PES 3950/PHYS 6950: Homework Assignment 1
Handed out:
Due in:
Wednesday January 21
Friday January 30, at the start of class at 3:05 pm sharp
Show all working and reasoning to receive full points.
Question 1 [5 points]
We made a simple estimate of the mass of a cell (radius = 4 µm) in class by assuming that
the cell has the same density of water. However, a more reasonable estimate is that the
density of the macromolecules of the cell (30% of total cell mass) is 1.3 times that of water.
As a result our estimate of the mass of a cell was a bit off. What was our error made by
treating the macromolecular density as the same as that of water?
Question 2 [15 points]
Imagine a biased (biased means that the random walker tends to move in one direction more
often than the other) walker starting at a given site on a one dimensional discrete lattice. At
each time step the walker can move a length L − l to the right or −L − l to the left, where
L > l. Let Dn be the position at time step n and calculate the following:
a) The average position after n steps < Dn >
b) The mean (average) square position after n steps < Dn2 > p
c) Use the relation n = t/∆t to obtain the typical distance < Dn2 > the biased random
walker has traveled. Besides the diffusion constant can you define a second constant? What
does this second constant represent?
Question 3 [10 points]
Consider the following simple model for a neuron. A long, thin cell of length L contains a
source of a particular protein at one end. The source maintains a concentration, C(L) = C0
of this protein at the end of the cell. The other end of the cell acts as a perfect sink, i.e.
C(0) = 0.
a)Solve for the equilibrium concentration along the length of the cell, considering a long,
thin neuron and taking the x direction to be along the length of the cell. You should get for
the equilibrium concentration: C(x) = C0 x/L.
b)Using this expression in conjunction with Fick’s law and the Stokes-Einstein relation, answer the following question:
A neuron is one meter long. Small vesicles with radius of R = 50.0 nm are observed moving
along the neurons with a net velocity of 400 mm per day. Can the observed vesicle speed be
explained by diffusion? Assume T=300 K and η = 1.00 × 10−3 Pa . s (cytoplasmic viscosity).
Hint: look at the dimensions
Question 4 [20 points] GRADUATE STUDENTS ONLY
In class we estimated the maximum radius of a spherical bacterium. In fact bacteria are
often in the shape of a cylinder. Assume an infinitely long cylindrical bacterium. Using 52
in cylindrical coordinates, you will find the maximum radius, r0 , for the bacterium using the
parameters given in class.
a) In equilibrium, show that the diffusion equation reduces to
r ∂C
= A, where A is a constant
∂r
b) Show that the solution to this has the form
ln(r/r0 )
C(r) = C0 ln(R
0 /r0 )
when we use the boundary conditions:
1) C(r) = C0 at some large distance R0
2) C(r) = 0 at r = r0 the surface of the bacterium
c) Using Fick’s law find the current density j and the net current entering a cylindrical
bacterium of length L.
d) Estimate the maximum r0 by calculating the O2 current required (as in class) and using
the result for the O2 current available calculated from part (c). At some point you will want
to use ln(R0 /r0 ) = 2 as an approximation.
To make a numerical estimate, use the following parameters given in class:
α = .020 mole/(kg-s) (metabolic rate)
C0 = 0.20 mole/m3
D for O2 in water = 1.0 × 10−9 m2 /s
Question 4