(a) A transcription factor binds a single DNA operator site with an unbound state of…

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 transcription factor binds a single DNA operator site with an unbound state of weight

1

and a bound state of weight

c/K_d

, where

c

is the free transcription-factor concentration and

K_d

its dissociation constant. Derive the occupancy

\theta(c)

from the two weights, and derive it again from the mass-action expression

K_d=[F][O]/[FO]

for free factor

F

and operator

O

, using conservation of total operator sites and holding the free factor concentration at

c

(factor in excess of the single site). Show the two agree, then evaluate

\theta

for

K_d = 8\,\mathrm{nM}

and

c = 12\,\mathrm{nM}

. (b) Two large mitochondrial-fragment complexes, modelled as rigid spheres of radius

R = 50\,\mathrm{nm}

, sit among inert Ficoll crowders modelled as impenetrable coils of radius

\delta = 6\,\mathrm{nm}

at volume fraction

\phi = 0.18

. Derive the overlap volume of the two exclusion shells (radius

R+\delta

) as a function of centre separation

d

, then take

d = 2R

and

R\gg\delta

to get

V_{\rm ov}\approx2\pi R\delta^2

at contact. Hence derive and evaluate the contact depletion free energy

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

, in

k_BT

and in

\mathrm{pN\,nm}

298\,\mathrm{K}

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