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Physiology 472/572 - Quantitative modeling of biological systems - 2011
Homework 2
1. a) For the following functions, find
∂f
∂2 f
and
:
∂x
∂x∂y
i) f ( x, y ) = x 2 + y 2
2
2
ii) f ( x, y ) = e − ( x + y ) / 2
iii) f ( x, y ) = (sin x)(cos y )
b) For the function in part i), sketch z = f(x,y) as a surface in xyz space for 0 ≤ x ≤ 2 and
0 ≤ y ≤ 2.
2. Fick's first law says that the diffusive flux of a solute is down its concentration gradient.
Concentration gradients are well defined for continuous quantities. The point of this exercise is
to consider whether the discrete nature of matter invalidates Fick's first law.
a) Assume that the sodium concentration in physiological saline is 150 mmol/l. Estimate
the dimensions of a cube of physiological saline that contains one million sodium ions.
b) Briefly describe how the results in a) can be used to justify the application of Fick's
first law even on sub-cellular scales.
3. Consider a cell composed of a (spherical) cell body to which is attached a long, thin, tubular
‘process’ (e.g., the axon of a neuron or the flagellum of a bacteria) of length ℓ cm. Suppose some
substance n (e.g. a metabolite), at a concentration cn, is generated in the cell body and diffuses
along the process with constant diffusivity D, creating a flux jn. In the process, the substance is
consumed at a uniform rate, αn mol/s per unit length. The continuity equation for the substance is
α
∂c n
∂j
= − n − n where A is the constant cross-sectional area of the process.
A
∂t
∂x
a) Combine the modified continuity equation above with Fick’s first law to obtain a
modified form of the diffusion equation that must be satisfied by cn.
b) Show that the solution of this equation in steady-state (∂cn/∂t = ∂jn/∂t = 0) is
c n ( x) =
αn
x 2 + a 0 x + b0
2 DA
and find the values of the constants a0 and b0 corresponding to the boundary conditions
cn(0) = C0 and jn(ℓ) = 0.
c) Show that the requirement for consumption at a uniform rate αn mol/(s·cm) along the
process sets an upper limit (if C0, D, A and αn are fixed) on the possible length ℓ of the
process, and find a formula for this upper limit.
(Continued on next page)
4. (Extra credit for undergraduates.) Assume that n0 mol/cm2 of hemoglobin with a diffusion
constant D = 0.7 × 10−6 cm2/s is placed in a trough of water in the plane x = 0 at time t = 0.
Assume that the concentration of hemoglobin depends on x and t only.
a) Show that for any fixed location xp that the maximum concentration occurs at
tm = xp2/2D.
b) How long does it take for the concentration to reach a maximum at x = ±2 cm?