(a) A chemoreceptor exists in two states: ligand-free, with statistical weight , and…

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture

Forces, Machines and Signals, Ch Free energy at work: states, weights and entropic forces

Problem

(a) A chemoreceptor exists in two states: ligand-free, with statistical weight

1

, and ligand-bound, with statistical weight

c/K_d

, where

c

is the free-ligand concentration and

K_d

the dissociation constant, both in the same units. Derive the occupancy

\theta(c)

(the probability the receptor is bound) from the ratio of weights. Then derive

\theta(c)

a second way, starting from the mass-action expression

K_d = [R][L]/[RL]

for receptor

R

and ligand

L

, using conservation of total receptor number and taking the free-ligand concentration to remain

c

throughout (the ligand is in large excess). Show the two results agree, then evaluate

\theta

for a chemotaxis receptor with

K_d = 2\,\mathrm{\mu M}

at a free attractant concentration

c = 8\,\mathrm{\mu M}

. (b) Two identical, rigid viral capsids of radius

R = 15\,\mathrm{nm}

sit in a solution of inert PEG crowders, modelled as impenetrable ideal coils of radius

\delta = 3\,\mathrm{nm}

at volume fraction

\phi = 0.20

. The centre of a crowder is excluded from a shell of radius

R+\delta

around each capsid's centre. Treating the overlap of the two exclusion shells as the lens formed by two spheres of radius

a = R+\delta

with centres separated by

d

, derive its volume as a function of

d

, then set

d = 2R

(capsids touching) and take

R \gg \delta

to show the overlap volume reduces to

V_{\rm ov} \approx 2\pi R\delta^2

. Hence derive the contact depletion free energy

F_{\rm dep} \approx -\tfrac32\phi(R/\delta)k_BT

and evaluate it in units of

k_BT

and in

\mathrm{pN\,nm}

298\,\mathrm{K}

(take

k_BT = 4.11\,\mathrm{pN\,nm}

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