A Markov Model of Cell Membrane Potassium Channel and Prediction of Channel Status

Received: Jun 9 th 2014 Revised: Jul 19 th 2014 Accepted: Jul 26 th 2014 The most cell electrophysiological models are not able to predict the channel state, which it can be very helpful in patient treatment. So in this research we intend to represent a model for predicting it based on Markov model for ion channel. To obtain the data, we used a software environment consistent with the cellular conditions. Next step is estimation the model parameters. In order to achieve this goal, there are some sub-step. First, it is essential to specify channel states. In addition, it is used a method of linearization of channel macroscopic current for states Distinction. After Distinction of different channel states, finding the stopping times in each state and calculating the model parameters by use of relations between continuoustime Markov systems is done. Then the probabilities of transfer from one state to another are calculated in terms of time and voltage. Consequently, we could find probability matrix of state changes of Markov, which made it possible to predict the channel state in different voltages. The results obviously show the dependence of model parameters to the voltage of two sides of channel. This method of modeling is able to predict channel state in each voltage. Furthermore, the predictions of channel states for 100 future states are shown. Assessment criteria for accuracy of this model, is measured by comparison between actual channel conductance that is obtained from macroscopic current of the software in different voltages and conductance that is obtained from the considered model. Keyword:


INTRODUCTION
Ion channels gates are responsible to regulate the channel permeability to pass the ions (1).Crossing of ions existing in the cytoplasm through cell membrane depends on a phenomenon called gating behavior.Gating behavior is originated from the act that the ability of ions from membrane gates depends on membrane voltage.The reason of gating phenomenon is that proteins covering membrane surface, change their direction because of changes in direction and intensity of electrical field.Changing in the direction of a molecule due to a certain voltage leads to opening a gate.
This phenomenon called activation (2).The same molecule may move in opposite direction under the effect of another voltage and cause that gate to get closed.This phenomenon called deactivation (3).Some pores have several types of proteins so that some of these proteins change their direct to open the gate affected by a certain voltage, but some other proteins again close the gate by often slower moving.
In the most cases, spatial structure of the gate molecule will change in response to electric potential between two sides of a cell membrane.Although there are a variety of models for cell membrane, Markov models are able to demonstrate the transition states and make it possible to evaluate the detailed of a process behavior.In the other word, Markov process is a random process that probability laws govern it (4), (12).

IJCB
Different parts of Markov process are: 1.
Set of states that may occur 2.
The initial state vector 3.
Probability matrix of state changing Previously proposed electrophysiological models for ion channels are not able to predict the status of channel (5), [6].Therefore, in this research we intended to represent a model for predicting it based on Markov model for ion channel.The most important advantage of this modeling method is the ability of predicting the channel statues in different voltages it can be very helpful in patient treatment.Besides, the predicted channel state for 100 future states will be shown In this research, the method of obtaining the model parameter and prediction of channel behavior will be completely explained.The results obviously show the dependence of model parameters to the voltage of two sides of channel.Such that increasing in voltage, leads to increasing in α coefficient and decreasing in voltage, leads to increasing in β coefficient.

RESEARCH METHOD
The method of predicting a process states that: If V is the initial condition vector and p is the probability matrix of state changing, the process state will change as follow [4]: New state=V×P Generally, the process state in step n th is calculated as follows: Step n=V×P n As know, Potassium Channel in Hodgkin-Huxley model is demonstrated as equation ( 1): (1) ) .( . 4  max By regarding to the term 4  n in this equation, which indicates there are four similar actuator gates, and Markov model in detail [4], [7].We can consider a model for potassium channel as shown in figure 1. (2) Where matrix Q is: In addition, as shown in equations ( 9) to (10), in continuous-time Markov processes, the Coefficient matrix has a relation with stopping times, which means if x(t) is Markov process and Ti is stopping time in state i th , and ) exp(

*
In equation (10) Vi is the stopping time parameter and pij shows the probability of channel state changing from state i th to state j th .Therefore, matrix Q is as equation ( 11): By equalization of two matrixes ( 8) and (11), model parameters can be obtained.
On the other hand, by rewriting equations as based on equations ( 12) and ( 13), [8] (12) (13) In Equation ( 14) is yielded: Because coefficients of models are obtained in before stage, now we can achieve to probability of changing state in each voltage.

RESULTS AND ANALYSIS
In this section, it is explained the results of research and at the same time is given the comprehensive discussion.

Data obtaining
In

Obtaining stopping time
After data obtaining, we should find channel stopping times in each state.One method to obtain stopping time is to divide the current linearly [4], [12].In this method, because the considered model has five different states, maximum obtained current of channel should be divided to five parts.The minimum interval is assigned to state C1 in which every four gates are closed and so on the next intervals are assigned to state C2, C3, C4 and maximum interval to state O in which every four channels are open.Table 1 shows this method done on potassium current.The data of channel macroscopic current is obtained in 45 mv voltage from COR software.Based on this method, we divide the output current to considered intervals and find the times proportional to each interval (the program of calculating the stopping time in each interval is written in MATLAB).The table elements are time in milliseconds.Having the obtained points from above table, next step is to find the equations of TC1, TC2, TC3, TC4 and TO [4].Based on Markov continuous-time process the form of them are exponential equations as shown in equation ( 15): Where, v i is obtained independently for each status of model.For example, in the voltage of 45mv and for the stopping time of TC1 we have: Solving above equation, we will obtain v i for TC1 in the voltage of 45mv.Similarly, we can repeat this step for other elements of table.

Model parameters
As stated, by equalization of two above Q matrix, the model parameters can be obtained as a function of voltage.In this paper, we applied the data of cellular Nobel 2000 model [10] and the minimum model of Nap i [1], [11] and obtained α parameter of them, as shown in fig. 3 and fig.4, respectively.Figure 3 and figure 4 show the dependency of α parameter on the voltage of two sides of potassium channel for data from cellular Nobel 2000 model and minimum model of Nap i , respectively.In these figures, vertical axis is the value of α parameter in velocity and horizontal axis is voltage in millivolts.As the figures show voltage increasing leads to increasing in the value of α parameter.This increment is because increasing in the voltage of two sides of channel leads to increasing in the channel macroscopic current.Therefore, the channel goes toward the state o of forward direction and then the velocity of α increases.  .In these figures vertical axis is the value of β parameter in velocity and horizontal axis is voltage in millivolts.As the figures show voltage decreasing leads to increasing in the value of β parameter.This increment is because decreasing in the voltage of two sides of channel leads to decreasing in the channel macroscopic current.Therefore, the channel goes toward the state C1 of backward direction and then the velocity of β increases.
After attaining the coefficients in each voltage and after solving the above Ordinary Differential Equations (ode), the probability of channel being in each state can be found in different voltages as depicted in fig.7 for cellular Noble 2000.As we can see from the last figures, although there are a few errors, this method of modeling has been able to simulate the macroscopic current of channel in different voltages.
Previously proposed electrophysiological models for ion channels cannot predict the status of channel.So in this research we wanted to represent a Markov model for ion channel.The most important advantage of this modeling method is the ability of predicting the channel statues in different voltages.For modeling an ion channel by using Markov model, at first it is necessary to obtain the data of evaluated channel.Because in this research , it is not possible to access directly to the cellular environment and do an experiment to reach the actual data , we had to use the data obtained from a software environment called COR, which is an environment, compatible with the cellular condition.After data obtaining, we should find the model parameters.In models based on Markov, there is a direct relation between model parameters and stopping times in different states of channel, so it is necessary to apply a method so that we can specify channel states.The applied method for states Distinction is the method of linearization of channel macroscopic current.Then, we would find the model parameters by using stopping times.After that, having the model parameters we can calculate the probability matrix of state changes in each voltage and then it is possible to predict the channel state in each voltage.In this article, to achieve result and to evaluate the accuracy of the represented method, we used the macroscopic current of potassium channel obtained from cellular Noble 2000 model and minimal model Nap i .As we see in the modeling results, increasing in the voltage leads to increasing in the probability of opening of the channel gates and the channel state goes toward state o.so the macroscopic current of channel will increase.These results are completely compatible with the existing data.It should be noted that in this research, just the effect of voltage between two sides of channel on the performance of gates was evaluated.Considering that, the performance of channel gates depends on various factors such as drug, the environment temperature, enthalpy, etc. we can study the effect of these factors on opening and closing of channel in future researches to represent a complete Markov model which is able to predict the channel behavior under the effect of various factors.The most important advantage of this method of modeling is the ability of predicting so that it can be very helpful in patient treatment.

Figure 1 .
Figure 1.Considered Markov model for shown potassium channel.C 1 to C 4 and O are states of gates.Where, C1: shows the state in which every four gates of channel are closed.C2: shows the state in which three gates of channel are closed and one is open.C3: shows the state in which two gates of channel are closed and two are open.C4: shows the state in which one gate of channel is closed and three others are open.O: shows the state in which every four gates of channel are open.With regard to the model, model equations are as follow [8]:

Figure 2 .
Figure 2. The channel macroscopic current in 40 mv voltage

Figure 7 .Figure 7 Figure 8 .Figure 9 .Figure 9 Figure 10 .
Figure 7. Probability of channel being in each state in different voltages.Horizontal axis is voltage in millivolts (for Noble 2000)

Figure 11 .Figure 11
Figure 11.Mean number of occurrence of different states in each voltage in 100 times prediction(for Noble 2000)

Figure 12 .Figure 13 .Figure 14 .
Figure 12.Mean number of occurrence of different states in each voltage in 100 times prediction.(for minimal

Figure 15 .
Figure 15.Simulation error for a potassium channel Figure 17.Simulation error for potassium channels

Table 1 -
Stopping Times in some voltages