Chapter 8 – Theoretical Biophysics 389
a
By treating each link as a position vector, and neglecting possible consequences
of interference between different parts of the chain, derive an expression for <x2>,
the mean square end-to-end distance.
b
Evaluate this expression in the limits that the contour length is much greater, and
much less, than b, and comment on both results.
8.6
A certain linear protein molecule was modeled as an ideal chain with 500 chain
segments each equal to the average alpha-carbon spacing for a single amino acid
carrying a charge of ±q at either end of each amino acid subunit where q is the unitary
electron charge. What is the end-to-end length relative to its rms unperturbed length
parallel to an E-field, of 12,000 V cm−1?
8.7
A protein exists in just two stable conformations, 1 and 2, and undergoes reversible
transitions between them with rate constants k12 and k21, respectively, such that the
free energy difference between the two states is ΔG.
a
Use detailed balance to estimate the probability that the system is in state 1 at
a given time t, assuming that initially the molecule is in state 1. A diferent pro
tein undergoes two irreversible transitions in conformation, 1 – 2 and 2 – 3, rate
constants k12 and k23, respectively.
b
What is the likelihood that a given molecule starting in state 1 initially will be in
state 3 at time t?
8.8
In an experiment, the total bending free energy of a liposome was measured at ~200
kBT and the polar heads tightly packed into an area equivalent to a circle of radius of
0.2 nm each, with the length from the tip of the head group to the end of the hydro
phobic tail measured at 2.5 nm. Estimate the free energy in Joules per phospholipid
molecule required to double the area occupied by a single head group.
8.9
A virus was fluorescently labeled and monitored in a living cell at consecutive sam
pling times of interval 33 ms up to 1 s. The rms displacement was calculated for two
such viral particles, giving values of [61, 75, 81, 95, 107, 112, 128, 131, 158, 167, 181,
176, 177, 182, 183, 178, 177, 180, 182, 184, 181, 179, 180, 178, 180, 182] nm and [59,
65, 66, 60, 64, 63, 58, 62, 63, 61, 64, 60, 59, 64, 62, 65, 61, 60, 63, 66, 62, 58, 60, 57,
61, 62] nm. What might this indicate about the virus diffusion?
8.10 A series of first-order biochemical reactions of molecules 1, 2, 3,…, n reacted irrevers
ibly as 1 ↔ 2 ↔ 3 ↔ … ↔ n with rate constants k1, k2, k3,…,kn, respectively.
a
What is the overall rate constant of the process 1 ↔ n?
b
What happens to this overall rate if the time scales for formation of one of the
intermediate molecules are significantly lower than the others?
c
What implications does this have for the local concentrations of the other inter
mediate molecules formed? (Hint: such a slow-forming intermediate transition is
often referred to as the “rate-limiting step.”)
8.11 Write down the Brownian diffusion equation for numbers of molecules n per unit
length in x that spread out by diffusion after a time t.
a
Show that a solution exists of the form n(x,t) = αt½exp(−x2/(4Dt)) and determine
the normalization constant a if there are ntot molecules in total present.
b
Calculate the mean expected values for <x> and <x2>, and sketch the solution of
the latter for several values of t > 0. What does this imply for the location of the
particles initially?
The longest cells in a human body are neurons running from the spinal cord to
the feet, roughly shaped as a tube of circular cross-section with a tube length
of ~1 m and a diameter of ~1 μm with a small localized cell body at one end
that contains the nucleus. Neurotransmitter molecules are synthesized in the
cell body but are required at the far end of the neuron. When neurotransmitter
molecules reach the far end of the neuron, they are subsequently removed by a
continuous biological process at a rate that maintains the mean concentration
in the cell body at 1 mM.
c
If the diffusion coefficient is ~1000 μm2s−1, estimate how many neurotransmitter
molecules reach the end of the cell a second due to diffusion.