Stem cells (SC): low frequency, not accessible to direct observation

provide inexhaustible supply of cells

Progenitor cells (PC): immediate downstream of SC, identifiable with cell surface markers, provide a quick proliferative response

Terminally differentiated cells: mature cells, represent a final cell type, do not divide

All cell types are susceptible to death

Renewing Cell Population

Example of renewing cell systems

• Development of Oligodendrocytes

• Kinetics of Leukemic cells

To: Stem cell

T1: Glial-restricted precursor (GRP)

T2: oligodendrocyte/type-2 astrocyte (O-2A)/oligodendrocyte precursor cell (OPC) (O-2A/OPC)

To: Leukemic stem cell (LSC)

T1: Leukemic progenitor (LP)

T2: Leukemic blast (LB)

Age-dependent Branching Process of Progenitor Cell Evolution without Immigration

• Evolution of an individual PC from birth to leavingEvery PC with probability p has a random life-time

with probability 1 - p it differentiates into another cell type

At the end of its life, every PC gives rise to v offsprings

v characterized by pgf

generally,

it takes a random time for differentiation to occur

• Stochastic processes Z(t), Z(t, x)

Z(t) ~ total number of PCs

Z(t, x) ~ number of PCs of

note that if then,

• pgfs of Z(t) , Z(t, x)

• Applying the law of total probability (LTP)

• Using notations:

with initial conditions:

• From (1) and (2):

with initial conditions :

• (3) is a renewal type equation with solution:

where,

and is the k-fold convolution

renewal function

Renewal-type Non-homogeneous Immigration• Let Y(t) be the number of PCs at time t with the same ev

olution of Z(t) in the presence of immigration

Y(t, x) number of PCs of

• pgfs of Y(t) and Y(t, x)

with initial conditions• time periods between the successive events of immig

ration: i.i.d, r.v.’s with c.d.f.

at any given t, number of immigrants is random with distribution

• pgf of number of immigrants at time t

mean number of immigrants at time t

• Applying LTP

(10)

(11)

with initial conditions:

•

(12)

(13)

with initial conditions: • Whenever is bounde

d, (12) has a unique solution which is bounded on any finite interval

• The solution is:

where is the renewal function

Neurogenic Cascade

d apoptosis rate

rate of G2M in

ANP2

differentiating into

NB

Modeling of Neurogenesis

11 11

12 12

13 13

21 21

22 22

23 23 23

31 31

32 32

33 33

4

0 1 0 0

0 1 0 0

0 2(1 ) 0 0

0 1 0 0

0 1 0 0

0 2(1 )(1 ) 0 2 (1 )

0 1 0 0

0 1 0

0 2(1 )

0

0 0 0

d d

d d

d d

d d

d d

m d d d

d d

d d

d d

d

The m matrix where mik is the expected number of progeny of type k at time t of a cell of type i

dij is the probability of its corresponding type of cells committed to apoptosis, is the chance that cell differentiated to NB directly from the phase G2M in the process of ANP2

232 (1 )d

Model Construction

• We obtain the fundamental solution of the model

where M is a matrix, with number of cells at time t in compartment j,

given the population was seeded by a single cell in compartment i

*

00

1

( ) [ ( , ) ] * [ ( , )]

= ( ( ))

tk

k

t

x

M t x G t x m I G x d

I m I G d

，

0

* )(*)(k

k GIGmM

under physiological conditions, the system is fed by a stationary influx of freshly activated ANPs, which may be represented by a Poisson process with constant intensity ω per unit of time, thus we obtain

~

11 11

~

12 12

~

13 13

~

21 21

~

22 22

~ ~ ~

23 23 23

~

31 31

~

32 32

~

33 33

4

1 0 0

1 0 0

1 2 0 0

1 0 0

1 0 0

1 2 0 2

1 0 0

1 0

1 2

1

0 1

d d

d d

d d

d d

d d

I m d d d

d d

d d

d d

d

To calculate the stationary distribution of M we need to derive the inverse matrix of I-m

The inverse of an upper triangular matrix is also an upper triangular matrix

1 2 3 4 5 6 7 8 9 1

3 5 6 7 8 92 42

1 1 1 1 1 1 1 1 1

3 5 6 7 8 943

2 2 2 2 2 2 2 2

5 6 7 8 944

3 3 3 3 3 3 3

5 6 7 8 95

4 4 4 4 4 4

6 7 8 916

5 5 5 5 5

7 8

6

1

0 1

0 1

0 1

0 1

0 1( )

0 1

A A A A A A A A A B C

A A A A A AA A BC

A A A A A A A A A

A A A A A AA BC

A A A A A A A A

A A A A AA BC

A A A A A A A

A A A A A BC

A A A A A A

A A A A BCI m

A A A A A

A A

A

97

6 6

8 98

7 7

99

8

4

0 1

0 1

0 1

0 0 1

AC

A A

A AC

A A

AC

A

d

~

1 11

~ ~

2 11 12

~ ~ ~

3 11 12 13

~ ~

4 11 21

~ ~

5 11 22

~ ~ ~

6 11 23

~ ~ ~

7 11 31

~ ~ ~

8 11 32

~ ~ ~

9 11 33

~ ~ ~

11 23

2

2

2

4

4

4

4

4

A d

A d d

A d d d

A d d

A d d

A d d

A d d

A d d

A d d

B d d

1 11

2 1 12

3 2 13

4 3 21

5 4 22

6 5 23

7 6 31

8 7 32

9 8 33

10 9 4

11 4

D d

D Ad

D A d

D A d

D A d

D A d

D A d

D A d

D A d

D A d

D Bd

11

11

11

221

11

332

11

443

11

554

11

665

10

776

10

887

9 9 108

1

1

1

1

1

1

1

1( )

ii

ii

ii

ii

ii

ii

ii

ii

C D

C DA

C DA

C DA

C DA

C DA

C DA

C DA

C D DA

Parameters in the inverse matrix are defined as

Example

• A computational example in simulating pulse labeling experiment with parameters

T1=10 hr, T2=8 hr, T3=4 hr, T4=96 hr, T5=1hr, dij=0.15, α=0.5

Total Number of Cells Labeled

0

500

1000

1500

2000

2500

3000

6h 18h 2d 4d 6d 8d 10d 12d

Time

Nu

mb

er

of C

ells

-500

0

500

1000

1500

2000

2500

2h 12h 1d 2d 3d 5d 7d 10d 15d 30d

BrdU total

QNPs/astros

ANPs

NBs

GCs

Number of cel l s l abel ed

0

500

1000

1500

2000

2500

3000

1 26 51 76 101 126 151 176 201 226 251 276

ti me/ hrs

# ce

lls NB

ANeuronANPs

Comparison suggests non-identical distribution of the mitotic cycle duration for amplifying neuroprogenitors (ANP) and unevenly distributed cell death rates