Stem cell self-renewal in regeneration and cancer: Insights from mathematical modeling

Dynamically adjusted cell fate decisions and resilience to mutant invasion during steady-state hematopoiesis revealed by an experimentally parameterized mathematical model

A major next step in hematopoietic stem cell (HSC) biology is to enhance our quantitative understanding of cellular and evolutionary dynamics involved in undisturbed hematopoiesis. Mathematical models have been and continue to be key in this respect, and are most powerful when parameterized experimentally and containing sufficient biological complexity. In this paper, we use data from label propagation experiments in mice to parameterize a mathematical model of hematopoiesis that includes homeostatic control mechanisms as well as clonal evolution. We find that nonlinear feedback control can drastically change the interpretation of kinetic estimates at homeostasis. This suggests that short-term HSC and multipotent progenitors can dynamically adjust to sustain themselves temporarily in the absence of long-term HSCs, even if they differentiate more often than they self-renew in undisturbed homeostasis. Additionally, the presence of feedback control in the model renders the system resilient against mutant invasion. Invasion barriers, however, can be overcome by a combination of age-related changes in stem cell differentiation and evolutionary niche construction dynamics based on a mutant-associated inflammatory environment. This helps us understand the evolution of e.g., TET2 or DNMT3A mutants, and how to potentially reduce mutant burden.

Mathematical modelling of stem and progenitor cell dynamics during ruxolitinib treatment of patients with myeloproliferative neoplasms

Introduction

The Philadelphia chromosome-negative myeloproliferative neoplasms are a group of slowly progressing haematological malignancies primarily characterised by an overproduction of myeloid blood cells. Patients are treated with various drugs, including the JAK1/2 inhibitor ruxolitinib. Mathematical modelling can help propose and test hypotheses of how the treatment works.

Materials and methods

We present an extension of the Cancitis model, which describes the development of myeloproliferative neoplasms and their interactions with inflammation, that explicitly models progenitor cells and can account for treatment with ruxolitinib through effects on the malignant stem cell response to cytokine signalling and the death rate of malignant progenitor cells. The model has been fitted to individual patients' data for the JAK2 V617F variant allele frequency from the COMFORT-II and RESPONSE studies for patients who had substantial reductions (20 percentage points or 90% of the baseline value) in their JAK2 V617F variant allele frequency (n = 24 in total).

Results

The model fits very well to the patient data with an average root mean square error of 0.0249 (2.49%) when allowing ruxolitinib treatment to affect both malignant stem and progenitor cells. This average root mean square error is much lower than if allowing ruxolitinib treatment to affect only malignant stem or only malignant progenitor cells (average root mean square errors of 0.138 (13.8%) and 0.0874 (8.74%), respectively).

Discussion

Systematic simulation studies and fitting of the model to the patient data suggest that an initial reduction of the malignant cell burden followed by a monotonic increase can be recapitulated by the model assuming that ruxolitinib affects only the death rate of malignant progenitor cells. For patients exhibiting a long-term reduction of the malignant cells, the model predicts that ruxolitinib also affects stem cell parameters, such as the malignant stem cells' response to cytokine signalling.

Homeostatic regulation of renewing tissue cell populations via crowding control: stability, robustness and quasi-dedifferentiation

To maintain renewing epithelial tissues in a healthy, homeostatic state, cell divisions and differentiation need to be tightly regulated. Mechanisms of homeostatic regulation often rely on crowding feedback control: cells are able to sense the cell density in their environment, via various molecular and mechanosensing pathways, and respond by adjusting division, differentiation, and cell state transitions appropriately. Here, we determine, via a mathematically rigorous framework, which general conditions for the crowding feedback regulation (i) must be minimally met, and (ii) are sufficient, to allow the maintenance of homeostasis in renewing tissues. We show that those conditions naturally allow for a degree of robustness toward disruption of regulation. Furthermore, intrinsic to this feedback regulation is that stem cell identity is established collectively by the cell population, not by individual cells, which implies the possibility of 'quasi-dedifferentiation', in which cells committed to differentiation may reacquire stem cell properties upon depletion of the stem cell pool. These findings can guide future experimental campaigns to identify specific crowding feedback mechanisms.

Stem Cell-Derived Extracellular Vesicles for Cancer Therapy and Tissue Engineering Applications

Recently, stem cells and their secretomes have attracted great attention in biomedical applications, particularly extracellular vesicles (EVs). EVs are secretomes of cells for cell-to-cell communication. They play a role as intercellular messengers as they carry proteins, nucleic acids, lipids, and therapeutic agents. They have also been utilized as drug-delivery vehicles due to their biocompatibility, low immunogenicity, stability, targetability, and engineerable properties. The therapeutic potential of EVs can be further enhanced by surface engineering and modification using functional molecules such as aptamers, peptides, and antibodies. As a consequence, EVs hold great promise as effective delivery vehicles for enhancing treatment efficacy while avoiding side effects. Among various cell types that secrete EVs, stem cells are ideal sources of EVs because stem cells have unique properties such as self-renewal and regenerative potential for transplantation into damaged tissues that can facilitate their regeneration. However, challenges such as immune rejection and ethical considerations remain significant hurdles. Stem cell-derived EVs have been extensively explored as a cell-free approach that bypasses many challenges associated with cell-based therapy in cancer therapy and tissue regeneration. In this review, we summarize and discuss the current knowledge of various types of stem cells as a source of EVs, their engineering, and applications of EVs, focusing on cancer therapy and tissue engineering.

Mathematics of neural stem cells: Linking data and processes

Adult stem cells are described as a discrete population of cells that stand at the top of a hierarchy of progressively differentiating cells. Through their unique ability to self-renew and differentiate, they regulate the number of end-differentiated cells that contribute to tissue physiology. The question of how discrete, continuous, or reversible the transitions through these hierarchies are and the precise parameters that determine the ultimate performance of stem cells in adulthood are the subject of intense research. In this review, we explain how mathematical modelling has improved the mechanistic understanding of stem cell dynamics in the adult brain. We also discuss how single-cell sequencing has influenced the understanding of cell states or cell types. Finally, we discuss how the combination of single-cell sequencing technologies and mathematical modelling provides a unique opportunity to answer some burning questions in the field of stem cell biology.

Why are cell populations maintained via multiple compartments?

We consider the maintenance of 'product' cell populations from 'progenitor' cells via a sequence of one or more cell types, or compartments, where each cell's fate is chosen stochastically. If there is only one compartment then large amplification, that is, a large ratio of product cells to progenitors comes with disadvantages. The product cell population is dominated by large families (cells descended from the same progenitor) and many generations separate, on average, product cells from progenitors. These disadvantages are avoided using suitably constructed sequences of compartments: the amplification factor of a sequence is the product of the amplification factors of each compartment, while the average number of generations is a sum over contributions from each compartment. Passing through multiple compartments is, in fact, an efficient way to maintain a product cell population from a small flux of progenitors, avoiding excessive clonality and minimizing the number of rounds of division en route. We use division, exit and death rates, estimated from measurements of single-positive thymocytes, to choose illustrative parameter values in the single-compartment case. We also consider a five-compartment model of thymocyte differentiation, from double-negative precursors to single-positive product cells.

Distinguishing between multiple mathematical models of neural stem cell quiescence and activation during age-related neural stem cell decline in neurogenesis

Stem cells are required for tissue maintenance and homeostasis during an organism's lifetime. Neural stem cells (NSCs) can be in an actively dividing state or in a quiescent state. The balance between stem cell quiescence and cycling activity determines the rate of neurogenesis. With age, more NSCs enter the quiescent state, while the total number of NSCs decreases. Here we reconsider an existing mathematical model of how neural stem cells switch between active and quiescent states from the point of view of control theory by considering the activation rate, self-renewal probability, and division rate as control parameters rather than as pre-defined functions. Our goal is to test whether those modifications to the basic model could explain the observed decline of neural stem cells with age better than Gomerzian time-dependent parameters, and compare the output from different model variants to experimental data from mice using AIC. We find that time-dependent activation rate provides the best fit to the activated cell fraction (ACF) of NSCs over time, but that other model variants with constant parameter values can better fit the total number of NSCs over time. We also consider an alternate model for NSCs with nonlinear feedback from progenitor cells that affect NSC parameters, and compare all models to experimental stem cell and progenitor data. However, all of the feedback models considered provide a worse fit to the experimental data. This suggests that when switching between active and quiescent stem cells is considered, a time-dependent linear model outperforms the integral feedback mechanism considered by other models of stem cell lineages. Fitting progenitor data for both the time varying models and feedback models indicates that four or five intermediate transit amplifying progenitor states are necessary. Our modeling suggests that in order to determine whether an increase in age-related neural stem cell quiescence is determined by by a decreasing stem cell activation rate or an increased stem cell depletion rate, additional experiments should be designed to explore whether or not depletion of the stem cell pool is occurring, and that a higher resolution time series for activated cell fraction (ACF) would be best to resolve this issue.

Local asymptotic stability of a system of integro-differential equations describing clonal evolution of a self-renewing cell population under mutation

In this paper we consider a system of non-linear integro-differential equations (IDEs) describing evolution of a clonally heterogeneous population of malignant white blood cells (leukemic cells) undergoing mutation and clonal selection. We prove existence and uniqueness of non-trivial steady states and study their asymptotic stability. The results are compared to those of the system without mutation. Existence of equilibria is proved by formulating the steady state problem as an eigenvalue problem and applying a version of the Krein-Rutmann theorem for Banach lattices. The stability at equilibrium is analysed using linearisation and the Weinstein-Aronszajn determinant which allows to conclude local asymptotic stability.

Computational Systems Biology of Morphogenesis

Extracting mechanistic knowledge from the spatial and temporal phenotypes of morphogenesis is a current challenge due to the complexity of biological regulation and their feedback loops. Furthermore, these regulatory interactions are also linked to the biophysical forces that shape a developing tissue, creating complex interactions responsible for emergent patterns and forms. Here we show how a computational systems biology approach can aid in the understanding of morphogenesis from a mechanistic perspective. This methodology integrates the modeling of tissues and whole-embryos with dynamical systems, the reverse engineering of parameters or even whole equations with machine learning, and the generation of precise computational predictions that can be tested at the bench. To implement and perform the computational steps in the methodology, we present user-friendly tools, computer code, and guidelines. The principles of this methodology are general and can be adapted to other model organisms to extract mechanistic knowledge of their morphogenesis.

Predicting Time to Relapse in Acute Myeloid Leukemia through Stochastic Modeling of Minimal Residual Disease Based on Clonality Data

Event-free and overall survival remain poor for patients with acute myeloid leukemia. Chemoresistant clones contributing to relapse arise from minimal residual disease (MRD) or newly-acquired mutations. However, the dynamics of clones comprising MRD is poorly understood. We developed a predictive stochastic model, based on a multitype age-dependent Markov branching process, to describe how random events in MRD contribute to the heterogeneity in treatment response. We employed training and validation sets of patients who underwent whole genome sequencing and for whom mutant clone frequencies at diagnosis and relapse were available. The disease evolution and treatment outcome are subject to stochastic fluctuations. Estimates of malignant clone growth rates, obtained by model fitting, are consistent with published data. Using the estimates from the training set, we developed a function linking MRD and time of relapse, with MRD inferred from the model fits to clone frequencies and other data. An independent validation set confirmed our model. In a third data set, we fitted the model to data at diagnosis and remission and predicted the time to relapse. As a conclusion, given bone marrow genome at diagnosis and MRD at or past remission, the model can predict time to relapse, and help guide treatment decisions to mitigate relapse.

Computational Reconstruction of Clonal Hierarchies From Bulk Sequencing Data of Acute Myeloid Leukemia Samples

Acute myeloid leukemia is an aggressive cancer of the blood forming system. The malignant cell population is composed of multiple clones that evolve over time. Clonal data reflect the mechanisms governing treatment response and relapse. Single cell sequencing provides most direct insights into the clonal composition of the leukemic cells, however it is still not routinely available in clinical practice. In this work we develop a computational algorithm that allows identifying all clonal hierarchies that are compatible with bulk variant allele frequencies measured in a patient sample. The clonal hierarchies represent descendance relations between the different clones and reveal the order in which mutations have been acquired. The proposed computational approach is tested using single cell sequencing data that allow comparing the outcome of the algorithm with the true structure of the clonal hierarchy. We investigate which problems occur during reconstruction of clonal hierarchies from bulk sequencing data. Our results suggest that in many cases only a small number of possible hierarchies fits the bulk data. This implies that bulk sequencing data can be used to obtain insights in clonal evolution.

Dynamic behavior and stabilization of brain cell reconstitution after stroke under the proliferation and differentiation processes for stem cells

Stem cells play a critical role in regulatory operations, overseeing tissue regeneration and tissue homeostasis. In this paper, a mathematical model is proposed and analyzed to study the impact of stem cell transplantation on the dynamical behavior of stroke therapy, which is assumed to be based on transplanting dead brain cells following a stroke. We transform the method of using hierarchical cell systems into a method of using different compartment variables by using ordinary differential equations, each of which elucidates a well-defined differentiation stage along with the effect of mature cells in improving the brain function after a stroke. Stem cells, progenitor cells, and the impacts of the stem cells transplanted on brain cells are among the variables considered. The model is studied analytically and solved numerically using the fourth-order Runge-Kutta method. We analyze the structure of equilibria, the ability of neural stem cells to proliferate and differentiate, and the stability properties of equilibria for stem cell transplantation. The model is considered to be stable after transplantation if the stem cells and progenitor cells differentiate into mature nerve cells in the brain. The results of the model analysis and simulation facilitate the identification of various biologically probable parameter sets that can explain the optimal time for stem cell replacement of damaged brain cells. Associating the classified parameter sets with recent experimental and clinical findings contributes to a better understanding of therapeutic mechanisms that promote the reconstitution of brain cells after an ischemic stroke.

Dynamical properties of feedback signalling in B lymphopoiesis: A mathematical modelling approach

Haematopoiesis is the process of generation of blood cells. Lymphopoiesis generates lymphocytes, the cells in charge of the adaptive immune response. Disruptions of this process are associated with diseases like leukaemia, which is especially incident in children. The characteristics of self-regulation of this process make them suitable for a mathematical study. In this paper we develop mathematical models of lymphopoiesis using currently available data. We do this by drawing inspiration from existing structured models of cell lineage development and integrating them with paediatric bone marrow data, with special focus on regulatory mechanisms. A formal analysis of the models is carried out, giving steady states and their stability conditions. We use this analysis to obtain biologically relevant regions of the parameter space and to understand the dynamical behaviour of B-cell renovation. Finally, we use numerical simulations to obtain further insight into the influence of proliferation and maturation rates on the reconstitution of the cells in the B line. We conclude that a model including feedback regulation of cell proliferation represents a biologically plausible depiction for B-cell reconstitution in bone marrow. Research into haematological disorders could benefit from a precise dynamical description of B lymphopoiesis.

Coordinated changes in cellular behavior ensure the lifelong maintenance of the hippocampal stem cell population

Neural stem cell numbers fall rapidly in the hippocampus of juvenile mice but stabilize during adulthood, ensuring lifelong hippocampal neurogenesis. We show that this stabilization of stem cell numbers in young adults is the result of coordinated changes in stem cell behavior. Although proliferating neural stem cells in juveniles differentiate rapidly, they increasingly return to a resting state of shallow quiescence and progress through additional self-renewing divisions in adulthood. Single-cell transcriptomics, modeling, and label retention analyses indicate that resting cells have a higher activation rate and greater contribution to neurogenesis than dormant cells, which have not left quiescence. These changes in stem cell behavior result from a progressive reduction in expression of the pro-activation protein ASCL1 because of increased post-translational degradation. These cellular mechanisms help reconcile current contradictory models of hippocampal neural stem cell (NSC) dynamics and may contribute to the different rates of decline of hippocampal neurogenesis in mammalian species, including humans.

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More proliferating hippocampal stem cells return to shallow quiescence with age

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Dormant stem cells enter deeper quiescence with age

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These changes drive the transition from developmental to adult neurogenesis

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Increasing degradation of ASCL1 protein by HUWE1 coordinates these changes

Thresholds for post-rebound SHIV control after CCR5 gene-edited autologous hematopoietic cell transplantation

Autologous, CCR5 gene-edited hematopoietic stem and progenitor cell (HSPC) transplantation is a promising strategy for achieving HIV remission. However, only a fraction of HSPCs can be edited ex vivo to provide protection against infection. To project the thresholds of CCR5-edition necessary for HIV remission, we developed a mathematical model that recapitulates blood T cell reconstitution and plasma simian-HIV (SHIV) dynamics from SHIV-1157ipd3N4-infected pig-tailed macaques that underwent autologous transplantation with CCR5 gene editing. The model predicts that viral control can be obtained following analytical treatment interruption (ATI) when: (1) transplanted HSPCs are at least fivefold higher than residual endogenous HSPCs after total body irradiation and (2) the fraction of protected HSPCs in the transplant achieves a threshold (76–94%) sufficient to overcome transplantation-dependent loss of SHIV immunity. Under these conditions, if ATI is withheld until transplanted gene-modified cells engraft and reconstitute to a steady state, spontaneous viral control is projected to occur.

Predicting pattern formation in embryonic stem cells using a minimalist, agent-based probabilistic model

The mechanisms of pattern formation during embryonic development remain poorly understood. Embryonic stem cells in culture self-organise to form spatial patterns of gene expression upon geometrical confinement indicating that patterning is an emergent phenomenon that results from the many interactions between the cells. Here, we applied an agent-based modelling approach in order to identify plausible biological rules acting at the meso-scale within stem cell collectives that may explain spontaneous patterning. We tested different models involving differential motile behaviours with or without biases due to neighbour interactions. We introduced a new metric, termed stem cell aggregate pattern distance (SCAPD) to probabilistically assess the fitness of our models with empirical data. The best of our models improves fitness by 70% and 77% over the random models for a discoidal or an ellipsoidal stem cell confinement respectively. Collectively, our findings show that a parsimonious mechanism that involves differential motility is sufficient to explain the spontaneous patterning of the cells upon confinement. Our work also defines a region of the parameter space that is compatible with patterning. We hope that our approach will be applicable to many biological systems and will contribute towards facilitating progress by reducing the need for extensive and costly experiments.

Global dynamics of healthy and cancer cells competing in the hematopoietic system

Stem cells in the bone marrow differentiate to ultimately become mature, functioning blood cells through a tightly regulated process (hematopoiesis) including a stem cell niche interaction and feedback through the immune system. Mutations in a hematopoietic stem cell can create a cancer stem cell leading to a less controlled production of malfunctioning cells in the hematopoietic system. This was mathematically modelled by Andersen et al. (2017) including the dynamic variables: healthy and cancer stem cells and mature cells, dead cells and an immune system response. Here, we apply a quasi steady state approximation to this model to construct a two dimensional model with four algebraic equations denoted the simple cancitis model. The two dynamic variables are the clinically available quantities JAK2V617F allele burden and the number of white blood cells. The simple cancitis model represents the original model very well. Complete phase space analysis of the simple cancitis model is performed, including proving the existence and location of globally attracting steady states. Hence, parameter values from compartments of stem cells, mature cells and immune cells are directly linked to disease and treatment prognosis, showing the crucial importance of early intervention. The simple cancitis model allows for a complete analysis of the long term evolution of trajectories. In particular, the value of the self renewal of the hematopoietic stem cells divided by the self renewal of the cancer stem cells is found to be an important diagnostic marker and perturbing this parameter value at intervention allows the model to reproduce clinical data. Treatment at low cancer cell numbers allows returning to healthy blood production while the same intervention at a later disease stage can lead to eradication of healthy blood producing cells. Assuming the total number of white blood cells is constant in the early cancer phase while the allele burden increases, a one dimensional model is suggested and explicitly solved, including parameters from all original compartments. The solution explicitly shows that exogenous inflammation promotes blood cancer when cancer stem cells reproduce more efficiently than hematopoietic stem cells.

The invasion of de-differentiating cancer cells into hierarchical tissues

Many fast renewing tissues are characterized by a hierarchical cellular architecture, with tissue specific stem cells at the root of the cellular hierarchy, differentiating into a whole range of specialized cells. There is increasing evidence that tumors are structured in a very similar way, mirroring the hierarchical structure of the host tissue. In some tissues, differentiated cells can also revert to the stem cell phenotype, which increases the risk that mutant cells lead to long lasting clones in the tissue. However, it is unclear under which circumstances de-differentiating cells will invade a tissue. To address this, we developed mathematical models to investigate how de-differentiation is selected as an adaptive mechanism in the context of cellular hierarchies. We derive thresholds for which de-differentiation is expected to emerge, and it is shown that the selection of de-differentiation is a result of the combination of the properties of cellular hierarchy and de-differentiation patterns. Our results suggest that de-differentiation is most likely to be favored provided stem cells having the largest effective self-renewal rate. Moreover, jumpwise de-differentiation provides a wider range of favorable conditions than stepwise de-differentiation. Finally, the effect of de-differentiation on the redistribution of self-renewal and differentiation probabilities also greatly influences the selection for de-differentiation.

How can a tissue such as the blood system or the skin, which constantly produces a huge number of cells, avoids that errors accumulate in the cells over time? Such tissues are typically organized in cellular hierarchies, which induce a directional relation between different stages of cellular differentiation, minimizing the risk of retention of mutations. However, recent evidence also shows that some differentiated cells can de-differentiate into the stem cell phenotype. Why does de-differentiation arise in some tumors, but not in others? We developed a mathematical model to study the growth competition between de-differentiating mutant cell populations and non de-differentiating resident cell population. Our results suggest that the invasion of de-differentiation is jointly influenced by the cellular hierarchy (e.g. number of cell compartments, inherent cell division pattern) and the de-differentiation pattern, i.e. how exactly cells acquire their stem-cell like properties.

Control of cell fraction and population recovery during tissue regeneration in stem cell lineages

Multicellular tissues are continually turning over, and homeostasis is maintained through regulated proliferation and differentiation of stem cells and progenitors. Following tissue injury, a dramatic increase in cell proliferation is commonly observed, resulting in rapid restoration of tissue size. This regulation is thought to occur via multiple feedback loops acting on cell self-renewal or differentiation. Models of ordinary differential equations have been widely used to study the cell lineage system. Prior modeling studies have suggested that loss of homeostasis and initiation of tumorigenesis can be contributed to the loss of control of these processes, and the rate of symmetric versus asymmetric division of the stem cells may also be altered. While most of the previous works focused on analysis of stability, existence and uniqueness of steady states of multistage cell lineage models, in this work we attempt to understand the cell lineage model from a different perspective. We compare three variants of hierarchical stem cell lineage tissue models with different combinations of negative feedbacks and use sensitivity analysis to examine the possible strategies for the cells to achieve certain performance objectives. Our results suggest that multiple negative feedback loops must be present in the stem cell lineage to keep the fractions of stem cells to differentiated cells in the total population as robust as possible to variations in cell division parameters, and to minimize the time for tissue recovery in a non-oscillatory manner.

Mathematical modeling of the impact of cytokine response of acute myeloid leukemia cells on patient prognosis

Acute myeloid leukemia (AML) is a heterogeneous disease. One reason for the heterogeneity may originate from inter-individual differences in the responses of leukemic cells to endogenous cytokines. On the basis of mathematical modeling, computer simulations and patient data, we have provided evidence that cytokine-independent leukemic cell proliferation may be linked to early relapses and poor overall survival. Depending whether the model of cytokine-dependent or cytokine-independent leukemic cell proliferation fits to the clinical data, patients can be assigned to two groups that differ significantly with respect to overall survival. The modeling approach further enables us to identify parameter constellations that can explain unexpected responses of some patients to external cytokines such as blast crisis or remission without chemotherapy.

Revealing age-related changes of adult hippocampal neurogenesis using mathematical models

New neurons are continuously generated in the dentate gyrus of the adult hippocampus. This continuous supply of newborn neurons is important to modulate cognitive functions. Yet the number of newborn neurons declines with age. Increasing Wnt activity upon loss of dickkopf 1 can counteract both the decline of newborn neurons and the age-related cognitive decline. However, the precise cellular changes underlying the age-related decline or its rescue are fundamentally not understood. The present study combines a mathematical model and experimental data to address features controlling neural stem cell (NSC) dynamics. We show that available experimental data fit a model in which quiescent NSCs may either become activated to divide or may undergo depletion events, such as astrocytic transformation and apoptosis. Additionally, we demonstrate that old NSCs remain quiescent longer and have a higher probability of becoming re-activated than depleted. Finally, our model explains that high NSC-Wnt activity leads to longer time in quiescence while enhancing the probability of activation. Altogether, our study shows that modulation of the quiescent state is crucial to regulate the pool of stem cells throughout the life of an animal.

Summary: New deterministic and stochastic mathematical models are proposed to investigate adult neurogenesis in young, old and perturbed hippocampus, and quantified using population-level and clonal experimental data.