Biomedical Engineering Reference
In-Depth Information
measure of the rate of transport relative to that of production. Setting x
¼
i
=
t, the
large-time behaviour for x
¼
O
ð
t
2
Þ
takes the form in (
2
) [note that (
1
) corresponds
to the large t limit of (
14
)] and the only novelty here concerns the regime
x
¼
O
ð
t
Þ
, wherein the Liouville-Green [as in (
9
)] implies
ot
þ
4t
2
sinh
2
of
ox
=
2t
þ
1
¼
0
;
of
so that
F
gP
þ
2t
2
ð
cosh
ð
P
=
t
Þ
1
Þþ
1
¼
0
;
g
þ
2t sinh
ð
P
=
t
Þ¼
0
and hence
F
¼
tg ln
1
þ
g
2
4t
2
2
2t
2
1
þ
g
2
4t
2
2
1
1
;
þ
g
2t
so that
F
1
þ
g
2
=
4 sg
!
0
;
F
tg ln
ð
g
=
t
Þ
tg
1
þ
2t
2
as
g
!þ1;
so the transfer by a finite distance between neighbouring cells that is associated
with discreteness leads to slower decay in the far field than occurs in the con-
tinuous case. More significant for what follows is that blow up occurs at a similar
rate over the range x
¼
O
ð
t
2
Þ
, i.e. diffusion suffices to drive many cells into an
upregulated state at a similar time.
The situation with the nonlinear case
du
i
dt
ð
t
Þ¼
t
2
ð
u
i
þ
1
ð
t
Þ
2u
i
ð
t
Þþ
u
i
1
ð
t
ÞÞ þ
u
i
ð
t
Þ1
\i\
þ1 ð
15
Þ
(where we could choose to scale t out) is markedly different: blow up occurs in
finite time and does so generically over three cells only [in the case of quadratic
nonlinearity adopted in (
15
)]—the central one of these (i
¼
0, say) has
t
!
t
c
u
0
ð
t
Þ
1
=ð
t
c
t
Þ
as
;
ð
16
Þ
as in (
4
), with transport having a negligible effect, while blow-up in the neigh-
bouring cells is driven by this one in the form
u
1
;
u
1
t
2
ln
ð
1
=ð
t
c
t
ÞÞ
t
!
t
c
as
;
exhibiting less dramatic blow up than (
16
). Thus upregulation occurs within a
small number of cells only; in practice the nonlinearity will saturate at high
concentrations, leading to upregulation propagating out from this initial 'quorate'
subpopulation.
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