Variables, compartments, and damage structuring.

We begin with the PDE system that underpins the analysis. The model captures the dual role of dedifferentiation. This role can either mitigate or exacerbate damage. The framework combines advection–reaction dynamics with nonlocal partitioning. It also incorporates feedback-regulated transitions between stem and TD cells.

The system consists of two populations: the stem cell P(t,x) and the TD cell W(t,x), where is time and is a structuring variable for cumulative damage. Stem cells divide with three possible outcomes: self-renewal with probability p₁, differentiation with probability p₂, or asymmetric division with probability p₃. The coupled PDEs governing stem and TD populations are

(1)

Here, and are constant damage accumulation rates, and the kernels encode nonlocal inheritance of damage during replication [35,36].

Division modes and dedifferentiation mechanisms.

Stem replication follows nonlocal partitioning rules. These rules ensure the conservation of total damage. There are three types of division in this model:

1. (i) Symmetric self-renewal (): this process assigns damage fractions and to the daughters. The sum of these fractions equals 1.
2. (ii) Symmetric differentiation (): this division assigns damage fractions and to the new cells. The total also equals 1.
3. (iii) Asymmetric division (): the stem daughter receives damage fraction . The TD daughter receives damage fraction . These fractions sum to 1.

Baseline choices are , , , , ensuring that each division conserves total damage. Asymmetric partitioning reflects experimental observations that the stem daughter inherits less damage than the TD daughter [37,38]. These rules are represented by δ-function transition kernels [34]:

Substituting these kernels into Eq (1) yields the advection-reaction system

(2)

Here, denote damage accumulation rates, is the stem replication rate, is the dedifferentiation rate, and is the damage-dependent TD cell death rate. The prefactors for follow from the substitution property of the δ-function:

(3)

The formulation above assumes that dedifferentiation preserves damage exactly. More generally, dedifferentiation can be modeled by a transition kernel with retention fraction . The kernel maps a TD cell with damage x^′ to a stem cell with reduced damage . The framework covers three cases: full retention with in Eq (2), partial repair with , and complete repair with . Substituting into Eq (1) yields the more general system:

(4)

For analytical tractability, we focused in this paper on the conservative full-retention case . In the Results section, we used numerical experiments to examine how partial repair with , including the representative case , alters damage accumulation and outcomes.

For comparison, consider a simpler advection-reaction system without partitioning:

(5)

which yields the same dynamics of total counts as Eq (2):

(6)

Here, are total number of stem and TD cells at time t, respectively. is the renewal fraction. These expressions follow by integrating the PDEs and using the boundary conditions described next.

Boundary conditions and feedback regulations.

Building on the PDE system above, we next introduce the boundary conditions and feedback terms that regulate growth and ensure biological plausibility. We impose a Dirichlet boundary condition

(7)

which mathematically specifies that the stem density vanishes at zero damage.

Feedback mechanisms are incorporated via negative regulation of stem replication and division probabilities by the TD population, consistent with experimental evidence [39–42]. In addition, we introduce the suppression of dedifferentiation by stem cell abundance [5]. Specifically,

(8)

(9)

(10)

(11)

Here, , , , are baseline values in the absence of feedback, k_i are regulation constants, and m_i are Hill exponents. Normalization gives

Therefore, p₃ is determined by normalization and therefore inherits feedback through p₁ and p₂.

As shown in Fig 1, stem cells undergo symmetric self-renewal, symmetric differentiation, or asymmetric division (Fig 1a). Partitioning ensures bounded damage dynamics (Fig 1b), whereas dedifferentiation pathways (Fig 1c) govern how damaged TD cells revert to the stem population.

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Fig 1. Schematic framework of the damage-structured stem cell lineage model, illustrating division modes, damage inheritance, and dedifferentiation pathways.

(a) Stem cell undergoes three distinct division types, including symmetric self-renewal, symmetric differentiation, and asymmetric division, with probabilities p₁, p₂, and p₃, respectively. The first type is symmetric self-renewal, which generates two stem daughters. The second type is symmetric differentiation, which generates two TD daughters. The last type is asymmetric division, which generates one stem and one TD daughter. Upon division, damage of the mother cell is distributed to offspring according to partitioning fraction in symmetric self-renewal, in symmetric differentiation, and in asymmetric division. (b) Comparison of damage evolution scenarios shows that nonlocal partitioning limits damage accumulation, resulting in a bounded stationary distribution. In the absence of partitioning, damage accumulates indefinitely due to constant aging rates and , leading to an unbound rightward drift of the damage distribution. (c) Three pathways allow TD cells to revert to the stem state: (i) constant dedifferentiation at a fixed rate ; (ii) threshold-dependent dedifferentiation (TDD), where dedifferentiation occurs only if damage exceeds a critical threshold x_c; and (iii) partial repair, where dedifferentiating cells undergo damage repair determined by the retention fraction .

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Biological interpretation of the mathematical model.

The terms in Eq (1) capture four biological processes. (i) Aging: both populations accumulate damage at constant rates and . (ii) Replication with partitioning: stem cells divide at a rate , with the levels of damage to the offspring determined by transition kernels. The three replication events are nonlocal. Transition kernels and describe how stem offspring with damage level x arise from a stem parent with damage x^′ during symmetric self-renewal and asymmetric division. Similarly, and specify how TD offspring with damage x arise from a stem parent with damage x^′ during symmetric maturation and asymmetric division. (iii) Apoptosis: TD cells die at rate , a nondecreasing function of x. Here, we interpret x as a normalized measure of cumulative damage, so larger x values correspond to proportionally higher death risks. For example, saturates lethality near . (iv) Dedifferentiation: TD cells revert to the stem population at a rate . A crucial modeling assumption is that the dedifferentiation term is local such that the process preserves the damage level x. This means that a TD cell with damage level x becomes a stem cell with the same damage level x.

The dedifferentiation rate is modeled in several ways: (i) as a constant parameter in the baseline model, (ii) as the baseline constant before regulation, (iii) as a feedback-regulated rate depending on the total stem population,

or (iv) a damage-dependent function in the TDD setting in Results section. We explicitly indicate in each case whether is constant, time-dependent, or damage-dependent. Analogous conventions apply to p₁, p₂, the renewal fraction f, the death rate δ, and the replication rate . Feedback is applied on p₁, p₂, , and . By contrast, the death rate δ is specified as either constant or damage-dependent. The explicit Hill-function forms of these feedback regulations are given in Eqs (8)–(11). For brevity, we sometimes use shorthand such as p₁(t) or to denote and , respectively.

The scaling in Eq (3) guarantees proper distribution of damage and mass conservation across daughter lineages. Replication with nonlocal partitioning recycles stem cells to lower-damage states, opposing the rightward drift from advection. The interplay of these forces produces stationary damage profiles confined within bounded ranges.

The advection terms in Eq (5) shift damage progressively toward larger x, producing unbounded mean damage. Thus, equilibrium in does not imply stationarity in the damage distribution. By contrast, the nonlocal partitioning in Eq (2) continuously recycles damage to lower states, yielding bounded stationary profiles once stabilize. Appendix details a two-step control strategy for Eq (6), which facilitates our parameter calibrations to achieve desired steady states in simulations.

Biologically, the boundary conditions in Eq (7) mean that no new stem cells enter with exactly zero damage—all stem cells arise from divisions of already-damaged cells. In transport terms, the boundary condition rules out any artificial influx of pristine cells at the left boundary of the damage domain.

We next give a model extension where damage repair during dedifferentiation is state-dependent.

Remark 1 (State-dependent repair during dedifferentiation). Activation of stem cell reprogramming machinery occurs during dedifferentiation. This process can trigger various repair mechanisms. These mechanisms include epigenetic resetting, enhanced proteostasis, or DNA repair. These repairs reduce the amount of damage returning to the stem population. To represent this mechanistically, we propose an extension where we replace the simple deterministic retention fraction ρ by a state-dependent retention function . Concretely, the dedifferentiation kernel

is generalized to

so that a TD cell with damage x^′ returns with reduced damage . Assume is a one-to-one function in x^′. The corresponding nonlocal dedifferentiation terms in the PDE become

(12)

Here, x^* is the unique number such that . The TD loss from dedifferentiation is the matching nonlocal term

When ρ is constant this reduces to Eq (4).

A biologically interpretable parametrization is

Here, is a monotone function that encodes how reprogramming and repair vary with damage level. For example, could be decreasing so that highly damaged TD cells trigger stronger reprogramming-mediated repair.

For analytical tractability, we retained the constant-ρ form in the paper, since it isolates the conservative limit where from the partial-repair case where . However, the state-dependent formulation above is straightforward to implement numerically by replacing the algebraic remap by the nonlocal integral . It also allows direct testing of hypotheses in which repair efficiency depends on damage.

Limitation and modeling justification.

Having established the governing PDEs together with boundary conditions and feedback regulations, we now examine the assumptions made when formulating the model. Our PDE system simplifies stem cell hierarchies to highlight the interplay between dedifferentiation and damage partitioning. Assumptions such as fixed transitions and constant drift provide a tractable foundation for studying long-term stability and aging dynamics. More elaborate hierarchies or repair mechanisms could refine quantitative thresholds, but would not alter the qualitative insights we draw.

First, stem cells P(t,x) are modeled as long-lived without explicit attrition or quiescence. While consistent with their relative longevity [26], apoptosis, senescence, or niche exit are well documented [8]. Stem attrition can be incorporated by adding a death term , which increases the replication burden required to offset loss. Quiescence can be represented by an additional compartment Q(t,x) with bidirectional transition rates , buffering stochastic loss and altering steady-state damage distributions. We explicitly incorporate stem cell attrition and quiescence in the Appendix. In either case, feedback regulation can be re-tuned to maintain homeostasis. The qualitative role of dedifferentiation—as a rescue mechanism versus an aging driver—should persist, but the parameter window for stability would be narrower.

Experimental studies have identified TA progenitor cells with biphasic behavior, i.e., early-phase amplification followed by late-phase extinction [43]. This continuous, gradual differentiation within the stem cell hierarchy has been documented to be involved with genetic and epigenetic changes [44]. However, TA cells are omitted to sharpen the analysis of dedifferentiation and damage within the stem → TD lineage. This two-compartment reduction enables tractable analysis. However, a three-compartment hierarchy from stem to TA to TD could capture how dedifferentiation interacts with TA proliferation and stem dormancy. Because TA cells divide rapidly [45], they may buffer or amplify damage fluxes and alter population ratios, potentially shifting the regimes where dedifferentiation stabilizes versus destabilizes tissue homeostasis. Exploring these effects is an important direction for future work. Nonetheless, the present two-compartment reduction remains valid for isolating the effects of dedifferentiation and partitioning, which are the central focus of this study. We analyze the role of an intermediate TA population and evaluate the robustness of our reduced stem–TD model in representing dedifferentiation dynamics in the Appendix. We assumed symmetric partitioning fractions with . Asymmetric choices alter higher moments of the damage distribution but preserve total damage. Parameter sweeps in the Results section confirmed their limited impact on long-term damage statistics.

We impose homogeneous Dirichlet boundary conditions with , which preclude inflow of undamaged cells. Because x is a continuous damage variable, the singleton has measure zero and does not affect population integrals. Moreover, positive partition fractions ensure that divisions never produce strictly undamaged progeny, even under highly asymmetric partitioning. Alternative Robin-type inflows can represent a sustained influx of minimally damaged cells [46,47]. Parameter sweeps in the Results section show that damage statistics remain robust under modest inflows.

Damage repair was omitted beyond dilution by partitioning. More general models could introduce an explicit repair mechanism that counteracts accumulation, such as leftward drift or a repair flux. Likewise, dedifferentiation was assumed to preserve damage exactly: a TD cell reverting to the stem population carries its current x, encoded by . This assumption is analytically tractable and conservative. If stability and the proposed control strategy hold under full retention, the same stability and control are expected to persist under partial repair. Including such a repair term in damage accumulation or dedifferentiation would lower effective damage loads and is expected to broaden the parameter regime in which dedifferentiation remains protective. Thus, while quantitative thresholds may shift, the qualitative dual role of dedifferentiation identified in this work should remain valid. More general formulations include retention kernels or TDD. Importantly, our numerical experiments in the Results section demonstrate that these mechanisms can substantially mitigate damage influx, further underscoring the robustness of the current framework.

Finally, feedback was restricted to symmetric divisions, with asymmetric division p₃ as a default stabilizing mechanism to maintain homeostasis [48,49]. This choice reduces parameter complexity but precludes direct regulation of all pathways. An alternative would allow each division probability to depend on TD cell feedback:

subject to . While most simulations here set p₃ = 0, incorporating asymmetric division is biologically relevant [50,51].

In the minimal feedback architecture, we regulate p₁, p₂, , and and set by normalization. We do not introduce an additional independent feedback law for p₃ because the constraint makes full independent regulation over-parameterized without additional biological data to identify distinct control pathways. Biologically, asymmetric division is often viewed as a baseline maintenance strategy, while tissue-level signals primarily modulate the renewal–differentiation balance. To verify that this modeling choice does not bias our conclusions, we performed parameter sweeps over p₃ in the Results section. The sweep results show that a moderate p₃ produces transient oscillations but leaves long-term damage statistics largely unchanged.

Another extension of the model that incorporates the feedback regulation of p₃ is the following. We set

Then truncate to [0,1] if needed.

The purpose of this subsection is to acknowledge biological simplifications introduced for tractability and to indicate how they could be relaxed in extended models. These structural limitations are distinguished from numerical and methodological limitations, which we revisit in the Discussion after presenting results. With the modeling framework in place, we turn to the numerical discretization and simulation methods used to analyze these equations.

Finite-volume upwind scheme.

We first describe the discretization, which provides the foundation for all subsequent simulations. A finite-volume upwind scheme in space, coupled with the forward Euler time-stepping scheme, is used to approximate the PDE system. Nonlocal partitioning terms are computed by linear interpolation, preserving total mass and maintaining second-order spatial accuracy. Because both the upwind discretization and Euler step are first-order, the overall method is first-order accurate, as later confirmed by grid refinement. Stability requires the usual CFL condition

(13)

which is enforced throughout.

We use C = 0.25 throughout. For manufactured-solution tests, we adopt a smaller C = 0.005 to suppress secondary numerical artifacts, such as phase error or dispersive oscillations, and to cleanly reveal truncation error. The domain is truncated to [0,A], with homogeneous Dirichlet conditions at x = 0 and outflow at x = A. Outflow is implemented using zero ghost values and upwind fluxes , which yield the discrete divergence . The combination of zero ghost values and upwind fluxes ensures that all mass leaving the domain is lost and none enters. These influx and outflux results are consistent with the absence of pristine inflows.

Conservation checks.

Two properties make the scheme suitable for biological applications. First, it is conservative: mass is preserved in the absence of source and sink terms, so any changes in total mass arise solely from reaction updates. Second, it is stable and first-order convergent, providing a reliable baseline for simulations. All computations were performed in MATLAB R2025a with Update 1 using only core functionality without additional toolboxes. For reproducibility, pseudocode for the update loop and interpolation of division integrals is provided in the Appendix. We also deposit all code supporting current research findings in the GitHub repository https://github.com/Louis-shuo-wang/PloS_one_stem.git. In addition, we also provide detailed difference equations, interpolation formulas, and truncation-error analysis in the Appendix section. Table 1 lists parameter values and ranges used in the subsequent analyses.

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Table 1. Model parameters used in simulations.

Values are either fixed baselines or ranges explored in sensitivity sweeps. All parameters are nondimensional.

https://doi.org/10.1371/journal.pone.0335163.t001

Non-negativity is enforced after each update. For conservation tests, , p₃ = 0, , , , and , on . Initial conditions are localized,

The resulting totals satisfy

so that stem counts are conserved while TD counts grow linearly. Reaction terms are switched off, and are varied. Numerical experiments reproduce the exact conservation laws in Fig 2. Fig 2a shows that, in the no-drift case with , stem counts remain constant, and TD counts grow linearly. Fig 2b shows that the same behavior is observed in the drift case with . The consistent numerical behaviors across no-drift and drift cases verify conservation and confirm the absence of spurious fluxes.

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Fig 2. Validation of the numerical scheme: verification of exact mass conservation under damage partitioning.

(a) Time evolution of total stem and TD cell populations initialized from localized distributions in the absence of drift with . The model assumes conditions of symmetric division and no TD cell loss. These conditions are defined by the parameters and . The analytical solution predicts a constant stem cell population. It also predicts a linear increase in the TD cell population. These findings are consistent with the equation . The numerical results confirm these dynamics, demonstrating that the finite-volume scheme preserves mass exactly. (b) The introduction of drift, , does not violate mass conservation under damage accumulation. The stem total number remains constant, and the TD total number maintains linear growth identical to the no-drift case. This confirms that the conservative finite-volume discretization function correctly without introducing spurious numerical fluxes or instability.

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With the numerical method established, we next verify the accuracy of the discretization through manufactured-solution tests before applying the scheme to biologically motivated simulations.

Manufactured-solution convergence tests.

To confirm the expected convergence properties, we next test the scheme against manufactured traveling-wave solutions. We tested the model against manufactured traveling-wave solutions. This test verifies the first-order accuracy of our discretization. These results establish a reliable computational platform. We use this platform to analyze how dedifferentiation influences damage dynamics within the model.

The manufactured traveling-wave profiles are such that the transport operator vanishes:

with and parameters on . Grid refinement uses with and the CFL number  = 0.005. Errors are measured in L¹ and norms for both stem and TD populations at T = 2. For these errors, we estimate the observed convergence orders,

Results are reported in Table 2 and Fig 3. Each halving of reduces error by roughly a factor of two, giving observed orders near one in all norms. High R² values confirm log–log linearity, verifying that the scheme is first-order accurate and conservative as designed.

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Fig 3. Log–log error versus grid spacing for manufactured traveling-wave solutions of the full PDE.

Parameters are . Time steps satisfy with the CFL number  = 0.005. The black dashed line indicates a reference curve with slope 1 for comparison. Both L¹ and errors for stem and TD populations decrease with slope , confirming first-order accuracy of the fully discrete scheme, including conservative remap.

https://doi.org/10.1371/journal.pone.0335163.g003

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Table 2. Discrete L¹ and errors for P and W at T = 2 under grid refinement, together with estimated convergence orders p and regression coefficients R².

Here N_x denotes the number of grid points and the time step. Errors decrease monotonically with N_x, and observed orders cluster near 1 in all norms, consistent with the scheme’s first-order truncation error. Slight deviations from exactly 1 arise from boundary/discretization effects. A conservative CFL number C = 0.005 was used here to suppress other artifacts. High R² values confirm excellent log–log linearity.

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The grid refinement test verifies that our upwind finite-volume and forward Euler discretization achieve the expected first-order accuracy in time and space, while preserving conservation. With the discretization validated, we next calibrate model parameters to biologically plausible ranges to ensure simulations reflect realistic stem–TD dynamics.

Parameter calibration strategy.

Having verified the numerical scheme, we now specify the parameters so that the simulated dynamics remain biologically grounded. The parameter choices in Table 1 are guided by tissue ratios, division rates, and previous modeling studies, with ranges selected to balance empirical plausibility and numerical stability.

Let and denote the numbers of TD and stem cells, respectively. Parameter values are summarized in Table 1 and are guided by prior modeling studies and biological evidence. TD cells typically outnumber stem cells across tissues. For example, keratinocyte stem cells make up only 4–7% of basal epidermal cells [52]. Stem cells are generally scarce relative to their differentiated progeny. We fix the ratio at 7 based on this scarcity. This value lies within a biologically reasonable range of 4 to 10, reported by previous work [34]. Damage is assumed to be partitioned symmetrically in self-renewal and differentiation, with . In asymmetric divisions, replication-induced errors are passed preferentially to differentiation-destined daughters [2], so that stem daughters inherit less damage, modeled by .

The stem cell division rate plays a central role in determining the TD-to-stem ratio, as established by Proposition 1. We conducted simulations using the damage-dependent death rate . These tests confirm a monotonic relationship between the ratio and . We therefore calibrate within the range 0.6785 to 1.1377 to keep the ratio close to 7. Damage accumulation rates and appear as drift terms and , and interact with nonlocal partitioning rules to produce bounded steady-state profiles. By normalizing the damage domain with x = 1 as an approximate upper bound, we select to keep steady-state distributions within this range. The chosen drift rates balance biological plausibility with numerical stability, since smaller drift rates also relax the CFL condition and improve computational efficiency.

Mortality is modeled either by a constant rate or by a linear, damage-dependent death function . The linear form yields moderate mortality across the biologically relevant range . Its effective average, , is comparable to constant-rate scenarios. The linear function provides a transparent baseline that captures progressive cell fragility and is consistent with observed increases in apoptosis and functional decline as damage accumulates. Empirical data often show exponential rises in dysfunction and cancer incidence with age [53]. We leave nonlinear death functions for future work, since the linear form suffices to isolate the effects of dedifferentiation, feedback, and partitioning.

Division probabilities satisfy . A parameter sweep shows that intermediate asymmetric-division probabilities p₃ induce transient overshoots and oscillations in TD-to-stem ratios. Steady-state damage distributions, however, remain largely unaffected; see Results section. To promote resilience and suppress oscillatory behavior, we restrict attention to p₃ = 0 in most simulations, thereby focusing on the balance between symmetric self-renewal and differentiation.

Regulation constants scale with steady-state population sizes according to Proposition 1. We impose the constraints and to reduce the degrees of freedom in the model. We set in most simulations, so that the influence of thresholding and repair during dedifferentiation can be examined in isolation. Finally, the Hill exponents are fixed at 2, following evidence that lower exponents suppress oscillations, whereas values generate unstable steady states in direct simulations [54].

These calibrated choices ensure that simulations remain within physiologically reasonable ranges. With both the numerical scheme and parameters in place, we now proceed to test baseline partitioning, dedifferentiation regimes, and parameter sensitivities.

Partitioning vs. no-partitioning: Homeostasis possible vs. damage runaway.

When intracellular damage is partitioned symmetrically or asymmetrically during stem divisions, the stem population can maintain a low-damage steady-state or homeostasis. By contrast, when partitioning is weak or absent, damage progressively accumulates in the stem population. Consequently, the system enters a damage-runaway regime characterized by steadily increasing mean damage and eventual loss of functional stem capacity. This dichotomy is robust across a wide range of division rates and initial damage distributions.

Constant dedifferentiation accelerates stem aging.

A persistent, non-negligible rate of dedifferentiation systematically raises the long-term damage burden in the stem population. In this process, the dedifferentiated TD cells import damage into the stem population, and the polluted stem mother cells segregate damage to the stem and TD daughter cells during division. The eventual result is the aging state in both the stem and TD populations. In simulations with constant dedifferentiation, the stem mean-damage increases faster and attains higher steady-state values than in corresponding no-dedifferentiation controls. The results indicate accelerated stem aging.

TD daughters filter damage and protect the stem population.

Dedifferentiation can have a specific threshold. Only TD cells with damage exceeding this threshold can dedifferentiate. These cells then revert to a stem cell state. In this case, dedifferentiation acts as a “detoxification loop." On the one hand, TD cells with high damage possess poor health and fail to perform tissue function effectively. Dedifferentiation allows them to transform into the stem state. They then quickly redifferentiate, go back into the TD population, and are eliminated by the damage-dependent death rate . On the other hand, this threshold prohibits a large portion of low-damage TD cells from dedifferentiating, protecting the stem population from the damage influx.

Partial repair during dedifferentiation mitigates the import of damage.

When dedifferentiation is accompanied by partial repair, the negative impact of dedifferentiation on stem damage is eliminated. To be specific, the damage carried by TD cells during dedifferentiation is reduced to a fraction. Thus, the overall damage imported on the stem population is lesser compared to the constant dedifferentiation case. As a result, damage repair during dedifferentiation is biologically plausible, and many mechanisms, such as epigenetic resetting during reprogramming, confirm our mathematical abstraction.

Constant vs. TDD.

A constant rate of dedifferentiation indiscriminately feeds TD cells back into the stem population, potentially importing damaged cells and accelerating stem aging. In contrast, experimental studies suggest that dedifferentiation is not uniform and can be triggered only when damage exceeds a threshold [16,55]. Therefore, TDD activates only when x>x_c:

TD cells are diverted from highly damaged cells back into the stem population. TDD could act as a detoxification loop, selectively protecting low-damage stem populations.

Simulations are performed on with

and initial condition . To preserve the consistent steady-state ratio of 7, the parameters are calibrated according to the scaling law in Proposition 1.

Under dynamic conditions, we track both population ratios and damage distributions. Since TD cells vastly outnumber stem cells [56], with reported TD-to-stem ratios of 4–10 [34], we fix the control ratio at 7 to anchor simulations. Both constant dedifferentiation and TDD converge to the stable TD-to-stem ratio of 7, but by different routes. Fig 5 shows the ratio dynamics under constant and TDD. Under constant dedifferentiation in Fig 5a, larger values yield smoother convergence. Under TDD in Fig 5b, however, they produce a transient overshoot, reflecting the delayed reentry of damaged TD cells after crossing the threshold x_c.

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Fig 5. Temporal evolution of the population ratio under different dedifferentiation rules.

(a) The system ratio exhibits overshoot before converging to the stable steady-state value under constant dedifferentiation rate. Higher dedifferentiation rates reduce overshoot magnitude and even eliminate overshoot when . (b) Dynamics under a threshold-dependent rule where only cells with damage over x_c can dedifferentiate. The system ratio still demonstrates overshoot before stabilizing at the steady-state value. Unlike the constant dedifferentiation case, larger dedifferentiation produces a transient overshoot. This overshoot arises because high-damage TD cells must first accumulate more damage, which must reach the threshold x_c. The cells then revert to the stem population, and this requirement creates a time delay. This delay temporarily inflates the TD population relative to the stem cell count.

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Fig 6 contrasts steady-state distributions under constant dedifferentiation and TDD. Under constant dedifferentiation in Fig 6a and 6b, increasing shifts and to the right, especially for stem cells, indicating an aging effect. The stem distribution also develops a heavier right tail as grows. In contrast, under TDD in Fig 6c and 6d, maxima remain nearly fixed across , and stem distributions stay centered at low damage levels.

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Fig 6. Impact of dedifferentiation strategies on steady-state damage distributions for stem and TD populations: constant vs. TDD.

(a-b) Steady-state damage profiles for stem and TD populations under a constant dedifferentiation rate. In this regime, dedifferentiation indiscriminately recycles cells from the TD population regardless of their damage level. This unrestricted flux of TD damage to the stem population shifts the stem (a) and TD (b) damage distributions toward higher damage states with heavier tails. (c-d) Profiles under a threshold-dependent rule where only TD cells with dedifferentiate. TDD restricts the reentry flux to only high-damage cells, preventing the aging of the stem population observed in the constant case. Both stem (c) and TD (d) cell populations effectively mitigate the aging effect induced by dedifferentiation. The mitigation ensures that the distribution maximum remains almost unchanged under different dedifferentiation rates. This mitigating effect maintains a youthful and low-damage distribution of cell populations. Distributions are shown for with and . The dedifferentiation threshold is set at x_c = 0.6.

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Table 3 quantifies these trends. Under constant dedifferentiation, stem averages nearly double from 0.1150 to 0.2385 and increases twofold from 0.0650 to 0.1250. By contrast, TD averages change only slightly from 0.2601 to 0.2709. The rightmost supports move in opposite directions: expands, whereas contracts. Under TDD, averages and maxima vary minimally, e.g., from 0.0600 to 0.0650, but the divergent shift in supports persists, with increasing and decreasing by comparable magnitudes.

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Table 3. Metrics of P and W at steady state under constant dedifferentiation and under TDD with x_c = 0.6.

Increasing under constant dedifferentiation strongly elevates stem averages and maxima, modestly affects TD metrics, and extends the stem right tail. Under TDD, stem averages vary weakly; TD metrics remain nearly constant. Supports diverge moderately with expanding and contracting in two scenarios.

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Together, these results show that constant dedifferentiation accelerates damage accumulation in stem cells but only modestly affects TD cells. The accelerated damage accumulation in stem cells produces an aging phenotype marked by sharply elevated stem averages. TDD suppresses the aging-inducing effect by stabilizing both averages and maxima, though the divergence in the rightmost supports remains. Thus, stem populations are intrinsically more sensitive to dedifferentiation perturbations, and threshold regulation provides a robust protective mechanism against dedifferentiation-induced aging. Since thresholds control which TD cells re-enter the stem population, we next test how partial repair alters damage burden and decouple dedifferentiation from the damage import.

Partial-repair dedifferentiation.

In many biological contexts, dedifferentiation is not a perfect reversal of TD-to-stem transition: cells may repair a fraction of accumulated damage during the process. We now numerically explore the generalized kernel introduced in Eq (4) with .

The parameters are set as in with initial data . To preserve the consistent steady-state ratio of 7, the parameters are calibrated according to the scaling law in Proposition 1.

Steady state damage distribution for stem and TD cells under repair-modulated dedifferentiation is demonstrated in Fig 7. Fig 7a shows the stationary stem damage distribution under partial repair, while Fig 7b shows the corresponding TD distribution. Compared to the full retention model, both distributions shift only slightly to the right as increases, confirming that repair mitigates the dedifferentiation-induced aging effect. In particular, the stem and TD maxima (, ) remain nearly constant, while the rightmost support of the stem compresses, producing shorter right tails.

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Fig 7. Mitigation of dedifferentiation-induced aging effect through partial repair mechanisms.

(a-b) Steady-state damage distribution profiles for stem (a) and TD (b) cell populations under a partial repair regime. In this regime, dedifferentiated cells retain only 50% of their accumulated damage with in before they revert to stem cells. Under the constant dedifferentiation setting in Fig 6, high dedifferentiation rates introduce excessive damage into the stem cell population and effectively induce aging in both cell populations. By contrast, the partial repair mechanism acts as a filter, mitigating the damage flux from each TD cell to the stem population. As the dedifferentiation rate increases, the distribution maxima of stem and TD populations remain pinned to low damage values and have negligible rightward shift. The right tail of the stem damage distribution widens while that of the TD damage distribution remains unchanged. This mitigation mechanism shows that even moderate repair capacity is sufficient to neutralize the aging cost associated with high lineage plasticity.

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Table 4 reports stationary metrics under partial repair. As increases, the averages and maxima grow only mildly with rising by less than 50% and by less than 40%. Averages and maxima remain well below the full retention values, which almost double. TD metrics are essentially invariant, with stable means, maxima, and rightmost support . These results show that partial repair interrupts damage transmission from TD to stem cells, thus weakening dedifferentiation-driven aging.

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Table 4. Summary of steady metrics under = 0.5.

Stem average and maxima rise only moderately with , well below full-retention levels; TD metrics remain essentially invariant.

https://doi.org/10.1371/journal.pone.0335163.t004

Repair during dedifferentiation prevents the stem population from accumulating damage rapidly. Even moderate repair fractions substantially mitigate the effects of aging, stabilizing tissue homeostasis. Unlike constant full-retention dedifferentiation, partial repair decouples flux from heavy damage carry-over. Partial repair also produces stem distributions with lower averages, smaller maxima, and shorter right tails, while leaving TD statistics nearly unchanged. Biologically, the protective effect of partial repair reflects evidence that reprogramming and dedifferentiation often trigger epigenetic remodeling, repair pathways, or metabolic resets. These processes suggest that dedifferentiation coupled with repair may protect the stem population by recycling cells without importing excess damage. Because threshold and repair strongly shape damage distribution, we next vary these control parameters x_c and ρ systematically.

Threshold sweep: Effect of varying x_c on overshoot and damage filtering.

We vary the threshold parameter x_c that triggers the dedifferentiation in the TDD rule. Biologically, x_c determines how damaged a TD cell must be before reentry into the stem population. Too low a threshold risks importing moderate-damage cells, while too high a threshold delays detoxification feedback.

Fig 5 shows that dedifferentiation suppresses overshoot at constant rates but amplifies it under TDD. Fig 8 shows that increasing x_c delays the overshoot, while its magnitude peaks at intermediate x_c. Fig 9 demonstrates the steady-state damage distributions for stem and TD cells across x_c sweeps. Fig 9a illustrates the stationary stem damage distributions, whose maxima remain aligned across thresholds. Fig 9b shows the corresponding TD distributions. At x_c = 0.2, TDD exacerbates aging via a rightward shift, whereas for thresholding progressively mitigates the aging effect by shifting the TD maxima left.

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Fig 8. Non-monotone transient dynamics of the TD-to-stem ratio under varying dedifferentiation thresholds at fixed .

The system ratio exhibits a transient overshoot before stabilizing at the steady-state value. In the low threshold regime where x_c = 0.2, the population ratio behaves similarly to the constant dedifferentiation case, showing transient overshoot before converging to the stable steady-state value. In the intermediate regime where x_c = 0.4, the overshoot magnitude peaks, reflecting a pronounced accumulation of TD cells before the stabilization of the population ratio. In the high regime where , the overshoot magnitude gradually diminishes and nearly disappears when x_c = 1.0. At this extreme value, the threshold approaches the effective maximum damage limit where death dominates, rendering the dedifferentiation flux negligible.

https://doi.org/10.1371/journal.pone.0335163.g008

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Fig 9. Sensitivity of steady-state damage distributions to the dedifferentiation threshold at fixed : the transition from aging to detoxification.

(a) For the steady-state damage profiles for the stem population across varying thresholds, the location of the distribution maxima remains nearly coincident for all x_c, indicating that the stem population’s damage-partitioning mechanism effectively buffers the impact of varying re-entry fluxes. (b) The steady-state profiles for the TD population show moderate sensitivity to x_c. A low threshold (x_c = 0.2) mimics indiscriminate constant dedifferentiation, coupling dedifferentiation with aging, and induces a rightward shift of TD distribution maxima. As the threshold increases, the TDD mechanism progressively mitigates the aging effect, shifting the TD distribution maxima slightly leftward and demonstrating the protective detoxification loop function.

https://doi.org/10.1371/journal.pone.0335163.g009

Quantitative comparisons in Table 5 reveal three strong dependencies on x_c. Stem mean damage decreases with , TD maxima shift left with , and the stem right tail extends with . Other metrics vary little. Overshoot peaks at x_c = 0.4 with R = 0.1317 and nearly vanishes at x_c = 1 with R = 0.0039.

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Table 5. Steady damage metrics for stem and TD populations and overshoot fraction R under TDD.

The stem average decreases with x_c, while TD maxima shift left. The overshoot fraction R attains its maximum at x_c = 0.4 and its minimum at x_c = 1.0.

https://doi.org/10.1371/journal.pone.0335163.t005

Overall, the sweep reveals a trade-off: small x_c values enable rapid dedifferentiation of TD cells, replenishing the stem population and dampening overshoot. At intermediate thresholds, dedifferentiation activates only after TD damage has accumulated, amplifying transients. At high thresholds, heavily damaged TD cells tend to die, suppressing overshoot. Therefore, TDD emerges as a protective mechanism: by restricting dedifferentiation to older TD cells, the process both reduces long-term stem aging and limits harmful transient overshoots.

Repair fraction sweep: Effect of varying ρ.

We next examine partial-repair dedifferentiation, where a TD cell returning to the stem population repairs only a fraction of its accumulated damage. The repair fraction models reprogramming or metabolic resetting that accompany dedifferentiation.

Fig 10 shows the overshoot fraction R as a function of ρ. The overshoot remains modest throughout the sweep but grows monotonically with ρ, rising from 0.0913 at to a maximum of 0.1103 at . Thus, greater retention of TD damage exacerbates the transient imbalance between the TD and stem populations. By contrast, repair with alleviates the overshoot by reducing the damage carried by dedifferentiated TD cells.

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Fig 10. Sensitivity of the TD-to-stem ratio to retention fractions {0.1,0.3,0.5,0.7,0.9} during dedifferentiation.

The magnitude of the transient overshoot grows steadily as repair efficiency decreases, increasing monotonically from 0.0913 at to 0.1103 at in Table 6. The repair mechanism during dedifferentiation induces a transient accumulation of TD cells. By contrast, the indistinguishable ratio curves across indicate negligible sensitivity of the ratio to damage retention fraction ρ.

https://doi.org/10.1371/journal.pone.0335163.g010

The corresponding steady-state distributions are plotted in Fig 11. Fig 11a shows the stem-cell distributions, which shift progressively rightward as ρ increases, while Fig 11b shows the TD distributions, which remain nearly invariant. These results confirm that repair affects stem quality more strongly than TD quality.

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Fig 11. Sensitivity of steady-state damage distributions for stem and TD populations to damage retention fraction .

(a) Steady-state damage distribution profiles for the stem population as a function of the retention fraction ρ. The distribution shifts progressively but slightly toward higher damage values as ρ increases from 0.1 to 0.9. This indicates that the stem population shows minor sensitivity to damage repair. (b) Corresponding damage profiles for the TD population. Unlike the stem population, the TD distribution remains nearly invariant across all values of ρ. This guarantees that the steady-state TD profile is primarily governed by its own internal dynamics rather than the damage repair mechanism. The sensitivity discrepancy between stem and TD populations indicates that repair primarily affects stem quality while leaving TD relatively unaffected.

https://doi.org/10.1371/journal.pone.0335163.g011

Table 6 quantifies these effects. The stem average rises monotonically from 0.0811 at to 0.1439 at , which shows a nearly 80% increase, while the TD average remains essentially constant between 0.2583–0.2622. Similarly, the stem maxima shift from 0.050 to 0.070, and the rightmost support nearly doubles from 0.6650 to 1.1900, whereas the TD metrics change only slightly. These numerical contrasts highlight that partial repair specifically protects the stem population from damage accumulation, without strongly perturbing the TD population.

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Table 6. Steady damage metrics for the stem and TD populations across ρ.

Stem average rises monotonically with % increase across the sweep, while the TD average remains nearly constant. Overshoot fraction R increases with ρ, attaining its maximum at .

https://doi.org/10.1371/journal.pone.0335163.t006

The repair-fraction sweep shows that even modest repair substantially protects the stem population from dedifferentiation-induced damage. Stem averages, maxima, and right tails all decrease under partial repair. Biological evidence supports this mechanism, as dedifferentiation is often accompanied by epigenetic remodeling or DNA repair. Such coupling enables safe recycling of cells without importing excessive damage into the stem population.

Having established these regulatory effects, we now test whether the conclusions hold under broader parameter changes, including division probabilities, boundary inflows, and partitioning asymmetry.

Probability sweep: Oscillations and overshoot.

In the baseline model, stem divisions were limited to symmetric self-renewal p₁ and differentiation p₂, with p₃ = 0. We now vary p₃ from 0.1 to 0.9 while maintaining such that .

Fig 12 shows the temporal ratio dynamics as p₃ increases. Allowing asymmetric division introduces oscillatory behavior not present when p₃ = 0. Both overshoot and oscillation amplitudes display a non-monotone dependence on p₃. They rise with increasing p₃, reach a maximum at intermediate values, and then decline. Larger p₃ also prolongs the duration of the oscillatory phase.

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Fig 12. Transient dynamics of the TD-to-stem cell population ratio under varying asymmetric division probabilities p₃.

Temporal trajectories of the TD-to-stem ratio are shown for a range of asymmetric division probabilities . To isolate the effect of division asymmetry, simulations assume balanced baseline symmetric renewal and differentiation rates defined by , with dedifferentiation inactive such that . The system exhibits transient overshoots and oscillations that behave non-monotonically with respect to p₃. Dynamic instability peaks at intermediate values of asymmetry where p₃ = 0.5 and declines at a high value where p₃ = 0.9. These behaviors indicate that strong asymmetric partitioning can dampen population fluctuations.

https://doi.org/10.1371/journal.pone.0335163.g012

The corresponding steady-state damage profiles are shown in Fig 13. Fig 13a shows that stem distributions remain robust to changes in p₃, while Fig 13b shows that TD distributions also shift only slightly. In particular, higher p₃ modestly mitigates aging in both populations.

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Fig 13. Robustness of steady-state damage distributions to variations in division asymmetry .

The stationary damage density profiles are plotted against the damage variable x for increasing asymmetric division probabilities p₃ to assess the sensitivity of lineage quality to partitioning strategies. (a) For the stem population, the damage distributions remain highly robust to changes in p₃, exhibiting only a marginal reduction in the distribution maxima as the probability of asymmetric division increases. (b) For the TD population, the distributions display a negligible rightward shift in the distribution maxima. The results indicate that the TD damage profile has limited sensitivity to p₃ and that asymmetric division is a secondary factor in damage control in both stem and TD populations.

https://doi.org/10.1371/journal.pone.0335163.g013

Table 7 quantifies these effects. Stem average , maxima , and rightmost support all slightly decrease as p₃ increases. The TD average also declines marginally, while the TD maxima shifts modestly rightward and remains invariant. By contrast, the overshoot fraction R is highly sensitive: the value peaks at R = 0.8198 for p₃ = 0.5 and declines to R = 0.1396 at p₃ = 0.9.

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Table 7. Steady metrics for P and W and overshoot fraction R across p₃.

Damage statistics vary modestly, whereas R changes strongly: the value peaks near p₃ = 0.5 and is minimized at p₃ = 0.9.

https://doi.org/10.1371/journal.pone.0335163.t007

Biologically, asymmetric division induces transient fluctuations in lineage balance but does not destabilize long-term tissue homeostasis. Since these oscillations leave damage distributions unaffected, this robustness allows us to fix p₃ = 0 in most simulations to avoid transient oscillations and focus on dedifferentiation and feedback. Because steady damage metrics are insensitive to p₃, we keep p₃ unregulated in the baseline model without loss of generality for the damage-focused conclusions.

Boundary sweep: Robustness to inflows.

The baseline simulations imposed homogeneous Dirichlet boundaries, excluding inflow of undamaged cells. To test robustness, we relax this assumption and introduce a Robin-type boundary at x = 0 that injects minimally damaged cells at rate g:

Biologically, a constant inflow g models continuous recruitment of new, nearly pristine cells from outside the modeled hierarchy. We sweep across the initial and boundary conditions:

Fig 14 shows the ratio dynamics. All nonhomogeneous inflows induce transient overshoot, with amplitudes of similar scale and a maximum at intermediate g. The corresponding steady-state damage profiles are shown in Fig 15. Fig 15a shows the stem distributions, which shift slightly rightward and sharpen as g decreases, while Fig 15b shows the TD distributions, which remain nearly invariant.

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Fig 14. Impact of inhomogeneous Dirichlet boundary conditions on the transient dynamics of the TD-to-stem cell ratio.

The temporal evolution of the population ratio is analyzed under varying inhomogeneous Dirichlet boundary conditions. These boundary conditions are defined by , with inflow values . These conditions simulate different influxes at the boundary x = 0. All tested boundary conditions result in comparable transient dynamics characterized by overshoots. The magnitude of this transient instability peaks at intermediate values of g. The peak suggests that moderate boundary inhomogeneities create the largest temporary perturbation in the lineage balance before the system settles into the steady state.

https://doi.org/10.1371/journal.pone.0335163.g014

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Fig 15. Sensitivity of steady-state damage distribution profiles to variations in boundary inflow magnitude g.

The stationary distributions of damage are plotted for the range of boundary inflow values to evaluate how sensitive the lineage quality is to the boundary conditions imposed at x = 0. (a) For the stem population, the damage distributions exhibit a slight rightward shift and increased sharpness with reduced peak values as the inflow parameter g decreases. (b) For the TD population, the damage profiles remain nearly invariant despite the variation in g. The invariance shows that the TD population is effectively insensitive to fluctuations in the boundary conditions applied to the cell population.

https://doi.org/10.1371/journal.pone.0335163.g015

Table 8 summarizes quantitative metrics. As g decreases from 5 to 1.5, stem averages decline modestly from 0.1011 to 0.0946 and rightmost supports contract from 0.8850 to 0.7200. By contrast, TD averages remain nearly fixed between 0.2572 to 0.2581 and supports remain unchanged. The overshoot magnitude R is non-monotone in g, peaking at intermediate inflows.

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Table 8. Steady metrics for P and W and overshoot R across boundary inflows.

Stem and TD damage metrics vary only slightly with g, and overshoot remains modest, peaking at intermediate g.

https://doi.org/10.1371/journal.pone.0335163.t008

Our results show that boundary inflows mainly affect transient dynamics and sharpen the stem damage profile. Long-term damage metrics remain unchanged, confirming that the model’s conclusions are robust to boundary conditions.

Partitioning parameter sweep: Robustness to asymmetric partitioning.

Finally, we vary the partitioning fractions that determine how damage is segregated between daughter cells at division. The baseline assumes symmetric partitioning for self-renewal with and differentiation with , and a mildly asymmetric split for asymmetric divisions with and . Experimental evidence, however, suggests that stem daughters often inherit disproportionately less damage than TD daughters. To systematically study different damage distribution rules, we first vary with , and then sweep with fixed .

Fig 16 shows the TD-to-stem ratio dynamics. Fig 16a corresponds to variations, while Fig 16b corresponds to variations. Across both sweeps, the trajectories are virtually indistinguishable. There is no overshoot or oscillation, demonstrating complete robustness in contrast to the strong sensitivities seen under TDD or when asymmetric-division probability varies.

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Fig 16. Robustness of transient population dynamics to variations in damage partitioning asymmetry.

The temporal evolution of the TD-to-stem ratio is plotted to evaluate the sensitivity of lineage dynamics to the parameters governing damage segregation during cell division. (a) Sensitivity to primary weights: trajectories for varying partitioning fractions . (b) Sensitivity to auxiliary weights: trajectories for varying auxiliary partitioning fractions . In both panels, the curves overlap almost perfectly. The overlap shows that the macroscopic population ratio dynamics are highly robust and largely insensitive to the specific mathematical tuning of the partitioning asymmetry.

https://doi.org/10.1371/journal.pone.0335163.g016

The stationary distributions in Fig 17 reinforce this robustness. Fig 17a and 17b show stem and TD distributions under variations, while Fig 17c and 17d show the corresponding distributions under variations. In all cases, stem and TD profiles are nearly identical, with no visible shifts in averages, maxima, or rightmost supports.

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Fig 17. Insensitivity of steady-state damage density profiles to partitioning asymmetry parameters.

Stationary damage distributions for the stem and TD populations are analyzed to determine the impact of damage segregation mechanics on lineage quality. (a,b) Varying primary weights: steady-state damage profiles for stem (a) and TD (b) cells under varying partitioning fractions . (c,d) Varying auxiliary weight: steady-state damage profiles for stem (c) and TD (d) cells under varying auxiliary fractions . Across all parameter sweeps, the damage distribution profiles remain virtually unchanged. This indicates that while partitioning asymmetry is a key biological feature of the model, its quantitative variation exerts only minor effects on the final damage distribution. The result contrasts with the strong sensitivities induced by dedifferentiation mechanisms, as seen in earlier figures.

https://doi.org/10.1371/journal.pone.0335163.g017

Table 9 quantifies this robustness. In the sweep, stem averages remain fixed near 0.059, TD averages near 0.163. Except for , maximum and rightmost supports vary only in the second decimal place, and overshoot fractions are negligible with . The γ sweep yields equally invariant metrics, with R<10⁻³ for all .

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Table 9. Steady damage metrics and overshoot R under variations in , , and .

For sweeps, ; for sweep, . Metrics are stable up to numerical precision; R<10⁻³.

https://doi.org/10.1371/journal.pone.0335163.t009

Partitioning asymmetry has a negligible impact on either transient or steady-state damage distributions. By contrast, TDD and division probabilities strongly reshape overshoots and oscillations. The model is therefore robust to moderate asymmetries: directing damage preferentially toward TD daughters enhances stem protection without destabilizing TD populations. Biologically, the preferential segregation of damage has been observed experimentally as a stabilizing mechanism in the long-lived stem population.

Collectively, these robustness checks show that the qualitative behaviors of the system persist across a broad parameter range, reinforcing the stability of our conclusions.