Structural Biochemistry/Membrane Proteins/Membrane Gradients and its Thermodynamics

Membrane Gradients and its Thermodynamics
The Second Law of Thermodynamics suggests that particles will naturally diffuse from an area of high concentration to an area of lower concentration. The potential energy or the free energy reserved in a concentration gradient can be mathematically represented. Since free energy is lowest when the distribution of molecules is even, the uneven concentration of particles is an environment with abundant energy. Energy must be added into the system to achieve the unequal distribution of molecules or to form a concentration gradient.

The quantity of energy that must be added can be accounted for by first considering an uncharged solute molecule. The free-energy difference in moving particles from side 1 (with a concentration of c1), to side 2 with a concentration of c2 can be represented by the following equation:

∆G =RT ln(c2/c1) = 2.303RTlog10(c2/c1)

in which R represents the ideal gas constant (8.314 x 10 - 3) and T is the temperature in units of kelvins.

The pictorial representation of the concentration gradient of an uncharged solute can be analyzed in the diagram below.



For the charged species, a mathematical and pictorial representation can be derived as well. The uneven distribution across the plasma membrane creates stored free energy that needs to be included in the formula because like charges will repel. The electrochemical potential (membrane potential) is the addition of the concentration and electrical factors. The free energy difference is

∆G =RT ln(c2/c1) + ZF∆V = 2.303RTlog10(c2/c1) + ZF∆V

where ∆V is the potential in volts across the plasma membrane, Z is the electrical charge of the transported species, and F stands for the Faraday constant (96.5 kJ/V. mol).

Note: the charged species across a membrane have the same charge as the transported ion.



∆G is positive if the transport process is active and ∆G negative represents the passive transport.

Membrane potential
The membrane potential of a cell is the electrical potential difference between the inside and outside of the cell. The potential is determined by the ion concentration between the inside and the outside. This is maintained by different membrane gradient.

Electrical and Concentration gradient help establish a cell's resting potential
The resting potential of a neuron has a charge of -70 mV, and this is called the electrical gradient. There is a constant exchange of ions between the cell and environment, and ions play a significant role in the resting potential include potassium, sodium, and chloride. The concentration and movement of ions is maintained by a protein pump, that pumps 3 K+ out for every 2 Na+ ion. This pump is called the sodium-potassium pump. There is higher concentration of Na+ outside the cell, and a higher concentration of K+ inside the cell. The mix between the concentration gradient and the electrical gradient cause the Na+ ions to have a tendency to move inside, and the K+ ions to move outside. The Na+ ions have a tendency to move in because of the charge difference, and the K+ have a tendency to move outside because of a concentration difference. When the equilibrium potential is reached, the K+ ions do not have a strong tendency to move out of the cell because of the charge difference. The inside of the cell is -70 mV, and further outflow of K+, despite the concentration gradient, will cause the cell to be even more negative. The equilibrium potential is the point when the electrical gradient and the concentration gradient have stabilized with respect to each other.

The resting potential can be calculated by Goldman’s equation, which is represented by

Em = RT/F ln[(Pk[K+]out + PNa[Na+]out + PCl[Cl-]in)/(Pk[K+]in + PNa[Na+]in + PCl[Cl-]out)]

The influx of K+ and Na+ into a cell affects the charge of a cell positively, while an influx of Cl- affects the charge of a cell negatively. The numerator represents the inside concentration of a cell, while the denominator represents the outside concentration. K+ and Na+ outside correspond Cl- inside, because of the opposite charges. P represents the permeability of the ion. Other ions affect the resting potential, but only these three ions are major contributors.

Reference
Biochemistry. 6th ed. New York : W. H. Freeman and Company, 2007. 352-353. Print.