Chapter 8 – Theoretical Biophysics  391

b

If the mRNA molecule is modeled as a GC with segment length of 4 nm, estimate

the Stokes radius of the whole molecule.

The longest cell in a human body is part of a group of nerve cells with axons

(tubes filled with cytoplasm) running from the spinal cord to the feet roughly

shaped as a cylinder of length of ~1 m and diameter ~1 μm.

c

Estimate the diffusion coefficient of the mRNA molecule in the axon, assuming

the viscosity of cytoplasm is ~30% higher than that of water. How long will it take

for the mRNA molecule on average to diffuse across the width of the axon, and

how long would it take to diffuse the whole length?

d

With reference to your previous answers, explain why some cells do not rely on

passive (i.e., no external energy input) diffusion alone to transport mRNA, and

discuss the other strategies they employ instead.

8.15 The prediction from the random telegraph model for the distribution of total number

of a given type of protein molecule present in each bacterial cell is for a gamma dis­

tribution (a distribution function that has a unimodal peak with a long tail). However,

if you measure the stoichiometry of certain individual molecular complexes, then

the shape of that distribution can be adequately fitted using a Gaussian distribution.

Explain.

8.16 Stepping rotation of the flagellar rotary motor of a marine species of bacteria was

powered by a flux of Na^+​ ions. The position of a bead attached to the flagellar fila­

ment was measured using back focal plane detection (see Chapter 6), which was

conjugated to the rotor shaft of the motor via a hook component. The hook can

be treated as a linear torsional spring exerting a torque Γ1 =​ κΔθ where κ is the

spring constant and Δθ the angle of twist, and the viscous drag torque on the bead

is Γ2 =​ γd(Δθ)/​dt where γ is the viscous drag coefficient. In reducing the external

Na^+​ concentration to below 1 mM, the motor can be made to rotate at one revo­

lution per second or less.

a

Estimate the maximum number of steps per revolution that might be observ­

able if κ =​ 1 × 10⁻¹⁹ N m rad⁻¹ and γ =​ 5 × 10⁻²² N m rad⁻¹ s. In the experiment

~26 steps per revolution were observed, which could be modeled as thermally

activated barrier hopping, that is, the local free energy state of each angular pos­

ition was roughly a parabolic-​shaped potential energy function but with a small

incremental difference between each parabolic potential.

b

If a fraction α of all observed steps are backward, derive an expression for the

sodium-​motive force, which is the electrochemical voltage difference across the

cell membrane due to both charge and concentration of Na^+​ ions embodied by

the Nernst equation (see Chapter 2), assuming each step is coupled to the transit

of exactly two Na^+​ ions into the cell.

c

If α =​ 0.2 and the internal and external concentrations of the driving ion are 12

and 0.7 mM, respectively, what is the voltage across the cell membrane? If a con­

stant external torque of 20 pN·nm were to be applied to the bead to resist the

rotation, what would be the new observed value of α?

8.17

a

Derive the Michaelis–​Menten equation for the rate of F1-​mediated hydrolysis

of ATP to ADP and inorganic phosphate in the limit of excess ATP, stating any

assumptions you make. How does this rate relate to the rate of rotation of the F1

γ subunit rotor shaft?

b

State with mathematical reasoning how applying a load opposing the rotation of

γ affects the rate of ATP hydrolysis?

c

Assuming the ATP-​independent rotation step of ~30°–​40° can be modeled by one

irreversible rapid step due to a conformational change in the F1·ADP complex

correlated with ATP hydrolysis with a rate constant k4, followed at some time

afterward by another rapid irreversible rotational step due to the release of the

ADP with a rate constant k5, derive an expression for the probability that the con­

version from the pre-​hydrolysis F1 · ADP state to the post-​ADP-​release F1 state