Download hw000 - WordPress.com

February 20, 2017
Biochemical Engineering
Due: March 15, 2017
HW 000 – Sperm Swimming
Consider the diffusion of ATP in the tail of a bull sperm. In the sperm tail or flagella, the ATP supplies the
energy which leads to the motion of the sperm. The ATP consumed in the tail must diffuse down the
length of the sperm tail from what is called the midpiece which is part of the sperm body and is where
the ATP is produced. This ATP diffusion rate down the tail must be sufficient to replace the ATP being
consumed in the tail section.
We can develop a model for the steady state diffusion of ATP in the bull sperm tail to determine how
much ATP is in the tail and then how much is in the midpiece. The sum of those would be the total
amount of ATP in a bull sperm. Experiments have shown that the total amount of ATP in a bull sperm is
about 130 to 280 x 10-18 moles ATP.
To develop our model for the diffusion and reaction of ATP in the tail, we can assume that the tail is a
straight cylinder with a uniform cross sectional area of A. The ATP is produced at a steady rate in the
midpiece (mp) and diffuses down the tail where it is hydrolyzed and provides energy for the sperm
motion. The figure below shows the geometry.
L
tail
x
Tail cross sectional area A
midpiece
x
x+x
Consumption of ATP within the tail section of the sperm occurs at a constant rate. We can let α
represent the rate of ATP hydrolysis per volume of tail tissue. We can also let D represent the diffusivity
of ATP in the tail tissue. Perform a shell balance over the tail and show that at steady state the following
differential equation describes the ATP concentration (C) in the tail, i.e.,
𝑑2 𝐶
𝑘
− (
) = 0
2
𝑑𝑥
𝐷𝐴𝐿
where k is the ATP consumption rate on the basis of a bull sperm tail, i.e., k = αAL.
The boundary conditions are that at the end of the tail, i.e., at x = L, we can assume that all of the ATP in
the tail has been consumed and that no ATP diffuses out from the tail tip. Hence, we have:
BC1 : x = L, C = 0
BC 2: x = L,
𝑑𝐶
𝑑𝑥
=0
Next, integrate the above differential equation for C and use the BC’s and show that C(x) is given by:
𝑘
𝑘
𝑘𝐿
𝐶(𝑥) = (2𝐷𝐴𝐿) 𝑥 2 − (𝐷𝐴) 𝑥 + (2𝐷𝐴)
Then, show that the total amount of ATP in the tail is given by:
𝐴𝑇𝑃𝑡𝑎𝑖𝑙 =
𝑘 𝐿2
6𝐷
We can obtain the concentration of ATP in the midpiece by finding C(x) at x = 0 and assuming the ATP is
uniformly at this concentration in the midpiece. Therefore, show that
𝑘𝐿
C(0) = 2 𝐷 𝐴
The amount of ATP in the midpiece is then C(0) times the volume of the midpiece, Vmp. So, show that the
total amount of ATP in the bull sperm is then given by:
𝐴𝑇𝑃𝑡𝑜𝑡𝑎𝑙 = 𝐴𝑇𝑃𝑡𝑎𝑖𝑙 + 𝐴𝑇𝑃𝑚𝑖𝑑𝑝𝑖𝑒𝑐𝑒 =
𝑉𝑚𝑝
𝑘𝐿 𝐿
( +
)
2𝐷 3
𝐴
Experiments have shown that the amount of oxygen consumed by a moving bull sperm is about 3.7 to
5.0 x 10-18 moles O2/sec. Also, about 6 ATP’s are formed for each molecule of oxygen consumed. The
length of a bull sperm tail (L) is 5 x 10-3 cm and the cross sectional area (A) of the tail is 3 x 10-10 cm2. The
volume of the midpiece (Vmp) in the bull sperm is 1.3 x 10-12 cm3. The diffusivity of ATP (D) in phosphate
buffer at 20 C is 4.16 x 10-6 cm2/sec. When corrected to 37 C, the diffusivity is 6 x 10-6cm2/sec. However,
ATP diffusion in the tail is like in a hydrogel with a water volume fraction of 0.75 and a tortuosity of 0.80.
Hence the ATP tail diffusivity is more like 3.6 x 10-6 cm2/sec.
Using these parameter values estimate the value of ATPtotal. Is the value of the estimated ATP in a bull
sperm copesetic with the experimental total amount of ATP in a bull sperm of about 130 to 280 x 10-18
moles ATP?