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.
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.
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 ). 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.
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; -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
-truncated kinetics illustrate the variability, and a tendency to slowly increase, of the failure rates prior to reaching the extinction transition.
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 and
months (horizontal axis). Calculations were performed using Algorithm 0 (compare Table S1 in Text S1). All exponents are
, thus falling into the region of anti-persistent behavior (defined by Hurst values
(light-yellow region)) and not into the region of persistent behavior (defined by
(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
. The value
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
values obtained from our experimental data were fitted to the line
as a function of lifespan
(gray solid line through the data). Goodness-of-Fit was determined using the Akaike Information Criterion (
). The parameter estimates were highly significant (intercept estimate
, standard error
, p-value
; slope estimate
, standard error
, p-value
). The extension of the fitted line to include lifespans
only serves visualization purposes, since we only considered HSCs with lifespans
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.
Figure 6.
Failures are Dissipated More Slowly in Shorter-lived HSCs than in Longer-lived HSCs.
A: We determined the dissipation rates (yellow dots; vertical axis) relative to the lifespan
(horizontal axis) using
for
-truncated failure rates. The lower bound
of the integrand was derived in Theorem 2. To highlight the general tendency in the data, not implying any dependencies of consecutive data points, we fitted the data to a non-linear model
(blue line; goodness-of-fit Akaike Information Criterion:
); parameter p-values
and
). Calculation of
by regressing to normal noise produced a slightly lower exponent
(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
(blue line; goodness-of-fit Akaike Information Criterion:
; parameter p-values
and
, 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
and
, respectively, as background.
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 months. Points
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
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
to the “terminal” absorbing point in region
(circled point). Region B is not visited by the dynamic trajectory and, therefore, empty.
Figure 8.
Breakdown of Mean-Reverting Behavior in the Failure Rate Kinetics of Long-term Repopulating HSCs.
A: An experimental failure rate kinetic (blue scatter-line plot; values 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
months. The realizations of the process were obtained using the iteration schema in eq 4. The same values of
,
and
as in the experimental data were used. For simplicity, the initial condition was set at
for
(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
increases slowly during the mean-reverting regime (raw moving average data (denoted “Moving Avg
”) 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
,
and
(p-values =
,
,
, respectively;
).