SME
Tissue Engineering
Cell–material interactions
Use this for “A surface carries 400 RGD ligands per μm². How far apart are they, and will cells form focal adhesions?”
Diagram
d
=
1000
÷
√density
=
1000
÷
√
400
=
50
nm
between
ligands
on
a
square
grid:
32
dots
on
0.08
μm²
is
400
per
μm².
d
≤
70
nm:
the
ring
of
radius
70
nm
around
one
ligand
takes
in
4
neighbours,
so
integrins
cluster
and
focal
adhesions
form
(purple).
This is an example. Type your own values to start a new problem.
d (nm)
=
1000
÷
σ
σₘᵢₙ
=
(1000
÷
dc (nm)
)
2
r
=
σ
÷
σₘᵢₙ
Formulas
d
=
1,000
÷
√
σ
50
=
1,000
÷
√
400
σₘᵢₙ
=
(
1,000
dc
)
2
204
=
(
1,000
70
)
2
r
=
σ
σₘᵢₙ
1.96
=
400
204
Assumptions
•
The ligands sit on a square grid, so each one has a square of side d to itself.
•
Integrins cluster into focal adhesions only when ligands are about 70 nm apart or closer: a measured threshold, not derived.
•
One square micrometre is 1000 nm by 1000 nm.
Step-by-step
WE KNOW
σ = 400 μm⁻², dc = 70 nm
FIND
spacing between ligands (d), least density for adhesions (σₘᵢₙ), density over the least (r)
Step 1 · Find spacing between ligands (d)
Use d = 1,000 ÷ √σ
Each
ligand
has
a
square
of
1
σ
to
itself;
its
side
is
the
spacing
(1
μm
is
1000
nm).
d
=
1,000
÷
√
400
d
=
1,000
20
d
=
50
nm
→
50
nm
≤
70
nm:
integrins
can
cluster,
so
focal
adhesions
form
Step 2 · Find least density for adhesions (σₘᵢₙ)
Use σₘᵢₙ = (1,000 ÷ dc)²
Ligands exactly at the threshold spacing: the fewest per square micrometre that still adhere.
σₘᵢₙ
=
(
1,000
70
)
2
σₘᵢₙ
=
14.286
2
σₘᵢₙ = 204 μm⁻²
Step 3 · Find density over the least (r)
Use r = σ ÷ σₘᵢₙ
How many times the least density the surface carries; 1 or more means adhesions form.
r
=
400
204
r
=
1.96
→
at
least
the
least
density:
focal
adhesions
form
Check
Put the numbers back in. Both sides should match.
50
=
1,000
÷
√
400
✓
204
=
(
1,000
70
)
2
✓
1.96
=
400
204
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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Use this for “Put these in order on a new scaffold: the cell spreads, proteins adsorb, integrins bind RGD, focal adhesions form.”