react in the same way.
Ion channels are distinguished from simple pores in a cell membrane by their ion
selectivity, permitting only ions of a certain size and charge to pass through. Some
channels permit relatively broad groups of ions to pass through, such as cations
(positively charged ions), while others may permit only a particular ion to pass
through, such as sodium. Ion selectivity depends on the diameter and shape of the
opening and the distribution of the charged amino acids of its lining. Ions which are
too large will not be able to fit into the opening, and ions which are too small will not
be able to come in close enough proximity to the oppositely charged amino acids to
interact with them and be transported through the channel.
Ion channels change randomly between open and closed conformations even when
environmental conditions which influence their gating are held constant. As the
appropriate environmental conditions change, the random behavior continues, but
with greatly changed probability. If the altered conditions tend to cause the channel
to stay open, the probability the channel will be in the open conformation greatly
increases. Environmental changes, such as the binding of a ligand or change in
voltage, do not determine whether an ion channel opens or closes, but instead have a
direct effect on the rate of openings and closings in a given time period.4
Insight into the detailed operation of ion channels comes from X-ray crystallographic
studies of a bacterial K+ channel, because the large quantity of channel protein
needed for crystallography could be obtained by growing vast numbers of bacteria.
The channel is formed by four molecular subunits that each cross the plasma
membrane twice. The channel’s pore is well suited for conducting K+ ions. The
narrowest part of the channel, the selectivity filter, is lined with carbonyl oxygen
atoms with a partial negative charge, forming transient binding sites for moving the
K+ ion through the channel. Larger cations cannot traverse the selectivity channel,
and smaller cations such as Na+ cannot enter the pore because its walls are too far
4
Bruce Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 406
apart to stabilize a dehydrated Na+ ion.5 Fig. 2 below shows a diagram of a bacterial
K+ ion channel with one of the four subunits removed.
Fig. 2 Bacterial K+ ion channel. (Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 404)
Function
As described above, ion channels differ in their gating mechanisms. The major gating
mechanisms are described below.
Voltage-gated
The environment in which most cells are in contact is salty, containing a relatively
high concentration of sodium and chloride ions, and much lower concentrations of
potassium, calcium and other ions. The plasma membrane of the cell provides a
barrier to these ions, thus maintaining their concentrations inside and outside the
cell at different levels. This creates an electrical membrane potential difference
between the interior and exterior of the cell which, combined with the differing ionic
concentrations, creates an electrochemical gradient across the membrane for the
5
Dale Purves et al., “The Molecular Structure of Ion Channels,” in Neuroscience, Second Edition,
http://www.ncbi.nlm.nih.gov/books/bv.fcgi?rid=neurosci.section.251
different ions. When the ion channels are open, ions flow down the gradients, either
into or out of the cell.
Membrane potential is governed by the permeability of specific ion channels to
specific ions. An ion channel transports only one ion or group of ions (for example,
cations, positively charged ions). Electricity in cells is carried by these ions, in the
form of cations or anions (negatively charged ions). The flow of ions across a cell
membrane is detectable as an electric current. An accumulation of ions of one charge
at the cell membrane which is not counterbalanced by the same amount of ions of
the opposite charge will result in a detectable accumulation of electrical charge, or
membrane potential. The membrane potential of a cell is controlled by the ion
channels themselves, by their opening and closing in response to stimuli. The
feedback loop of ion channel  membrane potential  ion channel is fundamental
to all electrical signaling in cells.
Voltage-gated Na+ channels play a key role in propagating nerve impulses along the
axon. When a nerve stimulus causes the membrane to depolarize to a threshold level,
voltage-gated Na+ channels open temporarily at the site of depolarization, which in
turn allows a flow of Na+ ions into the axon, further depolarizing the membrane,
thereby making the membrane potential less negative. This continues until the
membrane potential in the region has shifted from about -60 mV to about 40 mV. At
this point the electrochemical force driving more Na+ ions across the membrane and
into the axon is zero. The membrane potential causes the Na+ channels to switch to
their inactivated conformation, thereby blocking further entrance of Na+ ions in the
region. Voltage-gated K+ channels open and stay open until the membrane potential
returns to its resting state as K+ ions exit the cell, counteracting the net buildup of
anions outside the membrane. Once the membrane potential has returned to its
resting state, the Na+ channels resume their closed conformation. Fig. 3 below shows
the three conformations of the Na+ channel at different membrane potentials; Fig. 4
shows the propagation of an action potential along an axon.6
6
Bruce Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 412
Fig. 3 The three conformations of a voltage-gated Na+ ion channel. (Source: Alberts et al., Essential Cell
Biology, Second Edition, 2004, p. 413)
Fig. 4 Propagation of an action potential along the path of an axon. (Source: Alberts et al., Essential Cell
Biology, Second Edition, 2004, p. 407)
Mechanically Gated
The auditory hair cells, or stereocilia, in the inner ear rely on mechanically-gated ion
channels. The stereocilia are embedded in supporting cells which sit atop a basilar
membrane. Above them is the surface of a tectorial membrane. Sound vibrations
cause the basilar membrane to move up and down, tilting the stereocilia against the
tectorial membrane and causing the ion channels to open, allowing ions to flow into
the hair cells, and stimulating nerve cells underlying the hair cells. Fig. 5 shows the
mechanics of this process.
Fig. 5 Stress-activated ion channels in the inner ear. (Source: Alberts et al., Essential Cell Biology, Second
Edition, 2004, p. 408)
Ligand Gated
Ligand-gated ion channels are controlled by the binding of a molecule (the ligand) at a
receptor site in its structure. Transmitter-gated ion channels are a subclass of ligandgated channels. The neuroreceptors in the synapses of nerve axons are responsible for
transmitting nerve impulses across synapses, junctions between neurons and muscle
cells.
In humans, the neurotransmitter acetylcholine plays a key role in transmitting nerve
impulses across synapses. Acetylcholine is released from synaptic vesicles in presynaptic
nerve terminals in response to calcium (Ca2+) ions entering the terminal through voltagegated ion channels. These channels open in response to a change in the membrane
potential as a nerve impulse is propagated to the nerve terminal. The neurotransmitter is
released across the synapse, where it binds to receptor sites in ligand-gated ion channels
in postsynaptic cells. The binding of acetylcholine causes the ion channels to change
conformation and open, allowing ions to enter the cells. This in turn causes the
membrane potential of the postsynaptic cell to change. If the change in membrane
potential is large enough, it will trigger an action potential in the cell. Fig. 6 and 7 below
illustrate the steps in this process.7
Fig. 6 The conversion of an electrical signal to a chemical signal at a nerve synapse. (Source: Alberts et al.,
Essential Cell Biology, Second Edition, 2004, p. 418)
Fig. 7 The conversion of a chemical signal to an electrical signal at in a postsynaptic cell. (Source: Alberts et
al., Essential Cell Biology, Second Edition, 2004, p. 418)
7
Ibid, p. 418
Observation
Measuring changes in electrical current is the main method for studying ion
movements and ion channels in living cells. It is possible to detect and measure the
electric current flowing through a single ion channel, using the patch clamp
recording technique described above. With a small enough area of membrane in the
patch, a single ion channel will be present and can be tested. The current detected by
modern electrical instruments can be as low as a pico ampere (10-12 ampere). The fact
that the current of an ion channel switches off and on again randomly even as
conditions are held constant implies that the channel acts as an on/off switch,
opening and snapping shut in reaction to random thermal movements of molecules
in its environment. The recording of single channel currents shows two current levels
corresponding to the closed and open state respectively. Transitions between these
two states are very fast (fractions of a millisecond), and appear in the recordings as
rectangular jumps from one level to the other. The amplitude of the jump
corresponds to the current. Normally, the channels stay open for a fraction of a
second, allowing the flux of tens of thousands of ions through the pore. The activity
of the channel can be quantified as the fraction of time it stays in the open
conformation.8
Rate of ion transmission
The rate at which ions move across a membrane is determined by 1) the magnitude
of the electrochemical gradient; 2) the permeability and selectivity of the ion
channel; 3) the number of ion channels present per unit of membrane; and 4) the
proportion that are open at a given time.9 These factors will be described in detail
below.
Magnitude of the electrochemical gradient
Consider a system where there is a higher concentration of K+ ions outside the
membrane, due to a potassium salt ionizing into K+ cations and the oppositely
charged anions. The membrane contains ion channels which permit the flow of
8
Ibid., p. 406
9
Aidley and Stanfield, Ion Channels: Molecules in Action, p. 14
potassium ions but not the anions. When these ion channels open, they permit the
potassium ions to flow into cell along with their accompanying positive charge,
leaving the negatively charged anions outside, depleting the positive charge outside
the cell. This flow of ions reaches a point where the gradient driving the flow into the
cell is matched by an electrical gradient tending to push the cations back outside the
cell, and therefore reaching an electrochemical equilibrium. The Nerst equation
expresses this equilibrium quantitatively:
V = 62 log10(Co/Ci)
Where V is the membrane potential in millivolts and Co and Ci are the concentrations
of the ion outside and inside the cell.
Number of ion channels present per unit of membrane
The number of ion channels per unit of membrane has a direct relationship with the rate
of ions transmitted. Given a certain rate constant, r, of ions transmitted in unit time and
a certain number of ion channels, n, in a unit of membrane, the number of ions
transmitted per unit of membrane is r*n.
Proportion of channels open at a given time
The proportion of ion channels open at a given time figures into the relationship of the
number of ion channels open. Let o represent the number of open ion channels per unit
of membrane and c, the number closed. Then the total number of channels through
which ions can pass in a unit of membrane is o = n-c.
Conclusion
Ion channels play a key role in many biological functions important to the normal
functioning of the body. Due to their extremely small size, however, much of what we
know about them is gathered from indirect examination of their behavior using tools
such as patch clamp recording and stochastic methods. We will examine the use of some
of these stochastic methods in the next section.
Modeling
Mathematical models mimic behavior in the real world by representing a description of a
system, theory, or phenomenon that accounts for its known or inferred properties and
may be used for further study of its characteristics. Scientists rely on models to study
systems that cannot easily be observed through experimentation or to attempt to
determine the mechanism behind some behavior. “This process of translation into a
mathematical form can give a better handle for certain problems then would otherwise
be possible.”10
The type of model to be constructed relies on the type of system to be represented as well
as the question that is being asked. First the problem needs to be formulated, then all of
the known factors in the system and how they related to each other need to be
determined.11 There are two fundamentally different types of systems: deterministic and
stochastic. A deterministic system’s behavior relies on the preceding events or natural
laws. A stochastic system “evolves dynamically over time” and contains random
behavior.12
If the purpose of the model was to determine the outcome of throwing a die both a
deterministic model and a stochastic model could be constructed. It would be possible to
create a deterministic model that used the forces, the trajectory in the air, the tumbling
and bouncing, as well as imperfection's of dice and table to attempt to predict the
outcome. However, it would be far simpler to construct a stochastic model (with the six
possible outcomes having equal probability) since most of the parameters of the
deterministic model are not known, and the process of throwing cannot be controlled in
sufficient detail.13
An effective use of stochastic modeling consists of evaluating the likelihood that a
random event will occur. Predictions in stochastic systems depend on defining variables
that represent the system fluctuations. It is not possible to determine the specific value
10
Edward Beltrami, Mathematics for Dynamic Modeling (Boston: Academic Press Inc., 1987),
Xiii.
11
Ibid., 6-7
12
Barry L. Nelson, Stochastic Modeling Analysis & Simulation (New York: Dover Publications
Inc., 1995), 1.
13
Nikos Drakos and Ross Moore, (University of Leeds and Macquarie University, 1999) Available
at http://www.bio.vu.nl/thb/course/tb/tb/node28.html.
of these random variables but the probability of a given event occurring can be predicted
based on the variable.14
It is important to remember that a model cannot be used to predict outside of the system
it was designed to study. “This model will be a simplification and an idealization, and
consequently a falsification. It is to be hoped that the features retained for discussion are
those of greatest importance in the present state of knowledge.”15 In essence, “a
mathematical model is an abstract, simplified, mathematical construct related to a part
of reality and created for a particular purpose.”
Modeling Ion Channels
Extensive work by Colquhoun and Hawkes has focused on modeling ion-channels. Our
review focuses on their work for predicting single-ion channel behavior. The primary
aim of their single ion channel research consists of “gaining insight about the nature of
the reaction mechanism from experimental observations.” 16 They used single ion
channel records consisting of “amplitudes of the openings, the durations of the open and
shut periods, and the order in which the openings and shuttings occur”17 to derive the
model. Subsequent procedures were designed to evaluate the rate constants and answer
relevant biological questions.
The models were derived from experimental data. In order to get meaningful
information, observation needed to be measured for a sufficiently small number of ion
channels so that fluctuations around the average behavior were measured. But what
defines a sufficiently small membrane patch? Since the derivation is based on an
average it is necessary to determine the number of channels present in the record and for
the majority of the cases it is necessary to estimate the number of channels that are
present. The number of open channels, N, can be computed using the binomial
distribution as long as the number of channels is small. If the number of channels
14
Neil Gershenfeld, The Nature of Mathematical Modeling (Cambridge: Cambridge University
Press, 1999), 44.
15
A. M. Turing, The Chemical Basis of Morphogenesis (London: Philos. Trans. Royal Soc., Ser. B,
Vol. 237, 1952), 37-42.
16
David Coquhoun and Alan G. Hawkes, “The interpretation of single channel recordings”, In
Single-Channel Recording. Edited by Bert Sakmann and Erwin
Neher. New York : Plenum Press, c1983, p. 135.
17
Ibid., 141.
becomes too large the binomial distribution approaches a Poisson distribution and N is
indeterminate.18
Using these recordings the number of channels as well as the number of open channels
were determined using the Katz and Miledi method. The probability that an individual
channel is open can be calculated by the average number of open channels divided by the
total number of channels. “The individual molecules behave in an entirely random way
… the length of time for which the molecule stays in one of its (supposedly) discrete
states is quite random; the properties of the states themselves may be very constant.”19
Since what happens in the future relies only on the current system state and not on the
past history it can be mathematically represented using a homogeneous Markov
process.20 The Markov transition probabilities correspond to the rate constants for
ligand coupling and channel changes.
This variability must also be accounted for when calculating the transition rate for the
ion channel. Based on the law of mass action the reaction rate is proportional to
Determining the number of states within the system can be difficult. Typically the
experimental data reports just these two values, open or shut. This creates difficulties for
determining the actual number of states within the system. Long recordings are needed
since it is just a single molecule with random behavior.
Single ion channels exhibit two types of behaviors that are important for modeling
duration of state and the probability of transitions occurring.
Transition Probabilities
When attempting to determine the transition probability, the concern is not “how much
time elapses before the transition occurs but of where the transition leads when it
eventually does occur.”21 Two transition types exist that can be predicted in single ion
channel behavior: the number of oscillations within a burst and the probability that a
certain path of transitions will occur.
Physiologically ion channels tend to oscillate between an open and blocked state prior to
becoming shut. These oscillations are called bursts and by definition begin in an open
state and terminate in the shut state. Determining the probability of a chain of
18
Ibid., 163
19
Ibid., 144.
20
Ibid., 147
21
Ibid., 147.
transitions occurring within a burst can be calculated by the geometric distribution. The
probability that a certain transition occurs can be calculated by dividing the probability
that the specified transition occurs by the combined probability of any transition
occurring.
For example, in a three state system where the open state is an intermediate state
between a shut and a blocked state:
Define pob as the probability that the one channel will be blocked during each period then
1-pob represents the probability that it will shut. Since probability pob is independent of
past events the probability of success for any number of attempts, r, can be calculated by
the geometric distribution:
P(r) = (1-pob) (pob * pbo )(r-1)
The probability that the open state will become blocked is the probability of open moving
to blocked divided by the summed probability of open moving to blocked and open
moving to shut. For a transition to occur from the open state there are only two options
moving to blocked or moving to shut. Remaining in open is not considered a transition.
The probability that the burst has two openings, moves from open to blocked to open to
shut, can be computed by the product of these probabilities, (pob * pbo). The number of
oscillations within the burst is represented by r.22 The transition to the shut state is
represented by (1 – pob) since the open state can move either to the blocked or shut state
and the total probability must equal one.
This is similar to what occurs when calculating the probability of a path of transitions
occurring for the single ion channel. Since the events are independent the probability of
one event following another can be calculated by multiplying the probability of each
event. From conditional probability, P(AB) is P(A) * P(B) if A and B are independent.
The probability of transitioning from one state to another in a single step can be easily
calculated by using the one-step transition probability matrix which contains probability
of transitioning from one state to another in a single step. Determining the probability
consists of multiplying the matrix by itself for each step of the transition path. This can
be done for transition probabilities in the single ion channel since its behavior is
characterized as homogenous Markov. Markov chains are time homogeneous if the
transition probabilities from one state to another are independent of time.
22
Ibid., 147-150.
Three State Model
Once the number of states in the ion channel moves beyond open and shut determining
the state by looking at experimental data no longer works. Derivation of duration and
transition probabilities also becomes more complicated depending on the types of
transitions that can occur. Consider two three state models that both contain two shut
states: one which consists of an open state that can move to either a blocked or shut state
and one which consists of a shut state that can move to either an open or another shut
state.
The duration of the open state can be easily determined since the lifetime of any single
state has an average distribution 1 / (sum of transitions leading away from the state).23
There are two possible transitions out of the open state so the lifetime of the open state is
1 / ( + k+B). Determining the duration of the shut states becomes more difficult. The
shut state represented in the experimental data must be either the shut to open, average
1/ ’, or blocked to open, 1/k-B. However, since it is known that the shut states do not
directly communicate the overall distribution of the shut states is a proportional
relationship based on the frequency of transitions from open to each separate shut state.
What if the two shut states can communicate directly, which occurs in the Castillo and
Katz mechanism?
23
Ibid., 142.
This represents an ion channel that requires an agonist molecule to transition. Because
the there is no way to distinguish between the two shut states the proportional
relationship will not have a physiological significance. They are derived from the
quadratic equation.24
For both three state models the distribution of the number of bursts per opening is
geometric. The only difference occurs when computing the burst length. Calculating the
burst length results from summing the probability of the channel remaining in the open
state through the entire duration and the probability of the channel being open then
changing state but returning to the open state by time t.
P’11(t) = Prob (stays in burst throughout 0, t and open at t | open at time 0)
In the agonist model a burst cannot end directly from the open state, but must first travel
through AR. So the agonist model must account for the additional condition that once in
AR the burst ends.25
Computation of the Models
When looking at the simpler state models, it quickly becomes clear that each additional
state adds greatly to the equation complexity needed to predict the results. More
importantly, the person who derives the equation must take great care to include all of
the relevant material. “Furthermore, it is found that the approach given above is not
sufficiently general to allow analysis of some mechanisms that are of direct experimental
interest. In particular, mechanisms with more than one open state and/or cyclic
reactions cannot be analyzed by the relatively simple method used so far.”26 When there
24
David Coquhoun and Alan G. Hawkes, AQ-Matrix Cookbook, In Single-Channel Recording.
Edited by Bert Sakmann and Erwin Neher. 2nd ed. New York : Plenum Press, c1995, 154.
25
Ibid., 155.
26
David Coquhoun and Alan G. Hawkes, “The interpretation of single channel recordings”, In
Single-Channel Recording. Edited by Bert Sakmann and Erwin Neher. New York : Plenum
Press, c1983, 167.
are a vast number of routes the only practical approach is to describe the system within a
matrix. “This enables a single computer program to be written that will numerically
evaluate the predicted behavior of any mechanism, given only the transition rates
between the various states.”27
Transition rates between states can be specified in a transition matrix (Q), so that “ the
entry in the ith row and jth column (denoted qij) representing the transition rate from
state i to state j.”28 Deriving the probabilities for noise and relaxation analysis can also
be specified in a matrix (P).
P(t) = e Qt
In order to perform analysis of the single ion channel a probability is required that
predicts the likelihood that a specific subset of states is maintained thought the specified
time. Using the matrices, a subset of the states, which pertain to the relevant question,
are selected.
If it was important to determine the burst length of the matrix the submatrix QEE would
contain those states that are relevant to burst states.
Summary
Single ion channels are random systems and since they are random systems they must be
modeled using stochastic mechanisms. Using a transition matrix makes it possible to
write a general program for analyzing behavior of complex mechanisms.
27
Ibid., 168.
28
Ibid., 168.