Repopulation dynamics of single haematopoietic stem cells in mouse transplantation experiments: Importance of stem cell composition in competitor cells

Author’s Accepted Manuscript Repopulation dynamics of single hematopoietic stem cells in mouse transplantation experiments: importance of stem cell composition in competitor cells. Hideo Ema, Kouki Uchinomiya, Yohei Morita, Toshio Suda, Yoh Iwasa www.elsevier.com/locate/yjtbi

PII: DOI: Reference:

S0022-5193(16)00038-2 http://dx.doi.org/10.1016/j.jtbi.2016.01.010 YJTBI8491

To appear in: Journal of Theoretical Biology Received date: 22 January 2015 Revised date: 14 October 2015 Accepted date: 4 January 2016 Cite this article as: Hideo Ema, Kouki Uchinomiya, Yohei Morita, Toshio Suda and Yoh Iwasa, Repopulation dynamics of single hematopoietic stem cells in mouse transplantation experiments: importance of stem cell composition in competitor cells., Journal of Theoretical Biology, http://dx.doi.org/10.1016/j.jtbi.2016.01.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1

Repopulation dynamics of single hematopoietic stem cells in mouse transplantation experiments: importance of stem cell composition in competitor cells.

Hideo Ema1*, Kouki Uchinomiya2*, Yohei Morita3, Toshio Suda1, and Yoh Iwasa2

1

Department of Cell Differentiation, Sakaguchi Laboratories of Developmental Biology, Keio University School of Medicine 2 Department of Biology, Faculty of Sciences, Kyushu University 3

Leibniz Institute for Age Research, Fritz Lipmann Institute *, These authors contributed equally to this work. Corresponding authors: Kouki Uchinomiya Department of Biology, Faculty of Science, Kyushu University, Motooka 744, Fukuoka 819-0395, Japan Tel: +81 92-802-4299; E-mail address: [email protected] Hideo Ema, M.D. Institute of Hematology and Blood Diseases Hospital, Chinese Academy of Medical Sciences and Peking Union Medical College 288 Nanjing Road, Tianjin, 300020 China Tel: +86 22 2390 9176 E-mail address: [email protected]

1

2

Abstract The transplantation of blood tissues from bone marrow into a lethally irradiated animal is an experimental procedure that is used to study how the blood system is reconstituted by hematopoietic stem cells (HSC). In a competitive repopulation experiment, a lethally irradiated mouse was transplanted with a single HSC as a test cell together with a number of bone marrow cells as competitor cells, and the fraction of the test cell progeny (percentage of chimerism) was traced over time. In this paper, we studied the stem cell kinetics in this experimental procedure. The balance between symmetric self-renewal and differentiation divisions in HSC determined the number of cells which HSC produce and the length of time for which HSC live after transplantation. The percentage of chimerism depended on the type of test cell (long-, intermediate-, or short-term HSC), as well as the type and number of HSC included in competitor cells. We next examined two alternative HSC differentiation models, one-step and multi-step differentiation models. Although these models differed in blood cell production, the percentage of chimerism appeared very similar. We also estimated the numbers of different types of HSC in competitor cells. Based on these results, we concluded that the experimental results inevitably include stochasticity with regard to the number and the type of HSC in competitor cells, and that, in order to detect different types of HSC, an appropriate number of competitor cells needs to be used in transplantation experiments.

Keywords: Hematopoietic stem cells (HSC), cell cycle, self-renewal, differentiation, lifespan

2

3

1. Introduction Lifelong hematopoiesis is maintained by hematopoietic stem cells (HSC) that are capable of self-renewal and multi-lineage differentiation. Transplantation experiments reveal the ability of HSC to recover the whole blood system. Lethally irradiated mice survive for long periods if they receive both HSC and progenitor cells. HSC are required to sustain long-term hematopoiesis, while progenitor cells are required to supply mature blood cells immediately. HSC give rise to progenitor cells, but progenitor cells do not give rise to HSC. A competitive repopulation procedure (Micklem et al., 1972; Harrison, 1980) has been used in which test donor cells (test cells) are introduced into irradiated mice together with a number of bone marrow cells (competitor cells) that contain stem cells and progenitor cells. The repopulating activity of test cells is measured as a value relative to that of competitor cells (Harrison et al., 1993). The experimental procedure of single-cell competitive repopulation has been previously described (Ema et al., 2006). In brief, (1) mice are lethally irradiated, killing their own HSC; (2) one test cell and 2 10 5 bone marrow cells as standard competitor cells are transplanted into each mouse; and (3) cells derived from the test cell and competitor cells can be distinguished experimentally. After transplantation, blood is periodically sampled, and the numbers of cells originating from the test cell and competitor cells are counted by flow cytometry; (4) secondary transplantation is performed to confirm that self-renewal is induced by primary transplanted single cells. The fraction of HSC and their cell type composition usually remain unchanged between primary and secondary recipients (Morita et al., 2010; Ema et al., 2014). There have been some theoretical studies about the cell repopulation process. Roeder et al. (2008) discussed the engraftment of HSC based on Roeder & Loeffler (2002). They assumed that there are two different growth environments and that cells stochastically switch growth environments. They showed that the loss of repopulating capability and the 3

4

probability of an HSC to home to a stem cell-supporting niche environment can explain HSC heterogeneity. The results were not strongly dependent on the number of host HSC, as long as this number was small enough. Sieburg et al. (2011) also simulated cell repopulation using the cellular automata model, and reproduced the ballistic shape of cell repopulation. However, none of these studies have explicitly considered the influence of competitor cells in repopulation experiments, despite the fact that test cell activity is always measured in relation to the activity of competitor cells. In this paper, we used a mathematical model to assess the dynamics of clonal expansion of stem cells and progenitor cells in competitive settings. We then attempted to understand when, and to what extent, self-renewal and differentiation take place after single HSC transplantation, and how different patterns of repopulation kinetics can be established. We also evaluated the effect of competitor cells on repopulation kinetics of test cells to gain a better understanding of competitive repopulation outcomes. We concluded that the experimental results inevitably include stochasticity caused by the number and the type of HSC in competitor cells. Recently, the functional heterogeneity of HSC has been recognized (Muller-Sieburg et al., 2004; Dykstra et al., 2007; Challen et al., 2010; Morita et al., 2010; Yamamoto et al., 2013). Based on reconstitution time, HSC can be categorized as long-term (LT), intermediate-term (IT), and short-term (ST) (Osawa et al., 1996; Morrison et al., 1997; Yang et al., 2005; Benveniste et al., 2010). To explain the production of progenitor cells from these three types of HSC, two models were compared. In the one-step differentiation model, each of the three stem cell types can become progenitor cells in one step (Fig. 1a). In contrast, in the multi-step differentiation model, LT-HSC differentiate into IT-HSC, which then differentiate into ST-HSC, and only ST-HSC can differentiate into progenitor cells (Fig.1b). We also analyze the experimental data of single-cell transplantation based on our model.

4

5

2. One-step differentiation model We consider the situation in which stem cells divide at a constant rate. Let x be the number of stem cells and I be the mean interval between cell divisions or the length of time required for a newly divided cell to divide again. If a stem cell divides and results in two stem cells (symmetric self-renewal), the stem cell population, denoted by x, doubles every I and increases accordingly as x ( t ) = x ( 0) 2

t I

= x ( 0) exp ( t ln2 I ) . Note that the process

indicates a continuous-time Markovian transition where cell division occurs at a random time point with an exponential probability distribution with a mean time interval of I. In contrast, if cell division results in two progenitor cells (symmetric differentiation), the number of stem cells decreases by one at each cell division. Since the stem cell number becomes half in time interval I, it follows the equation x ( t ) = x ( 0) 2

t I

= x ( 0) exp ( t ln 2 I ) . These two equations

can be combined as follows:

dx ln 2 = x I dt

(1),

is the relative proliferation rate.

where

produces two stem cells, but

=1 if every cell division of a stem cell

= 1 if every cell division produces two progenitor cells. If

the cell division results in one stem cell and one progenitor cell (asymmetric cell division),

= 0 . In general, cell divisions are a mixture of these three types, and

1

1 is

satisfied. As time passes after transplantation, the environment containing the stem cells changes, and the interval of cell division changes over time, represented as I ( t ) , where I is dependent on the time after transplantation, t. The relative proliferation rate of stem cells

(t )

also changes with time. LT-, IT- and ST-HSC have been recently redefined (Ema et al., 2014). We distinguish

these three types of HSC using suffix i, in which i = LT, IT and ST. P indicates the number of 5

6

progenitor cells derived from the three stem cell types. Thus, we have the following equations:

dx j ln 2 = j (t ) x j , dt I ( t ) dP ln 2 = (1 dt j I (t ) where

(j = LT, IT and ST)

j

( t )) x j

(2a)

P.

(2b)

is the progenitor cell amplification rate and

is the turnover rate of progenitor

cells. The relative proliferation rate of type j stem cells

j

(t)

is defined considering three

different stem cell division modes. Stem cell division may be a symmetric division yielding two stem cells (symmetric self-renewal) with probability a j ( t ); asymmetric division yielding one stem cell and one progenitor cell (asymmetric self-renewal) with probability b j ( t ) ; and symmetric division yielding two progenitor cells (symmetric differentiation) with probability

c j ( t ) . These rates may differ between cell types j (j = LT, IT and ST) and with time after transplantation, t. Since one of the three occurs at cell division, a j ( t) + b j ( t ) + c j ( t ) =1. The relative stem cell proliferation rate j

j

(t)

is given by the following equation:

( t ) = a j ( t ) c j ( t ),

(3),

where j = LT, IT and ST.

2.1 Time dependence of relative proliferation rate and cell division interval Shortly after transplantation, cell division should be mostly symmetric self-renewal and j

(t )

should be a large value (

I j

). Cell division should then decrease with time. Specifically,

we assumed the relative rate of proliferation I j j (t ) =

F j

if I j

T2 T1 F j

(t

T1 ) +

I j

if

j

(t )

t T1 T1 < t T2

if

as follows:

T2 < t

6

(4).

7

The initial period of symmetric self-renewal continues for several weeks (T1). Then, the relative proliferation rate decreases linearly with time but stops decreasing at time T2, after which the relative proliferation rate is maintained at a low level,

F j

.

Symmetric self-renewal is likely to be induced immediately after transplantation and I j

is expressed as

= 1. For the numerical analysis, we assumed the following:

decreases from 1 to 0 as t increases from 0 to 2 months and thereafter IT

(t )

ST

(t )

(t )

= 0.0 (Fig. 2a);

decreases from 1 to -0.5 as t increases from 0 to 2 months and thereafter

(Fig. 2a); F ST

F LT

LT

F IT

= 0.5

decreases from 1 to -1 as t increases from 0 to 2 months and thereafter

= 1.0 (Fig. 2a). Cell division interval I(t) also changes with time (Fig. 2b). Therefore, we assumed

the following: cell division occurs most frequently in the beginning, indicating a short cell I

division interval ( I ) for the period 0 < t < T3 ; the length of the interval then starts linearly increasing with time. When it reaches a high of IF, the interval stops increasing. This can be written as follows:

II F

I (t ) =

if

I I ( t T3 ) + I F T4 T3 IF

t T3

I

if

T3 < t T4 if

(5).

T4 < t

Cell divisions are induced by environmental factors immediately after transplantation. All HSC types should receive the same stimuli at the same time. These I settings may differ between different HSC types, and in particular during a stable phase when most HSC are in a quiescent state. However, in this model, the cell-intrinsic regulation of the cell cycle was ignored for simplicity. Hereafter, we analyzed x j ( t ) and P ( t ) with the initial condition x j ( 0) =1 7

8

and P ( 0) = 0 , indicating transplantation of a single stem cell.

2.2 Cell growth from a single stem cell Next, we considered blood cells produced by a single stem cell. The trajectory depended on the type of test stem cell at t = 0. Most HSC isolated from bone marrow are in the G0 phase of the cell cycle (Bradford et al., 1997; Cheshier et al., 1999; Sudo et al., 2000), but start dividing immediately after transplantation into lethally irradiated mice (Noda et al., 2009). This can be interpreted that HSC divide more frequently after transplantation. As a standard condition, we assumed 10-fold increase in the cycle rate. In addition, we assumed that HSC cycle frequently for 1 month after transplantation, and that their cycling frequency decreases gradually and then returns to a normal state by 4 months: most HSC divide once per month (Bradford et al., 1997; Cheshier et al., 1999; Sudo et al., 2000). This can be expressed as T3 = 1 and T4 = 4. Immediately after transplantation, for standard conditions, we set I(t) at 0.1 for t < 1 month (initial phase), at 0.1 to 1.0 as t increased from 1 to 4 months (transition phase), and at 1.0 for t > 4 months (maintenance phase) (Fig. 2b). These assumptions are expressed using the parameters of Eq. (5) as II = 0.1, IF = 1.0, T3 = 1, and T4 = 4. The final value of the relative stem cell proliferation rate ( behavior over time. If

F j

F j

) determined the

< 0 , the right-hand side of Eq. (2a) is negative after a sufficient

period, and therefore, the number of HSC decreases and they eventually vanish. In contrast, if F j

> 0 , the number of HSC increase indefinitely with time. None of the

positive, and at least one value of

F j

F j

values are

is 0, reflecting the stationary state in which HSC

produce the same number of HSC and progenitor cells. This is achieved when the probability of producing two stem cells is the same as the probability of producing two progenitor cells,

8

9

or if HSC always divide asymmetrically. The present model does not distinguish between these two cases. As mentioned above, we assumed

F LT

= 0,

F IT

F ST

= 0.5 and

= 1.0 (Fig. 2a). As

a result, the number of stem cells x j ( t ) increased when one or more LT-HSC was given (Fig. 2c) and the number of progenitor cells was maintained after overshooting (Fig. 2d). In contrast, the number of stem cells increased transiently and then decreased if no LT-HSC was given. Then, the number of progenitor cells was decreased after overshooting (Fig. 2d). When the number of HSC became constant, using Eq. (2b), the kinetics of the progenitor cells derived from a single LT-HSC is calculated as shown in Appendix A. The

(

)

* F number of progenitor cells P ( t ) converges to a constant ln 2 ×x I × , where x * is the

number of LT-HSC in the stationary state in which

LT

(t )

and I ( t ) become constant

(Appendix A). The parameter-dependent results are shown in Appendix B. The most influential parameters are those affecting the magnitude of

LT

and the cell division interval in the

period with the rapid symmetric cell division and the length of that period. These results imply that small differences in the initial HSC condition alter significantly the final number of cells.

2.3 Competitive repopulation In single-cell competitive repopulation experiments, a test cell and 2 ´ 105 competitor cells were mixed and transplanted into a lethally irradiated mouse. Competitor cells contained one or more HSC because mice receiving lethal irradiation could survive when only competitor cells were transplanted. The number of progenitor cells was calculated as P(t) from Eq. (2b) by setting x j (0) = 1, x k (0) = 0, ( k

j), and P(0) = 0, and was denoted as Pˆ ( i) ( t ) . 9

10

Let Ptest ( t ) be the number of progenitor cells produced by test cells at time t:

Pˆ ( LT ) ( t ) , Pˆ ( IT ) ( t ), or Pˆ ( ST ) ( t ). The progenitor cells originating from the test cell were genetically marked and distinguishable from competitor cells. Competitor cells initially included xci stem cells of type i, and Pc progenitor cells. The number of Pc progenitor cells decreased with survival ability exp[

t ] . Progenitor cells Pˆ ( i) ( t ) were generated from

xci stem cells (i = LT-, IT- and ST-HSC). The total number of progenitor cells originating xci Pˆ (i) ( t ) + Pc e t .

from competitor cells was expressed as follows: P ( t ) = i

Progenitor cells derived from test cells increased with those derived from competitor cells in recipient mice after transplantation. The percentage of chimerism

(t)

was defined

as the percentage of test cell-derived progenitor cells among all progenitor cells in recipient mice at time t and was calculated as follows:

Ptest ( t )

(t ) =100 j

x Pˆ j c

( j)

(t ) + Pce

t

+ Ptest ( t )

.

(6).

The percentage of chimerism was calculated using the numbers of progenitor cells with the assumption that the number of mature blood cells can be represented by the number of progenitor cells in the competitive repopulation settings. The trajectory of

(t )

depended

on the stem cell type of the test cell and the stem cell types and numbers of the competitor cells. Due to the nature of experiment, the precise number of stem cells included in competitor cells could not be controlled. Therefore, we considered the case in which LT-HSC number x cLT , IT-HSC number x cIT and ST-HSC number x cST were included in the initial cell population of competitor cells. This was represented as

(x

LT c

, x cIT , x cST ). Let P0c be the

number of progenitor cells initially present in the competitor cells. We assumed P0c = 2000 in 2 ´ 105 bone marrow cells (Ema et al., 2006). 10

11

Blood cells produced by a mixture of the three HSC types generated trajectories represented as a weighted sum of three solutions each starting from a single stem cell. We first considered the case in which the competitor cells included a single stem cell (LT-, IT- or ST-HSC):

(x

LT c

, x cIT , x cST ) was

(1, 0, 0) , (0,1, 0)

or

(0, 0,1) ,

respectively. Figure 2d

illustrates the number of progenitor cells produced by a single stem cell, and the three curves correspond to the cases when the stem cell was a LT-, IT- or ST-HSC. We always assumed that competitor cells were 2 ´ 105 bone marrow cells consisting mostly of progenitor cells that decrease rapidly after transplantation (Fig. 2d). Figure 3 illustrates the percentage of chimerism originating from test cells, as shown in Eq. (6). The type of test cell is indicated by the suffix LT, IT or ST; thus, the three functions are

LT

,

IT

, and

ST .

i) When no stem cell was included in the competitor cells Figure 3a shows the case when the competitor cells had no HSC:

(x

LT c

, xcIT , xcST ) = ( 0, 0, 0) .

The percentage of chimerism converged to 100% eventually without competitor cells: LT

( )=

IT

( )=

ST

( ) =100 . There were only test cell-derived blood cells in mice.

ii) When one ST-HSC was included in the competitor cells Figure 3b shows the results when the competitor cells had one ST-HSC but neither LT nor IT HSC cells:

(x

LT c

became 100%:

, xcIT , xcST ) = ( 0, 0,1). The percentage of chimerism by a single LT- or IT-HSC LT

( )=

IT

( ) =100 .

However, if the test cell was a ST-HSC, the

dynamics of progenitor cells from a test cell and those derived from a ST-HSC in competitor

11

12

cells were exactly the same. Therefore, equation

ST

ST

(t )

converged to 50, as represented by the

( ) = 50 .

iii) When one IT-HSC was included in the competitor cells Figure 3c shows the results when competitor cells had only one IT-HSC:

(x

LT c

, xcIT , xcST ) = ( 0,1, 0). In this case,

LT

(t )

became 100:

LT

( ) =100

LT because Pˆ ( ) ( t )

remained positive, while the progenitor cells derived from an IT-HSC in competitor cells decreased and vanished eventually. If the test cell was an IT-HSC, progenitor cells derived from the test cell and progenitor cells derived from an IT-HSC from the competitor cells behaved similarly. Therefore, test cell was a ST-HSC,

ST

ST

(t )

IT

(t )

decreased and converged to 50%:

IT

( ) = 50 . If the

first increased but then decreased and eventually reached 0:

( ) = 0.

iv) When one LT-HSC was included in competitor cells Figure 3 (d) shows the result when one LT-HSC was included in competitor cells:

(x

LT c

LT

, xcIT , xcST ) = (1, 0, 0). If the test cell was a LT-HSC, the chimerism converged to 50:

( ) = 50 . If the test cell was an IT-HSC or ST-HSC, the percentage of progenitor cells

first increased and then decreased and eventually converged to 0, as follows: IT

( )=

ST

( ) = 0.

The final value of the percentage of chimerism reveals the type of test cell when only a single stem cell is included in competitor cells. The results are summarized in Table 1a.

12

13

v) When two or more stem cells are included in competitor cells Using a similar logic, we can derive the cases in which two or more stem cell types are included in the competitor cells at transplantation. The predictions of the model regarding the final chimerism

value are summarized in Table 1b. The results depended on the type

of the test cell (LT, IT and ST-HSC) and the competitor cell composition. We classified the composition of competitor cells into the following three groups: [L] one or more LT-HSC are included, [I] no LT-HSC is included but one or more IT-HSC are included and [S] neither a LT-HSC nor an IT-HSC is included but one or more ST-HSC are included. Chimerism becomes 100% when the test cell is a LT-HSC and when the competitor cell population is either [I] or [S], or when the test cell is an IT-HSC and the competitor cell population is [S]. Chimerism

converges to a value between 0% and 100% when the test cell is a LT-HSC

and the competitor cell population is [L], when the test cell is an IT-HSC and the competitor cell population is [I], or when the test cell is a ST-HSC and the competitor cell population is [S]. Finally, chimerism

converges to 0% when the test cell is an IT-HSC and the

competitor cell population is [L], or when the test cell is a ST-HSC and the competitor cell population is [L] or [I].

3. Multi-step differentiation model In the model studied above, all three types of stem cells can differentiate directly into progenitor cells, and there is no transition or differentiation among the types (Muller-Sieburg et al., 2012). Whether the one-step differentiation model or other models are more accurate remains debatable (Muller-Sieburg et al., 2004; Dykstra et al., 2007; Challen et al., 2010; Morita et al., 2010; Benz et al., 2012 ). In this study, we considered an alternative model of cell differentiation in which progenitor cells are only produced from ST-HSC, which are differentiated from IT-HSC, which are, in turn, differentiated from LT-HSC, as illustrated in Figure 1b. 13

14

We assumed that a single LT-HSC produces two LT-HSC with probability aLT ( t ), one LT-HSC and one IT-HSC with probability bLT ( t ), and two IT-HSC with probability

cLT ( t ) . Next, we defined the relative proliferation rate of LT-HSC as

LT

Likewise, the relative IT- and ST-HSC proliferation rates were defined as

( t) = aLT (t ) cLT ( t). IT

(t)

and

ST

( t ),

respectively. Similar to the one-step differentiation model, the dynamics of HSC and progenitor cells are written as follows (Appendix C):

dx LT ln2 = dt I( t )

( t ) x LT ,

(7a)

dx IT 1 LT ( t ) ( t) = ln2 IT x IT + x LT dt I( t) I(t )

(7b)

dx ST 1 IT ( t ) (t) = ln2 ST x ST + x IT dt I( t ) I( t )

(7c)

dP ln2 = × (1 dt IST ( t )

(7d).

LT

ST

( t )) x IT

P

In this hematopoietic repopulation process, the trajectory of the number of progenitor cells

Pˆ ( j ) ( t ) was defined as the solution from Eq. (7); if the initial condition was a single test cell with type j (j = LT, IT or ST), then the percentage of chimerism

(t )

can be calculated in a

manner similar to Eq. (6). In the multi-step differentiation model, the number of progenitor cells produced was greater than in the one-step differentiation model (Fig. 4a and 4b). However, competitive repopulation results represented by chimerism were very similar between the two models (Fig. 4c), suggesting that identifying the different differentiation pathways solely through competitive repopulation analysis is difficult (Ema et al., 2014). In the multi-step differentiation model, for the number of progenitor cells to be maintained at a positive value, LT-HSC division must produce identical numbers of LT- and IT-HSC (

F LT

must be 0), IT-HSC division must produce more ST-HSC than IT-HSC (

14

F IT

is

15

negative), and ST-HSC division must produce more progenitor cells than ST-HSC (

F ST

must

be negative) (Appendix D).

4. Comparison of experimental results with a mathematical model The experimental results of a single-cell transplantation reported by Morita et al. (2010) were re-analyzed using the model developed in this paper. Methods of single-cell sorting and transplantation have been described in detail (Ema et al., 2006; Morita et al., 2010). In brief, single CD150high, CD150med or CD150lowCD34-Kit+Sca-1+Lin- cells from B6-Ly5.1 mice were mixed with 2 ´ 105 bone marrow cells from B6-Ly5.1/5.2 F1 mice and transplanted into lethally irradiated B6-Ly5.2 mice. To compare our model analysis with experimental data, we focused on neutrophil repopulation kinetics because neutrophils are extremely short-lived and their long-term repopulation levels accurately reflect stem cell activity (Jordan and Lemischka, 1990; Forsberg et al., 2006; Sieburg et al., 2011). Figure 5 (a) shows the data of neutrophil reconstitution after single CD150lowCD34-Kit+Sca-1+Lin- cells were transplanted. In the case of ST-HSC, reconstitution typically took place for a few months. The percentage of chimerism converged to 0% after 6 months for all cases; however, the mice still survived. When a low level of the percentage of chimerism persisted over 6 months, this type of stem cells, by definition, belonged to IT-HSC. Figure 5 (b) shows the simulations of these ST-HSC and IT-HSC where the conditions described for

IT

and

ST

in Figure 3(d) were used. Figure 5 (c) shows the data of

neutrophil reconstitution after single CD150highCD34-Kit+Sca-1+Lin- cells were transplanted. A typical LT-HSC, an alternative type of IT-HSC and a latent type of HSC which exhibits delayed reconstitution after transplantation (Morita et al., 2010) are demonstrated. Figure 5

15

16

(d) shows that LT-HSC but neither alternative type of IT-HSC nor latent HSC could be simulated well where the conditions of y LT and y IT in Figure 3(c) were used. Both alternative type of IT-HSC and latent HSC were not often observed among HSC. In order to simulate these rare HSC, we attempted to modify

(t )

and I(t) conditions.

Resultant simulations of IT-HSC and latent HSC are shown in Figure 6. These delayed onset types of reconstitution were simulated by prolonged self-renewal with low rate of division. These analyses suggest that some types of HSC might be regulated differently from ST-HSC and LT-HSC.

16

17

5. Discussion There is no existing data showing the kinetics of a single HSC after transplantation because currently available methods do not permit their division or fate to be tracked in vivo. To address this issue, we developed a mathematical model for single HSC dynamics, and applied these dynamics to analyze the repopulation process by HSC with competitive settings. We demonstrated that chimerism depends on the activity and lifespan of HSC from both test and competitor cells (Fig. 3, Table 1) if HSC activity is considered to be a blood lineage repopulating activity and HSC lifespan is considered to be the length of repopulating time after transplantation. Total blood cell production depends on the number of symmetric self-renewal divisions experienced by an individual HSC, and in particular, self-renewal that occurs immediately after transplantation, as demonstrated in the case of a LT-HSC with a relative proliferation rate

I LT

(Fig. A1). Whether some cells remain for a sufficiently long time

depends on the final relative proliferation rate,

F j

. In our model,

j

may differ between

individual HSC types. Presumably, quantitative changes in the levels or interactions of core transcription factors modulate the HSC fate decision parameter (Graf and Enver, 2009) that is possibly represented by

j

. How the lifespan of HSC is determined is poorly understood.

However, this study suggests that the balance of symmetric self-renewal and differentiation is the key element that distinguishes LT-, IT- and ST-HSC from one another. How often HSC divide, I I (Fig. A2), and how long symmetric self-renewal persists, (T1), (Fig. A3) are crucial for the rate of blood cell production. Small differences in properties soon after transplantation may result in pathological conditions. In our model, cell division is influenced by extracellular signals and change over the time after transplantation, and all HSC types undergo similar cell divisions under proliferation stress. Dormant HSC, which rarely go through the cell cycle, have been reported (Wilson et al., 2008; Takizawa et al., 17

18

2011; Qiu et al., 2014). For such HSC, we may have to postulate a very long cell division interval during aging. It is difficult to determine the absolute activity of HSC. HSC activity has been detected only by comparison to competitor cells. The chimerism of test cells depends both on the type of test cell and the types and numbers of stem cells included in the competitor cells (Table 1a, b). Our models showed that the percentage of chimerism given by a test LT-HSC is reduced, but remains positive if competitor cells include some LT-HSC (Table 1b). In contrast, when the test cell is either IT-HSC or ST-HSC, the chimerism can reach 0%, depending on the presence of LT-HSC or IT-HSC (Table 1b). It is possible to estimate the number of LT-, IT- and ST-HSC in competitor cells using Table 1b. In this case, the influence of HSC in the upper classes was considered minimal because the three types of HSC have different progenitor production onsets. For instance, progenitor production of LT-HSC is often significantly delayed compared to that of ST-HSC. In our previous studies (Morita et al., 2010; Yamamoto et al., 2013), most LT-HSC showed >95% chimerism, suggesting that the 2 x 105 bone marrow cells utilized barely contain any LT-HSC. IT-HSC showed approximately 50% chimerism on average, suggesting that 2 x 105 bone marrow cells often contain one IT-HSC. ST-HSC, on average, showed 10% chimerism, suggesting that 2 x 105 bone marrow cells contain nine ST-HSC. These HSC numbers should change according to a Poisson distribution, and because the number of ST-HSC is significantly greater than those of LT- and IT-HSC, the number of competitor cells should be adjusted based on which HSC is of interest. In this study, we assumed that most LT-, IT- and ST-HSC were homogeneous populations. However, we sometimes observed the percentage of chimerism between 50 and 100%. A large variation in the percentage of chimerism was also found. Some IT-HSC and latent HSC may basically differ from ST-HSC and LT-HSC. If this is the case, these HSC are

18

19

generated independent of LT-HSC. Nevertheless, each type of HSC seems, to some extent, functionally heterogeneous. Many types of stem and progenitor cells have been identified (Akashi et al., 2000; Katsura and Kawamoto, 2001; Adolfsson et al., 2005). However, their relationship in the context of differentiation pathways remains unclear (Ema et al., 2014). The results of our analysis show that competitive repopulation is similar between the one-step and multi-step differentiation models (Fig. 4c). Therefore, whether different HSC types give rise to progenitor cells directly or indirectly is difficult to determine in competitive settings. HSC are likely to follow multiple pathways as they differentiate into mature blood cells. Defining these pathways requires novel techniques allowing for the tracking of single HSC as they differentiate into each mature blood cell type. The stochasticity of HSC composition in competitor cells was the focus of this work. For simplicity, we adopted a deterministic model for cell population growth. However the demographic stochasticity might be important especially when a population size is small. If a HSC may die with some probability, the lineage of a cell may go extinct with a positive probability, even if the population is expected to grow on average. The demographic stochasticity for analysis of cell population growth will be a very important theme of the future theoretical study of competitive repopulation. Our models are simple, but are useful in clarifying the interpretation of repopulation experiments, which are essential for understanding the molecular regulation of self-renewal and differentiation in normal and leukemic stem cells.

19

20

Acknowledgements We thank Ben MacArthur and Aled O’Neill for critical reading of the manuscript. This work was supported in part by Grants-in-Aid for Scientific Research (A) and (C), Grants-in-Aid for Scientific Research on Innovative Areas in Japan and the European Union’s Seventh Framework Program (FP7/2007-2013) under grant agreement number 306240 (SyStemAge), Grant-in-Aid from the Japanese Society for the Promotion of Science to K.U. (13J01212), and a Grant-in-Aid for Scientific Research (B) 15H004423 of Japan Society for the Promotion of Science to Y. I.

20

21

Appendix A Kinetics of progenitor cells when the number of HSC is constant After a sufficiently long time ,

(t )

j

and I ( t ) become constant values,

and I F ,

F j

respectively. Then, Eq. (2a) can be rewritten as follows: F

dx j

= ln 2 × Fj x j dt I

(A.1), F

which is solved as x j = x * exp ×n 2 × Fj ( t t *) where x * is the number of HSC at time t * , I which is the time at which both

j

(t )

and I ( t ) become constant. If progenitor cells are

derived from a single HSC of type j, Eq. (2b) becomes the following:

dP ln 2 = (1 dt IF

F j

)x

*

(A.2),

P

which is solved as *

P ( t ) = P exp

ln 2 (1

(t t ) + ln 2 × *

F j

F j

)x

*

F

exp ln 2 × Fj ( t t * ) I

+ IF ×

exp

(t t ) ÷÷, *

where P * is the number of progenitor cells derived from j-HSC after t * . If the number of F j

HSC is maintained at a positive constant,

must be 0, indicating the HSC is of LT-type

and the following equation can be written as:

ln 2 ×x* P (t ) = F I × Hence, P converges to

P

*

ln 2 ×x* ÷exp IF ×

ln 2 ×x* I F ×

j

(t t ) . *

after a sufficiently long time.

21

(A.3).

22

Appendix B Parameter dependence The most influential parameters are those related to the magnitude of

LT

, the cell division

interval in the period with fast symmetric cell division and the length of that period. The dependence of the overall model behavior on these parameters is described below.

i) Initial When

LT I LT

value

is smaller (Fig. A1a), HSC produce progenitor cells earlier, and the total number

of HSC and progenitor cells produced in the entire period become smaller (Fig. A1b and A1c).

ii) Initial value of cell division interval, I A large I I implies a longer time interval for successive cell divisions during the initial I period of fastest HSC proliferation (Fig. A2a). As I becomes smaller, the number of HSC

and progenitor cells becomes smaller (Fig. A2b and A2c).

iii) Length of period for symmetric cell division A large T1 implies a longer period of fast LT-HSC proliferation in which all cell divisions are symmetric (self-renewal). As T1 becomes larger (Fig. A3a), the number of HSC and progenitor cells increase dramatically (Fig. A3b and A3c). These analyses suggest the number of self-renewal divisions in the early phase of transplantation exerts the greatest effect on progenitor cell production.

22

23

Appendix C Multi-step differentiation model When considering the probability of cell division, aj(t), bj(t) and cj(t), and the interval of cell division, I(t), the kinetics of HSC and progenitor cells can be written as follows:

dxLT ln 2 = ( aLT (t ) cLT (t )) xLT , dt I (t )

(C.1a)

( aIT (t ) bIT (t )) x + ( bLT (t ) + 2cLT (t )) x , dxIT = ln 2 IT LT dt I (t ) I (t )

(C.1b)

( aST (t ) cST (t )) x + ( bIT (t ) + 2cIT (t )) x , dxST = ln 2 ST IT dt I (t ) I (t )

(C.1c)

ln 2 dP = × ( bST (t ) + 2cST (t )) xIT I (t ) dt

(C.1d).

P.

Since a j ( t ) , b j ( t ) and c j ( t ) are probabilities, a j ( t ) + bj ( t ) + c j ( t ) =1 holds, and using Eq. (3), Eqs. (C.1) can be rewritten as Eqs. (7) in the manuscript.

23

24

Appendix D Stable state of the multi-step differentiation model In the stable state of the multi-step differentiation model, we assumed that a constant number of HSC and progenitor cells exist. If the number of HSC and progenitor cells is constant, Eq. (7) becomes 0 after a sufficiently long time, as shown below:

dxLT ln 2 = dt I (t )

(t ) xLT ® 0 ,

(D.1a)

é (t ) 1 LT ( t ) ù dxIT = ln 2 ê IT xIT + xLT ú ® 0 dt I (t ) úû êë I ( t )

(D.1b)

é (t ) 1 IT ( t ) ù dxST = ln 2 ê ST xST + xIT ú ® 0 dt I (t ) úû êë I ( t )

(D.1c)

ln 2 dP (1 - j ST (t ))x IT - dP ® 0 =l× dt I (t )

(D.1d).

LT

In Eq. (D.1a), xLT is not 0. Therefore,

LT

( )=0

and Eq. (D.1b) becomes the following:

é (t ) ù dxIT 1 = ln 2 ê IT xIT + xLT ú ® 0 dt I ( t ) úû êë I ( t ) in which

IT

Likewise,

( )=

ST

(D.2),

xLT xIT . Since both xLT , l and xIT , l are positive,

( ) = ( x IT ,l + x LT ,l ) x ST ,l < 0

IT

( ) is negative.

holds. From Eq. (D.1d), the number of

progenitor cells in stable state can be calculated as follows:

P= If

ST , l

(t )

l ln 2 (1 - j ST (t ))x IT × d I (t )

(D.3).

is negative, Eq. (D.3) is positive, indicating that several progenitor cells exist.

24

25

References

Adolfsson, J., Mansson, R., Buza-Vidas, N., Hultquist, A., Liuba, K., Jensen, C. T., Bryder, D., Yang, L., Borge, O. J., Thoren, L. A., Anderson, K., Sitnicka, E., Sasaki, Y., Sigvardsson, M., Jacobsen, S. E., 2005. Identification of Flt3+ lympho-myeloid stem cells lacking erythro-megakaryocytic potential a revised road map for adult blood lineage commitment. Cell 121, 295-306, doi:10.1016/j.cell.2005.02.013. Akashi, K., Traver, D., Miyamoto, T., Weissman, I. L., 2000. A clonogenic common myeloid progenitor

that

gives

rise

to

all

myeloid

lineages.

Nature

404,

193-7,

doi:10.1038/35004599. Benveniste, P., Frelin, C., Janmohamed, S., Barbara, M., Herrington, R., Hyam, D., Iscove, N. N., 2010. Intermediate-term hematopoietic stem cells with extended but time-limited reconstitution potential. Cell Stem Cell 6, 48-58, doi:S1934-5909(09)00619-5 [pii] 10.1016/j.stem.2009.11.014. Benz, C., Copley, M. R., Kent, D. G., Wohrer, S., Cortes, A., Aghaeepour, N., Ma, E., Mader, H., Rowe, K., Day, C., Treloar, D., Brinkman, R. R., Eaves, C. J., 2012. Hematopoietic stem cell subtypes expand differentially during development and display distinct lymphopoietic programs. Cell Stem Cell 10, 273-83, doi:10.1016/j.stem.2012.02.007. Bradford, G. B., Williams, B., Rossi, R., Bertoncello, I., 1997. Quiescence, cycling, and turnover in the primitive hematopoietic stem cell compartment. Exp Hematol 25, 445-53. Challen, G. A., Boles, N. C., Chambers, S. M., Goodell, M. A., 2010. Distinct hematopoietic stem cell subtypes are differentially regulated by TGF-beta1. Cell Stem Cell 6, 265-78, doi:10.1016/j.stem.2010.02.002. Cheshier, S. H., Morrison, S. J., Liao, X., Weissman, I. L., 1999. In vivo proliferation and cell cycle kinetics of long-term self-renewing hematopoietic stem cells. Proc Natl Acad Sci U S A 96, 3120-5. Dykstra, B., Kent, D., Bowie, M., McCaffrey, L., Hamilton, M., Lyons, K., Lee, S. J., Brinkman, R., Eaves, C., 2007. Long-term propagation of distinct hematopoietic differentiation programs in vivo. Cell Stem Cell 1, 218-29. Ema, H., Morita, Y., Suda, T., 2014. Heterogeneity and hierarchy of hematopoietic stem cells. Exp Hematol 42, 74-82 e2, doi:10.1016/j.exphem.2013.11.004. Ema, H., Morita, Y., Yamazaki, S., Matsubara, A., Seita, J., Tadokoro, Y., Kondo, H., Takano, H., Nakauchi, H., 2006. Adult mouse hematopoietic stem cells: purification and single-cell assays. Nat Protoc 1, 2979-87. Forsberg, E. C., Serwold, T., Kogan, S., Weissman, I. L., Passegue, E., 2006. New evidence supporting megakaryocyte-erythrocyte potential of flk2/flt3+ multipotent hematopoietic progenitors. Cell 126, 415-26, doi:S0092-8674(06)00858-0 [pii]

25

26 10.1016/j.cell.2006.06.037. Graf,

T.,

Enver,

T.,

2009.

Forcing

cells

to

change

lineages.

Nature

462,

587-94,

doi:10.1038/nature08533. Harrison, D. E., 1980. Competitive repopulation: a new assay for long-term stem cell functional capacity. Blood 55, 77-81. Harrison, D. E., Jordan, C. T., Zhong, R. K., Astle, C. M., 1993. Primitive hemopoietic stem cells: direct assay of most productive populations by competitive repopulation with simple binomial, correlation and covariance calculations. Exp Hematol 21, 206-19. Jordan, C. T., Lemischka, I. R., 1990. Clonal and systemic analysis of long-term hematopoiesis in the mouse. Genes Dev 4, 220-32. Katsura,

Y.,

Kawamoto,

H.,

2001.

Stepwise

lineage

restriction

of

progenitors

in

lympho-myelopoiesis. Int Rev Immunol 20, 1-20. Micklem, H. S., Ford, C. E., Evans, E. P., Ogden, D. A., Papworth, D. S., 1972. Competitive in vivo proliferation of foetal and adult haematopoietic cells in lethally irradiated mice. J Cell Physiol 79, 293-8, doi:10.1002/jcp.1040790214. Morita, Y., Ema, H., Nakauchi, H., 2010. Heterogeneity and hierarchy within the most primitive hematopoietic stem cell compartment. J Exp Med 207, 1173-82. Morrison, S. J., Wandycz, A. M., Hemmati, H. D., Wright, D. E., Weissman, I. L., 1997. Identification of a lineage of multipotent hematopoietic progenitors. Development 124, 1929-39. Muller-Sieburg, C. E., Sieburg, H. B., Bernitz, J. M., Cattarossi, G., 2012. Stem cell heterogeneity: implications

for

aging

and

regenerative

medicine.

Blood

119,

3900-7,

doi:10.1182/blood-2011-12-376749. Muller-Sieburg, C. E., Cho, R. H., Karlsson, L., Huang, J. F., Sieburg, H. B., 2004. Myeloid-biased hematopoietic stem cells have extensive self-renewal capacity but generate diminished lymphoid progeny with impaired IL-7 responsiveness. Blood 103, 4111-8. Noda, S., Ichikawa, H., Miyoshi, H., 2009. Hematopoietic stem cell aging is associated with functional decline and delayed cell cycle progression. Biochem Biophys Res Commun 383, 210-5, doi:10.1016/j.bbrc.2009.03.153. Osawa, M., Hanada, K., Hamada, H., Nakauchi, H., 1996. Long-term lymphohematopoietic reconstitution by a single CD34-low/negative hematopoietic stem cell. Science 273, 242-5. Qiu, J., Papatsenko, D., Niu, X., Schaniel, C., Moore, K., 2014. Divisional history and hematopoietic stem cell function during homeostasis. Stem Cell Reports 2, 473-90, doi:10.1016/j.stemcr.2014.01.016. Roeder, I., Loeffler, M., 2002. A novel dynamic model of hematopoietic stem cell organization based on the concept of within-tissue plasticity. Exp Hematol 30, 853-61. Roeder, I., Horn, K., Sieburg, H. B., Cho, R., Muller-Sieburg, C., Loeffler, M., 2008. Characterization and quantification of clonal heterogeneity among hematopoietic stem

26

27 cells: a model-based approach. Blood 112, 4874-83, doi:10.1182/blood-2008-05-155374. Sieburg, H. B., Rezner, B. D., Muller-Sieburg, C. E., 2011. Predicting clonal self-renewal and extinction of hematopoietic stem cells. Proc Natl Acad Sci U S A 108, 4370-5, doi:10.1073/pnas.1011414108. Sudo, K., Ema, H., Morita, Y., Nakauchi, H., 2000. Age-associated characteristics of murine hematopoietic stem cells. J Exp Med 192, 1273-80. Takizawa, H., Regoes, R. R., Boddupalli, C. S., Bonhoeffer, S., Manz, M. G., 2011. Dynamic variation in cycling of hematopoietic stem cells in steady state and inflammation. J Exp Med 208, 273-84, doi:10.1084/jem.20101643. Wilson, A., Laurenti, E., Oser, G., van der Wath, R. C., Blanco-Bose, W., Jaworski, M., Offner, S., Dunant, C. F., Eshkind, L., Bockamp, E., Lio, P., Macdonald, H. R., Trumpp, A., 2008. Hematopoietic stem cells reversibly switch from dormancy to self-renewal during homeostasis and repair. Cell 135, 1118-29, doi:10.1016/j.cell.2008.10.048. Yamamoto, R., Morita, Y., Ooehara, J., Hamanaka, S., Onodera, M., Rudolph, K. L., Ema, H., Nakauchi, H., 2013. Clonal analysis unveils self-renewing lineage-restricted progenitors generated

directly

from

hematopoietic

stem

cells.

Cell

154,

1112-26,

doi:10.1016/j.cell.2013.08.007. Yang, L., Bryder, D., Adolfsson, J., Nygren, J., Mansson, R., Sigvardsson, M., Jacobsen, S. E., 2005. Identification of Lin(-)Sca1(+)kit(+)CD34(+)Flt3- short-term hematopoietic stem cells capable of rapidly reconstituting and rescuing myeloablated transplant recipients. Blood 105, 2717-23, doi:10.1182/blood-2004-06-2159.

27

28

Table 1 The final percentage value of the progenitor cells originating from the test cell (i.e., chimerism),

( ) . The left column shows the type of test cell (LT-, IT- or ST-HSC). The top

row indicates the stem cell composition of the competitor cells when: (a) a single stem cell is included in competitor cells (others are progenitor cells) or (b) multiple cells are included in competitor cells. Asterisk (*) indicates any non-zero number. (a) Stem cells included in competitor cells

Test cell

(1, 0, 0)

(0, 1, 0)

(0, 0, 1)

LT

50

100

100

IT

0

50

100

ST

0

0

50

(b) Stem cells included in competitor cells ( x cLT , *, *)

(0, x cIT , *)

(0, 0, x cST )

x cLT 1

x cIT 1

x cST 1

LT

100 (1+ x cLT )

100

100

IT

0

100 (1+ x cIT )

100

ST

0

0

100 (1+ x cST )

Test cell

28

29

Figure Legends Figure 1. A schematic representation of mathematical models for single HSC dynamics. (a) One-step differentiation model. A stem cell is maintained in G0 phase for time interval

I ( t ) and then enters the cell cycle. A stem cell gives rise to stem cells or progenitor cells. (b) Multi-step differentiation model. A LT-HSC gives rise to LT- and IT-HSC. An IT-HSC gives rise to IT- and ST-HSC. A ST-HSC gives rise to ST-HSC and progenitor cells.

Figure 2. Numerical simulations for single HSC dynamics.

(a) Changing of

j

. (b)

Changing of I. (c) Dynamics of HSC. Numbers of LT-HSC ( xLT ), IT-HSC ( x IT ) and ST-HSC ( xST ) are shown in solid, broken and dotted lines, respectively. (d) Dynamics of progenitor cells. Numbers of progenitor cells derived from LT-HSC ( Pˆ ( LT ) ), progenitor cells derived ST from IT-HSC ( Pˆ ( IT ) ), progenitor cells derived from ST-HSC ( Pˆ ( ) ), and progenitor cells

c contained in competitor cells ( P ) are shown in solid, broken, dotted, and gray lines,

respectively. xLT ( 0) =1.0 , xIT ( 0) =1.0 , xST ( 0) =1.0 . Parameters were set as follows:

P0 ( 0) = 2000 ,

I LT

I IT

=1.0 ,

=1.0 ,

I ST

=1.0 ,

T2 = 2.0 , I I = 0.1, I F =1.0 , T3 =1.0 , T4 = 4.0 ,

F LT

= 0.0 ,

F IT

= 0.5 ,

= 50 , and

F ST

= 1.0 , T1 = 0.0 ,

= 3.0 .

Figure 3. Numerical simulations for competitive repopulation model. The percentages of chimerism by LT-HSC (

LT

), IT-HSC (

IT )

and ST-HSC (

ST )

are shown in solid, broken

and dotted lines, respectively. (a) Competitor cells contained no HSC. (b) Competitor cells contained one ST-HSC. (c) Competitor cells contained one IT-HSC. (d) Competitor cells contained one LT-HSC. Parameters were set as in Figure 2.

Figure 4. Numerical simulations for the multi-step differentiation model.

29

30

(a) Dynamics of LT-HSC ( xLT ), IT-HSC ( x IT ) and ST-HSC ( xST ) are shown in solid, broken and dotted lines, respectively. (b) Dynamics of progenitor cells derived from LT-HSC, IT-HSC and ST-HSC are shown in solid, broken and dotted lines, respectively. (c) Comparison of chimerism when the test cell was an LT-HSC between one-step and multi-step differentiation models. The percentages of chimerism in multi-step and one-step differentiation models are shown in black and gray lines, respectively. Competitor cells

(

)

LT IT ST containing one LT-HSC: xc , xc , xc = (1, 0, 0) , one IT-HSC:

(

(x

LT c

, xcIT , xcST ) = ( 0, 1, 0) and

)

LT IT ST one ST-HSC: xc , xc , xc = ( 0, 0, 1) are shown in solid, broken and dotted lines,

respectively. Parameters were set as in Figure 2.

Figure 5. Comparison of competitive repopulation models with experimental data. (a) Experimental results from the transplantation of single CD150lowCD34-Kit+Sca-1+Lincells. (b) Simulations with

IT

and

ST

from Figure 3(d). (c) Experimental results from

transplantation of single CD150highCD34-Kit+Sca-1+Lin- cells. (d) Simulations with y LT and

y IT from Figure 3(c).

Figure 6. Simulation of latent HSC and alternative type of IT-HSC Experimental results of latent HSC and of alternative type of IT-HSC are shown in circles and triangles, respectively. Lines show the simulations of percentage of chimerism when test cell had different properties. Solid line is when test cell was LT-HSC ( took longer (T2=10). Broken line is when test cell was IT-HSC (

F LT F IT

= 0.0 ) and self-renewal

= 0.5) and self-renewal

took longer (T2=8). In both cases, initial division of a test cell was less frequent (II=0.27) than that of HSC in competitor cells. Competitor cells contained one of each HSC type

(x

LT c

, xcIT , xcST ) = (1, 1, 1) . Other parameters were the same as shown in Figure 2.

30

31 I LT

Figure A1. A small decrease in

significantly reduces the number of stem cells and

progenitor cells. (a) Time courses of

LT

with

I LT

=1.0 ,

I LT

= 0.95 and

I LT

= 0.9 are shown in solid,

broken and dotted lines, respectively. (b) The number of HSC with and

I LT

I LT

=1.0 ,

I LT

= 0.95

= 0.9 are shown in solid, broken and dotted lines, respectively. (c) The number of

progenitor cells with

I LT

=1.0 ,

I LT

= 0.95 and

I LT

= 0.9 are shown in solid, broken and

dotted lines, respectively. Other parameters were set as in Figure 2.

Figure A2. A small increase in the initial cell division interval value decreases HSC and progenitor cell numbers. (a) Solid, broken and dotted lines illustrate time change of I when I I = 0.1, I I = 0.15 and

I I = 0.2 , respectively. (b) Solid, broken and dotted lines represent the numbers of HSC when I I = 0.1, I I = 0.15 and I I = 0.2 , respectively. (c) Solid, broken and dotted lines represent the numbers of progenitor cells when I I = 0.1, I I = 0.15 and I I = 0.2 , respectively. Other parameters are as in Figure 2.

Figure A3. Prolonged duration of highest self-renewal rate. (a) Solid, broken and dotted lines represent time changes of

LT

when T1 = 0.0 , T1 = 0.5

and T1 =1.0 , respectively. (b) Solid, broken and dotted lines represent the number of HSC when T1 = 0.0 , T1 = 0.5 and T1 =1.0 , respectively. (c) Solid, broken and dotted lines represent the number of progenitor cells when T1 = 0.0 , T1 = 0.5 and T1 =1.0 , respectively. Other parameters are as in Figure 2.

31

32

The English in this document has been checked by at least two professional editors, both native speakers of English. For a certificate, please see:

http://www.textcheck.com/certificate/Pu2gfT Highlights We study a mathematical model for competitive repopulation experiment of HSCs.

The test cell type and the composition of competitor cells determine the kinetics. One-step differentiation and multistep differentiation give very similar kinetics. Experimental results inevitably include stochasticity with the HSC in competitor. Detecting different types of HSC, an appropriate number of competitor cells are needed.

32

!"

!"#$

&'()*+,

%

#"

$"

"%$

!"

#"

%

$"

!"#$%&'(

Relative proliferation rate: φj

Number of stem cells: xj

&%

!"

#$!"

%!!"

&$!"

'!!"

!"

)-+

!"#

%$!"%

!&#

%$&%

&#

%$"%

"#

%$)*+

xST

xLT

("

xIT

'%

)%

&" '" Month (t) Months(t)

Month (t) Months(t)

(%

φ IT

)"

*%

φ ST

φ LT

#!"

"&%

!"

!"

$!!!") $('! #('! #!!!")

%('! %!!!")

&!!!") &('!

'#!!!") '#('! '!('! '!!!!")

'$!!!") '$('!

!"

).+

!"

!#$"

!#%"

!#&"

!#'"

("

(#$"

),+

'%!!!") '%('!

Mean period of cell cycle: I Number of progenitor cells

Pc ST Pˆ ( )

#"

$"

IT Pˆ ( )

LT Pˆ ( )

Month (t) Months(t)

&"

$" %" Month (t) Months(t)

%"

&"

'"

'!"

(!"

!"

#!"

$!"

%!"

&!"

'!!"

)-+

!"

#!"

$!"

%!"

&!"

'!!"

)*+

!"#$%&'(

ψj j Chimerism:'ψ 
!"#$%&#'$(

Chimerism:'ψ ψj j 
!"#$%&#'$(

!"

!"

#"

#"

%"

Months(t) Month (t)

$"

ψ ST

ψ IT

ψ LT

$" %" Month (t) Months(t)

&"

&"

'!"

'!"

ψj j Chimerism:'ψ 
!"#$%&#'$(

Chimerism:'ψ ψjj 
!"#$%&#'$(

!"

!"

#!"

$!"

%!"

&!"

'!!"

!"

).+

!"

#!"

$!"

%!"

&!"

'!!"

),+

ψ ST #"

#"

ψ LT

ψ IT

ψ ST

%"

Months(t) Month (t)

$"

ψ LT

$" %" Month (t) Months(t)

ψ IT

&"

&"

'!"

'!"

)*+

!"

#!"

$!"

%!"

&!"

'!!"

)-+

!"

#!!"

$!!"

%&!!"

%'!!"


!"#$%&#'$( ψ LTLT Chimerism.'ψ

Number of stem cells.'xj

!"#$%&'(

!"

(x

!"

LT c

#" '" Month (t) Months(t)

LT c

#"

(x

LT c

$"

$" %" Month (t) Months(t)

&"

, xcIT , xcST ) = ( 0, 1, 0 )

, xcIT , xcST ) = (1, 0, 0 )

(x

, xcIT , xcST ) = ( 0, 0, 1)

&"

x LT (t )

x IT (t )

xST (t )

'!"

%!"

Number of progenitor cells.'P" !"

#!!!!"

$!!!!"

%!!!!"

),+

!"

#"

ST Pˆ ( )

$" %" Month (t) Months(t)

IT Pˆ ( )

LT Pˆ ( )

&"

'!"

)*

+)*

,))*

2!4

)*

)*

+)*

,))*

234

6#78&%*+

Percentage of neutrophils derived from test cell

Percentage of neutrophils derived from test cell

)*

-*

-*

.*

/*

Months(t)

/*

Months(t)

.*

0*

0*

,)*

,)*

1"#$%&#'$(*ψ 
!"#$%&#'$( (ψ j Chimerism:*ψ 
!"#$%&#'$( ( ψj j

!"

#!"

$!!"

294

)*

+)*

,))*

254

!"

)*

ψ ST

%"

-*

ψ IT /*

'" Months(t)

&"

ψ ITI

ψ LT

Months(t)

.*

("

0*

$!"

,)*


!"#$%&#'$

)*

-)*

.)*

/)*

0)*

,))*

)*

-*

/*

Months(t)

.*

6#78&%*/

0*

,)*

Number of progenitor cells/,P

Relative proliferation rate/,φLT

&%

!"

) $!!!" $('! ) #('! #!!!"

) %!!!" %('!

) &!!!" &('!

) '#('! '#!!!" ) '!('! '!!!!"

) '$('! '$!!!"

) '%('! '%!!!"

!%#

!"#

%$!"%

!&#

%$&%

&#

%$"%

"#

%$!"#

&'()*+,-.

!"

#"

)%

$" %" Month (t) Months(t)

I = 0.9 φ LT

Months(t) Month (t)

(%

I φ LT = 0.8

'%

&"

'!"

"&%

I φ LT = 1.0

*%

Number of LT-HSCs/,xLT !"

#!!"

$!!"

%!!"

&!!"

'!!"

(!!"

!$#

!"

$"

("

Months(t) Month (t)

&"

I φ LT = 0.8

I φ LT = 0.9

I φ LT = 1.0

)"

#!"

Number of progenitor cells/,P

Mean period of cell cycle/,I

!"

) $!!!" $('! ) #('! #!!!"

) %!!!" %('!

) &!!!" &('!

'#('! '#!!!") '!('! '!!!!")

'$('! '$!!!")

!"

!"

'%('! '%!!!")

!%#

!"

!#$"

!#%"

!#&"

!#'"

("

(#$"

!"#

&'()*+,-.

#"

$"

$" %" Month (t) Months(t)

I = 0.15

I

I I = 0.1

%" &" Month (t) Months(t)

&"

I I = 0.2

'"

'!"

(!"

Number of LT-HSCs/,xLT !"

#!!"

$!!"

%!!"

&!!"

'!!"

(!!"

!"

!$#

$"

&" (" Month (t) Months(t)

I I = 0.15

I I = 0.1

)"

#!"

I I = 0.2

Number of progenitor cells/,P

!"#

%$!"%

!&#

%$&%

&#

%$"%

"#

%$!"

#!!!!" #($!)

$%!!!!" $%($!)

$&($!) $&!!!!"

!%#

Relative proliferation rate/,φLT

!"#

&'()*+,-.

!"

&%

%"

'" #" Month (t) Months(t)

T1 = 0.5

T1 = 1.0

(% )% Month (t) Months(t)

T1 = 0.0

'%

*%

&"

"&%

$!"

Number of LT-HSCs/,xLT

*!!!" )!!!" (!!!" '!!!" &!!!" %!!!" $!!!" #!!!" !" !"

!$#

$"

T1 = 0.0

&" (" Month (t) Months(t)

T1 = 0.5

T1 = 1.0

*"

#!"