Question 11.1: Calculate the time rate of change of proteoglycans within th...
Calculate the time rate of change of proteoglycans within the synovial fluid, if their diffusion is only restricted by the presence of glycosaminoglycans (GAGs) within the synovial space. Assume that the molecular weight for the proteoglycan of interest is 300 kDa and its density is 1.45 g/mL. The effective radius for the GAGs is 0.5 nm and the volume fraction is 0.008. The free diffusion of proteoglycan in water is 1.1E – 7 cm²/s. The permeability of the proteoglycan through the synovial membrane is 1E – 6 cm/s. Assume that the degradation of the proteoglycan is negligible. The formation rate of proteoglycans within the synovial membrane and the cartilage are 6E – 9 mg/(cm²s) and 3E – 7 mg/(cm²s), respectively. The cross-sectional areas of the synovial membrane and the cartilage are 29.5 mm² and 13.5 mm², respectively. The concentration gradient of the proteoglycan is 0.13 mg/mL.
First it will be necessary to calculate the effective radius of the proteoglycan:
a= \left(\frac{3 \ast 300 kDa}{4\pi \ast (1.45 g/mL)(6.02_{E}23)}\right)^{1/3}= 4.35 nm
As a side note, Avogadro’s constant converts Daltons into grams. Now we can calculate the restricted diffusion coefficient:
D_{i}= (1.1_{E} – 7 cm^{2}/s)e^{(-\sqrt{0.008})(1 + \frac{4.35 nm}{0.5 nm})}= 4.62_{E} – 8cm^{2}/s
To calculate the time rate of change of the proteoglycan, we need to know the flux of the proteoglycans out of the synovial membrane. The flux can be formulated from
J= p\Delta cA \frac{D_{i}}{D}= (1_{E} – 6 cm/s)(0.13 mg/mL)(29.5 mm^{2})(\frac{4.62_{E} – 8 cm^{2}/s}{1.1_{E} – 7 cm^{2}/s})= 0.000058 mg/hr
Therefore, the time rate of change of proteoglycans within the synovial fluid will be
\frac{\partial (Vc)}{\partial t}= r_{s}A_{s} + r_{c}A_{c} – d – J
= (6E -9 mg/cm²s)(29.5 mm²) + (3E -7 mg/cm²s)(13.5 mm²) – 0 – 0.000058 mg/hr
= 0.000094 mg/hr