(a) A single-site calcium sensor has an empty state of weight and a calcium-bound…

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 single-site calcium sensor has an empty state of weight

1

and a calcium-bound state of weight

c/K_d

, where

c

is the free

\mathrm{Ca^{2+}}

concentration and

K_d

its dissociation constant. Derive the occupancy

\theta(c)

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

K_d = [S][\mathrm{Ca}]/[S\cdot\mathrm{Ca}]

with conservation of total sensor and free calcium held at

c

(calcium in excess). Show the two agree, then evaluate

\theta

for a sensor with

K_d = 0.5\,\mathrm{mM}

in a microdomain where the free

\mathrm{Ca^{2+}}

concentration is

c = 1.5\,\mathrm{mM}

. (b) Two synaptic vesicles, modelled as rigid spheres of radius

R = 20\,\mathrm{nm}

, are suspended among inert dextran crowders modelled as impenetrable coils of radius

\delta = 4\,\mathrm{nm}

at volume fraction

\phi = 0.12

. By finding the volume of overlap of the two exclusion shells of radius

R+\delta

(as the lens of two spheres separated by

d

), then setting

d=2R

and taking

R\gg\delta

, derive

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

at contact, and hence derive and evaluate the contact depletion free energy

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

k_BT

and in

\mathrm{pN\,nm}

298\,\mathrm{K}

(

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

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