Stem cell control, oscillations, and tissue regeneration in spatial and non-spatial models

Licensing and niche competition in spermatogenesis: mathematical models suggest complementary regulation of tissue maintenance

To maintain and regenerate adult tissues after injury, division and differentiation of tissue-resident stem cells must be precisely regulated. It remains elusive which regulatory strategies prevent exhaustion or overgrowth of the stem cell pool, whether there is coordination between multiple mechanisms, and how to detect them from snapshots. In Drosophila testes, somatic stem cells transition to a state that licenses them to differentiate, but remain capable of returning to the niche and resuming cell division. Here, we build stochastic mathematical models for the somatic stem cell population to investigate how licensing contributes to homeostasis. We find that licensing, in combination with differentiation occurring in pairs, is sufficient to maintain homeostasis and prevent stem cell extinction from stochastic fluctuations. Experimental data have shown that stem cells are competing for niche access, and our mathematical models demonstrate that this contributes to the reduction in the variability of stem cell numbers but does not prevent extinction. Hence, a combination of both regulation strategies, licensing with pairwise differentiation and competition for niche access, may be needed to reduce variability and prevent extinction simultaneously.

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.

Regulation of stem cell dynamics through volume exclusion

The maintenance and regeneration of adult tissues rely on the self-renewal of stem cells. Regeneration without over-proliferation requires precise regulation of the stem cell proliferation and differentiation rates. The nature of such regulatory mechanisms in different tissues, and how to incorporate them in models of stem cell population dynamics, is incompletely understood. The critical birth-death (CBD) process is widely used to model stem cell populations, capturing key phenomena, such as scaling laws in clone size distributions. However, the CBD process neglects regulatory mechanisms. Here, we propose the birth-death process with volume exclusion (vBD), a variation of the birth-death process that considers crowding effects, such as may arise due to limited space in a stem cell niche. While the deterministic rate equations predict a single non-trivial attracting steady state, the master equation predicts extinction and transient distributions of stem cell numbers with three possible behaviours: long-lived quasi-steady state (QSS), and short-lived bimodal or unimodal distributions. In all cases, we approximate solutions to the vBD master equation using a renormalized system-size expansion, QSS approximation and the Wentzel–Kramers–Brillouin method. Our study suggests that the size distribution of a stem cell population bears signatures that are useful to detect negative feedback mediated via volume exclusion.

Mathematical modelling identifies conditions for maintaining and escaping feedback control in the intestinal epithelium

The intestinal epithelium is one of the fastest renewing tissues in mammals. It shows a hierarchical organisation, where intestinal stem cells at the base of crypts give rise to rapidly dividing transit amplifying cells that in turn renew the pool of short-lived differentiated cells. Upon injury and stem-cell loss, cells can also de-differentiate. Tissue homeostasis requires a tightly regulated balance of differentiation and stem cell proliferation, and failure can lead to tissue extinction or to unbounded growth and cancerous lesions. Here, we present a two-compartment mathematical model of intestinal epithelium population dynamics that includes a known feedback inhibition of stem cell differentiation by differentiated cells. The model shows that feedback regulation stabilises the number of differentiated cells as these become invariant to changes in their apoptosis rate. Stability of the system is largely independent of feedback strength and shape, but specific thresholds exist which if bypassed cause unbounded growth. When dedifferentiation is added to the model, we find that the system can recover faster after certain external perturbations. However, dedifferentiation makes the system more prone to losing homeostasis. Taken together, our mathematical model shows how a feedback-controlled hierarchical tissue can maintain homeostasis and can be robust to many external perturbations.

Effect of feedback regulation on stem cell fractions in tissues and tumors: Understanding chemoresistance in cancer

While resistance mutations are often implicated in the failure of cancer therapy, lack of response also occurs without such mutants. In bladder cancer mouse xenografts, repeated chemotherapy cycles have resulted in cancer stem cell (CSC) enrichment, and consequent loss of therapy response due to the reduced susceptibility of CSCs to drugs. A particular feedback loop present in the xenografts has been shown to promote CSC enrichment in this system. Yet, many other regulatory loops might also be operational and might promote CSC enrichment. Their identification is central to improving therapy response. Here, we perform a comprehensive mathematical analysis to define what types of regulatory feedback loops can and cannot contribute to CSC enrichment, providing guidance to the experimental identification of feedback molecules. We derive a formula that reveals whether or not the cell population experiences CSC enrichment over time, based on the properties of the feedback. We find that negative feedback on the CSC division rate or positive feedback on differentiated cell death rate can lead to CSC enrichment. Further, the feedback mediators that achieve CSC enrichment can be secreted by either CSCs or by more differentiated cells. The extent of enrichment is determined by the CSC death rate, the CSC self-renewal probability, and by feedback strength. Defining these general characteristics of feedback loops can guide the experimental screening for and identification of feedback mediators that can promote CSC enrichment in bladder cancer and potentially other tumors. This can help understand and overcome the phenomenon of CSC-based therapy resistance.

Modelling stem cell ageing: a multi-compartment continuum approach

Stem cells are important to generate all specialized tissues at an early life stage, and in some systems, they also have repair functions to replenish the adult tissues. Repeated cell divisions lead to the accumulation of molecular damage in stem cells, which are commonly recognized as drivers of ageing. In this paper, a novel model is proposed to integrate stem cell proliferation and differentiation with damage accumulation in the stem cell ageing process. A system of two structured PDEs is used to model the population densities of stem cells (including all multiple progenitors) and terminally differentiated (TD) cells. In this system, cell cycle progression and damage accumulation are modelled by continuous dynamics, and damage segregation between daughter cells is considered at each division. Analysis and numerical simulations are conducted to study the steady-state populations and stem cell damage distributions under different damage segregation strategies. Our simulations suggest that equal distribution of the damaging substance between stem cells in a symmetric renewal and less damage retention in stem cells in the asymmetric division are favourable strategies, which reduce the death rate of the stem cells and increase the TD cell populations. Moreover, asymmetric damage segregation in stem cells leads to less concentrated damage distribution in the stem cell population, which may be more robust to the stochastic changes in the damage. The feedback regulation from stem cells can reduce oscillations and population overshoot in the process, and improve the fitness of stem cells by increasing the percentage of cells with less damage in the stem cell population.

Development of Aspirin-Inducible Biosensors in Escherichia coli and SimCells

A simple aspirin-inducible system has been developed and characterized in Escherichia coli by employing the P_sal promoter and SalR regulation system originally from Acinetobacter baylyi ADP1. Mutagenesis at the DNA binding domain (DBD) and chemical recognition domain (CRD) of the SalR protein in A. baylyi ADP1 suggests that the effector-free form, SalR_r, can compete with the effector-bound form, SalR_a, binding the P_sal promoter and repressing gene transcription. The induction of the P_sal promoter was compared in two different gene circuit designs: a simple regulation system (SRS) and positive autoregulation (PAR). Both regulatory circuits were induced in a dose-dependent manner in the presence of 0.05 to 10 µM aspirin. Overexpression of SalR in the SRS circuit reduced both baseline leakiness and the strength of the P_sal promoter. The PAR circuit forms a positive feedback loop that fine-tunes the level of SalR. A mathematical simulation based on the SalR_r/SalR_a competitive binding model not only fit the observed experimental results in SRS and PAR circuits but also predicted the performance of a new gene circuit design for which weak expression of SalR in the SRS circuit should significantly improve induction strength. The experimental result is in good agreement with this prediction, validating the SalR_r/SalR_a competitive binding model. The aspirin-inducible systems were also functional in probiotic strain E. coli Nissle 1917 and SimCells produced from E. coli MC1000 ΔminD These well-characterized and modularized aspirin-inducible gene circuits would be useful biobricks for synthetic biology.IMPORTANCE An aspirin-inducible SalR/P_sal regulation system, originally from Acinetobacter baylyi ADP1, has been designed for E. coli strains. SalR is a typical LysR-type transcriptional regulator (LTTR) family protein and activates the P_sal promoter in the presence of aspirin or salicylate in the range of 0.05 to 10 µM. The experimental results and mathematical simulations support the competitive binding model of the SalR/P_sal regulation system in which SalR_r competes with SalR_a to bind the P_sal promoter and affect gene transcription. The competitive binding model successfully predicted that weak SalR expression would significantly improve the inducible strength of the SalR/P_sal regulation system, which is confirmed by the experimental results. This provides an important mechanism model to fine-tune transcriptional regulation of the LTTR family, which is the largest family of transcriptional regulators in the prokaryotic kingdom. In addition, the SalR/P_sal regulation system was also functional in probiotic strain E. coli Nissle 1917 and minicell-derived SimCells, which would be a useful biobrick for environmental and medical applications.

A computational model of feedback-mediated hematopoietic stem cell differentiation in vitro

Hematopoietic stem cells (HSCs) play an important physiological role as regulators of all blood and immune cell populations, and are of clinical importance for bone marrow transplants. Regulating HSC biology in vitro for clinical applications requires improved understanding of biological inducers of HSC lineage specification. A significant challenge for controlled HSC expansion and differentiation is the complex network of molecular crosstalk between multiple bone marrow niche components influencing HSC biology. We describe a biology-driven computational approach to model cell kinetics in vitro to gain new insight regarding culture conditions and intercellular signaling networks. We further investigate the balance between self-renewal and differentiation that drives early and late hematopoietic progenitor populations. We demonstrate that changing the feedback driven by cell-secreted biomolecules alters lineage specification in early progenitor populations. Using a first order deterministic model, we are able to predict the impact of media change frequency on cell kinetics, as well as distinctions between primitive long-term HSCs and differentiated myeloid progenitors. Integrating the computational model and sensitivity analyses we identify critical culture parameters for regulating HSC proliferation and myeloid lineage specification. Our analysis suggests that accurately modeling the kinetics of hematopoietic sub-populations in vitro requires direct contributions from early progenitor differentiation along with the more traditionally considered intermediary oligopotent progenitors. While consistent with recent in vivo results, this work suggests the need to revise our perspective on HSC lineage engineering in vitro for expansion of discrete hematopoietic populations.

Effect of cellular de-differentiation on the dynamics and evolution of tissue and tumor cells in mathematical models with feedback regulation

Tissues are maintained by adult stem cells that self-renew and also differentiate into functioning tissue cells. Homeostasis is achieved by a set of complex mechanisms that involve regulatory feedback loops. Similarly, tumors are believed to be maintained by a minority population of cancer stem cells, while the bulk of the tumor is made up of more differentiated cells, and there is indication that some of the feedback loops that operate in tissues continue to be functional in tumors. Mathematical models of such tissue hierarchies, including feedback loops, have been analyzed in a variety of different contexts. Apart from stem cells giving rise to differentiated cells, it has also been observed that more differentiated cells can de-differentiate into stem cells, both in healthy tissue and tumors, aspects of which have also been investigated mathematically. This paper analyses the effect of de-differentiation on the basic and evolutionary dynamics of cells in the context of tissue hierarchy models that include negative feedback regulation of the cell populations. The models predict that in the presence of de-differentiation, the fixation probability of a neutral mutant is lower than in its absence. Therefore, if de-differentiation occurs, a mutant with identical parameters compared to the wild-type cell population behaves like a disadvantageous mutant. Similarly, the process of de-differentiation is found to lower the fixation probability of an advantageous mutant. These results indicate that the presence of de-differentiation can lower the rates of tumor initiation and progression in the context of the models considered here.

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.

Multi-scale modeling of APC and [Formula: see text]-catenin regulation in the human colonic crypt

Stem cell renewal and differentiation in the human colonic crypt are linked to the [Formula: see text]-catenin pathway. The spatial balance of Wnt factors in proliferative cells within the crypt maintain an appropriate level of cellular reproduction needed for normal crypt homeostasis. Mutational events at the gene level are responsible for deregulating the balance of Wnt factors along the crypt, causing an overpopulation of proliferative cells, a loss of structure of the crypt domain, and the initiation of colorectal carcinomas. We formulate a PDE model describing cell movement and reproduction in a static crypt domain. We consider a single cell population whose proliferative capabilities are determined by stemness, a quantity defined by intracellular levels of adenomatous polyposis coli (APC) scaffold protein and [Formula: see text]-catenin. We fit APC regulation parameters to biological data that describe normal protein gradients in the crypt. We also fit cell movement and protein flux parameters to normal crypt characteristics such as renewal time, total cell count, and proportion of proliferating cells. The model is used to investigate abnormal crypt dynamics when subjected to a diminished APC gradient, a scenario synonymous to mutations in the APC gene. We find that a 25% decrease in APC synthesis leads to a fraction of 0.88 proliferative, which is reflective of normal-appearing FAP crypts. A 50% drop in APC activity yields a fully proliferative crypt showing a doubling of the level of stemness, which characterizes the initial stages of colorectal cancer development. A sensitivity analysis of APC regulation parameters shows the perturbation of factors that is required to restore crypt dynamics to normal in the case of APC mutations.

Cellular Hierarchy as a Determinant of Tumor Sensitivity to Chemotherapy

Chemotherapy has been shown to enrich cancer stem cells in tumors. Recently, we demonstrated that administration of chemotherapy to human bladder cancer xenografts could trigger a wound-healing response that mobilizes quiescent tumor stem cells into active proliferation. This phenomenon leads to a loss of sensitivity to chemotherapy partly due to an increase in the number of tumor stem cells, which typically respond to chemotherapy-induced cell death less than more differentiated cells. Different bladder cancer xenografts, however, demonstrate differential sensitivities to chemotherapy, the basis of which is not understood. Using mathematical models, we show that characteristics of the tumor cell hierarchy can be crucial for determining the sensitivity of tumors to drug therapy, under the assumption that stem cell enrichment is the primary basis for drug resistance. Intriguingly, our model predicts a weaker response to therapy if there is negative feedback from differentiated tumor cells that inhibits the rate of tumor stem cell division. If this negative feedback is less pronounced, the treatment response is predicted to be enhanced. The reason is that negative feedback on the rate of tumor cell division promotes a permanent rise of the tumor stem cell population over time, both in the absence of treatment and even more so during drug therapy. Model application to data from chemotherapy-treated patient-derived xenografts indicates support for model predictions. These findings call for further research into feedback mechanisms that might remain active in cancers and potentially highlight the presence of feedback as an indication to combine chemotherapy with approaches that limit the process of tumor stem cell enrichment. Cancer Res; 77(9); 2231-41. ©2017 AACR.

Near Equilibrium Calculus of Stem Cells in Application to the Airway Epithelium Lineage

Homeostatic maintenance of tissues is orchestrated by well tuned networks of cellular signaling. Such networks regulate, in a stochastic manner, fates of all cells within the respective lineages. Processes such as symmetric and asymmetric divisions, differentiation, de-differentiation, and death have to be controlled in a dynamic fashion, such that the cell population is maintained at a stable equilibrium, has a sufficiently low level of stochastic variation, and is capable of responding efficiently to external damage. Cellular lineages in real tissues may consist of a number of different cell types, connected by hierarchical relationships, albeit not necessarily linear, and engaged in a number of different processes. Here we develop a general mathematical methodology for near equilibrium studies of arbitrarily complex hierarchical cell populations, under regulation by a control network. This methodology allows us to (1) determine stability properties of the network, (2) calculate the stochastic variance, and (3) predict how different control mechanisms affect stability and robustness of the system. We demonstrate the versatility of this tool by using the example of the airway epithelium lineage. Recent research shows that airway epithelium stem cells divide mostly asymmetrically, while the so-called secretory cells divide predominantly symmetrically. It further provides quantitative data on the recovery dynamics of the airway epithelium, which can include secretory cell de-differentiation. Using our new methodology, we demonstrate that while a number of regulatory networks can be compatible with the observed recovery behavior, the observed division patterns of cells are the most optimal from the viewpoint of homeostatic lineage stability and minimizing the variation of the cell population size. This not only explains the observed yet poorly understood features of airway tissue architecture, but also helps to deduce the information on the still largely hypothetical regulatory mechanisms governing tissue turnover, and lends insight into how different control loops influence the stability and variance properties of cell populations.

Tissue stability is the basic property of healthy organs, and yet the mechanisms governing the stable, long-term maintenance of cell numbers in tissues are poorly understood. While more and more signaling pathways are being discovered, for the most part it remains unknown how they are being put together by different cell types into complex, nonlinear, hierarchical control networks that, on the one hand, reliably maintain constant cell numbers, and on the other hand, quickly adjust to oversee the robust response to tissue damage. Theoretical approaches can fill the gap by being able to reconstruct the underlying control network, based on the observations about the aspects of cellular dynamics. We argue that while many hypothetical networks may be capable of basic cell lineage maintenance, some are much more efficient from the viewpoint of variance minimization. Thus, we developed a new methodology that can test various control networks for stability, variance, and robustness. In the example of the airway epithelium that we highlight, it turns out that the evolutionary selected, actual architecture coincides with the mathematically optimal solution that minimizes the fluctuations of cell numbers at homeostasis.

Characterizing inhibited tumor growth in stem-cell-driven non-spatial cancers

Healthy human tissue is highly regulated to maintain homeostasis. Secreted negative feedback factors that inhibit stem cell division and stem cell self-renewal play a fundamental role in establishing this control. The appearance of abnormal cancerous growth requires an escape from these regulatory mechanisms. In a previous study we found that for non-solid tumors if feedback inhibition on stem cell self-renewal is lost, but the feedback on the division rate is still intact, then the tumor dynamics are characterized by a relatively slow sub-exponential growth that we called inhibited growth. Here we characterize the cell dynamics of inhibited cancer growth by modeling feedback inhibition using Hill equations. We find asymptotic approximations for the growth rates of the stem cell and differentiated cell populations in terms of the strength of the inhibitory signal: stem cells grow as a power law t(1/k+1),and the differentiated cells grow as t(1/k), where k is the Hill coefficient in the feedback law regulating cell divisions. It follows that as the tumor grows, undifferentiated cells take up an increasingly large fraction of the population. Implications of these results for specific cancers including CML are discussed. Understanding how the regulatory mechanisms that continue to operate in cancer affect the rate of disease progression can provide important insights relevant to chronic or other slow progressing types of cancer.

Stem cell regulation: Implications when differentiated cells regulate symmetric stem cell division

We use a mathematical model to show that if symmetric stem cell division is regulated by differentiated cells, then changes in the population dynamics of the differentiated cells can lead to changes in the population dynamics of the stem cells. More precisely, the relative fitness of the stem cells can be affected by modifying the death rate of the differentiated cells. This result is interesting because stem cells are less sensitive than differentiated cells to environmental factors, such as medical therapy. Our result implies that stem cells can be manipulated indirectly by medical treatments that target the differentiated cells.

An agent-based model of cancer stem cell initiated avascular tumour growth and metastasis: the effect of seeding frequency and location

It is very important to understand the onset and growth pattern of breast primary tumours as well as their metastatic dissemination. In most cases, it is the metastatic disease that ultimately kills the patient. There is increasing evidence that cancer stem cells are closely linked to the progression of the metastatic tumour. Here, we investigate stem cell seeding to an avascular tumour site using an agent-based stochastic model of breast cancer metastatic seeding. The model includes several important cellular features such as stem cell symmetric and asymmetric division, migration, cellular quiescence, senescence, apoptosis and cell division cycles. It also includes external features such as stem cell seeding frequency and location. Using this model, we find that cell seeding rate and location are important features for tumour growth. We also define conditions in which the tumour growth exhibits decremented and exponential growth patterns. Overall, we find that seeding, senescence and division limit affect not only the number of stem cells, but also their spatial and temporal distribution.

Mathematical Modelling as a Tool to Understand Cell Self-renewal and Differentiation

Mathematical modeling is a powerful technique to address key questions and paradigms in a variety of complex biological systems and can provide quantitative insights into cell kinetics, fate determination and development of cell populations. The chapter is devoted to a review of modeling of the dynamics of stem cell-initiated systems using mathematical methods of ordinary differential equations. Some basic concepts and tools for cell population dynamics are summarized and presented as a gentle introduction to non-mathematicians. The models take into account different plausible mechanisms regulating homeostasis. Two mathematical frameworks are proposed reflecting, respectively, a discrete (punctuated by division events) and a continuous character of transitions between differentiation stages. Advantages and constraints of the mathematical approaches are presented on examples of models of blood systems and compared to patients data on healthy hematopoiesis.

Interactions and tradeoffs between cell recruitment, proliferation, and differentiation affect CNS regeneration

Regeneration of central nervous system (CNS) lesions requires movement of progenitor cells and production of their differentiated progeny. Although damage to the CNS clearly promotes these two processes, the interplay between these complex events and how it affects a response remains elusive. Here, we use spatial stochastic modeling to show that tradeoffs arise between production and recruitment during regeneration. Proper spatial control of cell cycle timing can mitigate these tradeoffs, maximizing recruitment, improving infiltration into the lesion, and reducing wasteful production outside of it. Feedback regulation of cell lineage dynamics alone however leads to spatial defects in cell recruitment, suggesting a novel, to our knowledge, hypothesis for the aggregation of cells to the periphery of a lesion in multiple sclerosis. Interestingly, stronger chemotaxis does not correct this aggregation and instead, substantial random cell motions near the site of the lesion are required to improve CNS regeneration.

Robustness of differentiation cascades with symmetric stem cell division

Stem cells (SCs) perform the task of maintaining tissue homeostasis by both self-renewal and differentiation. While it has been argued that SCs divide asymmetrically, there is also evidence that SCs undergo symmetric division. Symmetric SC division has been speculated to be key for expanding cell numbers in development and regeneration after injury. However, it might lead to uncontrolled growth and malignancies such as cancer. In order to explore the role of symmetric SC division, we propose a mathematical model of the effect of symmetric SC division on the robustness of a population regulated by a serial differentiation cascade and we show that this may lead to extinction of such population. We examine how the extinction likelihood depends on defining characteristics of the population such as the number of intermediate cell compartments. We show that longer differentiation cascades are more prone to extinction than systems with less intermediate compartments. Furthermore, we have analysed the possibility of mixed symmetric and asymmetric cell division against invasions by mutant invaders in order to find optimal architecture. Our results show that more robust populations are those with unfrequent symmetric behaviour.

Clonal selection and therapy resistance in acute leukaemias: mathematical modelling explains different proliferation patterns at diagnosis and relapse

Recent experimental evidence suggests that acute myeloid leukaemias may originate from multiple clones of malignant cells. Nevertheless, it is not known how the observed clones may differ with respect to cell properties, such as proliferation and self-renewal. There are scarcely any data on how these cell properties change due to chemotherapy and relapse. We propose a new mathematical model to investigate the impact of cell properties on the multi-clonal composition of leukaemias. Model results imply that enhanced self-renewal may be a key mechanism in the clonal selection process. Simulations suggest that fast proliferating and highly self-renewing cells dominate at primary diagnosis, while relapse following therapy-induced remission is triggered mostly by highly self-renewing but slowly proliferating cells. Comparison of simulation results to patient data demonstrates that the proposed model is consistent with clinically observed dynamics based on a clonal selection process.