Harvard-MIT Division of Health Sciences and Technology
HST.410J: Projects in Microscale Engineering for the Life Sciences, Spring 2007
Course Directors: Prof. Dennis Freeman, Prof. Martha Gray, and Prof. Alexander Aranyosi
HST.410J/6.07J
Lecture 5
February 22, 2007
Concentration at a point�
in space and time
volume V
amount of substance in V�
concentration c(x,t) = lim
V
V →�0
Flux at a point�
in space and time
window�
area A
φ(x,t)
amount of substance flowing�
flux φ(x,t) = lim through test window A in Δt�
A →�0
AΔt
Δt→�0
Fick's First Law
Figure from Weiss, T. F. Cellular Biophysics,
Vol. I. Cambridge, MA: MIT Press, 1996.
Courtesy of MIT Press. Used with permission.
Random Walk Model
• number of solute particles << number of solvent particles
• motion of solute determined by collisions with solvent
(ignore solute-solute interactions)
• focus on 1 solute particle, assume motions of others
are statistically identical
Every τ seconds, solute particle gets hit by solvent particle.
In response, solute particle is equally likely to move +l or −l.
τ = mean free time; l = mean free path
0
l
2l
x
3l
0
1
1
1
1
n
1
2
1
−3
3
−2
−1
2
1
3
0
m
1
+1
+2
3
Figures from Weiss, T. F. Cellular Biophysics,
Vol. I. Cambridge, MA: MIT Press, 1996.
Courtesy of MIT Press. Used with permission.
+3
1
0.8
1
0.6
0.8
0.4
0.6
0.2
0.4
0
0
W(m,n)
1
2
3
4 n
0.2
0
-7
5
-5
-3
6
-1 1
m
3
7
5
7
Fick's First Law
Figure from Weiss, T. F. Cellular Biophysics,
Vol. I. Cambridge, MA: MIT Press, 1996.
Courtesy of MIT Press. Used with permission.
How long till half the solute diffuses to |x|>x1/2
x1/2 ~ 2
<
3
2Dt
–3
–2
–1
0
50%
68%
95%
99%
c(x,t) =
1
2
3
x
2Dt
~
2
2Dt >
x1/2
3
~ 2
4
2Dt >
x1/2
9
2
~ x1/2 ≡� t
t >
1/2
D
no
−x2/(4Dt)
e
4πDt
Importance of Scale
2
t1/2 =
x1/2
D
; D = 10
−5
cm2
+
s for small solutes (e.g., Na )
x1/2
membrane sized�
10 nm�
cell sized�
10 µm�
dime sized
10 mm
t1/2
1�
µsec�
10
1�
sec�
10
105 sec ≈ 1 day