# Calculating the ionic strength of a histidine solution

I'm trying to repeat the calculation of the ionic strength of a solution containing histidine as a buffer I recently read in a paper. In their paper, the authors calculate the ionic strength of a 20 mM solution of histidine at pH 5.8 to be 7.7 mM. They also state that the average charge of Histidine at this pH is equal to + 0.61.

For my calculation I used to Henderson-Hasselbalch equation to calculate the distribution of charged species at the pH specified,

pH = np.arange(0.1, 14, 0.001)

def macroscopic(pH, pka1, pka2, pka3):
return ((-1)**2 + 0 * 10**(pka1 - pH) + (1)**2 * 10**(pka1+pka2 - 2*pH) + (2)**2 * 10**(pka1+pka2+pka3 - 3*pH)) / (1+10**(pka1 - pH) + 10**(pka1+pka2 - 2*pH) + 10**(pka1+pka2+pka3 - 3*pH))
def DDD(pH, pka1, pka2, pka3):
return 1 / (1+10**(pka1 - pH) + 10**(pka1+pka2 - 2*pH) + 10**(pka1+pka2+pka3 - 3*pH))
def PDD(pH, pka1, pka2, pka3):
return 10**(pka1 - pH) / (1+10**(pka1 - pH) + 10**(pka1+pka2 - 2*pH) + 10**(pka1+pka2+pka3 - 3*pH))
def PPD(pH, pka1, pka2, pka3):
return 10**(pka1+pka2 - 2*pH) / (1+10**(pka1 - pH) + 10**(pka1+pka2 - 2*pH) + 10**(pka1+pka2+pka3 - 3*pH))
def PPP(pH, pka1, pka2, pka3):
return 10**(pka1+pka2+pka3 - 3*pH) / (1+10**(pka1 - pH) + 10**(pka1+pka2 - 2*pH) + 10**(pka1+pka2+pka3 - 3*pH))'''


Where the values of pKa1, 2 and 3, used by the authors, are equal to 9.17, 6.0 and 1.82 respectively.

I then calculate the ionic strenght of the solution according to

def ionicstrength(pH, pka1, pka2, pka3):
return 0.5*(DDD(pH, 9.17, 6.0, 1.82)*BuffermMConc*(-1)**2 + PDD(pH, 9.17, 6.0, 1.82)*BuffermMConc*(0)**2 + PPD(pH, 9.17, 6.0, 1.82)*BuffermMConc*(1)**2 + PPP(pH, 9.17, 6.0, 1.82)*BuffermMConc*(2)**2)
#def averagecharge(pH, pka1, pka2, pka3)


where the variable BuffermMConc is equal to the mM concentration of the buffer, in this case 20.

When I plot the results I get the following graph,

Where the dashed line represents the ionic strength and the coloured lines the distribution of histidine species. For pH 5.8 I obtain an ionic strength of 6.13 mM, which compares to the ionic strength of 7.7mM determined by the authors. The authors also report the average charge of histidine, which is identical to the value I determined. This would indicate that the distribution of charged species is probably correct in my calculation. I cant figure out why my ionic strength calculation is different.

Any help would be greatly appreciated.