New York University

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Probabilistic Model of Ion Channel Behavior

Name

Institution

Introduction

A.L Hodgkin and A.F. Huxley successfully developed a mathematical model that

demonstrates the actual action potential of nerve cells found in a squid giant axon. The two

researchers used both analytics and experiments to develop the models. Though the

researchers used squid giant in their modelling, their model can be generalized for both

unicellular and higher organisms. The squid giant axon was more appropriate for modelling

because its large diameter of 0.5mm is sufficient enough to run experiments compared to

other species, hence making experimentation and subsequent modeling easy and more

accurate.

Figure 1: Equivalent Circuit of Hodgkin-Huxley Model

From the model of figure 1, the cell membrane mainly comprises of lipid bilayers. C

is the capacitance of the cell due to the separation of the liquid bilayers. K. R and Na

represent the conductance. The conductance shows the movement of ions across the cell

membrane. Before ion channels were discovered, scientists believed that the movement of

ions across cell membranes was aided by gating particles that allowed or rejected the

entrance of ions. Each ion was assumed to have its own specific gating particles. Sodium ion

was assumed to have two gating particles – m and h. m was assumed to be an activation

particle while n was an inactivation particle. Potassium ion was assumed to have a single

gating particle, n. n functioned as an activation particle. The circuit of figure 1 was also

assumed to have leaking current [1].

Figure 2: Ions Inside and Outside the Cell Membrane

From figure 2, it is apparent that Na ions are more concentrated outside the cell than

inside the cell. On the other hand, K ions are more concentrated inside the cell than outside

the cell. The difference in ion concentration causes an electrical action potential across the

cell membrane which can be considered as a voltage source. On another hand, the gating

particles control the amount of voltage and current that flows across the cell membrane in

either direction, rejecting or allowing the movement of ions. Comparing the two types of

ions, sodium ions have a higher rate of flow and concentration into the cell than potassium

ions which flow out of the cell. In fact, the difference in rates of flow of these two types of

ions is the source of voltage. The inner side of the cell is usually negative. Additionally, the

concentration of Na ions is higher than the concentration of K ions, resulting in the large

movement of Na ions across the membrane to the inside to neutralize the cell. This generates

a voltage.

Apparently, gating particles control the movement of ions, with each ion having a

particular probability of activation. The probability values depend on the amount of voltage

source across the cell membrane. The probability values represent an average value of

different experiments that were conducted in form of activation or inactivation of ion

movement for each particular ion.

Figure 3: Probability of Current Flow of Ions at Different Instances

From figure 3, every ion particle represents the channel of the ion. The probabilities

of activation of each ion channel are shown in figure 3. The probability of open-channel

shows the possibility that a channel is open to the ion at each given instant. Consequently, the

amount of time a channel is open is determined by this probability value over many different

experiments. Time trajectories of different channels also vary according to the cell setting’s

un-specificity. In this report, a model will be developed using the results of Hodgkin and

Huxley. This model will help in understanding the body reaction of cells. That said, the

model developed can go a long way in finding cures for neurological diseases by

characterizing the biological body responses as a probabilistic model.

Methods

The Probabilistic Behavior of a Single Potassium Ion Channel

The first step of building the model was to simulate the probabilistic behavior of a

single potassium ion channel when a step voltage is applied to it. In this experiment, the

probability of a potassium ion channel being open was determined at different values of the

voltage step held for adequate time to estimate the gating variable. Dr. Alfrey’s steady state

graph representing voltage and probability was used for this purpose. The graph shows the

gating particles for sodium and potassium as shown below.

Figure 4: Dr. Alfrey Steady State Graph

From figure 4, it is apparent that the activation of K gating particle ranges from 40mV to 100mV with a probability of activation of less than 1. Within this voltage range,

intervals of 20 are chosen between voltage points to measure the probabilities of activation.

From the graph, the curves of gating particles m and n have discontinuities. For K, the

discontinuity is at 101 (10mV) where the probability is 0.4754. The table below summarizes

the probabilities of K ion activation at different voltages.

Voltage(mV)

-20

0

P [activation

0.0891 0.318

20

40

60

80

100

0.619

0.806

0.895

0.938

0.962

of n particle]

Table 1: Probabilities of K ion Activation at Different Voltage Levels

For the model to be valid, certain requirements must be met. Probabilities provide

analytical values for a valid model. Table one provides reference probabilities for the

developed model. To determine probabilities, one should pick a value between 0 and 1. If the

value is greater than the reference probability value, the value is assigned as a closed state

with a reference value of 0 pA. if the value is less than the reference voltage, it is assigned as

an open state probability value and the current must be close to 2pA. The probability

distribution function is obtained as the area under the curve over the specified range out of 1

or the total probability area. The fact that the model is a continuous function makes it difficult

to sum up the area under the curve. As such, since the graph of figure 3 is PDF, calculating

the area under each curve is impossible and sum the total areas under the curve. However,

one can add the areas under each curve into a single curve to obtain the probability of K

gating particle activated, open.

Addition of probabilities is essential in explaining why the pickup values should be

smaller than the reference value, as this would produce a probability close to the actual one.

Obviously, the probability value changes every time it is chosen. Therefore, the process of

choosing the probability values should be repeated numerous times to obtain a more accurate

result. One value should be chosen any time the code is run and should be represented in the

form of either 0pA or 2 pA. The below formula can be used to calculate the probability of K

ion gating particle.

𝑃[𝑜𝑝𝑒𝑛 𝑠𝑡𝑎𝑡𝑒] =

# 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑏𝑒𝑖𝑛𝑔 𝑖𝑛 𝑜𝑝𝑒𝑛 𝑠𝑡𝑎𝑡𝑒

# 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑖𝑡 ℎ𝑎𝑠 𝑏𝑒𝑒𝑛 𝑟𝑢𝑛

.

𝐸𝑞. 1

This method is especially easier to understand if the code is run a small number of

times such as 5 times with a final value of 8pA. With this value, the channel is open 4 times

and close only once. Alternatively, one can add 2pA each time the channel is open then

dividing the value by 2 and multiplying by it with the assumption that the channel opens each

time it is run. Likewise, the code must be run many times, in this case 1000 times, to obtain

more accurate results. Since the model is applied in biological applications, physiological

responses must be included. 10ms is enough time for testing the activation of the K gating

particle as suggested by a K channel activation study that produced the most accurate

simulation with a frequency of 20,000 Hz. This frequency resulted in proper physiological

activation of the ion channel. The inverse of frequency gives the incremental time, which in

this case is 0.5ms [2]. Once an assumption is made on the time range, there are current points

matching each time point from the model in order to plot the model. These points are

considered on an individual basis and later considered in a bigger range in an attempt to

obtain matching step voltage with higher probability for potassium ions gating particle to be

in active state. It is essential to note that the experiment is entirely random and to obtain

matching with actual values, reference points are used to calculate the percentage error to

ensure the error is below 5% for the model to be good. The equation below is used to

calculate the percentage error.

%𝑒𝑟𝑟𝑜𝑟 = |

𝑃[𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒] − 𝑃[𝑚𝑜𝑑𝑒𝑙]

| ∗ 100.

𝑃[𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒]

𝐸𝑞. 2

P[reference] represent the steady state data points of figure 4. P[model] are data points from

the model. More than 10 points were obtained and plotted. The curve tool in Matlab was then

used to obtain voltage points for each.

The Probabilistic Behavior of The Total Potassium Ion Channels in a Cell

The states of the K ion gating particle (both activation and inactivation) depends on

the size of the membrane potential. The gating particles are assumed to be independent of

each other since they have independent probabilities. Markov chain method is used to add up

all the probabilities of potassium gating particles of a cell into one current.

Reference points are further used to determine whether the gating particle activation

is in open state or closed state. The value of the current of each potassium ion gating particle

changes each time the time increases and then sum all the gating particles up into one value

which represents the behavior of all the potassium ion gating particles of a cell. In addition,

the unit area is need to be calculated in order to find the numbers of the gating particles. In

this case, the surface area is calculated in a squid giant axon according to the HodgkinHuxley model. As earlier indicated, the squid giant axon has a diameter of 0.5mm. It has a

length of 10*104µm and the surfa...