388 Questions

◾Water is very important in molecular simulations and may be approximated as

an explicit solvent, which is very accurate but computationally expensive, or as

an implicit solvent that does not simulate details of individual models but enables

longer time scales to be simulated for the same computational effort.

◾The mechanics of biopolymers can be modeled using pure entropic forces, with

improved modeling incorporating additional effects of enthalpic forces and

excluded volumes.

◾Reaction–​diffusion and Navier–​Stokes analysis can often be reduced in com­

plexity for modeling important biological processes to generate tractable models,

for example, for characterizing molecular motor translocation of tracks, pattern

formation in cell and developmental biology, and the fluid flow around biological

particles.

◾Bayesian inference is an enormously powerful statistical tool that enables us to use

prior knowledge to infer correct models of biological processes.

◾PCA and wavelet analysis are powerful computational tools to enable automated

recognition and classification of images.

◾Localization microscopy data from fluorescence imaging can be objectively quan­

tified to assess molecular stoichiometry and kinetics.

QUESTIONS

8.1

To model an interaction between a particular pair of atoms, masses m1 and

m2, separated by a distance r an empirical potential energy function was used,

U(r) =​ (3/​r⁴ − 2/​r)α, where α is a constant.

a

Draw the variation of U with r.

b

Show that there is a stable equilibrium separation, and calculate an expression

for it.

c

Calculate an expression for the resonant frequency about this equilibrium

position.

8.2

What are the particular advantages and limitations of QM, MM, MC, and CG

simulations?

8.3

Compare and contrast the techniques that can be used to reduce the computational

time in molecular simulations.

8.4

Consider a typical single-​molecule refolding experiment on a short filamentous pro­

tein of a molecular weight of ~14 kDa performed by holding the ends between two

beads held in two low stiffness optical tweezers and allowing the tethered peptide to

spontaneously refold against the imposed trapping force.

a

Estimate the number of peptide bonds in the short protein.

b

Each peptide bond has two independent bond angles called phi and psi, and

each of these bond angles can be in one of three stable conformations based on

Ramachandran diagrams (see Chapter 2). Estimate roughly how many different

conformations the protein can have.

c

If the unfolded protein refolds by exploring each conformation rapidly in ~1 ps

and then subsequently exploring the next conformation if this was not the true

(most stable) folded state, estimate the average length of time taken before it finds

the correct stable folded conformation.

d

In practice, unfolded proteins in cells will refold over a time scale of microseconds

up to several seconds, depending on the protein. Explain why proteins in cells do

not refold in the exploration manner described earlier. What do they do alterna­

tively? (Hint: the best current estimate for the age of the universe is ca. 14 billion

years.)

8.5

An ideal FJC consists of n rigid links in the chain each of length b that are freely

hinged where they join up.