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The authors have declared that no competing interests exist.

Conceived and designed the experiments: CEMS. Performed the experiments: GC. Analyzed the data: HBS. Wrote the paper: HBS CEMS.

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.

All hematopoietic stem cells (HSCs) are characterized by the capacities to produce all blood cell-types by differentiation 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 population size. We broadly find that one condition explains our experimental data: repair mechanisms are a priori imperfect 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.

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

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

Several hypotheses have been developed to identify and explain how HSCs limit their lifespan. The generation-age hypothesis

According to Hayflick's hypothesis

Yet, HSCs and other stem cells, express telomerase

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

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

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

Mathematically, the lifespan of populations has been addressed in manufacturing, engineering, actuarial and biological applications of reliability theory

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.

We obtained repopulation data experimentally by transplanting single HSCs into ablated mice as described previously

We previously showed that any HSC will eventually fail to repopulate all mature cell populations

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

Statistical analysis of 38 repopulation kinetics ascertains that the time-to-failure estimator is unbiased and almost efficient (compare

The time-to-failure

In systems theory, reliability is defined as a conditional probability (

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

A–D: Four types of kinetics were calculated (compare

Using the repopulation data of 38 HSC clones with lifespans ranging from

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 populations 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

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 (

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 preserved 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

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 (

Application of the method of symbolic time series comparison in

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

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

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;

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

We had previously shown that past values in HSC repopulation kinetics predict future values

Plotted are the Hurst exponents (plot symbol: blue triangles; values vertical axis) of the failure rate kinetics of HSCs with lifespans

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

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

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

We showed numerically that the weighted sum of the variance-adjusted rate of change plus the standardized rate of each

Our numerical analyses showed that

In Theorem 2, we proved that the deterministic repopulation kinetic

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

We first asked what the properties of

A: We determined the dissipation rates

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

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

The mathematical analysis also predicted that, during the initial expansion period of a clone, quantitated by

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:

The half-life data (compare

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

An important observation common to all failure rate kinetics is that, near the end of clonal life, the failure rates strongly increase. When we compared actual failure rate kinetics with kinetics generated by simulation using the experimentally derived parameters in

Phase space plot of the failure rate kinetic of a long-term repopulating HSC with long lifespan of

The phase space trajectory of an HSC's failure rate kinetic starts in the initial engraftment regime (

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 (

A: An experimental failure rate kinetic (blue scatter-line plot; values

Specifically, we asked whether the mean of the experimental failure rates is a constant or changes in time. Analysis of the

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.

Clonal hematopoiesis begins with a single HSC and ends with its loss after months to years

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

Daughter HSCs of successive generations may, thus, carry an increasing failure load - consistent with previous experimental findings on the aging of HSCs

We showed that the failure rate is stable, but “noisy”. This “noise”, properly classified using rescaled range, or Hurst, analysis

Our reliability analysis of HSCs has implications on aging in HSCs

Freshly explanted BM cells were transplanted in limiting dilution into lethally irradiated CD45 congenic hosts exactly as described

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).

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

Though load and strength can be deterministic, it is advantageous to consider the general case where the strength-load inequality (compare

The failure probability is defined as the conjugate probability:

The rate of system failure, or failure rate, is determined by the ratio of the failure density to the system's reliability, provided that

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 conditions. From these measurements, one attempts to form a discrete empirical distribution as a time series

Statistical distribution fitting with appropriate goodness-of-fit measures may identify a suitable closed-form model

Time series representing the evolution of discrete failure probabilities and failure densities can be obtained by point-wise application of

We computed discrete failure rates as the ratio of density to reliability, as introduced in

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

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

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,

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

In Theorem 1, we showed that

We wanted to know the order of magnitude of the “initial damage load”

We wanted to know under which conditions on time

We now define

Since

With some algebraic substitutions,

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

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

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We thank the unknown reviewers for valuable suggestions that resulted in improvements in the presentation of this paper.

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