Biomedical Engineering Reference
In-Depth Information
λ
+
eigenvalue
d
, as one can see in the equations of system (
20
).
As shown in [
39
], the first eigenvalue
λ
is then in fact
is then solution of the following equation,
which in population dynamics is referred to, in the 1-phase case (
I
λ
=
1) with no
death term, as Euler-Lotka's equation
+
∞
I
i
=
1
1
2
=
e
−
0
K
i
→
i
+
1
(
ξ
)
d
ξ
e
−
λ
x
d
x
K
i
→
i
+
1
(
x
)
(21)
0
Integrating the first equation of System (
19
) along its characteristics [
119
], we can
in the stationary case with no death rate derive the formula
e
−
0
K
i
→
i
+
1
(
ξ
)
d
ξ
n
i
(
t
+
x
,
x
)=
n
i
(
t
,
0
)
.
This can be interpreted in the following way: the probability for a cell which entered
phase
i
at time
t
to stay for at least an age duration
x
in phase
i
is given by
e
−
0
K
i
→
i
+
1
(
ξ
)
d
ξ
P
(
τ
i
≥
x
)=
The time
τ
i
spent in phase
i
is thus a random variable on
R
+
, with probability density
function
f
i
given by
e
−
0
K
i
→
i
+
1
(
ξ
)
d
ξ
d
x
d
P
τ
i
(
x
)=
f
i
(
x
)
d
x
=
K
i
→
i
+
1
(
x
)
.
or equivalently:
f
i
(
x
)
K
i
→
i
+
1
(
x
)=
ξ
.
(22)
−
0
f
i
(
ξ
)
1
d
7.4
FUCCI (Cell Cycle) Reporters to Identify Model
Parameters
FUCCI (for fluorescence ubiquitination-based cell cycle indicator) is a recent cell
imaging technique that allows tracking progression within the cell cycle of an
individual cell [
125
] after hybridisation with fluorescent proteins of cell cycle phase
characteristic indicators (geminin and Cdt1) involved in the ubiquitination, i.e.,
natural degradation, of the actual proteic determinants of evolution in the cell cycle.
We used FUCCI data that consisted in time series of the intensity of red and green
fluorescence emitted by individual NIH 3T3 cells (mouse embryonic fibroblasts)
within an in vitro homogeneous population proliferating without any control in a
liquid medium. This allowed us to measure the time an individual cell spent in the
G
1
phase and in the phases
S
M
of the cell cycle. Our data consisted of cell cycle
phase durations from 55 proliferating cells. Note that is fully justified in our case to
assume
K
i
→
i
+
1
(
/
G
2
/
since these experimental conditions correspond to
cells proliferating in a completely independent manner, without any communication
t
,
x
)=
K
i
→
i
+
1
(
x
)
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