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Volume-5, Issue-6, December-2015
International Journal of Engineering and Management Research
Page Number: 263-269
Pharmacodynamic Model of Ligand Gated Ion Channel Concomitant with
Cell Signalling Pathways
Ashis Kumar Das1, Suman Halder2
1,2
Department of Electrical Engineering, NationalInstitute of Technology, Durgapur, INDIA
ABSTRACT
In this present paper the authors endeavored to show the effect of neuromuscular drug on the ligand channels via cell signaling pathways. It is well known that during surgery neuromuscular drug is administered to achieve muscle relaxation. Thus the response of the ligand gated channels along with acetylcholine receptor concomitant with cell signaling pathways due to neuromuscular drugs are simulated with help of MATLAB. It has been investigated that due to increment in doses of neuromuscular drug the frequency at which power spectrum density (PSD) occurs, decreases very slowly.
Keywords—Ligand binding channels, Neuromuscular drug, Power Spectrum Density, Signaling Pathways.
I.
INTRODUCTION
Ligand-gated ion channels (LGICs) are a group of transmembrane ion channel proteins which open to allow ions such as Na+, K+,Ca2+, or Cl− to pass through the membrane in response to the binding of a chemical messenger (i.e. a
ligand
), such as a neurotransmitter [1,2]. These proteins are typically composed of at least two different domains: a transmembrane domain which includes the ion pore, and an extracellular domain which includes the ligand binding location (an allosteric binding site). The function of such receptors located at synapses is to convert the chemical signal of presynaptically released neurotransmitter directly and very quickly into a postsynaptic electrical signal. LGICs are classified into three superfamilies which lack evolutionary relationship: Cys-loop receptors, Ionotropic glutamate receptors andATP-gated channels. LGICs can be contrasted
with metabotropic receptors (which use second messenger activated ion channels), voltage-gated ion channels (which
open and close depending on membrane potential), and stretch-activated ion channels (which open and close depending on mechanical deformation of the cell membrane) [2, 3]. The prototypic ligand-gated ion channel is the nicotinic acetylcholine receptor. It consists of a
pentamer of protein subunits (typically α, α, β, γ, δ), with two binding sites for acetylcholine (one at the interface of each alpha subunit). When the acetylcholine binds it alters the receptor's configuration and causes the constriction in the pore of approximately 3 angstroms to widen to approximately 8 angstroms so that ions can pass through.
This pore allows Na+ ions to flow down
their electrochemical gradient into the cell. With a sufficient number of channels opening at once, the inward flow of positive charges carried by Na+ ions depolarizes the postsynaptic membrane sufficiently to initiate an
Genes in this family provide instructions for making ligand-gated ion channels [4]. These channels span the membrane that surrounds cells and form a hole (pore) through which positively or negatively charged atoms (ions), such as sodium, potassium, calcium, or chloride ions, can flow. The flow of ions is triggered by the attachment (binding) of proteins called ligands. Each channel is "opened" by the binding of a particular ligand. The ligands that open ligand-gated ion channels are called action potential. In this present work the authors tried to show the variations of action potentials at neuronal axon with the application of neuromuscular drug in the presynaptic neuron i.e., how the neuromuscular drug exerts its effects on the acetylcholine-nicotinic receptor binding and finally varies the characteristics of action potential of axon and signal transmission.
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neurotransmitters because they transmit signals in the brain. Genes in the ligand-gated ion channels family can be divided into several groups based on the ligand that binds to the channel. Nicotinic acetylcholine receptors are triggered by the binding of a chemical called acetylcholine. Serotonin type 3 (5-HT3) receptors are triggered by
serotonin. GABAA receptors are triggered by
gamma-aminobutyric acid (GABA) [5]. Ionotropic glutamate receptors are triggered by glutamate. Glycine receptors are triggered by glycine [6]. Ionotropic purinergic receptors are triggered by a chemical called adenosine triphosphate (ATP) [7]. And zinc-activated channels are triggered by zinc. Ligand-gated ion channels play an important role in the nervous system. The channels are primarily found in nerve cells (neurons) in the brain, spinal cord, and muscles, and they function to control the transmission of nerve impulses from neuron to neuron or neuron to muscle. The flow of positively charged ions into the cell turns on (excites) nerve impulses. In contrast, the flow of negatively charged ions into the cell blocks (inhibits) nerve impulses to prevent the nervous system from being overloaded with too many signals. Nicotinic acetylcholine receptors, 5-HT3 receptors, and ionotropic glutamate
receptors allow the flow of positively charged sodium, potassium, and, sometimes, calcium ions, which excites nerve signaling. Similarly, ionotropic purinergic receptors allow calcium ions to cross the cell membrane, exciting nerve signaling [8]. GABAA
Dendrite
Nucleus
Cell body Neck of axon
Axon Myelin Sheath
Terminal button and vesicles containing
neurotransmitter and glycine receptors allow the flow of negatively charged chloride ions, and so these channels inhibit nerve signaling. Signaling in the brain through ligand-gated ion channels is important for many neurological functions, including learning, memory formation, movement of muscles, and reflexes.
III.
ACTION OF NERVE
Figure 1: Nerve cell of mammal
Figure1 shows the structure of typical nerve cell. The nucleus of the cell is found in the large cell body situated at one end of the neuron. Small arms (dendrites) radiate from the cell body and receive messages from other neurons. These messages either stimulate or de-stimulate
the neuron. The cell body ‘collects’ the sum total of these messages. Ion channels are selective for different ions.
There are cationic ion channels for Na+, K+, and Ca2+ions.
When these channels are open, they are generally excitatory and lead to depolarization of the cell. Assuming that the overall stimulation is great enough, an electrical signal is fired down the length of the neuron (the axon). When a signal is received from the axon, the vesicles merge with the cell membrane and release their neurotransmitter into the gap between the neuron and the
target cell (synaptic gap) [9]. The neurotransmitter binds to
de-265
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stimulate the neuron. On opening of few sodium ion channels the sodium ions would flow into the cell, as a result, the electric potential would become less negative. This is known as depolarization and results in a stimulation of the neuron. Ion channels are controlled by the neurotransmitters released by communicating neurons. The neurotransmitters bind with their receptors and this leads to the opening or closing of ion channels. Such ion channels are known as ligand gated ion channels. For example, acetylcholine controls the sodium ion channel,
whereas γ-aminobutyric acid (GABA) and glycine control
chloride ion channels. The resulting flow of ions leads to a localized hyperpolarization or depolarization in the area of the ion channel. The cell body collects and sums all this information such that the neck of the axon experiences an overall depolarization or hyperpolarization depending on the sum total of the various excitatory or inhibitory signals
received. Figure 2 shows cell membrane of the axon also
has sodium and potassium ion channels, but they are
different in character from those in the cell body [10]. The
axon ion channels are not controlled by neurotransmitters, but by the electric potential of the cell membrane. Therefore, they are known as voltage-gated ion channels. The sodium ion channels located at the junction of the nerve axon with the cell body are the crucial channels as they are the first channels to experience whether the cell body has been depolarized or hyperpolarized. If the cell body is strongly depolarized then a signal is fired along the neuron. A specific threshold value has to be reached before this happens, however. If the depolarization from the cell body Resting state Overall membrane potential from cell body -+ -+ -+ -+-+-+-- ----++ ++ ++ Cell body Neck -+ -+ -+ -+-+-+-- ----++ ++ ++ Cell
body
-+ -+ -+ -+-+-+--
----++ ++ ++
Cell
body
-+ -+ -+-+-+- --
--- ++ ++ ++
Cell body Hyperpolarization
of cell body by inhibitory nerves
Weak depolarization of cell body by excitory nerves
Strong depolarization of cell body by excitory nerves Overall membrane potential from cell body Neck Axon Overall membrane potential from cell body Neck Axon Overall membrane potential from cell body + + - +-Hyperpolarization of axon neck:
• K+ channels open
• Na+ channels closed
Weak depolarization: • Na+ ion channels open slightly • Electric potential reduced but not reversed
Strong depolarization: • Na+ ion channels open • Reversal of polarity of axon
• Action potential generated • Message sent down axon
Neck Axon Axon
Figure 2: Hyperpolarization and depolarization effects at the neck of action
is weak, only a few sodium channels open up and the depolarization at the neck of the axon does not reach that threshold value. The sodium channels then re-close and no
signal is sent. With stronger depolarization, more sodium
channels open up until the flow of sodium ions entering the axon becomes greater than the flow of potassium ions leaving it. This results in a rapid increase in depolarization, which, in turn, opens up more sodium channels, resulting in very strong depolarization at the neck of the axon. The flow of sodium ions into the cell increases dramatically, such that it is far greater than the flow of potassium ions out of the axon, and the electric potential across the membrane is reversed, such that it is positive inside the cell and negative outside the cell. This process lasts less than a millisecond before the sodium channels re-close and sodium permeability returns to its normal state. More potassium channels then open and permeability to potassium ions increases for a while to speed up the return to the resting state. The process is known as an action potential and can only take place in the axon of the neuron.
IV.
DESCRIPTION OF MODEL
Here the authors have fabricated the MATLAB model for depicting the action of neuron. Mainly three ligands (neurotransmitters) are chosen for modelling that how the neurotransmitter binds with its receptor in the synaptic gap. Nicotinic acetylcholine receptors are triggered by the binding of a chemical called acetylcholine. Serotonin type 3 (5-HT3) receptors are triggered by
serotonin and Ionotropic glutamate receptors are triggered by glutamate. Nicotinic acetylcholine receptors, 5-HT3 receptors, and ionotropic glutamate receptors allow
the flow of positively charged sodium, potassium, and, sometimes, calcium ions, which excites nerve signaling. Similarly, ionotropic purinergic receptors allow calcium ions to cross the cell membrane, exciting nerve signaling.
GABAA
Nicotinic acetylcholine
receptor binding
5-HT3 receptor
binding
Opening of Na+/ K+ ion
channel
Opening of Na+/ K+ ion
channel Signalling pathway synaptic vesicles Release of serotonin Signal from Action + ionotropic glutamate receptor binding Opening of Na+/ K+ ion
channel Stimulate/ destimulate neuron Signalling pathway Stimulate/ destimulate neuron Signalling pathway Signal transmitted to junction of the nerve axon with the cell body Release of acetylcholine Release of glutamate Stimulate/ destimulate neuron
and glycine receptors allow the flow of negatively charged chloride ions, and so these channels inhibit nerve signaling [11]. Here in this proposed model the authors have considered the effects of acetylcholine, Serotonin and glutamate because they are associated with sodium and
Figure 3: Ligand-receptor binding model
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the model is transmitted to axon of the neuron. The cell membrane of the axon also has sodium and potassium channel but they are different in character from those in the cell body. The axon ion channels are not controlled by neurotransmitters, but by the electric potential of the cell membrane [14, 15]. Here in this proposed model the authors simulated the action potential in the axon of neuron and the variation of the same with the change in time constant of ligand-receptor binding at the synaptic cleft.
IV.
METHODS
By analyzing the estimated sodium conductance
(GNa) and potassium conductance (GK) from voltage
clamp pulses of various amplitudes and durations, Hodgkin and Huxley were able to obtain a complete set of nonlinear empirical equations that described the action potential [12,13,16]. Simulations using these equations accurately describe an action potential in response to a wide variety of stimulations. Before presenting these equations, it is important to qualitatively understand the sequence of events that occur during an action potential. The start of an action potential begins with a depolarization above threshold that causes an increase in
GNa
Figure 4: Change in Na
and results in inward sodium ion current.
+
Figure 5: Change in K
conductance with time
+
The sodium ion current causes a further depolarization of the membrane, which then increases the sodium ion current. This continues to drive membrane
potential (V
conductance with time
m) to the Nernst potential or equilibrium potential for sodium ion. As shown in Figure 4, GNa is a function of both time and voltage, which peaks and then falls to zero. During the time it takes for GNa to return to zero, GK continues to increase, which hyperpolarizes the cell membrane and drives Vm from ENa toward EK. The
increase in GK results in an outward K+current. The K+ current causes further hyperpolarization of the membrane, which then increases K+ current. This continues to drive
Vm to the Nernst potential for K+
Figure 6: V
, which is below resting potential.
m, GNa, and GK during an action potential
Figure 6, illustrates the changes in Vm, GNa, and
GKduring an action potential.
The empirical equation used by Hodgkin and Huxley to model GNa and GK is of the form
𝐺𝐺(𝑡𝑡) = (𝐴𝐴+𝐵𝐵𝑒𝑒−𝐶𝐶𝑡𝑡)𝐷𝐷 (1)
Values for the parameters A, B, C, and D were estimated from the voltage clamp data that were collected on the squid giant axon. Not evident in Eqn. (1) is the voltage dependence of the conductance channels. The voltage dependence is captured in the parameters as described in this section. In each of the conductance models, D is selected as 4 to give a best fit to the data.
The potassium conductance waveform shown in Figure 5 is described by a rise to a peak while the stimulus is applied. This aspect is easily included in a model of GK by using the general Hodgkin-Huxley expression as follows.
𝐺𝐺
𝐾𝐾=
𝐺𝐺
𝐾𝐾𝑛𝑛
4(2)
where 𝐺𝐺𝐾𝐾 is maximum K+ conductance and n is thought of
as a rate constant and given as the solution to the following differential equation:
𝑑𝑑𝑛𝑛
𝑑𝑑𝑡𝑡 =𝛼𝛼𝑛𝑛(1− 𝑛𝑛)− 𝛽𝛽𝑛𝑛𝑛𝑛 (3)
where
𝛼𝛼𝑛𝑛 = 0.01 𝑉𝑉+10 𝑒𝑒𝑉𝑉+1010 −1
(4)
𝛽𝛽𝑛𝑛 = 0.125𝑒𝑒 𝑉𝑉
80 (5)
𝑉𝑉=𝑉𝑉𝑟𝑟𝑟𝑟− 𝑉𝑉𝑚𝑚 (6)
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Vrp is the membrane potential at rest without any membrane stimulation. Note that V is the displacement from resting potential and should be negative. Clearly, GK is a time dependent variable, since it depends on Eq. (3), and a voltage-dependent variable, since n depends on voltage because of αnand βn.
The sodium conductance waveform in Figure 4 is described by a rise to a peak and a subsequent decline. These aspects are included in a model of GNa as the product of two functions, one describing the rising phase and the other describing the falling phase, and modelled as
𝐺𝐺𝑁𝑁𝑁𝑁 =𝐺𝐺𝑁𝑁𝑁𝑁𝑚𝑚3ℎ (7)
where 𝐺𝐺𝑁𝑁𝑁𝑁 is maximum Na+conductance, and m and h are thought of as rate constants and given as the solutions to the following differential equations:
𝑑𝑑𝑚𝑚
𝑑𝑑𝑡𝑡 =𝛼𝛼𝑚𝑚(1− 𝑚𝑚)− 𝛽𝛽𝑚𝑚𝑚𝑚 (8)
where
𝛼𝛼𝑚𝑚= 0.1 𝑉𝑉+25
𝑒𝑒𝑉𝑉+2510 −1 (9)
𝛽𝛽𝑚𝑚 = 4𝑒𝑒
𝑉𝑉
18 (10) and
𝑑𝑑ℎ
𝑑𝑑𝑡𝑡 =𝛼𝛼ℎ(1− ℎ)− 𝛽𝛽ℎℎ (11)
𝛼𝛼ℎ= 0.07𝑒𝑒 𝑉𝑉
20 (12)
𝛽𝛽ℎ= 1
𝑒𝑒𝑉𝑉+3010 +1
(13)
m describes the rising phase and h describes the falling phase of GNa. The units for the 𝛼𝛼𝑖𝑖′𝑠𝑠 and 𝛽𝛽𝑖𝑖′𝑠𝑠 in Eqn. (3), (8), and (11) are ms-1, while n, m, and h are dimensionless and range in value from 0 to 1.
Outside
Inside Membrane
Cmy
Cmem
Cmem
Rmy
Rmem
GL GK GNa
EL EK ENa
Vm
I IK INa
+
Figure 7: Circuit model of a section of mammal axon with leaky myelin sheath
Figure 7 shows a model of the cell membrane that is stimulated via an external stimulus current I coming from the cell body of the neuron due to ligand-receptor binding. This current is appropriate for simulating action potentials. Applying Kirchhoff’s current law at the cytoplasm yields
𝐼𝐼
=
𝐺𝐺
𝐾𝐾(
𝑉𝑉
𝑚𝑚− 𝐸𝐸
𝐾𝐾) +
𝐺𝐺
𝑁𝑁𝑁𝑁(
𝑉𝑉
𝑚𝑚− 𝐸𝐸
𝑁𝑁𝑁𝑁) +
(𝑉𝑉𝑚𝑚𝑅𝑅−𝐸𝐸𝐿𝐿 𝐿𝐿)+
𝑉𝑉𝑚𝑚𝑅𝑅𝑚𝑚𝑚𝑚+𝑅𝑅𝑚𝑚 𝑒𝑒𝑚𝑚
+
𝐶𝐶𝑡𝑡𝑡𝑡𝑡𝑡𝑁𝑁𝑡𝑡
𝑑𝑑𝑉𝑉𝑚𝑚
𝑑𝑑𝑡𝑡
(14) where GK and GNa are voltage time dependent conductance given by Eqs. (2) and Eqs (7), 𝑅𝑅𝐿𝐿= 1
𝐺𝐺𝐿𝐿 is the resistance of
leakage channel, Rmy and Rmem are the leaky myelin sheath resistance and axolemma (membrane) resistance respectively. 𝐶𝐶𝑡𝑡𝑡𝑡𝑡𝑡𝑁𝑁𝑡𝑡 =𝐶𝐶𝑚𝑚𝑒𝑒𝑚𝑚 + 𝐶𝐶𝑚𝑚𝑚𝑚𝐶𝐶𝑚𝑚𝑒𝑒𝑚𝑚
𝐶𝐶𝑚𝑚𝑚𝑚+𝐶𝐶𝑚𝑚𝑒𝑒𝑚𝑚, where Cmem and
Cmy are the leaky myelin sheath capacitances and
axolemma (membrane) capacitance respectively.
To find Vm during an action potential, four differential equations (Eqs. (3), (8), (11), and (14)) and six algebraic equations (𝛼𝛼𝑖𝑖′𝑠𝑠 and 𝛽𝛽𝑖𝑖′𝑠𝑠 in Eqs (4), (5), (9), (10), (12) and (13) need to be solved. Since the system of equations is nonlinear due to the n4 and m3 conductance terms, an analytic solution is not possible. To solve for Vm, it is therefore necessary to simulate the solution. By constructing MATLAB model for the ligand-receptor binding and by MATLAB script for the voltage gated axon of neuron, we have simulated and found solution for the action potential. The action potential of the simulated neuron Vm
The parameter values used in the simulation were based on empirical results from Hodgkin and Huxley, Jacob E. Smit et al[16].
IV.
RESULTS AND OBSERVATIONS
In this proposed MATLAB model, a number of simulations have been carried out to show the change in characteristics of the waveform of action potential with change in time constant of the signalling pathways of ligand-receptor binding blocks. At the same time the authors are tried to show the of power spectral density for each simulation. The power spectral density (PSD) of any signal gives the frequency at which the signal gives maximum amount of power. If any neuromuscular drug is administered on the nicotinic acetylcholine receptor binding pathway, it may increase the ligand-receptor binding time and subsequently it will change the characteristics of action potential and it can be verified by the PSD. To demonstrate the action of neuromuscular drug the authors increase the time constant of the nicotinic-acetylcholine receptor binding pathway gradually. The increment of time constant may be analogous to percentage of concentrations of drugs. Figure 8 and Figure 9shows the action potential and PSD of the model without application of any drug.
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Figure 8: Action potential of the model without application of any drug
Figure 9: PSD of the model without application of any drug
It shows that the action potential has its maximum power of 0.00341 unit at 38Hz. Now we apply the neuromuscular drug at a rate of increased concentration, it increases the binding time of ligand (neurotransmitter) with its receptor and simultaneously the characteristics of action potential and PSD change. This changes can has shown in Figure 10 and in TABLE I.
a
b
c
d
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f
Figure 10: Action potential (a,c,e) and PSD (b,d,f) of simulated neuron on applications of increased percentage
of concentration of neuromuscular drug.
TABLE II
EFFECTS OF NEUROMUSCULAR DRUGS ON ACTION
POTENTIALS Percentage(%) of drug
concentration administered
Power Spectral Density (PSD)of action potential
0 0.00341 at 38Hz
20 0.00373 at 37Hz
40 0.00335 at 37Hz
100 0.00400 at 36Hz
V.
CONCLUSION
The authors have investigated the how the PSD frequency changes with the concentration of the neuromuscular drug. It has been shown by the authors that as the concentration of neuromuscular drug increase the PSD frequency decreases very slowly being highest at 38 Hz with no doses and lowest at 36 Hz with 100% concentration. The authors have come into conclusion that during surgery to achieve optimum muscle relaxation PSD frequency of action potential should be minimum at 36 Hz
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