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Lifespan Differences in Hematopoietic Stem Cells are

Due to Imperfect Repair and Unstable Mean-Reversion

Hans B Sieburg*, Giulio Cattarossi, Christa E. Muller-Sieburg

Stem Cell and Regenerative Medicine Program, The Sanford-Burnham Medical Research Institute, La Jolla, California, United States of America

Abstract

The life-long supply of blood cells depends on the long-term function of hematopoietic stem cells (HSCs). HSCs are

functionally defined by their multi-potency and self-renewal capacity. Because of their self-renewal capacity, HSCs were

thought to have indefinite lifespans. However, there is increasing evidence that genetically identical HSCs differ in lifespan

and that the lifespan of a HSC is predetermined and HSC-intrinsic. Lifespan is here defined as the time a HSC gives rise to all

mature blood cells. This raises the intriguing question: what controls the lifespan of HSCs within the same animal, exposed

to the same environment? We present here a new model based on reliability theory to account for the diversity of lifespans

of HSCs. Using clonal repopulation experiments and computational-mathematical modeling, we tested how small-scale,

molecular level, failures are dissipated at the HSC population level. We found that the best fit of the experimental data is

provided by a model, where the repopulation failure kinetics of each HSC are largely anti-persistent, or mean-reverting,

processes. Thus, failure rates repeatedly increase during population-wide division events and are counteracted and

decreased by repair processes. In the long-run, a crossover from anti-persistent to persistent behavior occurs. The cross-over

is due to a slow increase in the mean failure rate of self-renewal and leads to rapid clonal extinction. This suggests that the

repair capacity of HSCs is self-limiting. Furthermore, we show that the lifespan of each HSC depends on the amplitudes and

frequencies of fluctuations in the failure rate kinetics. Shorter and longer lived HSCs differ significantly in their pre-

programmed ability to dissipate perturbations. A likely interpretation of these findings is that the lifespan of HSCs is

determined by preprogrammed differences in repair capacity.

Citation: Sieburg HB, Cattarossi G, Muller-Sieburg CE (2013) Lifespan Differences in Hematopoietic Stem Cells are Due to Imperfect Repair and Unstable Mean-

Reversion. PLoS Comput Biol 9(4): e1003006. doi:10.1371/journal.pcbi.1003006

Editor: Mark S. Alber, University of Notre Dame, United States of America

Received September 21, 2012; Accepted February 8, 2013; Published April 18, 2013

Copyright: ? 2013 Sieburg et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This work was supported by the National Institutes of Health grants DDK48015 and AG023197 to CEMS. The funders had no role in study design, data

collection and analysis, decision to publish, or preparation of the manuscript.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: hsieburg@sanfordburnham.org

Introduction

Adult tissue stem cells, such as hematopoietic stem cells (HSCs),

are distinguished from mature cells by the ability to generate all

mature cell-types of a particular tissue (multi-potency). To

generate mature cells, HSCs differentiate into cells of lower

potency. The resulting loss of stem cells must be compensated for

by self-renewal, i.e. cell divisions which preserve the multi-

potential differentiation capacity of the ancestral HSC. The

reliability with which HSCs can transfer their identity and

maintain self-renewal upon proliferation has been of keen interest

to the field [1,2]. Important questions are: Are daughter HSCs ‘‘as

good as old’’ after self-renewal? How often can individual HSCs

self-renew? Do different HSCs have different self-renewal capac-

ities? What controls the fidelity of self-renewal? These questions

remain incompletely understood.

Because of their extensive self-renewal capacity, HSCs were

initially thought to be immortal. This view was supported by the

observation that populations of HSCs could be serially transplant-

ed for a very long period of time - exceeding the normal lifespan of

the donor [3,4]. However, when HSCs were examined on the

clonal level, extensive heterogeneity in lifespan was revealed [5–7].

A detailed analysis of a large panel of HSCs showed that the

lifespan of individual HSCs is mathematically predictable [8].

HSCs with lifespans from 10 to nearly 60 months were found

side-by-side in the same donor [8], indicating that the lifespan is

pre-determined on the level of each HSC. Because lifespans of

single transplanted HSCs are predictable from few initial values of

their repopulation kinetic, the lifespan is a programmed HSC-

specific property [8]. The population dynamics, therefore, predict

that the molecular machinery which preserves self-renewal, will

ultimately fail.

Several hypotheses have been developed to identify and explain

how HSCs limit their lifespan. The generation-age hypothesis [9]

states that for every cell division, an HSC loses some quality that is

referred to as ‘‘stemness’’.

According to Hayflick’s hypothesis [10], the probability that

somatic cells produce viable daughter cells which can themselves

divide, decreases as the number of divisions increases. The

decrease might be caused by progressive telomere shortening [10].

Hence, an extension of Hayflick’s hypothesis predicts that stem cell

self-renewal capacity should be self-limiting at the level of

individual HSCs.

Yet, HSCs and other stem cells, express telomerase [11–14].

This enzyme repairs telomere damage and, thus, aids in preserving

genomic integrity. Thus, telomere shortening alone is unlikely to

explain a limited lifespan of HSCs. Indeed, mice that have been

homozygously ablated for telomerase activity show only mild

effects and need to be severely stressed to reveal deficiencies in the

hematopoietic system [15]. Potentially in line with these findings in

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mice are clinical data. It was suggested that telomerase expression

declines in the long-run and may be a cause for late bone marrow

transplant failure [16]. Declining telomerase expression may act in

conjunction with the high stressor load imposed by the many co-

morbidities affecting transplant patients [17].

Another proposal suggested that, in conjunction with oxidative

stresses, high levels of reactive oxygen species (ROS) could be a

damaging force acting on the long-term repopulating capacity of

HSCs [18,19]. The corresponding restoring force is provided by

Forkhead box class O (FoxO) transcription factors. FoxO

transcription factors increase the expression of genes whose

products blunt the effects of elevated ROS [20–22]. That different

sources of self-renewal failures could be causally co-dependent is

suggested by findings that oxidative stress could shorten telomeres

[23].

Along-side genome stability, the preservation of epigenetic

patterning is an important prerequisite to reliably produce

functional daughter HSCs upon self-renewal. It has been

suggested that both maintenance and de novo methylation are

needed to maintain epigenetic stability [24]. The expression levels

of DNA methyltransferases [25,26] responsible for maintenance

(DNMT1) and de novo methylation (DNMT3a and DNMT3b)

could be important for restoring HSC multi-potency [27,28].

Quantitative work has suggested that small failures may accumu-

late over time in the DNMT1 pathways leading to the loss of

maintenance methylation and, ultimately, epigenetic stability [29].

Yet, neither of these mechanisms and hypotheses explain how

HSCs with different lifespans co-exist in a single host.

It was suggested that HSCs could preserve their functional

integrity over long periods of time by alternating between two

states, called resting or quiescent, and active, respectively [30–32].

This idea poses that intermittent transitions to quiescence could

provide the time needed to minimize the detrimental effects of

repeated DNA replication and other stresses on the HSC

population as a whole [33]. Elegant mathematical models of this

idea have been formulated [34–36]. Surprisingly, quiescence may

leave HSCs more vulnerable to mutations following DNA repair

[37]. Quiescent and active HSCs may use different DNA repair

mechanisms and the restoring pathway used by quiescent HSCs

may lead to higher differentiation probabilities following re-

activation. Never-the-less, when HSC quiescence was inhibited by

the expression of the Wnt inhibitor Dickkopf-1, the HSC pool

exhausted prematurely [38]. This suggests that periods of rest in

the niche are essential for controlling HSC lifespan - supporting

the idea that repair is necessary to maintain HSC lifespan.

Mathematically, the lifespan of populations has been addressed

in manufacturing, engineering, actuarial and biological applica-

tions of reliability theory [39]. Reliability theory was first

developed as a quality control tool to predict the time-to-failure

- the manufacturing term for lifespan - of manufactured goods to

determine warranty times. When examined as a population, the

lifespan of manufactured goods proceeds through well-defined

phases (Figure 1). First, a decline in population size is found, which

is interpreted as failure due to factory error. Second, there is a

period of little change, known as the useful phase of the population

of goods. Thereafter, the population size declines again, this time

caused by age-related failure of essential machine components

(wear-out phase). The second phase can be prolonged, if goods are

repaired. If repairs occur repeatedly, the useful life will be

extended, yet the population of goods will fail in the end, because

of a general deterioration of many essential parts. In biology,

failure theory has been applied to develop general laws of aging

and longevity [40–42], respector-ligand dissociation [43], or

genome instability [44]. Here, we show that the principles of

reliability and failure can be exploited to craft a new model of

HSC self-renewal suggesting that HSCs differ a priori in the

number of (self-)repair cycles they can undergo.

Results

Repopulation Time-to-Failure

We obtained repopulation data experimentally by transplanting

single HSCs into ablated mice as described previously [45–48].

The donor HSCs and the host type mice differed in the allelic

forms of the Cluster of Differentiation 45 (CD45) antigen. The

CD45 antigen is expressed in most hematopoietic cells. This

allowed us to follow the mature progeny derived from the

transplanted HSCs by staining white blood cells for the donor-type

marker. Mice were analyzed every other month and the percent of

white blood cells that stained for the donor-type marker were

recorded (% donor-type cells). Together, all the data points form

the repopulation kinetic as a time series (compare Algorithm 1,

Input specification). Because all cell populations were derived from

a single HSC, the repopulation kinetic of the clone represents the

total repopulation capacity of the original HSC.

We previously showed that any HSC will eventually fail to

repopulate all mature cell populations [8] (also compare summary

Figure 2). The time period until the multi-potential repopulation

capacity fails was called the HSC lifespan. Since the lifespan marks

the loss of multi-potency, it is the time-to-failure of a stem cell

clone as a system [39]. We showed previously that the time-to-

failure of individual HSCs is mathematically predictable with great

accuracy from few initial measurements of the repopulation kinetic

[8]. Hence, the time-to-failure (lifespan) is a deterministic property

of individual HSCs, but can be treated as a (Gumbel-distributed)

random variable at the level of the HSC compartment.

Statistical analysis of 38 repopulation kinetics ascertains that the

time-to-failure estimator is unbiased and almost efficient (compare

Text S1). This means that the repopulation kinetic of a clone

provides near optimal information about the time-to-failure. Of

note, the time-to-failure can be modeled as a power law of the

Author Summary

All hematopoietic stem cells (HSCs) are characterized by

the capacities to produce all blood cell-types by differen-

tiation and to maintain their own population through self-

renewal divisions. Every individual HSC, therefore, can

generate a complete blood system, or clone, conveying

oxygenation and immune protection for a limited time.

The time for which all mature blood cell-types can be

found in a clone is called the lifespan. Interestingly, HSCs

with different lifespans co-exist in the same host. We

address the unresolved question: what controls the

lifespan of HSCs of the same genotype exposed to the

same environment? Here, we use a new approach to multi-

scale modeling based on reliability theory and non-linear

dynamics to address this question. Large-scale fluctuations

in the experimental failure rate kinetics of HSC clones are

identified to predict small-scale, genome level, events of

deep penetrance, or magnitudes that approach popula-

tion size. We broadly find that one condition explains our

experimental data: repair mechanisms are a priori imper-

fect and do not improve, nor deteriorate, during the

lifespan. As a result, progressively ‘‘worse-than-old’’

genome replicates are generated in self-renewal. A likely

interpretation of our findings is that the lifespan of adult

HSCs is determined by epigenetically pre-programmed

differences in repair capacity.

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proliferative capacity of the clone (compare Text S1, eq 1). From

the traditional interpretation of a power law relationship (for

example, see [49]), we can expect that reliability and failure rate

analyses of the clonal time-to-failure are unaffected by the time

scale on which repopulation data were obtained. Therefore, the

systems reliability approach can be applied to repopulation

kinetics obtained using different time scales (for example, [50]).

Repopulation Reliability

The time-to-failure T, or lifespan, of an HSC, is a deterministic

quantity that measures the time until at least one mature cell

population is no longer repopulated. T, therefore, equals the time

to the first, and last, failure of multi-potency in all, not just single,

clonal HSCs.

In systems theory, reliability is defined as a conditional

probability ([39]; also compare Materials & Methods (M&M),

section ‘‘Reliability Theory’’). Specifically, a system is said to

operate reliably, if its strength is likely to exceed its load at a future

time, given that the system has operated within specifications up to

the present time.

The strength of a clonal system lies in the self-renewal and

multi-potential differentiation capacities of its HSC population.

Here, the term ‘‘capacity’’ can be given the rigorous quantitative

meaning of ‘‘obtainable work’’. Clonal experiments measure the

amount of ‘‘realized work’’, i.e. how much of the strength has

actually been transformed into new HSCs (by self-renewal) and

mature cells (by differentiation) over time, given load.

Therefore, we could use the repopulation kinetics to identify the

rate at which ‘‘work’’ is performed to quantitate clonal reliability.

Specifically, we defined the clonal repopulation reliability as the

normalized area under the curve of the repopulation rate kinetic

(compare Algorithm 1, Line 8; also see Figure 3, A and B). Of

note, the clonal reliability can be estimated even if only few initial

repopulation data are known, since an HSC’s lifespan is

Figure 1. Failure Rate Kinetics of Machine and Clonal Blood

Cell Populations. Schematic representations of the failure rate

kinetics (vertical axis) of two systems over time (horizontal axis):

A. Population of machines; B. Population of cells derived from a single

long-term repopulating hematopoietic stem cell (HSC). In reliability

theory, it is thought that three major phases describe the shape of the

failure rate function (the black curve in Parts A and B was generated for

demonstration purposes using appropriate mathematical functions).

A: For populations of machines, the ‘‘bathtub’’ shape is thought to

begin with a ‘‘wear-in’’ phase (yellow). During ‘‘wear-in’’, factory

defective items are flushed out. The population, there-after, reaches

the so-called ‘‘useful life’’ period (blue), where failure rates are

minimized. The ‘‘bathtub’’ is completed by the third phase of ‘‘wear-

out’’ (red), where many essential parts fail in an increasingly larger

number of machines. B: The biology of clonal stem cell populations may

lead to a different assembly of phases, generating a different shape of

the failure rate curve. Unlike for machines, the clonal population creates

itself during expansion. Consequently, a direct analog of ‘‘wear-in’’ may

not exist, or may be short, and not characterized by failure rate

decrease. The ‘‘useful life’’ period may, therefore, extend to the start of

the failure rate curve. ‘‘Wear-out’’ may occur for reasons similar to

machine populations, i.e. through the accumulation of failures in an

increasingly larger number of HSCs. The present paper uses an

interdisciplinary approach combining the analysis of experimental data,

mathematical reasoning and computer simulation to determine the

actual shape of clonal failure rates and make predictions about the

dynamical mechanisms responsible for failure accumulation and clonal

extinction. The goal of this approach is to find new experimentally

testable hypotheses about how stem cells autonomously control their

growth using ‘‘built-in’’ failure as a passive mechanism against

cancerous proliferation.

doi:10.1371/journal.pcbi.1003006.g001

Algorithm 1. Algorithm for calculating the discrete time

series of reliabilities failure probabilities failure densities

and failure rates for batches of raw kinetics. Numbers to

the left of the listing indicate line numbers. These are used

to reference particular computations in the main narrative.

The general notation X½k : w? denotes the elements X½k?

through X½w? of a list X. The symbol D denotes the

successive differences operator defined for a list of length l

by Dfu1,...,ulg:~funz1{un: 1ƒnƒlg. The function

Sum(X) sums the elements of a list X.

input

1ƒnƒN(i)g of lengths N(i)wNmin, 1ƒiƒM. t(i):~

ft(i)

scale, and by R(i):~fR(i)

measured experimentally. Nminw1 is a minimum lower

bound on time series size needed to conduct meaningful

analyses.

output: A list of lists W(i)of time series: Time scale

(symbolically: t(i)), Process Rate (r(i)), Reliability (S(i)), Failure

Probability (F(i)), Failure Density (f(i)), Failure Rate (Z(i)), for

1ƒiƒM.

: A batch of

M

kinetics

R(i):~f(t(i)

n,R(i)

n) :

n: 1ƒnƒN(i)g with t(i)

nvt(i)

n: 1ƒnƒN(i)g the i-th kinetic

nz1we denote the i-th time

1 for i/1 to M do

2process the i-th repopulation kinetic R(i);

3

r(i),S(i),F(i),f(i),Z(i)/ 60;

4

t(i)/R(i)½1?;

5

R(i)/R(i)½2?;

6

r(i)/R(i)=t(i);

//determine instantaneous rates;

7

r(i)

8

S(i)/r(i)

9

F(i)/1{S(i);

// failure probability;

10

f(i)/DF(i)=Dt(i);

// failure density

11

Z(i)/f(i)=S(i);

// failure rate;

12

W(i)/(t(i),r(i),S(i),F(i)),f(i),Z(i));

// clear all lists;

// extract time scale;

// extract process measurements;

k,N/ Sum (r(i)½k : N(i)?);

nz1,N(i)=r(i)

1,N(i);

// reliability;

// reliability profile

of i-th kinetic;

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predictable from the first few measurements of its repopulation

kinetic [8].

Using the repopulation data of 38 HSC clones with lifespans

ranging from w10 to nearly 60 months (compare Figure S1 in

Text S1), we first determined the repopulation rate kinetics for the

whole clone (parameters shown in Table S1 of Text S1) and also

for all major subpopulations of T cells, B cells and myeloid cells

(representative example shown in Figure 3, A; calculation:

Algorithm 1, Line 6). Next, we used the clone data to calculate

the respective reliabilities (Figure 3, B; calculation: Algorithm 1,

Line 8). An additional curve can be inferred, if the clonal structure

is considered in a so-called ‘‘common cause’’ model of clonal

reliability. A common cause model poses that, in a multi-

component system, the unknown reliability of a central component

can be approximately determined by the relationship of all

remaining system components to the system’s reliability ([39],

pp. 217–222).

Since all mature lineages derive from HSCs, a mature cell

population and the HSC population can be viewed as serially

connected system components. On the other hand, the popula-

tions of mature cells appear connected in parallel, since the failure

of one such population does not imply the failure of all. The serial

connectivity is a mathematical way of representing multi-potency.

Hence, an inferred reliability structure model should generate the

reliability kinetic of the clonal HSC population.

Considering the HSC population a ‘‘common cause’’ with

reliability denoted SH, we posed that the clonal reliability (denoted

SC) is connected to the reliabilities of the T cell, B cell and myeloid

Figure 2. The Life of A Hematopoietic Stem Cell. A: Limited lifespan: When a monoclonal hematopoietic system is initiated by transplanting a

single HSC (dark blue sphere), it expands to a pool of clonal HSCs through self-renewal (cluster of blue spheres). This pool distributes through the

organism. HSCs differentiate to generate mature cells of all lineages (shown as magenta, orange, green, light-blue spheres). This process depends on

the intrinsic properties of the founder HSC [63,65]. The overall output of mature cells in blood (measured in %-donor type cells; vertical axis (not

shown in the figure)) over time (horizontal axis labelled ‘‘Lifespan’’) is indicated by the black curve. For all normal HSCs, this kinetic has a ballistic

shape, thus indicating that a clone’s ability to produce mature cells of all major lineages (the lifespan) is limited. The lifespan is mathematically

predictable with high accuracy from few initial points of the repopulation kinetic [8]. B: Programmed Lifespan: When daughter HSCs derived from a

single ancestral HSC are transplanted into separate hosts, the repopulation kinetics are very similar (modified from [2]). In particular, all daughter

HSCs become extinct at the same time [8]. This suggested that the lifespan is epigenetically fixed (programmed) and heritable in self-renewal. C:

Lifespan Diversity: The relialogram illustrates that when HSCs are sampled from bone marrow, lifespans of different durations are found [2,66,67].

Therefore, the length of time for which HSCs can repopulate an ablated host varies according to the epigenetic programs of individual HSCs.

doi:10.1371/journal.pcbi.1003006.g002

Figure 3. Reliability and Failure Kinetics of a Long-lived Long-

term Repopulating HSC. A–D: Four types of kinetics were calculated

(compare Algorithm 1) from experimental kinetics for all clonal cell

populations together (black), and the myeloid (green), T lymphocyte

(red), and B lymphocyte (gray) cell populations, separately. In the

representative example shown, notation is as in Algorithm 1 (applied to

a single kinetic, i.e. batch size M~1). Also shown are the respective

kinetics for the population of clonal hematopoietic stem cells (HSCs;

blue). Since population data are difficult to obtain for stem cells directly,

the HSC-related kinetics were inferred from the other data. This was

accomplished by first predicting the reliability (Part B, blue curve) using

the structure balance eq 1 and, then, deriving the other kinetics (blue

curves for C, D, then A) with the methods of Algorithm 1.

doi:10.1371/journal.pcbi.1003006.g003

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cell populations (denoted ST, SB, SM, respectively) by:

1{(1{SH)(1{SC):t1{(1{ST)(1{SB)(1{SM)

ð1Þ

Eq 1 is a balance formula expressing the conservation of system

structure over the life of a clone. It states that two parallel structure

models of a clone behave similarly for all times t (indicated by the

notation :t). The first model (left-hand side, eq 1) considers the

HSC populationand the clone as a parallel reliability structure. The

second model (right-hand side, eq 1) quantitates the reliability of the

major mature cell-types as a parallel structure. We applied eq 1 to

calculate the values SH(tn) at discrete time points tngiven by our

data. The inferred kinetics are shown as blue curves in Figure 3.

The common cause argument suggests that the HSC population

reliability closely resembles the reliability of the clone, modulo a

time lag that is small compared to the lifespan (Figure 3, B). This

outcome is consistent with our previous conclusions, obtained by

different methods that the information contained in the repopu-

lation time series predicts HSC behavior [8].

Repopulation Failure Probability and Failure Density

By definition, the reliability is a forward looking measure that

predicts the chances that a system will continue to operate

according to its specifications for some time into the future,

provided that it has operated reliably in the past. For HSCs,

reliable operation means that their main characteristics, i.e. self-

renewal and multi-potential differentiation capacities, are pre-

served when these cells divide. What are the chances of unreliable

operation?

In systems theory, unreliability is defined by the conjugate

probability of the reliability (for computation compare Algorithm 1,

Line 9). Specifically, the probability that the system fails, or failure

probability, equals 1 minus the probability that it can continue to

operate as before. Hence, the repopulation failure probability, or

repopulation ‘‘unreliability’’, is a cumulative probability function

[39] (high/low values indicate high/low likelihood of failure). Its

graphical representation is an S-shaped curve whose shape is a

horizontally flipped image of the corresponding reliability kinetic

(Figure 3).

The failure probability gives rise to the failure density. The

latter is a probability density in the usual sense and defined by the

rate with which the failure probability changes over time (small

values mean little change, high values mean lots of change). We

determined the failure probabilities and densities for all clonal

populations (Figure 3, C and D; Algorithm 1, Line 10). Because we

could predict the reliability kinetics of the HSC population

(previous section; also Figure 3, B), we could predict an

approximate shape for the repopulation failure density of the

HSC population, as well.

Application of the method of symbolic time series comparison in

[51] shows that the relationship between the failure densities of the

subpopulations of a clone changes over time (Figure 3 C).

Specifically, the repopulation failure densities of the HSC

population and some mature cell populations becomes more

similar (increasingly converge to the same symbol sequence (data

not shown)), as the lifespan is approached. By contrast, the

densities lack similarity at the beginning of clonal life. To clarify

this observation, we conducted a more detailed analysis of the

long-range failure dynamics of HSC clones with different lifespans.

Repopulation Failure Rate Kinetics

In systems theory, the failure rate provides information about

how system failure occurs as a function of system load acting

against system strength over time [39]. Since the future behavior

of HSCs is largely pre-programmed [8,46], the failure rate kinetic

informs about how strength, or capacity, is transformed into new

HSCs (by self-renewal) and mature cells (by differentiation) over

time.

The failure rate is defined as the ratio of failure density divided

by the probability of reliable operation (which equals the negative

rate of change over time of the logarithm of the reliability [39];

also compare M&M, ‘‘Reliability Theory’’). We applied this

definition to the reliability kinetics of the clone, each mature

subpopulation and the predicted reliability of the HSC pool to

obtain the respective failure rate kinetics (Figure 3, D and

Algorithm 1, Line 11). The main observation is that, for all HSC

clones, the failure rate kinetics of all populations, and the clone as

a whole, increase sharply towards the end of clonal life. The

overall behavior is valid for HSCs of all lifespans (Figure 4, A, C,

E; also compare Theorems 1 and 2, and Lemma 1). The onset of

the increase, which we called extinction transition, coincides with

the onset of the increase in the predicted failure rate kinetic of the

HSC population. However, the HSC failure rate increases more

rapidly than those of the mature cell populations (Figure 3, D).

This suggests that clonal extinction is due to an event that affects

the HSC population as a whole and, likely, synchronously.

Repopulation Failure Rates are Mean-Reverting

To better understand the failure behavior of the clonal HSC

population before the extinction transition, we looked at truncated

failure rate kinetics. We defined a failure rate kinetic as (j,k)-

truncated, if j§0 elements are removed from the beginning and

k§0 from the end of the time series, respectively. Hence, the full

Figure 4. Failure Rate Kinetics of Long-term Repopulating

HSCs. To facilitate visualization, failure rate kinetics (vertical axes: each

colored line-scatter curve represents the failure rates of the total output

of an individual HSC) were displayed in three non-classifying groups

(rows A–B, C–D, E–F) and at two levels of resolution (full kinetics in

column A, C, E; (0,2)-truncated kinetics in column B, D, F) over time

(horizontal axes). The full kinetics show that the failure rates increase

strongly as the lifespan is approached. We called this behavior the

‘‘extinction transition’’. The (0,2)-truncated kinetics illustrate the

variability, and a tendency to slowly increase, of the failure rates prior

to reaching the extinction transition.

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time series is (0,0)-truncated, while (0,k)-truncation means

removing the last kw0 elements of the time series. For all

repopulation kinetics in our database (compare Figure 4 for

representative examples of different lifespans; also see Figure S1 of

Text S1), we analyzed the behavior of the (0,2)-truncated failure

rates (Figure 4, B, D, F).

We had previously shown that past values in HSC repopu-

lation kinetics predict future values [8]. In other words, memory

of past behaviors influences the long-term behavior of repop-

ulation kinetics. We now asked, if memory effects could be

shown in the failure rate kinetics. To find out, we calculated the

Hurst exponent [49,52,53] of each (0,2)-truncated failure rate

kinetic (Figure 5; compare section ‘‘Computation of Hurst

Exponent’’ in M&M; for values compare Table S1 of Text S1).

The Hurst exponent method is a statistical approach for finding

long-term memory patterns in a time series. Evidence of

memory is defined by the inequality H=1=2, where H denotes

the Hurst exponent. If 0ƒHv0:5, the behavior of a time-series

is characterized by a pattern of reversions to a mean, where

decreases/increases are followed by increases/decreases. Such a

pattern is called ‘‘anti-persistent’’ behavior, which is considered

stronger the closer H is to 0. When 0:5vHƒ1, past time series

values influence future values either only in the upward, or

downward, direction, i.e. increases/decreases follow increases/

decreases. This pattern is aptly called ‘‘persistent behavior’’.

Persistence is considered stronger the closer H is to 1. Values

close to H~1=2 are interpreted as evidence that no relationship

exists between past and future values of a time series. Since we

had previously shown that past values of HSC repopulation

kinetics predict future values [8], we could hypothesize that we

might find values of H significantly different from 1=2 in failure

rate kinetics.

To properly conducted Hurst exponent analysis, for example,

using a standard approach such as the rescaled range or R/S

method, knowledge of the average behavior of all sufficiently large

segments of the data is required. Because clonal repopulation

kinetics have a deterministic core behavior of ballistic shape [8],

we could determine averages based on a deterministic failure rate

kinetic for each clone (compare Theorem 1). Plotting the

experimental failure rate data together with the deterministic

failure rate kinetic showed that the former alternate around the

latter (data not shown). Using the additional information available

in the framework of HSCs, we could overcome the problem of

small time series, similar to approaches in financial market

analyses, where standard methods require modifications [54,55].

Specifically, we used the ballistic trend of repopulation data as a

domain-specific mean in the Hurst approach, instead of the

uniform mean usually applied. For all clones, the Hurst exponents

had median value~0:346 (compare Figure 5; Wilcoxon test,

significantdifferenceofthe

p~8:9|10{6). Therefore, the values of H indicated that the

failure rate kinetics of HSCs show anti-persistent memory

behavior.

Anti-persistence describes a long-range memory behavior of a

time series [52,53], where increases in value are followed by

decreases and decreases by increases, as opposed to increases/

decreases following increases/decreases, as would be the case for

persistent behavior. Using the traditional interpretation, the anti-

persistent behavior in the failure rate kinetic of an HSC indicates

that the ‘‘noise’’ of the failure rate data follows a long-term

pattern that is informative about the biology of HSCs. This

pattern suggested the presence of a mean-reverting process in the

context of failure kinetics. Therefore, we next considered the

possibility that mean-reverting behavior of the failure rate could,

median~0:346

to

H~1=2,

biologically, indicate the effects of repair mechanisms acting to

decrease the failure rate following increases. In this model,

fluctuations in the failure rates, as obtained from measurements

of clonal repopulation kinetics, are viewed as indicators for the

successive interaction of failure generation and repair, a marked

anti-persistent behavior. What is missing, is a quantitative

rationale for repair - acting at the cell level, but derived from

cell population data.

Iterative Model of Mean-reverting HSC Failure Rates

Because we found evidence of mean-reversion, we asked how

the (0,2)-truncated failure rates would fit to realizations of the

proto-type mean-reverting process, the Ornstein-Uhlenbeck pro-

cess [56,57]. The benefits of linking the HSC failure rates to this

process are: (a) A quantitative rationale for repair, in the form of a

failure dissipation rate; (b) An iterative model for simulating (0,2)-

truncated failure rate kinetics based on clonal repopulation data.

We showed numerically that the weighted sum of the variance-

adjusted rate of change plus the standardized rate of each (0,2)-

truncated failure rate kinetic could be regressed to the rate of

change of noise in the data. Mathematically:

Figure 5. Hurst Exponents of the Failure Rate Kinetics of Long-

term Repopulating HSCs. Plotted are the Hurst exponents (plot

symbol: blue triangles; values vertical axis) of the failure rate kinetics of

HSCs with lifespans w10 and v60 months (horizontal axis). Calcula-

tions were performed using Algorithm 0 (compare Table S1 in Text S1).

All exponents are v0:5, thus falling into the region of anti-persistent

behavior (defined by Hurst values 0ƒHv0:5 (light-yellow region)) and

not into the region of persistent behavior (defined by 0:5vHƒ1 (light-

pink region (only displayed up to 0.7 to enhance visibility of the data))).

Our previous results [8] that past values of an HSC’s repopulation kinetic

predict future values, had suggested the hypothesis that Hurst

exponents of the failure rates would either be greater, or less, than

H~0:5. The value H~0:5 is traditionally interpreted as ‘‘no memory’’

of past behavior in future behavior (horizontal line marked ‘‘no

memory’’). The data shown then suggest that, mechanistically, anti-

persistence plays a role in controlling clonal growth. The H values

obtained from our experimental data were fitted to the line

L(T)~0:411{0:0023T as a function of lifespan T (gray solid line

through the data). Goodness-of-Fit was determined using the Akaike

Information Criterion (AIC~{71:98). The parameter estimates were

highly significant (intercept estimate&0:411, standard error&0:017, p-

value~2:25|10{14;slopeestimate&{0:00228,

error&0:00057, p-value~9:5|10{4). The extension of the fitted line

to include lifespans ƒ10 only serves visualization purposes, since we

only considered HSCs with lifespans Tw10 months. The negative slope

of the linear fit predicts that anti-persistent behavior in the failure rate

kinetics is more pronounced for longer-lived long-term repopulating

HSCs than for shorter-lived long-term repopulating HSCs.

doi:10.1371/journal.pcbi.1003006.g005

standard

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Page 7

Ut:~1

s

DZt

Dtzh

Zt{m

s

??

*DWt

ð2Þ

Both sides of the similarity eq 2 can be determined from the data.

DWt, on the right-hand side, represents the discrete rate of change

of noise isolated from the data (for normally distributed noise, W is

called a ‘‘Wiener process’’). m and s denote the mean and standard

deviation of the (0,2)-truncated failure rate kinetic, respectively. h

denotes a positive weight parameter, called the ‘‘dissipation rate’’.

Because the dissipation rate occurs in the context of failure

kinetics, we interpreted h as a quantitative indicator of repair

activity - implying that repair could be modeled mechanistically as

a dissipation of failure. Eq 2 is equivalent to:

DZt

Dt*{h Zt{m

ðÞzsDWt

ð3Þ

Eq 3 is a discrete form of the Ornstein-Uhlenbeck stochastic

differential equation [56,57]. Its solution, called Ornstein-Uhlen-

beck process, is well-known as the prototypical mean-reverting

process.

Our numerical analyses showed that Ut=s

fit of the form y(t) :~(1{e{ht)=2h. The values of the weight h,

calculated from experimental data, were specific for individual

HSCs. Furthermore, we found that in distribution, Ut=s

similar to y(t)N(0,1), where the second factor denotes the

standardized normal distribution with mean 0 and standard

deviation 1.

In Theorem 2, we proved that the deterministic repopulation

kinetic [8] gives rise to a deterministic differential equation for the

failure rate. This equation is formally similar to eq 3, without the

noise term. We considered eq 3, therefore, as the approximate

Ornstein-Uhlenbeck representation for the (0,2)-truncated exper-

imental failure rate data.

Together, these findings independently support the prediction,

established earlier by Hurst analysis that the truncated failure rate

kinetic of an HSC is mean-reverting. We could conclude that the

(0,2)-truncated failure rate kinetics can be simulated by an

Ornstein-Uhlenbeck model [58,59] using the iterative scheme:

ðÞ2had a non-linear

ðÞ2is

Ztz1~Zte{hDtzm(1{e{hDt)zs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2h

(1{e{2hDt)

r

N(0,1) ð4Þ

We now asked, if eq 4 could be a model for the full, not only the

(0,2)-truncated failure rate kinetics. To answer this question in

light of the extinction transition seen in the data (sharp increase of

failure rates near the end of clonal life), we needed to explain how

the mean-reversion property may break down. This required a

closer look at the behavior of the parameters m, s and, in

particular, h, the quantitative indicator of population level repair

activity through dissipation of failure.

Failure Dissipation and Imperfect HSC Repair

We first asked what the properties of h are. The Ornstein-

Uhlenbeck formulation (eq 3) allowed us to assess HSC repair

efficiency from our clonal repopulation data. We broadly defined

‘‘repair’’ as the total of repair mechanisms available to HSCs. The

dissipation rate h quantitates the strength of the restoring forces as

the rate with which the system variable, in our case the failure rate,

reverts toward an average behavior m. We, therefore, could

consider the dissipation of failures as evidence of repair activity.

h quantifies how rapidly the clonal system reverts back in the

direction of m-equilibrium. Analysis of the dissipation rate for our

experimental data showed that h depends on the lifespan T.

Specifically, determining each hTby fitting to a noisy process (eq

2) and plotting the results as points (T,hT), suggested that the

dissipation rate has a non-linearly increasing tendency when

increasing, but fixed, lifespans are considered (compare Figure 6,

A; green curve). Explicitly, we found:

hT~0:063T0:67

ð5Þ

(compare Figure 6, A; green curve; goodness-of-fit: Akaike

Information Criterion

AIC~{73:17;

2:26|10{8and 8:1|10{11, respectively). It is important to

understand that eq 5 represents a tendency, not a dependency,

among the dissipation rates of repopulation kinetics for indepen-

dent time-to-failure values T. However, going back to a remark at

the beginning, the time-to-failure is a function of load which, as

noted, comprises two components relating to peripheral demand,

and demands due to disease or injury, respectively. Biologically,

load may affect, in parallel, all accessible HSCs in the HSC

compartment of a single host. Therefore, if T is parametrized by

load, eq 5, and the eq 6 below, may be interpreted as a power law,

dependent on load exposure. This suggested that h may be

dependent on ‘‘running’’ time, as well.

To find out, we used an analytical approach that took

advantage of the deterministic behavior of HSC repopulation

kinetics. In Theorem 2, we showed that an analytical definition of

h can be found as a function of T and t, denoted hT(t), based on

the ballistic model of repopulation kinetics developed by us

previously. To see if behavior similar to eq 5 can be found

analytically, we averaged hT(t), defined by hT:~ÐT{2

where integration extends over bounds that are analogous to

(0,2)-truncation of failure rates (t0is explicitly given in Theorem

2). We found that the dissipation rates derived analytically or

from data have similar properties (compare Figure 6, A). As

before (compare eq 5), we fitted the resulting set of hTvalues to a

non-linear model:

parameterp-values

t0

hT(u)du,

hT~0:113T0:53

ð6Þ

(compare Figure 6, A; blue curve; goodness-of-fit: Akaike

InformationCriterion

AIC~{93:87;

6:54|10{11and 5|10{18, respectively). Both approaches show

the same tendency of the dissipation rate to increase as a point-

wise function of T. Hierarchical cluster analysis showed that the

(lifespan, dissipation rate) data (T,hT) separate into three clusters.

Thedissipationratesof

(T(1),h(1))~(16, 0:47), (T(2),h(2))~(30, 0:72) and (T(3),h(3))~

(47, 0:87) are significantly different (Wilcoxon Test, Bonferroni

correctedp-values:

p1,3~6:11|10{6,

p1,2~3:75|10{5).

Together, our findings suggested that HSCs of different

lifespans may differ in their ability to utilize repair mechanisms.

We used the formulae developed in the proof of Theorem 1 to

compare the dynamics of failure generation and failure dissipation

(repair). As stated above, we found that, for the deterministic

repopulation kinetics, the dissipation rate is a function of time t

and the lifespan T, i.e. h~hT(t), for 0ƒtƒT. hT(t) equals the

negative logarithmic derivative of the failure density function. As

shown in the proof of Theorem 2, Z(t)whT(t) for all twt0, with t0

that can be calculated from data. This means that, throughout

parameterp-values

thethree clustercentroids

p2,3~2:74|10{4,

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Page 8

most of the lifespan period, failures are generated at a higher rate

than they are dissipated. Therefore, we could use the inequality

Z(t)whT(t) to quantitatively define ‘‘imperfect repair’’ and to

ascertain that repair activity is largely insufficient to compensate

for the rates at which failures are generated. The analytical

treatment supported the notion that failures should accumulate in

the long-run.

The mathematical analysis also predicted that, during the initial

expansion period of a clone, quantitated by 0ƒtƒt0, repair

capacity exceeds the rate at which failures are generated.

Quantitatively, Z(t)ƒhT(t) for 0ƒtƒt0. In particular, our theory

predicts that each HSC starts ‘‘its’’ clone with a non-zero ‘‘initial

damage load’’ Z(0) [40–42]. The ‘‘initial damage load’’ can be

calculated for each repopulation kinetic. Its precise biological

meaning and, particularly, its developmental origins, will need to

be determined experimentally. The switch from higher repair

capacity to Z(t)whT(t) may be attributable to the diluting effects

of clone size on the programmed, epigenetically heritable, repair

capacity of the founder HSC.

Why Some HSCs Live Longer Than Others

Evidence favoring the hypothesis that shorter lived HSCs may

be less efficient in dissipating the effects of failures than longer

lived HSCs, can be biologically interpreted by placing the

dissipation rate in a time-context. The time-context is given by

the half-life of the dissipation rate. The half-life is defined by:

1=2~log(2)

h

ð7Þ

The half-life data (compare Figure 6, B) show a tendency, where

failure rate increases dissipate more rapidly in long-term

repopulating HSCs with higher lifespans T. This holds indepen-

dently of the manner in which h values were obtained, either

analytically or from data. Mathematically:

1=2(hT)~11:57T{0:729

ð8Þ

(goodness-of-fit: Akaike Information Criterion AIC~{93:86;

parameter p-values 5:69|10{8and 1:67|10{15, respectively).

Hierarchical cluster analysis showed that the (lifespan, half-life) data

(T,

1=2(hT)) separate into three clusters. The half-lives of the three

cluster centroids (T(1),

(25,1:1), (T(3),

(Wilcoxon Test; Bonferroni corrected p-values: p1,3~3:1|10{4,

p2,3~1:83|10{6, p1,2~3:51|10{5). Like eq’s 5 and 6, the

relationship eq 8 must be interpreted with care as T is a time-to-

failure variable and should not be confused with a continuum.

The data show larger failure dissipation rate half-lives for HSCs

with shorter lifespans, while smaller half-lives associate with longer

life. A possible interpretation is that repairs may occur less frequently

in shorter lived HSCs than in longer lived HSCs. According to

Theorem 1 shorter lived HSCs may have to counteract higher initial

damage loads (as suggested by the higher initial values of Z(0) for

smaller lifespan values). Together, these findings may explain the

largerandlonger-lasting‘‘peaks’’and‘‘valleys’’seeninthefailurerate

kinetics of shorter lived HSCs (Figure 4).

1=2(T(1)))~(11,2), (T(2),

1=2(T(3)))~(47,0:7) are significantly different

1=2(T(2)))~

Break-down of Mean-Reversion and Clonal Extinction

An important observation common to all failure rate kinetics is

that, near the end of clonal life, the failure rates strongly increase.

Figure 6. Failures are Dissipated More Slowly in Shorter-lived

HSCs than in Longer-lived HSCs. A: We determined the dissipation

rates hT(yellow dots; vertical axis) relative to the lifespan T (horizontal

axis) using hT:~ÐT{2

the general tendency in the data, not implying any dependencies of

consecutive data points, we fitted the data to a non-linear model

hT~0:113T0:53(blue line; goodness-of-fit Akaike Information Criterion:

AIC~{93:87); parameter p-values 6:54|10{11and 5|10{18).

Calculation of hT by regressing to normal noise produced a slightly

lower exponent T0:67(fitted curve indicated by green line). B: Half-lives

of dissipation rates (yellow dots; vertical axis) relative to individual

lifespans (horizontal axis). To highlight the general tendency in the data,

we fitted the data to a non-linear model

line; goodness-of-fit Akaike Information Criterion: AIC~{93:86;

parameter p-values 5:69|10{8and 1:67|10{15, respectively). The

model of half-lives obtained from experimental data (green line) is

shown for comparison. In both graphics A and B, we used contour plots

of the respective data sets (T,ht) and (T,1=2(hT)), respectively, as

background.

doi:10.1371/journal.pcbi.1003006.g006

t0

hT(t)dt for (0,2)-truncated failure rates. The

lower bound t0of the integrand was derived in Theorem 2. To highlight

1=2(T)~11:57T{0:729(blue

Lifespan Differences in Hematopoietic Stem Cells

PLOS Computational Biology | www.ploscompbiol.org8April 2013 | Volume 9 | Issue 4 | e1003006

Page 9

When we compared actual failure rate kinetics with kinetics

generated by simulation using the experimentally derived param-

eters in eq 4, we noticed that the terminal behavior of the

experimental failure rate departed significantly from the mean-

reverting characteristic of the simulated rates (for more detailed

discussion see below). This suggested that regime-breaking may

characterize the extinction transition. ‘‘Regimes’’ is standard

terminology to describe disjoint regions in the phase space of a

dynamical system such that transitions between the regions are

rare (compare Figure 7 for a representative example). We, thus,

analyzed the regimes of the phase space of experimental HSC

failure rate kinetics.

The phase space trajectory of an HSC’s failure rate kinetic starts

in the initial engraftment regime (Figure 7, region E). Here, it

remains for up to 3 months of the HSC’s life. Then, the trajectory

escapes to the region where failure rates are governed by mean-

reversion (Figure 7, region OU). It remains in this regime for most

of the clonal life - first contracting, then slowly expanding. At some

point, the phase space trajectory transitions to a third regime

(Figure 7, region T), where the clone becomes extinct. Therefore,

regime-breaking is associated with clonal extinction.

We next established the conditions under which the mean-

reverting regime breaks. As discussed above, comparison of the

experimental failure rate kinetics with realizations of the iterated

Ornstein-Uhlenbeck process (eq 4) showed that the simulated

process diverges from the experimental data in the terminal

regime (Figure 8, Part A). The simulated failure rates continue as

before, but the experimental failure rates rise sharply. This

behavior could be changed, and brought closer to the experimen-

tal data, when a further constraint was added.

Specifically, we asked whether the mean of the experimental

failure rates is a constant or changes in time. Analysis of the (2,0)-

truncated failure rate kinetics of each clone using moving averages

showed that the parameter m, the mean failure rate, increases in a

well-defined pattern that occurs in all kinetics (compare an

example in Figure 8, B). Though this pattern of increasing mean is

present in the raw failure rate data, the change there is subtle, but

cumulative. It is enhanced and, thus, becomes more recognizable

if moving averages are used (for short lifespans only small windows

are needed; larger windows (up to 5 months) are required for

longer lifespans). Using non-linear fitting, we found that the

succession of moving averages, labelled ? m mt, increases over time as:

? m mt*DT?{tD{1

ð9Þ

The critical limit T?is slightly smaller than the time-to-failure, or

lifespan, T of the clone. This result obtained from data analysis has

an analytical counterpart. In Theorem 1, we showed that the

ballistic model of HSC repopulation kinetics [8] leads to a

deterministic formula for the failure rate function of any HSC

clone, such that, according to Lemma 1, Z(t)*DT{tD{1in the

Figure 7. Failure Rate Phase Space Regimes of Long-term

Repopulating HSCs. Phase space plot of the failure rate kinetic of a

long-term repopulating HSC with long lifespan of T~57 months.

Points (Zt,Ztz1) in the plot (red) represent successive failure rates

calculated every 2 months. To facilitate visualization, regions were

separated by dotted lines. After initial expansion (region E, circled

point), the kinetic transitions (blue arrow) into a regime (region OU),

where it remains for most of clonal life. The behavior in region OU is

governed by an Ornstein-Uhlenbeck iterative process (compare eq 4).

The end of clonal life is indicated by the transition (black arrow) from

region OU to the ‘‘terminal’’ absorbing point in region T (circled point).

Region B is not visited by the dynamic trajectory and, therefore, empty.

doi:10.1371/journal.pcbi.1003006.g007

Figure 8. Breakdown of Mean-Reverting Behavior in the Failure

Rate Kinetics of Long-term Repopulating HSCs. A: An experi-

mental failure rate kinetic (blue scatter-line plot; values Z(t) vertical

axis) compared to the kinetics of 100 realizations (thin red lines) of an

Ornstein-Uhlenbeck process over the lifespan period (horizontal axis) of

a clone with lifespan T~39 months. The realizations of the process

were obtained using the iteration schema in eq 4. The same values of m,

s and h as in the experimental data were used. For simplicity, the initial

condition was set at Z(0)~0 for t~0 (equivalent to assuming a load-

free transplant). The important observation is that without additional

conditions on the Ornstein-Uhlenbeck process, the expected behavior

of the kinetic generated from data (blue curve) will not occur. B: The

moving average (vertical axis; window size=6) of the same failure rate

kinetic as in Part A (blue line-scatter curve) reveals that the parameter m

increases slowly during the mean-reverting regime (raw moving

average data (denoted ‘‘Moving Avg SZTt’’) are in black). The slow

increase changes to rapidly increasing failure rates at around 82% of the

lifespan. Both behaviors combine into the model of equation 9 with

parameters A&0:48, B&3:13 and T?&37:48 (p-values=5|10{5,

4:6|10{20, 9:8|10{13, respectively; R2~0:98).

doi:10.1371/journal.pcbi.1003006.g008

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Page 10

limit t?T. Therefore, the analytically derived result matches the

behavior extracted from data in eq 9.

We could, thus, conclude that as more self-renewals occur,

repair mechanisms must revert to larger failure rates means in

daughter HSCs. Biologically, the increasing mean suggests that

damage accumulates over successive generations, likely due to

imperfect repair.

Discussion

Clonal hematopoiesis begins with a single HSC and ends with

its loss after months to years [4–8]. During this period of time - the

lifespan -, the genetic/epigenetic program of the original HSC is

replicated to many daughter HSCs. In HSC self-renewal,

discrepancies between replicates should be vanishingly small.

Yet, clones extinguish after a limited time, suggesting that self-

renewal may not be perfectly reliable in the long-run.

Working from repopulation kinetics, we needed to develop

quantitative measures of reliability, failure and repair that capture

events of magnitudes that approach population size. Our data

show that increases in failure rates associate with increasing

population-wide failure loads, and decreases relate to dissipative

effects of the collective repair strength. Together, the fluctuation

patterns of the failure rate kinetics characterize the summary

dynamics of microscopic failure and repair events at the

macroscopic level of clonal HSC populations.

Our approach does not depend on particular failure sources and

repair mechanisms. An advantage, therefore, is that our current

understanding of stem cell Omics is not limiting. Rather, reliability

theory could open new avenues for interpreting longitudinal

network data. For example, we make the prediction that repair

capability stays approximately constant through the lifespan of an

individual HSC. Constancy does not mean that individual repair

mechanisms are unaffected by failure inducing processes. Instead,

the function of a deteriorating repair mechanism could be taken

over by an alternative pathway.

We predict that the repopulation failure rate kinetics stay at low

levels for a long time, but will never revert to zero failure rate. This

supports the conclusion that failures are continuously generated,

but are never completely cleared. Indeed, failure rates increase

slowly, indicating that failures accumulate. Evidence of failure

accumulation is not only seen in our experimental data. It also

followed by mathematical proof (M&M section; Theorem 2) for

the deterministic failure rate kinetics derived from our previously

developed ballistic model [8].

Daughter HSCs of successive generations may, thus, carry an

increasing failure load - consistent with previous experimental

findings on the aging of HSCs [60]. As shown here, increasing

failure load coupled with constant repair capacity is key to

explaining the differences in lifespans of HSCs in a single host in

the absence of strong extrinsic perturbations (e.g., expression of

the Wnt inhibitor Dickkopf 1 [38]). These predictions could be

tested experimentally using longitudinal genome studies compar-

ing genetic networks in myeloid-biased (My-bi) and lymphoid-

biased (Ly-bi) HSCs [47]. The former are typically longer lived

than the latter. Reliability analysis may aid in predicting universal

check-points (such as t0 in Theorem 2) to meaningfully time

longitudinal genome studies of My-bi and Ly-bi HSCs. So far, the

genome of My-bi HSCs has only been mined at isolated instances

[47,61]. Timing of genomic analyses is experimentally challeng-

ing, since HSCs are rare relative to mature blood cell

populations.

We showed that the failure rate is stable, but ‘‘noisy’’. This

‘‘noise’’, properly classified using rescaled range, or Hurst, analysis

[53] shows how stability is generated. The experimental failure

rate kinetics alternate irregularly around the deterministic failure

rate component, which we used as the mean value in our rescaled

range analysis of the data. We found that the Hurst exponent is

v0:5 for the failure rate kinetics of all long-term repopulating

HSCs. HSCs with lifespans greater than one mouse life have

smaller exponents than for lifespans below one mouse life. Using

non-linear time series theory, we could conclude that an anti-

persistent, or mean-reverting, regime governs the time period

where failure rates are at a minimum. The implication is that the

interaction between failure-generating mechanisms, such as HSC

self-renewal divisions, and failure-dissipating processes, or repairs,

creates a self-organized failure-repair equilibrium. An HSC clone

exits the equilibrium state, and becomes extinct shortly there-after,

when failure load has accumulated sufficiently to surpass repair

capacity.

Our reliability analysis of HSCs has implications on aging in

HSCs [2,45,62] and in other non-homogenous, hierarchically

organized, cell systems. The elegant general theory of aging

developed by the Gavrilovs [40–42] - tested extensively for

populations of organisms but less for those of cells [43,44] - poses

that aging systems have three properties: redundancy, initial

damage load, and redundancy depletion. For HSCs, redundancy

emerges over time as a function of self-renewals. The declining

quality of genome replicates, due to imperfect repair as predicted

here, quantifies the rate of redundancy depletion in HSCs. The

mathematical results of this paper predict that the failure dissipation

rate determined from our data provides a quantitative measure of

‘‘progressive damage load’’ starting from an HSC-specific ‘‘initial

damage load’’. However, what constitutes ‘‘initial damage load’’ in

the biology of HSCs, and what its sources are, must first be

investigated experimentally - primarily to address the question of

how HSCs are programmed during early development.

Materials and Methods

Clonal Analysis

Freshly explanted BM cells were transplanted in limiting

dilution into lethally irradiated CD45 congenic hosts exactly as

described [45–47,63]. Each host received on average 0.2–0.5

HSCs together with 26transplanted BM as a source of radio-

protecting cells. Mice were bled in regular intervals and the %

myeloid and lymphoid cells amongst the donor-type cells were

measured by Flow cytometry. All experiments were approved by

the IACUC.

Software

We used Mathematica version 8.0.1 (Wolfram Research, Inc)

for numerical mathematics and computer simulations. R version

2.12.2 and Instat version 3 (GraphPad, Inc) were used for all

statistical analyses. Figures were generated using Mathematica

version 8.0.1 and edited using GIMP version 2.8.3. The

manuscript was written in LaTeX2e using GNU Emacs version

23.3.50 (Free Software Foundation).

Reliability Theory

The subject of reliability theory is to determine the length of

time for which a system is capable of bearing ‘‘load’’ given its

material ‘‘strength’’. Mathematically, system reliability versus

unreliability at time t is quantitated by the respective inequalities

between strength K(t) and load L(t):

K(t)§L(t) versus K(t)vL(t)

ð10Þ

Lifespan Differences in Hematopoietic Stem Cells

PLOS Computational Biology | www.ploscompbiol.org 10 April 2013 | Volume 9 | Issue 4 | e1003006

Page 11

Strength and load are non-negative functions of time, i.e. K(t)§0

and L(t)§0. A system’s time-to-failure T is defined as the earliest

point in time for which load exceeds strength, i.e.:

T :~minft : K(t)vL(t)gð11Þ

For repairable systems, multiple occurrences of K(t)vL(t) are

possible, only to be reset by the repair process to some level of

reliable operations, i.e. K(t)§L(t). Hence, the operation of a

repairable system will generate a sequence of time-to-failure

values. By contrast, in an unrepairable system, the first time-to-

failure is also the last.

Though load and strength can be deterministic, it is advanta-

geous to consider the general case where the strength-load

inequality (compare eq 10) is subject to uncertainty. Hence,

system reliability is usually defined by a probability measure P½...?

for events Twt (colloquially: ‘‘time-to-failure not yet reached’’):

S(t) :~P½Twt?ð12Þ

The explicit reference to the strength-load inequality K(t)wL(t)

can be suppressed due to the definition of the time-to-failure (eq

11). The definition of probabilities implies that 0ƒS(t)ƒ1.

The failure probability is defined as the conjugate probability:

F(t) :~P½Tƒt?~1{S(t)

ð13Þ

The rate of change of the failure probability over time is used to

define the failure density:

f(t) :~F’(t) :~dF

dt(t)

ð14Þ

This definition requires that the reliability S(t) is a differentiable

function of time t.

The rate ofsystemfailure,orfailure rate,isdetermined bythe ratio

of the failure density to the system’s reliability, provided that S(t)w0:

Z(t) :~f(t)

S(t)

ð15Þ

Due to the properties of the reliability and the failure density, it

follows that Z(t)§0 for all permissible t. Using eqs 14 and 13 in eq

15showsthatthefailureratecanalsoberepresentedbythederivative

of the negative logarithm of the reliability.

Computation of Reliability and Failure Measures

In practical applications, a system’s reliability is determined

based on field measurements of a particular system variable

associated with well-defined, system-specific, time-to-failure con-

ditions. From these measurements, one attempts to form a discrete

empirical distribution as a time series f(ti,Si) : 1ƒiƒNg, where

tivtiz1, normalized such that 0ƒSiƒ1, and

(compare Algorithm 1).

Statistical distribution fitting with appropriate goodness-of-fit

measures may identify a suitable closed-form model S(t) (such as

in eq 12), so that S(ti)%Si. In this case, a system’s reliability and

failure rate evolution can be described using probability,

dynamical systems theory, and stochastic processes. We show in

the main narrative that the failure rate of the repopulation kinetics

of hematopoietic stem cells follow an Ornstein-Uhlenbeck

stochastic process.

PN

i~1Si~1

Time series representing the evolution of discrete failure

probabilities and failure densities can be obtained by point-wise

application of eqs 13 and 14 (as used in Algorithm 1). In the case of

the failure density, the differences operator is used to approximate

differentiation. This operator is defined by

D(f(ti,Fi),(tiz1,Fiz1)) :~Fiz1{Fi

tiz1{ti

,for 1ƒiƒN{1

ð16Þ

We computed discrete failure rates as the ratio of density to

reliability, as introduced in eq 15. This approach is computation-

ally more efficient and more transparent than using the numerical

derivative of the negative logarithm of the reliability sequences.

Computation of Hurst Exponent

To make predictions about the long-term reliability of water

reservoirs, Harold Hurst introduced a new statistic, called the

rescaled range. This statistic is determined by forming the ratio of

the difference between the water level extremes over a long time

period, called the range, to the standard deviation from the mean

water inflow over the same time period, but using sub-divisions of

the time scale into smaller segments [64].

Because of its generality, the rescaled range, or R/S, statistic

can be used in many different contexts - with appropriate

reinterpretation of time scales and measured entities. Hurst’s

original finding, expanded by Mandelbrot and collaborators

[49,52,53], was that, in the limit, the average rescaled range over

time periods of increasing size n, behaves like a fixed power of n:

SfrgK=sdKg1ƒKƒnT*CnH, n??

ð17Þ

H is called the Hurst exponent. An algorithm for determining the

rescaled range statistic and the empirical Hurst estimate is given

below (Algorithm 2). Explanations for the notation used above are

given there. Implementations of the R/S and other methods for

estimating the Hurst exponent are available in the statistical

programming language R.

In applications, the benefit of determining the rescaled range

sequence is that we can analyze and interpret data that have no

characteristic scale. This is sometimes interpreted as lacking bias

introduced by specific measurement scales. The Hurst exponent,

0ƒHƒ1, measures the smoothness of (self-similar) time series

based on the asymptotics of the rescaled range sequence.

0:5vHv1 indicates a persistent, or trend-reinforcing, time series.

In this case, increases (decreases) of time series values are followed

by increases (decreases). This trending increases as H?1.

0vHv0:5 holds for mean-reverting, or anti-persistent, time

series. In this case, deviations from the mean lead to reversal of the

time series values towards a long-term mean. The ‘‘strength’’ of

the mean reversion increases as H?0. Geometrically, anti-

persistent time series will appear more jagged as H?0. The

value H~0:5 is considered indicative for lack of correlation in the

time series: Any values do not inform about future values. In the

case of the hematopoietic system, we showed previously that past

values of the clonal repopulation kinetics are predictive for future

values [8].

Exact Reliability & Failure Kinetics: Theorem 1

We wished to identify the failure rate kinetics under noise-less

conditions. To do this, we could use the deterministic model of

clonal repopulation kinetics, denoted R(t) below, that we had

Lifespan Differences in Hematopoietic Stem Cells

PLOS Computational Biology | www.ploscompbiol.org 11April 2013 | Volume 9 | Issue 4 | e1003006

Page 12

developed previously [8] based on experimentally obtained clonal

repopulation data.

Theorem 1.

Consider the ballistic kinetic R(t) :equalsb:t{a:ta,

with parameters b, a, and a such that 0vavb, aw1. Let Tw0 be such

that R(T)~0. Define the rate r(t) :~R(t)=t. The failure rate Z(t) of the

reliability

S(t) :~p(t,T)=p(0,T),

t1vt2, equals:

where

p(t1,t2) :~Ðt2

t1r(u)du,

Z(t)~

1{ t=T

ð

T{t 1{

Þa{1

1

a

1{1

a

?? ? ?

(t=T)a{1

??

ð18Þ

and satisfies:

Z(0)~

a

a{1

T{1

and

Z(t)??, t?T

ð19Þ

Z(0) is defined as ‘‘initial damage load’’ and depends only on the system

parameters.

Proof. From the definition of R(t) follows r(t)~b{a:ta{1.

Then:

p(t1,t2)~b(t2{t1){(a=a)(ta

2{ta

1)

ð20Þ

from which follows that p(0,T)~a((a{1)=a)Ta. Therefore, using

Ta{1~b=a, the function S(t) has the form:

S(t)~1z

1

a{1

t

T

? ?a

{

a

a{1

t

T

? ?

ð21Þ

It follows from eq 21 that S(T)~0 and S(0)~1. Furthermore,

integration shows that we can express S(t) in a normalized form,

since:

1

2

1{1

a2

??1

T

ðT

0

S(u)du~1

ð22Þ

Since F(t) :~1{S(t) (eq 13), we get:

F(t)~

a

a{1

t

T

? ?

{

1

a{1

t

T

? ?a

ð23Þ

It follows that F(0)~0 and F(T)~1. Differentiation with respect

to t and using eq 14 shows:

f(t)~

a

a{1

1

T

1{

t

T

? ?a{1

??

ð24Þ

One notes that 0ƒf(t)v1 for Twa=(a{1)w1. Using eq 15, we

find:

Z(t)~

1{ t=T

ð

T{t 1{

Þa{1

1

a

1{1

a

??? ?

(t=T)a{1

?

ÞT{1. If we write eq 25 in the

?

ð25Þ

It follows that Z(0)~ a=(a{1)

form Z(t)~U(t)=V(t), we see that U(t)?0 and V(t)?0 in the

limit t?T. The numerator and denominator are differentiable, so

that with the substitution q(t) :~(t=T)a{1:

ð

U’(t)~{

a{1

t

??

q(t)

ð26Þ

V’(t)~{ 1{q(t)

ðÞð27Þ

from which follows:

U’(t)

V’(t)~

a{1

t

??

q(t)

1{q(t)

??

ð28Þ

It follows from de l’Hopital’s rule that Z(t)??, as t approaches T

from below.

Asymptotic Behavior near the Lifespan: Lemma 1

In Theorem 1, we showed that Z(t)?? in the time limit

approaching the lifespan. The following result details the

asymptotic growth properties of the deterministic failure rate

kinetic.

Lemma 1.

The asymptotic behavior of the repopulation failure rate near

the lifespan T is:

Z(t)*

1

T{t

ð29Þ

where, symbolically, the relationship h(t)*g(t) between two functions h(t)

and g(t) indicates similarity in the limit.

Proof. Series expansion of Z(t) near T shows:

Z(t)~

2

T{tz2{a

3T

z(5{a)(2{a)

8T2

(t{T)zO(t{T)2

ð30Þ

Algorithm 2. Algorithm for calculating the rescaled

range statistic and estimating the Hurst exponent H. The

notations used are the same as in Algorithm 1. In addition

the functions Max() and Min() refer to the maximum or

minimum of a data set A respectively. The notation SAT

denotes the uniform sample mean of A. The function

LinModFit(U*V) linearly regresses the predictor data V

on the response data U with * being used as in the

statistical programming language R to express a potential

relationship between data. Kminw1 is a minimum lower

bound on time series size to conduct meaningful

analyses.

input: A time series X :~f(tn,jn) : 1ƒnƒNg of length N,

with tnvtnz1, 1ƒnƒN.

output: A sequence RS of rescaled range values of X and

a number H representing an estimate of the Hurst

exponent.

1 for K/Kminto N do

2

SXTK/ Sum(X½1 : K?:f½1 : K?); //K-th

respect to PDF f;

3

Y(m,K) / Sum (X½1 : m?{SXTK);

deviation from mean;

4

rgK

/

1ƒmƒK); // range;

5

sdK/ Sum((X½1 : m?{SXTK)2); //standarddeviation;

6

RSK/rgK=sdK;

// K-th rescaled range;

7 RS/fRSK: KminƒKƒNg;

8 H/ LinModFit (log(SRS½p : pzn?T)*log(n);Kminƒp

vpznƒN);

meanwith

//cumulative

Max (Y(m,K);1ƒmƒK){ Min (Y(m,K);

//rescaled range sequence;

Lifespan Differences in Hematopoietic Stem Cells

PLOS Computational Biology | www.ploscompbiol.org12April 2013 | Volume 9 | Issue 4 | e1003006

Page 13

Gross Value Distribution of Failure Rates: Lemma 2

We wanted to know the order of magnitude of the ‘‘initial

damage load’’ Z(0) in relation to the lifespan T. In particular, is

there a lower bound on T?

Lemma 2.

For

Tw5

we

Z(t)§1=2 for t§T{4.

Proof. We have for Tw2:

have

Z(0)v1=2. Furthermore:

a

a{1

1

Tv1=2uaw

T

T{2

ð31Þ

The claim follows from the condition av2. Furthermore, using

the asymptotic expansion in the proof of Lemma 1, we get:

Z(t)w

2

T{tw1=2utwT{4

ð32Þ

from which the second claim follows.

Failure Growth and Dissipation: Theorem 2

We wanted to know under which conditions on time t the

deterministic failure rate kinetic Z(t) increases and decreases. To

find answers, we determined explicitly the rate of change of the

noise-less failure rate kinetic derived in Theorem 1. Throughout,

we used the simplified notation dtG :~(d=dt)G(t) for the

derivative of a differentiable function G of variable t.

Theorem 2.

The rate of change dtZ of the failure rate Z(t) has the

form:

dtZ~{h(t)Z(t)zZ2(t)

ð33Þ

with a time-dependent dissipative ‘‘frictional force’’ term h(t)w0. The latter is

given explicitly by the logarithmic rate of change of the failure density:

h(t) :~{dtlog(f(t)).Furthermore,

(2=a{1)1=(a{1):T, implying that Z(t) is strictly increasing for twt0.

Furthermore, Z(t)wh(t) for twt0.

Remark.

Z(t)wh(t) for twt0can be interpreted as ‘‘imper-

fect’’ effects of the dissipative force. This motivated our

quantitative definition of ‘‘imperfect repair’’.

Proof. We write Z(t)~f(t)v(t) in the form of an intensity with

v(t) :~1=S(t). Differentiating, we get:

dtZw0

for

twt0:~

dtZ~dtf:v(t)zf(t):dtv

ð34Þ

Since dtv~{1=S(t)2:S’(t)~Z(t):v(t), it follows that:

dtZ~dtf:v(t)zf(t):Z(t):v(t)

ð35Þ

~(dtfzf(t)Z(t)):v(t)

ð36Þ

This expression can be rewritten once more to yield:

dtZ~(dtlog(f(t))zZ(t)):Z(t)

ð37Þ

where, explicitly:

dtlog(f(t))~(a{1)

1

tz

1

T

(t=T)a{t

??

ð38Þ

Since aw1 and the second factor is always negative, it follows that

dtlog(f(t))v0 for 0vtvT. Furthermore, dtlog(f(t))?{? for

t?T and dtlog(f(t))?{? for t?0.

We now define h(t) :~{dtlog(f(t)) to obtain the general

expression for the slope of the failure rate:

dtZ~{h(t)Z(t)zZ2(t)

ð39Þ

with the time-dependent positive ‘‘dissipation’’, or retarding

‘‘frictional force’’, term h(t). The latter is given explicitly by the

logarithmic rate of change of the failure density as shown above.

Since Z(t)w0, dtZw0 when Z(t)wh(t) or dtZv0 when

Z(t)vh(t). Using the definitions of h(t) and Z(t), Z(t)wh(t) is

equivalent to f2(t):v(t)w{dtf

calculation shows that

or f2(t)w{dtf:S(t). Direct

{dtf~a

T2

T

t

? ?2{a

ð40Þ

Using eqs 21 and 24 from the proof of Theorem 1,

f2(t)w{dtf:S(t) becomes:

a

a{1

?

1

T

1{

t

T

? ?a{1

? ?a

????2

? ?

wa

T2

?

T

t

? ?2{a:

1z

1

a{1

t

T

{

a

a{1

t

T

ð41Þ

or, equivalently:

a

a{1

?

??21

t

T

T2

1{

t

T

? ?a{1

?

?

?2

w

a

a{1

??1

T2

T

t

? ?2{a

{a

? ?

z a{1z

t

T

? ?a

??

ð42Þ

which is equivalent to:

a

a{1

?

With the substitution X :~t=T, the latter becomes an inequality

between two functions denoted h1(t) and h2(t), respectively:

?

1{

t

T

? ?a{1

??2

t

T

? ?2{a

wa 1{t

T

??

{ 1{

t

T

? ?a

??

ð43Þ

h1(X) :~

a

a{1

??

1{Xa{1

??2X2{aw

a 1{X

ðÞ{ 1{Xa

ðÞ~: h2(X)

ð44Þ

The function h2(X) on the right-hand side of inequality 44 is

positive for Xv1, has the value h2(0)~a{1 at 0, and decreases

strictly monotonically to h2(1)~0. The function h1(X) on the left-

hand side of inequality 44 also satisfies h1(X)§0 with h1(1)~0,

but h1(0)~0. Some manipulation shows that h1(X) is monotone

increasingfor

Xv(2=a{1)1=(a{1)

Xw(2=a{1)1=(a{1). Taking the second derivative of h2 and

resubstitution shows that h1takes a maximum at:

anddecreasingfor

t0:~

2

a{1

??1=(a{1)

T

ð45Þ

and the inequality 44 is satisfied starting in a neighborhood of t0,

i.e. t§t0{e. It follows that dtZw0, at least from t0on.

Lifespan Differences in Hematopoietic Stem Cells

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Page 14

With some algebraic substitutions, eq 33 can be formally

rewritten as a deterministic form of the Langevin equation.

Specifically:

Corollary.

With Z(t), h(t) having the same meaning as in Theorem

2, we can write

dtZ(t)~h(t)(m{Z(t))dtzs:dtW(t)

with W(t) :~Ðt

Proof. Replace {h(t)Z(t) by h(t)(0{Z(t)) for the exact Z(t).

Write

Z2(t)~dt

W(t) :~Ðt

Statistical Analyses

To analyze the (lifespan, dissipation rate) and (lifespan, half-life)

data, denoted (T,hT) and (T,1=2), respectively, in the main

narrative, we first standardized the data by subtracting the sample

mean and dividing by the sample standard deviation column-wise.

We used hierarchical cluster analysis on the standardized data to

determine the number of clusters and their centroids. Confirma-

tory analysis was conducted using a standard partitioning

approach.

ð46Þ

0Z2(u)du interpreted as a mean-squared variation

of Z(t).

Ðt

0Z2(u)duzZ2(0).Define thefunction

0Z2(u)du to replace Z2(t) by dtW(t).

Supporting Information

Text S1

to-failure, or lifespan, of the repopulation capacity of clonal long-

term repopulating hematopoietic stem cells is statistically unbiased,

and almost efficient relative to a Gumbel distribution of maximum

extremes. Together, these properties indicate that the Gumbel

distribution makes best use of the information obtainable from

serial repopulation experiments. Table S1 contains experimentally

determined HSC lifespans, growth and decline rates, and the

Hurst exponents calculated using Algorithm 2 (compare Materials

& Methods). Figure S1 shows the shapes of clonal repopulation

time series obtained experimentally by measuring the percent

donor-type cells in blood after transplanting a single HSC.

(PDF)

It is first shown that the empirical estimator of the time-

Acknowledgments

We thank the unknown reviewers for valuable suggestions that resulted in

improvements in the presentation of this paper.

Author Contributions

Conceived and designed the experiments: CEMS. Performed the

experiments: GC. Analyzed the data: HBS. Wrote the paper: HBS CEMS.

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