The Nernst Equation - Single Ion Species
Electrochemical gradients drive countless physiological processes. Let's focus on one the majority of you will be familiar with, the action potential, a tenet of high school level biology. The standard cell maintains an electrochemical gradient via sodium and potassium ions dissolved in aqueous environments. The best way to remember the model, is to think of a French fry, a potassium rich potato sprinkled with sodium on the outside. This trend remains true in the standard cell. A higher concentration of sodium exists outside the cell, and a higher concentration of Potassium ions inside the cell.
Luigi Galvani first birthed the idea that electricity powers the biological system, and Walther Nernst derived an equation to help describe the basis for the theory. The Nernst equation, allows us to calculate the membrane potential of a given cell, generated by a concentration gradient of ions separated by a membrane permeable only to a specific ion. This permeable membrane was later evidenced in the form of a particular transmembrane protein, the voltage gated sodium channel, and another transmembrane protein called a sodium-potassium pump. Together, they allow differential trafficking of sodium and potassium ions normally pushed away by a structural phospholipid bi-layer with hydrophilic and hydrophobic regions. The net result of this enforced gradient, is part of the biological electricity Galvani first hypothesized.
Today, we're going to write a program to calculate the expected membrane potential of a human cell using the Nernst Equation.
We only use the Nernst equation for single ions, for multiple ions see the Goldman-Hodgkin-Katz equation.
For our first model, we'll use the Nernst in it's raw form, but asking the user to enter all non-constant variables, calculate our potential, and plot our imagined recording.
import numpy as np
import matplotlib.pyplot as pltFaraday = 96485
R = 8.314class Nernst(): def temp(self):
"""
:args: none
:return: Temperature in Kelvins
"""
C = float(input("Temperature in Celsius"))
K = C + 273.15
return K def charge(self):
"""
:args: none
:return: charge of ionic species
"""
v = int(input("Charge"))
return v def innerConcentration(self):
"""
:args: none
:return: Concentration of ion species inside cellular membrane, in millimolars
"""
inner = float(input("Inner Concentration (mM)"))
return inner def outerConcentration(self):
"""
:args: none
:return: Concentration of ion species outside of cellular membrane, in millimolars
"""
outer = float(input("Outer Concentration (mM)"))
return outer def equilibrium(self, T, z, Xi, Xo):
"""
:param T: Temperature in Kelvins
:param z: Valence of ionic species
:param Xi: Inner membrane concentration
:param Xo: Outer membrane concentration
:return: Membrane potential for single species
"""
GasIons = (R*T)/(z*Faraday)
Concentrations = np.log(Xo/Xi)
membranePotential = GasIons*Concentrations
return membranePotential def viz(self, volts):
plt.scatter([0,1,2,3], [0, 0, volts, volts])
plt.plot([0,1,2,3], [0, 0, volts, volts])
plt.ylim(-100,100)
plt.xlim(0,3)
xlabs = np.arange(0, 3, 1.0)
bins = ['','Channel Closed', 'Channel Open','']
plt.xticks(xlabs, bins)
plt.text(2, volts+5, str(round(volts, 2)))
plt.title("Membrane Potential Single Ion Species")
plt.xlabel("Membrane Permeability")
plt.ylabel("Voltage (mV)")
plt.show()if __name__ == "__main__":
Nernst = Nernst()
temperature = Nernst.temp()
charge = Nernst.charge()
inner = Nernst.innerConcentration()
outer = Nernst.outerConcentration()
membrane = Nernst.equilibrium(temperature, charge, inner, outer)
print("Result: " + str(membrane))
Nernst.viz(membrane*1000)[25 Degrees Celsius, -1 Ionic Charge, 10 mM Inner Concentration, 100 mM Outer Concentration]
Let's modify what we have. Rather than prompting the user to enter the attributes, let's assume there is a file containing the relevant data, where each row contains information unique to a particular chemical ion. We'll then automate the process, allowing the generation of a new plot for each recorded ion in the dataset. We'll also augment our global constants to include a standard mammalian temperature already in kelvins. We'll also set a global constant temperature we can use to standardize our mammalian cell attributes.
Let's look at the full script:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
# Global Constants
Faraday = 96485
R = 8.314
Temp = 310.15
class Nernst():
def equilibrium(self, T, z, Xi, Xo):
"""
:param T: Temperature in Kelvins
:param z: Valence of ionic species
:param Xi: Inner membrane concentration
:param Xo: Outer membrane concentration
:return: Membrane potential for single species
"""
GasIons = (R*T)/(z*Faraday)
Concentrations = np.log(Xo/Xi)
membranePotential = GasIons*Concentrations
return membranePotential
def viz(self, volts, ion):
plt.scatter([0,1,2,3], [0, 0, volts, volts])
plt.plot([0,1,2,3], [0, 0, volts, volts])
plt.ylim(-100,100)
plt.xlim(0,3)
xlabs = np.arange(0, 3, 1.0)
bins = ['','Channel Closed', 'Channel Open','']
plt.xticks(xlabs, bins)
plt.text(2, volts+5, str(round(volts, 2)))
plt.title("Membrane Potential Single Ion Species: "+str(ion))
plt.xlabel("Membrane Permeability")
plt.ylabel("Voltage (mV)")
plt.show()
if __name__ == "__main__":
file = pd.read_csv('C:\\Users\\ajh\\Desktop\\ions.txt',
names=['name', 'charge', 'temperature', 'inner', 'outer'],
dtype={'name':str, 'charge':int, 'temperature':int, 'inner':int, 'outer':float},
skiprows=1)
df = pd.DataFrame(file)
Nernst = Nernst()
for index,row in file.iterrows():
potential = Nernst.equilibrium(Temp,row.charge,row.inner,row.outer)
Nernst.viz(potential*1000, df.name[index])
Great! Let's continue refining our script, and develop a cleaner presentation for our audience. We're going to plot all recordings onto a single plot with colors and a legend to distinguish our observations.
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
# Global Constants
Faraday = 96485
R = 8.314
Temp = 310.15
class Nernst():
def equilibrium(self, T, z, Xi, Xo):
"""
:param T: Temperature in Kelvins
:param z: Valence of ionic species
:param Xi: Inner membrane concentration
:param Xo: Outer membrane concentration
:return: Membrane potential for single species
"""
GasIons = (R*T)/(z*Faraday)
Concentrations = np.log(Xo/Xi)
membranePotential = GasIons*Concentrations
return membranePotential
def viz(self, volts):
plt.scatter([0,1,2,3], [0, 0, volts, volts])
plt.plot([0,1,2,3], [0, 0, volts, volts])
plt.ylim(-100,100)
plt.xlim(0,3)
xlabs = np.arange(0, 3, 1.0)
bins = ['','Channel Closed', 'Channel Open','']
plt.xticks(xlabs, bins)
plt.text(2, volts+5, str(round(volts, 2)))
plt.title("Membrane Potential Single Ion Species")
plt.xlabel("Membrane Permeability")
plt.ylabel("Voltage (mV)")
plt.legend(['Sodium', 'Chloride', 'Potassium'])
if __name__ == "__main__":
file = pd.read_csv('C:\\Users\\ajh\\Desktop\\ions.txt',
names=['name', 'charge', 'temperature', 'inner', 'outer'],
dtype={'name':str, 'charge':int, 'temperature':int, 'inner':int, 'outer':float},skiprows=1)
df = pd.DataFrame(file)
Nernst = Nernst()
for index,row in file.iterrows():
potential = Nernst.equilibrium(Temp,row.charge,row.inner,row.outer)
Nernst.viz(potential*1000)
plt.show()