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New York University Probabilistic Model of Ion Channel Behavior Analysis Paper

New York University

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paraphrasing each sentence for the whole report even the sentence that are describing each pictures.


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Probabilistic Model of Ion Channel Behavior Introduction: A.L Hodgkin and A.F. Huxley were two scientists who were successfully able to come up with a mathematical model that illustrates the real action potential of nerve cells that found in a squid giant axon. They were able to approach these models using different techniques that include analytics and experiments. They only used squid giant to model their model but the model can also represents both unicellular and higher organisms. A.L Hodgkin and A.F. Huxley mainly used and focused on the squid giant axon since its diameter is large enough, about 0.5 mm, to run the experiments on that time comparing to other species and that made their work much easier and more accurate. Figure 1, this shows an electrical circuit that represents the model of Hodgkin-Huxley. As shown in figure 1, the circuit basically shows the cell membrane which mainly consists of lipid bilayers. The C term represents the capacitance of the cell which is the separation of the lipid bilayers of the membrane while the R, K, and Na act as the conductance, inverse of resistance, which show the movement of the ions across the cell membrane and this was done before the discovery of ions’ channels. In that time, they assumed that the ions across the membrane through gating particles that allow or reject the entrance of the ions. For each ion, both K and Na, they assumed that each ion has its own gating particles. For sodium ions, Na, they assumed that it has two gating particle: m as an activation particle and h as an inactivation particle and at the same time they only assumed one gating particle for potassium ions, n, which acts as an activation particle. Since the circuit is not perfect, they assumed that there is a leaking current [1]. Figure 2, this figure shows the ions inside and outside the cell membrane. As shown in figure 2 at resting state, sodium ions have higher concentration in the extracellular than the intracellular where at the same state, potassium ions are the opposite, have higher concentration in the intracellular than the extracellular. Since the concentration of each ion is spread unequally inside and outside the cell, this causes a difference in the electrical potential of the cell and this can be considered as a voltage source. From this inequality, the gating particles are the main characters that control the amount of the voltage and current that flow in both directions by allowing or rejecting them. Basically, the flow of the sodium ions from extracellular to the inner side of the cell has much higher rate and concentration than the reverse flow of the potassium out of the cell and this is the main source for the cell voltage. The intracellular is usually negative; the concentration of sodium ions much higher than the potassium ions in the organisms, and this explains the huge movement of the sodium ions toward it to neutralize the cell which in turn generate the voltage source. As it has been said, the gating particles are the controllers of the movement of the ions and each one has different value of probability to be activated. These probabilities are dependent on the value of the voltage across the cell and it might increase or decrease according to the change of the voltage. The found probabilities are an average of different experiments that have been done according to the activation or inactivation of the movement of each ion. Figure 3, the figure shows the probabilities of currents' flow of both ions at different time Here, each ion particle represents the ion's channel. Figure 3 above shows the probabilities of each ion's channel to be activated, open, at different time which depend on the voltage input. Its open-channel probability determines the possibility that it is open at each instant and so the proportion of time it is open should approach this probability over a large number of experiments. At the same time, the individual time trajectory can be different from an ion channel to the other ones and that can be explained as cell settings' un-specificity. Here, the main goal is to develop a model using A.L Hodgkin and A.F. Huxley results and this can help to understand more about the cells' body reactions. Basically, this model can be helpful to find some 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 model starts with simulating the probabilistic behavior of a single potassium ion channel in response to a step of voltage. The experiment begins by finding the potassium ion's channel's probability of being open at given values of a voltage step that has been held for enough time that can approximate the average response of the gating variable. The best way was to use the steady state graph, provided by Dr.Alfrey, which represents the voltage being held for a period of time and read the probability at a given range of voltage step. The graph shows the probability of the two gating particles of sodium ions activation, h and m, and the gating particle of potassium ions, n and it is shown below. Gating Variables at Steady State as Functions of Potential 1 0.9 Gating variables at steady state 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -100 -80 -60 -40 -20 0 20 Potential (mV) 40 60 80 100 Figure 4, the graph shows the probability of each ion particle activation over a range of voltage. The activation of potassium gating particle, n, starts at – 40 mV and it goes all the way up to 100 mV without reaching a probability of 1 of being activated. The range is chosen to start from the beginning of the activation, -40 mV, to the end of the graph, 100 mV with a difference of 20 between each two voltage points using the data from the shown graph. There is a discontinuity in the curves of m and n gating particles. The discontinuity of potassium ion gating particle is located at point number 101, where the voltage is 10 mV and this can be approximated by the average of the sum of the probabilities at -9 mV and 11 mV which gives a probability of .4754. Voltage(mV) P[activation of n particle] -20 0 0.0891 0.318 20 0.619 40 0.806 60 0.895 80 0.938 100 0.962 Table 1: demonstrates the probability of activation of potassium ion at different voltage values. The model has to meet some requirements in order to be considered as a valid model. A probability is the analytical value for a valid model. Table1 is used here as reference probabilities for this model. The start is to pick up a number between 0 to1, in order to be considered a correct probability, and if it is greater than the reference value, this can be assigned as a closed state with current value of 0 pA. On the other hand, if the picked up value is less than the reference value, then it is considered to be in open state and the current should get close to 2 pA. The PDF is a known range is calculated as the area under the curve of this specific range out of 1 or the total probability area and this explains why the above analysis has been said. However, it is very complicated to sum up the areas under a given range for this model since it is a continuous function. Figure 3 is an example of a PDF and it is obvious that it is almost impossible to calculate the area of each curve and then sum all the areas of the curves. Nonetheless, it is possible to add up all the areas under each curve into only one curve and this gives the probability for a potassium gating particle to be activated, open. The idea of adding up the probabilities into one is a helpful step and it can explain why the pickup value should be smaller than the reference value from the table above since it would produce a close probability to the actual one. The probability value would change each time it being chosen and from this it is supposed to be run for enough times in order to be obtain a better and more accurate result. Practically, only one of two values should be picked up for each time the created code run and they can be represented as the current values mentioned above, 0 pA and 2 pA. The probability of potassium ion gating particle to be activated, open, can be calculated using the common way of dividing the number of the gating particle being activated out of total number of times the code run by the total number of times the code run and this shown in the equation below: 𝑃 𝑂𝑝𝑒𝑛 π‘†π‘‘π‘Žπ‘‘π‘’ = # π‘œπ‘“π‘‘π‘–π‘šπ‘’π‘  𝑏𝑒𝑖𝑛𝑔 𝑖𝑛 π‘œπ‘π‘’π‘› π‘ π‘‘π‘Žπ‘‘π‘’ , # π‘œπ‘“ π‘‘π‘–π‘šπ‘’π‘  𝑖𝑑 β„Žπ‘Žπ‘  𝑏𝑒𝑖𝑛𝑔 π‘Ÿπ‘’π‘› πΈπ‘ž. 1 An easier way to show this is when the code has be run for five times with a finally value of 8 pA, then this shows that it was open for four times while it was only close for once. Another way is to think about is to add 2 pA each time it has been open dividing that by two multiplying it by assuming it has been open each time it has been run. Again, the code had to be run for many times, here one thousand, to make the results more accurate and logical since the picked up number is random. In addition of these criteria, physiological response should be included since the goal is to apply this work toward a biological application. The experiment was estimating 10 ms to be enough time to test potassium ion gating particle activation and this has made according to a potassium channel activation study that found the most accurate simulation was the best with a frequency of 20,000 Hz. The ion channel was activated in a proper physiological manner when using this frequency. Thus by taking the inverse of this frequency, the time increment can be established as 0.05 ms [2]. After the time range is assumed, there are current points matching each time point from the model in order to plot the model. The points here are considered individually and the next parts of this project they are going to be considered in a bigger range in order to find the matching step voltage with higher probability for the potassium ions gating particle to be in active state. This experiment is completely random and in order to find how it matching with actual value, reference points, the percent error is calculated to ensure they are ≀5% in order to be considered a good model. The error percent was calculated as values: %π‘’π‘Ÿπ‘Ÿπ‘œπ‘Ÿ = β”‚ 𝑃 π‘Ÿπ‘’π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’ βˆ’ 𝑃 π‘šπ‘œπ‘‘π‘’π‘™ 𝑃 π‘Ÿπ‘’π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’ β”‚ βˆ— 100 , πΈπ‘ž. 2 P[reference] is a point from the steady state data points, figure 4, and P[model] is a point from the model. After having more than ten points to compare, they were plotted and the curve tool was used in Matlab in order to find the voltage point for each. The Probabilistic Behavior of The Total Potassium Ion Channels In a Cell: The activation and inactivation states of the gating particle of potassium ion directly depend on how much membrane potential is there. Each potassium ion gating particle is assumed to be independent of each other since the probability of being activated or inactivated of each one of them doesn't have any effect on the other ones. Since the goal here is to sum up all the probabilities of potassium gating particles of a cell into one current, the best way to model this problem is to use Markov chain method. The model is again to use the reference points where the gating particle considered being activated, open state, when it is smaller than the reference point, otherwise it is considered to be inactivated, 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. The surface area is calculated in a squid giant axon since A.L Hodgkin and A.F. Huxley model used it. As mentioned in the introduction, the diameter of the squid giant axon is about 0.5 mm or 500 Β΅m and its length is 10*104 Β΅m the surface is considered to be a cylindrical with a formula of [3]: 𝐴 = 2πœ‹π‘Ÿβ„Ž + 2πœ‹π‘Ÿ ! = 2πœ‹ . 25 βˆ— 10! 10 βˆ— 10! + 2πœ‹ 0.25 βˆ— 10! ! = 15.75 βˆ— 10! Β΅π‘š! The number of potassium gating particles found in a squid giant axon is given as 30 per 1 Β΅m2. From the area of the squid giant axon and the number of the potassium gating particles, the number of the gating particle in the whole cell can be calculated as follows: π‘›π‘’π‘šπ‘π‘’π‘Ÿπ‘  π‘œπ‘“ 𝑛 π‘π‘Žπ‘Ÿπ‘‘π‘–π‘π‘™π‘’π‘  = π΄π‘Ÿπ‘’π‘Ž βˆ— #𝑠 π‘œπ‘“ 𝑛 π‘π‘’π‘Ÿ 𝑒𝑛𝑖𝑑 π‘Žπ‘Ÿπ‘’π‘Ž = 15.75 βˆ— 10! βˆ— 30 = 4.72 βˆ— 10! π‘π‘œπ‘‘π‘Žπ‘ π‘ π‘–π‘’π‘š π‘”π‘Žπ‘‘π‘–π‘›π‘” π‘π‘Žπ‘Ÿπ‘‘π‘–π‘π‘™π‘’π‘  The found number of potassium ion grating particles is acceptable to some extend comparing to the physiological value. To be more precise with results, the percent error was calculated again but for the whole cell. This has been done by comparing the model current with one that is given as potassium current equations and inputting membrane potential. The values for the potassium gating particles were given and it was plotted and compared with the matching model point in order to calculate the percent error. As in the first portion of this project, equation 2 was used to calculate the percent error and the average was within 10% which was one of the requirements. After these steps, a random number was picked using the rand function in Matlab and that was the same as in the previous part. This part is different than the first for finding the probability for each state and here it was done using integration of the given derivative of the potassium ion gating particles, !". The resulted formula is !" shown below: 𝑛 𝑑 = 𝛼! 𝛼! + 𝛽! βˆ— 1 βˆ’ 𝑒! !! !!! ! , πΈπ‘ž. 3 Equation 3 represents basically the probability of a potassium ion gating particle to be activated at a certain time. In addition, from the given equations, the IK ,𝑔! 𝑛! 𝑉! βˆ’ 𝐸! , has n in the 4th degree and this suggests having 4 transitions and in turn this shows that there are 5 states between being activated or inactivated. The n4 is here is considered to be the reference value that the model compared with and it follows the same idea as it is explained in the beginning of this part and in the first part too. Each n out of the four represents one out of the four transitions and each one represent a probability, shown below in figure 5. Figure 5 shows the n transitions along with their states. Figure 6 shows the movments of the particles into different states with alpha and beta rates The transitions are calculated by using Eq.3 above where the alpha and beta terms represents the dependent rates of voltage and these shows how the particles move from a state to the next and this has been shown in figure 6 above. The state on the right side of figure 6 shows the open state where the one on the left shows the closed state. The transition from the open state to close state has a beta rate where the other direction has an alpha rate. From the calculated number of potassium ion gating particles, 4.72 billion, the Matlab should be run with same number. On the other hand, there is no way for Matlab to be able to run with this number three thousand should a good number to be run with. The model was compared with actual one and .85 Β΅A is used here as a dc shift for the model. Also, the action potential is used here as the only reference value to be involved in the calculation of the current of the model since it is the needed voltage to show that the potassium ion gating particles are fully activated, open. The resistance of the axon of giant squid is 104 Ξ© per 1 cm [4]. The length of the axon is mentioned earlier as 10 cm which gives a resistance value of 1000 Ξ© and then its inverse would be multiplied by the voltage that is in use minus the action potential. Here the equation can be written as follows: 𝐼!"!#$ = 1 βˆ— 𝑉! βˆ’ 60 + 0.85 1000 The results that described below are good enough but it needs more work in order to make it more physiologically reasonable. In this model, the action potential was used all the time while the number of the potassium gating particles didn't too. Here, the idea is to take in consideration that the current doesn’t take only two values which in turn change the membrane potential voltage and this would be much more physiologically reasonable since the rate of the ions flow changes. The current and the voltage are linearly propositional to each other since the resistance of the gating particles kept the same. The resistance of the gating particle is represented here by the area of the cell membrane along with proteins and fats associated with it. These two factors do not change over time which allows this relationship between the current and the voltage. In addition, the rise time has some errors to calculate by comparing with reference points. The rise time is described here as the needed time for the current to get close to 70% of the steady state value. Again, the same equation of percent error, Eq.2, was used to find how off the model from the reference points and it should be within 10%. The code was not run due to some problems that were not figure out. The results should be similar for each value of the voltage and that is described in the results section in details. In order to accomplish better results, more knowledge must be gained in order to make the results to the best degree of perfection. Moreover, the single potassium ion channel was much easier to design and satisfy the given requirements since only two possibilities were there so there is no need to any more restrictions or requirements to model the equations for it. For the second part, most of the requirements were tested but some of them didn't come up with desired results. A good idea is to use the knowledge from the previous semester class, BME 344, of using correctors to satisfy any requirement given. That can be done by evaluating the results each time it runs and then test it with corrector and if it doesn't come with wanted value the code would run again until it work. However, this cannot be done using Matlab due to the restrictions it has. Another way is to get the accurate values of the currents and voltages that the axon passes and uses and find the actual number of the potassium ion channels. Indeed, considering only one type of channels individually cannot come up with good results because they collaborate with each other and they affect each other work and mechanism. Finally, Results & Discussion: The Probabilistic Behavior of a Single Potassium Ion Channel : Before start finding the probability for the potassium ion gating particle of being activated, open, at different values of membrane potential voltage, there was a need to calculate the percent error to make sure that it is within 5% to satisfy one of the requirements. The percent error was calculated for each point, from -20mV to 100 mV with step size of 20, and then averaged to get the mean percent error. The percent error was found to be 2.65% which is more than enough for this case. After the satisfying the error percent requirement, the Matlab code was run with 7 steady state values of membrane potential voltage. In this case, the model is set to accept only two values of current that 2 pA for activating, open state, and 0 pA for inactivating, closed state. The code was run and the results are shown below for each one of the steady state membrane potential voltage value. Current, pA A Single Potassium Ion Gating Particle In Response To Voltage of 20 mV And 40 mV 3 2 1 0 0 1 2 3 4 5 ...
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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[π‘œπ‘π‘’π‘› π‘ π‘‘π‘Žπ‘‘π‘’] =

# π‘œπ‘“ π‘‘π‘–π‘šπ‘’π‘  𝑏𝑒𝑖𝑛𝑔 𝑖𝑛 π‘œπ‘π‘’π‘› π‘ π‘‘π‘Žπ‘‘π‘’
# π‘œπ‘“ π‘‘π‘–π‘šπ‘’π‘  𝑖𝑑 β„Žπ‘Žπ‘  𝑏𝑒𝑒𝑛 π‘Ÿπ‘’π‘›

.

πΈπ‘ž. 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.
%π‘’π‘Ÿπ‘Ÿπ‘œπ‘Ÿ = |

𝑃[π‘Ÿπ‘’π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’] βˆ’ 𝑃[π‘šπ‘œπ‘‘π‘’π‘™]
| βˆ— 100.
𝑃[π‘Ÿπ‘’π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’]

πΈπ‘ž. 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*104Β΅m and the surfa...

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