Spatial dynamics of multistage cell lineages in tissue stratification

A model about regulation on three division modes of stem cell

We construct a multi-stage cell lineage model for cell division, apoptosis and movement. Cells are assumed to secrete and respond to negative feedback molecules which act as a control on the stem cell divisions (including self-renewal, asymmetrical cell division (ACD) and differentiation). The densities of cells and molecules are described by coupled reaction-diffusion partial differential equations, and the plane wavefront propagation speeds can be obtained analytically and verified numerically. It is found that with ACD the population and propagation of stem cells can be promoted but the negative regulation on self-renewal and differentiation will work slowly. Regulatory inhibition on differentiation will inversely increase stem cells but not affect the population and wave propagation of the cell lineage. While negative regulation on self-renewal and ACD will decrease the population of stem cells and slow down the propagation, and even drive stem cells to extinction. Moreover we find that inhibition on self-renewal has a strength advantage while inhibition on ACD has a range advantage to kill stem cells. Possible relations to model cancer development and therapy are also discussed.

A multiscale chemical-mechanical model predicts impact of morphogen spreading on tissue growth

The exact mechanism controlling cell growth remains a grand challenge in developmental biology and regenerative medicine. The Drosophila wing disc tissue serves as an ideal biological model to study mechanisms involved in growth regulation. Most existing computational models for studying tissue growth focus specifically on either chemical signals or mechanical forces. Here we developed a multiscale chemical-mechanical model to investigate the growth regulation mechanism based on the dynamics of a morphogen gradient. By comparing the spatial distribution of dividing cells and the overall tissue shape obtained in model simulations with experimental data of the wing disc, it is shown that the size of the domain of the Dpp morphogen is critical in determining tissue size and shape. A larger tissue size with a faster growth rate and more symmetric shape can be achieved if the Dpp gradient spreads in a larger domain. Together with Dpp absorbance at the peripheral zone, the feedback regulation that downregulates Dpp receptors on the cell membrane allows for further spreading of the morphogen away from its source region, resulting in prolonged tissue growth at a more spatially homogeneous growth rate.

A cell model about symmetric and asymmetric stem cell division

We construct a multi-stage cell lineage model including self-renewal, apoptosis, cell movement and the symmetrical/asymmetrical division of stem cells. The evolution of cell populations can be described by coupled reaction-diffusion partial differential equations, and the propagating wavefront speeds can be obtained analytically and verified by numerical solutions of the equations. The emphasis is on the effect of symmetric/asymmetric division of stem cells on the population and propagating dynamics of cell lineage. It is found that stem cells' asymmetric cell division (ACD) can move the phase boundary of the homogenous solution of the system. The population of the cell lineage will be promoted in presence of ACD. The concentration of stem cells increases with ACD but that of differentiated daughter cells decreases with ACD. In addition, it is found that the propagating speed of the stem cells can be evaluated with ACD. When the daughter cells move fast to a new space, stem cells can catch them up through increasing ACD. Our results may suggest a mechanism of collective migration of cell lineage through cooperation between ACD of stem cells and fast diffusion of the daughter cells.

Spatial dynamics of feedback and feedforward regulation in cell lineages

Feedback mechanisms within cell lineages are thought to be important for maintaining tissue homeostasis. Mathematical models that assume well-mixed cell populations, together with experimental data, have suggested that negative feedback from differentiated cells on the stem cell self-renewal probability can maintain a stable equilibrium and hence homeostasis. Cell lineage dynamics, however, are characterized by spatial structure, which can lead to different properties. Here, we investigate these dynamics using spatially explicit computational models, including cell division, differentiation, death, and migration / diffusion processes. According to these models, the negative feedback loop on stem cell self-renewal fails to maintain homeostasis, both under the assumption of strong spatial restrictions and fast migration / diffusion. Although homeostasis cannot be maintained, this feedback can regulate cell density and promote the formation of spatial structures in the model. Tissue homeostasis, however, can be achieved if spatially restricted negative feedback on self-renewal is combined with an experimentally documented spatial feedforward loop, in which stem cells regulate the fate of transit amplifying cells. This indicates that the dynamics of feedback regulation in tissue cell lineages are more complex than previously thought, and that combinations of spatially explicit control mechanisms are likely instrumental.

Regulatory feedback effects on tissue growth dynamics in a two-stage cell lineage model

Identifying the mechanism of intercellular feedback regulation is critical for the basic understanding of tissue growth control in organisms. In this paper, we analyze a tissue growth model consisting of a single lineage of two cell types regulated by negative feedback signaling molecules that undergo spatial diffusion. By deriving the fixed points for the uniform steady states and carrying out linear stability analysis, phase diagrams are obtained analytically for arbitrary parameters of the model. Two different generic growth modes are found: blow-up growth and final-state controlled growth which are governed by the nontrivial fixed point and the trivial fixed point, respectively, and can be sensitively switched by varying the negative feedback regulation on the proliferation of the stem cells. Analytic expressions for the characteristic timescales for these two growth modes are also derived. Remarkably, the trivial and nontrivial uniform steady states can coexist and a sharp transition occurs in the bistable regime as the relevant parameters are varied. Furthermore, the bistable growth properties allows for the external control to switch between these two growth modes. In addition, the condition for an early accelerated growth followed by a retarded growth can be derived. These analytical results are further verified by numerical simulations and provide insights on the growth behavior of the tissue. Our results are also discussed in the light of possible realistic biological experiments and tissue growth control strategy. Furthermore, by external feedback control of the concentration of regulatory molecules, it is possible to achieve a desired growth mode, as demonstrated with an analysis of boosted growth, catch-up growth and the design for the target of a linear growth dynamic.

Multiple morphogens and rapid elongation promote segmental patterning during development

The vertebrate hindbrain is segmented into rhombomeres (r) initially defined by distinct domains of gene expression. Previous studies have shown that noise-induced gene regulation and cell sorting are critical for the sharpening of rhombomere boundaries, which start out rough in the forming neural plate (NP) and sharpen over time. However, the mechanisms controlling simultaneous formation of multiple rhombomeres and accuracy in their sizes are unclear. We have developed a stochastic multiscale cell-based model that explicitly incorporates dynamic morphogenetic changes (i.e. convergent-extension of the NP), multiple morphogens, and gene regulatory networks to investigate the formation of rhombomeres and their corresponding boundaries in the zebrafish hindbrain. During pattern initiation, the short-range signal, fibroblast growth factor (FGF), works together with the longer-range morphogen, retinoic acid (RA), to specify all of these boundaries and maintain accurately sized segments with sharp boundaries. At later stages of patterning, we show a nonlinear change in the shape of rhombomeres with rapid left-right narrowing of the NP followed by slower dynamics. Rapid initial convergence improves boundary sharpness and segment size by regulating cell sorting and cell fate both independently and coordinately. Overall, multiple morphogens and tissue dynamics synergize to regulate the sizes and boundaries of multiple segments during development.

In segmental pattern formation, chemical gradients control gene expression in a concentration-dependent manner to specify distinct gene expression domains. Despite the stochasticity inherent to such biological processes, precise and accurate borders form between segmental gene expression domains. Previous work has revealed synergy between gene regulation and cell sorting in sharpening borders that are initially rough. However, it is still poorly understood how size and boundary sharpness of multiple segments are regulated in a tissue that changes dramatically in its morphology as the embryo develops. Here we develop a stochastic multiscale cell-base model to investigate these questions. Two novel strategies synergize to promote accurate segment formation, a combination of long- and short-range morphogens plus rapid tissue convergence, with one responsible for pattern initiation and the other enabling pattern refinement.

Potential Osteoinductive Effects of Hydroxyapatite Nanoparticles on Mesenchymal Stem Cells by Endothelial Cell Interaction

Nano-hydroxyapatite (nano-HA) has attracted substantial attention in the field of regenerative medicine. Endothelial cell (EC)-mesenchymal stem cell (MSC) interactions are necessary for bone reconstruction, but the manner in which nano-HA interacts in this process remains unknown. Herein, we investigated the cytotoxicity and osteoinductive effects of HA nanoparticles (HANPs) on MSCs using an indirect co-culture model mediated by ECs and highlighted the underlying mechanisms. It was found that at a subcytotoxic dose, HANPs increased the viability and expression of osteoblast genes, as well as mineralized nodules and alkaline phosphatase production of MSCs. These phenomena relied on HIF-1α secreted by ECs, which triggered the ERK1/2 signaling cascade. In addition, a two-stage cell-lineage mathematical model was established to quantitatively analyze the impact of HIF-1α on the osteogenic differentiation of MSCs. It demonstrated that HIF-1α exerted a dose-dependent stimulatory effect on the osteogenic differentiation rate of MSCs up to 1500 pg/mL, which was in agreement with the above results. Our data implied that cooperative interactions between HANPs, ECs, and MSCs likely serve to stimulate bone regeneration. Furthermore, the two-stage cell-lineage model is helpful in vitro system for assessing the potential influence of effector molecules in bone tissue engineering.

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.

STOCHASTIC DYNAMICS OF CELL LINEAGE IN TISSUE HOMEOSTASIS

During epithelium tissue maintenance, lineages of cells differentiate and proliferate in a coordinated way to provide the desirable size and spatial organization of different types of cells. While mathematical models through deterministic description have been used to dissect role of feedback regulations on tissue layer size and stratification, how the stochastic effects influence tissue maintenance remains largely unknown. Here we present a stochastic continuum model for cell lineages to investigate how both layer thickness and layer stratification are affected by noise. We find that the cell-intrinsic noise often causes reduction and oscillation of layer size whereas the cell-extrinsic noise increases the thickness, and sometimes, leads to uncontrollable growth of the tissue layer. The layer stratification usually deteriorates as the noise level increases in the cell lineage systems. Interestingly, the morphogen noise, which mixes both cell-intrinsic noise and cell-extrinsic noise, can lead to larger size of layer with little impact on the layer stratification. By investigating different combinations of the three types of noise, we find the layer thickness variability is reduced when cell-extrinsic noise level is high or morphogen noise level is low. Interestingly, there exists a tradeoff between low thickness variability and strong layer stratification due to competition among the three types of noise, suggesting robust layer homeostasis requires balanced levels of different types of noise in the cell lineage systems.

Multiscale modeling of layer formation in epidermis

The mammalian skin epidermis is a stratified epithelium composed of multiple layers of epithelial cells that exist in appropriate sizes and proportions, and with distinct boundaries separating each other. How the epidermis develops from a single layer of committed precursor cells to form a complex multilayered structure of multiple cell types remains elusive. Here, we construct stochastic, three-dimensional, and multiscale models consisting of a lineage of multiple cell types to study the control of epidermal development. Symmetric and asymmetric cell divisions, stochastic cell fate transitions within the lineage, extracellular morphogens, cell-to-cell adhesion forces, and cell signaling are included in model. A GPU algorithm was developed and implemented to accelerate the simulations. These simulations show that a balance between cell proliferation and differentiation during lineage progression is crucial for the development and maintenance of the epidermal tissue. We also find that selective intercellular adhesion is critical to sharpening the boundary between layers and to the formation of a highly ordered structure. The long-range action of a morphogen provides additional feedback regulations, enhancing the robustness of overall layer formation. Our model is built upon previous experimental findings revealing the role of Ovol transcription factors in regulating epidermal development. Direct comparisons of experimental and simulation perturbations show remarkable consistency. Taken together, our results highlight the major determinants of a well-stratified epidermis: balanced proliferation and differentiation, and a combination of both short- (symmetric/asymmetric division and selective cell adhesion) and long-range (morphogen) regulations. These underlying principles have broad implications for other developmental or regenerative processes leading to the formation of multilayered tissue structures, as well as for pathological processes such as epidermal wound healing.

Epidermal morphogenesis, which occurs during the second half of embryogenesis, is the developmental process that generates a skin permeability barrier essential for terrestrial survival. Defects with this barrier are associated with common skin disorders such as atopic dermatitis. Study of mechanisms that control epidermal development and differentiation is therefore highly relevant to human health. Motivated by recent experimental observations on the role of Ovol transcription factors in regulating epidermal development, we developed a multiscale model to investigate the underlying mechanisms responsible for epidermal layer formation and homeostasis. We report that regulation of proliferation and differentiation by Ovol plays an important role in epidermal development. In addition, our computational analysis shows that asymmetric cell division, selective cell adhesion, and morphogen regulation work in a synergetic manner to produce the well-stratified epidermal layers. Taken together, our results demonstrate that robust epidermal morphogenesis involves a balance between proliferation and differentiation, and an interplay between short- and long-range spatial control mechanisms. This principle may also be applicable to other complex systems of tissue development or regeneration.

Hydrodynamics of stratified epithelium: Steady state and linearized dynamics

A theoretical model for stratified epithelium is presented. The viscoelastic properties of the tissue are assumed to be dependent on the spatial distribution of proliferative and differentiated cells. Based on this assumption, a hydrodynamic description of tissue dynamics at the long-wavelength, long-time limit is developed, and the analysis reveals important insights into the dynamics of an epithelium close to its steady state. When the proliferative cells occupy a thin region close to the basal membrane, the relaxation rate towards the steady state is enhanced by cell division and cell apoptosis. On the other hand, when the region where proliferative cells reside becomes sufficiently thick, a flow induced by cell apoptosis close to the apical surface enhances small perturbations. This destabilizing mechanism is general for continuous self-renewal multilayered tissues; it could be related to the origin of certain tissue morphology, tumor growth, and the development pattern.

Quantifying Lgr5-positive stem cell behaviour in the pyloric epithelium

Using in-vivo lineage tracing data we quantified clonal expansion as well as proliferation and differentiation of the Lgr5-positive stem cell population in pyloric gastric glands. Fitting clone expansion models, we estimated that there are five effective Lgr5-positive cells able to give rise to monoclonal glands by replacing each other following a pattern of neutral drift dynamics. This analysis is instrumental to assess stem cell performance; however, stem cell proliferation is not quantified by clone expansion analysis. We identified a suitable mathematical model to quantify proliferation and differentiation of the Lgr5-positive population. As expected for populations in steady-state, the proliferation rate of the Lgr5-positive population was equal to its rate of differentiation. This rate was significantly faster than the rate at which effective cells are replaced, estimated by modelling clone expansion/contraction. This suggests that the majority of Lgr5-positive cell divisions serve to renew epithelial cells and only few result in the effective replacement of a neighbour to effect expansion to the entire gland. The application of the model under altered situations with uncoupled differentiation and proliferation was demonstrated. This methodology represents a valuable tool for quantifying stem cell performance in homeostasis and importantly for deciphering altered stem cell behaviour in disease.

Regulatory effects on the population dynamics and wave propagation in a cell lineage model

We consider the interplay of cell proliferation, cell differentiation (and de-differentiation), cell movement, and the effect of feedback regulations on the population and propagation dynamics of different cell types in a cell lineage model. Cells are assumed to secrete and respond to negative feedback molecules which act as a control on the cell lineage. The cell densities are described by coupled reaction-diffusion partial differential equations, and the propagating wave front solutions in one dimension are investigated analytically and by numerical solutions. In particular, wavefront propagation speeds are obtained analytically and verified by numerical solutions of the equations. The emphasis is on the effects of the feedback regulations on different stages in the cell lineage. It is found that when the progenitor cell is negatively regulated, the populations of the cell lineage are strongly down-regulated with the steady growth rate of the progenitor cell being driven to zero beyond a critical regulatory strength. An analytic expression for the critical regulation strength in terms of the model parameters is derived and verified by numerical solutions. On the other hand, if the inhibition is acting on the differentiated cells, the change in the population dynamics and wave propagation speed is small. In addition, it is found that only the propagating speed of the progenitor cells is affected by the regulation when the diffusion of the differentiated cells is large. In the presence of de-differentiation, the effect on down-regulating the progenitor population is weakened and there is no effect on the propagation speed due to regulation, suggesting that the effect of regulatory control is diminished by de-differentiation pathways.

Models, measurement and inference in epithelial tissue dynamics

The majority of solid tumours arise in epithelia and therefore much research effort has gone into investigating the growth, renewal and regulation of these tissues. Here we review different mathematical and computational approaches that have been used to model epithelia. We compare different models and describe future challenges that need to be overcome in order to fully exploit new data which present, for the first time, the real possibility for detailed model validation and comparison.

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.

Deciphering principles of morphogenesis from temporal and spatial patterns on the integument

BACKGROUND

How tissue patterns form in development and regeneration is a fundamental issue remaining to be fully understood. The integument often forms repetitive units in space (periodic patterning) and time (cyclic renewal), such as feathers and hairs. Integument patterns are visible and experimentally manipulatable, helping us reveal pattern formative processes. Variability is seen in regional phenotypic specificities and temporal cycling at different physiological stages.

RESULTS

Here we show some cellular/molecular bases revealed by analyzing integument patterns. (1) Localized cellular activity (proliferation, rearrangement, apoptosis, differentiation) transforms prototypic organ primordia into specific shapes. Combinatorial positioning of different localized activity zones generates diverse and complex organ forms. (2) Competitive equilibrium between activators and inhibitors regulates stem cells through cyclic quiescence and activation.

CONCLUSIONS

Dynamic interactions between stem cells and their adjacent niche regulate regenerative behavior, modulated by multi-layers of macro-environmental factors (dermis, body hormone status, and external environment). Genomics studies may reveal how positional information of localized cellular activity is stored. In vivo skin imaging and lineage tracing unveils new insights into stem cell plasticity. Principles of self-assembly obtained from the integumentary organ model can be applied to help restore damaged patterns during regenerative wound healing and for tissue engineering to rebuild tissues. Developmental Dynamics 244:905-920, 2015. © 2015 Wiley Periodicals, Inc.

Stochastic control of proliferation and differentiation in stem cell dynamics

In self-renewing tissues, cell lineages consisting of stem cell and classes of daughter cells are the basic units which are responsible for the correct functioning of the organ. Cell proliferation and differentiation in lineages is thought to be mediated by feedback signals. In the simplest case a lineage is comprised of stem cells and differentiated cells. We create a model where stem cell proliferation and differentiation are controlled by the size of cell populations by means of a negative feedback loop. This two-dimensional Markov process allows for an analytical solution for the mean numbers and variances of stem and daughter cells. The mean values and the amounts of variation in cell numbers can be tightly regulated by the parameters of the control loop.

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.

A mathematical-biological joint effort to investigate the tumor-initiating ability of Cancer Stem Cells

The involvement of Cancer Stem Cells (CSCs) in tumor progression and tumor recurrence is one of the most studied subjects in current cancer research. The CSC hypothesis states that cancer cell populations are characterized by a hierarchical structure that affects cancer progression. Due to the complex dynamics involving CSCs and the other cancer cell subpopulations, a robust theory explaining their action has not been established yet. Some indications can be obtained by combining mathematical modeling and experimental data to understand tumor dynamics and to generate new experimental hypotheses. Here, we present a model describing the initial phase of ErbB2⁺ mammary cancer progression, which arises from a joint effort combing mathematical modeling and cancer biology. The proposed model represents a new approach to investigate the CSC-driven tumorigenesis and to analyze the relations among crucial events involving cancer cell subpopulations. Using in vivo and in vitro data we tuned the model to reproduce the initial dynamics of cancer growth, and we used its solution to characterize observed cancer progression with respect to mutual CSC and progenitor cell variation. The model was also used to investigate which association occurs among cell phenotypes when specific cell markers are considered. Finally, we found various correlations among model parameters which cannot be directly inferred from the available biological data and these dependencies were used to characterize the dynamics of cancer subpopulations during the initial phase of ErbB2⁺ mammary cancer progression.

Fluctuations of cell population in a colonic crypt

The number of stem cells in a colonic crypt is often very small, which leads to large intrinsic fluctuations in the cell population. Based on the model of cell population dynamics with linear feedback in a colonic crypt, we present a stochastic dynamics of the cell population [including stem cells (SCs), transit amplifying cells (TACs), and fully differentiated cells (FDCs)]. The Fano factor, covariance, and susceptibility formulas of the cell population around the steady state are derived by using the Langevin theory. In the range of physiologically reasonable parameter values, it is found that the stationary populations of TACs and FDCs exhibit an approximately threshold behavior as a function of the net growth rate of TACs, and the reproductions of TACs and FDCs can be classified into three regimens: controlled, crossover, and uncontrolled. With the increasing of the net growth rate of TACs, there is a maximum of the relative intrinsic fluctuations (i.e., the Fano factors) of TACs and FDCs in the crossover region. For a fixed differentiation rate and the net growth rate of SCs, the covariance of fluctuations between SCs and TACs has a maximum in the crossover region. However, the susceptibilities of both TACs and FDCs to the net growth rate of TACs have a minimum in the crossover region.

Mathematical model of adult stem cell regeneration with cross-talk between genetic and epigenetic regulation

Adult stem cells, which exist throughout the body, multiply by cell division to replenish dying cells or to promote regeneration to repair damaged tissues. To perform these functions during the lifetime of organs or tissues, stem cells need to maintain their populations in a faithful distribution of their epigenetic states, which are susceptible to stochastic fluctuations during each cell division, unexpected injury, and potential genetic mutations that occur during many cell divisions. However, it remains unclear how the three processes of differentiation, proliferation, and apoptosis in regulating stem cells collectively manage these challenging tasks. Here, without considering molecular details, we propose a genetic optimal control model for adult stem cell regeneration that includes the three fundamental processes, along with cell division and adaptation based on differential fitnesses of phenotypes. In the model, stem cells with a distribution of epigenetic states are required to maximize expected performance after each cell division. We show that heterogeneous proliferation that depends on the epigenetic states of stem cells can improve the maintenance of stem cell distributions to create balanced populations. A control strategy during each cell division leads to a feedback mechanism involving heterogeneous proliferation that can accelerate regeneration with less fluctuation in the stem cell population. When mutation is allowed, apoptosis evolves to maximize the performance during homeostasis after multiple cell divisions. The overall results highlight the importance of cross-talk between genetic and epigenetic regulation and the performance objectives during homeostasis in shaping a desirable heterogeneous distribution of stem cells in epigenetic states.

Temporal control of self-organized pattern formation without morphogen gradients in bacteria

Diverse mechanisms have been proposed to explain biological pattern formation. Regardless of their specific molecular interactions, the majority of these mechanisms require morphogen gradients as the spatial cue, which are either predefined or generated as a part of the patterning process. However, using Escherichia coli programmed by a synthetic gene circuit, we demonstrate here the generation of robust, self-organized ring patterns of gene expression in the absence of an apparent morphogen gradient. Instead of being a spatial cue, the morphogen serves as a timing cue to trigger the formation and maintenance of the ring patterns. The timing mechanism enables the system to sense the domain size of the environment and generate patterns that scale accordingly. Our work defines a novel mechanism of pattern formation that has implications for understanding natural developmental processes.

Principles of regulation of self-renewing cell lineages

Identifying the exact regulatory circuits that can stably maintain tissue homeostasis is critical for our basic understanding of multicellular organisms, and equally critical for identifying how tumors circumvent this regulation, thus providing targets for treatment. Despite great strides in the understanding of the molecular components of stem-cell regulation, the overall mechanisms orchestrating tissue homeostasis are still far from being understood. Typically, tissue contains the stem cells, transit amplifying cells, and terminally differentiated cells. Each of these cell types can potentially secrete regulatory factors and/or respond to factors secreted by other types. The feedback can be positive or negative in nature. This gives rise to a bewildering array of possible mechanisms that drive tissue regulation. In this paper, we propose a novel method of studying stem cell lineage regulation, and identify possible numbers, types, and directions of control loops that are compatible with stability, keep the variance low, and possess a certain degree of robustness. For example, there are exactly two minimal (two-loop) control networks that can regulate two-compartment (stem and differentiated cell) tissues, and 20 such networks in three-compartment tissues. If division and differentiation decisions are coupled, then there must be a negative control loop regulating divisions of stem cells (e.g. by means of contact inhibition). While this mechanism is associated with the highest robustness, there could be systems that maintain stability by means of positive divisions control, coupled with specific types of differentiation control. Some of the control mechanisms that we find have been proposed before, but most of them are new, and we describe evidence for their existence in data that have been previously published. By specifying the types of feedback interactions that can maintain homeostasis, our mathematical analysis can be used as a guide to experimentally zero in on the exact molecular mechanisms in specific tissues.

Model on cell movement, growth, differentiation and de-differentiation: reaction-diffusion equation and wave propagation

We construct a model for cell proliferation with differentiation into different cell types, allowing backward de-differentiation and cell movement. With different cell types labeled by state variables, the model can be formulated in terms of the associated transition probabilities between various states. The cell population densities can be described by coupled reaction-diffusion partial differential equations, allowing steady wavefront propagation solutions. The wavefront profile is calculated analytically for the simple pure growth case (2-states), and analytic expressions for the steady wavefront propagating speeds and population growth rates are obtained for the simpler cases of 2-, 3- and 4-states systems. These analytic results are verified by direct numerical solutions of the reaction-diffusion PDEs. Furthermore, in the absence of de-differentiation, it is found that, as the mobility and/or self-proliferation rate of the down-lineage descendant cells become sufficiently large, the propagation dynamics can switch from a steady propagating wavefront to the interesting situation of propagation of a faster wavefront with a slower waveback. For the case of a non-vanishing de-differentiation probability, the cell growth rate and wavefront propagation speed are both enhanced, and the wavefront speeds can be obtained analytically and confirmed by numerical solution of the reaction-diffusion equations.

Numerical Methods for Two-Dimensional Stem Cell Tissue Growth

Growth of developing and regenerative biological tissues of different cell types is usually driven by stem cells and their local environment. Here, we present a computational framework for continuum tissue growth models consisting of stem cells, cell lineages, and diffusive molecules that regulate proliferation and differentiation through feedback. To deal with the moving boundaries of the models in both open geometries and closed geometries (through polar coordinates) in two dimensions, we transform the dynamic domains and governing equations to fixed domains, followed by solving for the transformation functions to track the interface explicitly. Clustering grid points in local regions for better efficiency and accuracy can be achieved by appropriate choices of the transformation. The equations resulting from the incompressibility of the tissue is approximated by high-order finite difference schemes and is solved using the multigrid algorithms. The numerical tests demonstrate an overall spatiotemporal second-order accuracy of the methods and their capability in capturing large deformations of the tissue boundaries. The methods are applied to two biological systems: stratified epithelia for studying the effects of two different types of stem cell niches and the scaling of a morphogen gradient with the size of the Drosophila imaginal wing disc during growth. Direct simulations of both systems suggest that that the computational framework is robust and accurate, and it can incorporate various biological processes critical to stem cell dynamics and tissue growth.

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

Normal human tissue is organized into cell lineages, in which the highly differentiated mature cells that perform tissue functions are the end product of an orderly tissue-specific sequence of divisions that start with stem cells or progenitor cells. Tissue homeostasis and effective regeneration after injuries requires tight regulation of these cell lineages and feedback loops play a fundamental role in this regard. In particular, signals secreted from differentiated cells that inhibit stem cell division and stem cell self-renewal are important in establishing control. In this article we study in detail the cell dynamics that arise from this control mechanism. These dynamics are fundamental to our understanding of cancer, given that tumor initiation requires an escape from tissue regulation. Knowledge on the processes of cellular control can provide insights into the pathways that lead to deregulation and consequently cancer development.

Stem cell niche structure as an inherent cause of undulating epithelial morphologies

The spatial organization of stem cells into a niche is a key factor for growth and continual tissue renewal during development, sustenance, and regeneration. Stratified epithelia serve as a great context to study the spatial aspects of the stem cell niche and cell lineages by organizing into layers of different cell types. Several types of stratified epithelia develop morphologies with advantageous, protruding structures where stem cells reside, such as rete pegs and palisades of Vogt. Here, multistage, spatial cell lineage models for epithelial stratification are used to study how the stem cell niche influences epithelial morphologies. When the stem cell niche forms along a rigid basal lamina, relatively regular morphologies are maintained. In contrast, stem cell niche formation along a free-moving basal lamina may prompt distorted epithelial morphologies with stem cells accumulating at the tips of fingerlike structures that form. The correspondence between our simulated morphologies and developmental stages of the human epidermis is also explored. Overall, our work provides an understanding of how stratified epithelia may attain distorted morphologies and sheds light on the importance of the spatial aspects of the stem cell niche.

Engineered cell-cell communication and its applications

Over the past several decades, biologists have become more appreciative of the fundamental role of intercellular communication in natural systems spanning prokaryotic biofilms to eukaryotic developmental systems and neurological networks. From an engineering perspective, the use of cell-cell communication provides an opportunity to engineer more complex and robust functions using cellular components. Indeed, this strategy has been adopted in synthetic biology in the creation of diverse gene circuits that program spatiotemporal dynamics in one or multiple populations. Gene circuits such as these may offer insights regarding basic biological questions and motifs or serve as a basis for novel applications.

A reaction-diffusion mechanism influences cell lineage progression as a basis for formation, regeneration, and stability of intestinal crypts

Background

Colon crypts, a single sheet of epithelia cells, consist of a periodic pattern of stem cells, transit-amplifying cells, and terminally differentiated cells that constantly renew and turnover. Experimental evidence suggests that Wnt signaling promotes and regulates stem cell division, differentiation, and possible cell migrations while intestinal BMP signaling inhibits stem cell self-renewal and repression in crypt formation. As more molecular details on Wnt and BMP in crypts are being discovered, little is still known about how complex interactions among Wnt, BMP, and different types of cells, and surrounding environments may lead to de novo formation of multiple crypts or how such interactions affect regeneration and stability of crypts.

Results

We present a mathematical model that contains Wnt and BMP, a cell lineage, and their feedback regulations to study formation, regeneration, and stability of multiple crypts. The computational explorations and linear stability analysis of the model suggest a reaction–diffusion mechanism, which exhibits a short-range activation of Wnt plus a long-range inhibition with modulation of BMP signals in a growing tissue of cell lineage, can account for spontaneous formation of multiple crypts with the spatial and temporal pattern observed in experiments. Through this mechanism, the model can recapitulate some distinctive and important experimental findings such as crypt regeneration and crypt multiplication. BMP is important in maintaining stability of crypts and loss of BMP usually leads to crypt multiplication with a fingering pattern.

Conclusions

The study provides a mechanism for de novo formation of multiple intestinal crypts and demonstrates a synergetic role of Wnt and BMP in regeneration and stability of intestinal crypts. The proposed model presents a robust framework for studying spatial and temporal dynamics of cell lineages in growing tissues driven by multiple signaling molecules.

Multispecies model of cell lineages and feedback control in solid tumors

We develop a multispecies continuum model to simulate the spatiotemporal dynamics of cell lineages in solid tumors. The model accounts for protein signaling factors produced by cells in lineages, and nutrients supplied by the microenvironment. Together, these regulate the rates of proliferation, self-renewal and differentiation of cells within the lineages, and control cell population sizes and distributions. Terminally differentiated cells release proteins (e.g., from the TGFβ superfamily) that feedback upon less differentiated cells in the lineage both to promote differentiation and decrease rates of proliferation (and self-renewal). Stem cells release a short-range factor that promotes self-renewal (e.g., representative of Wnt signaling factors), as well as a long-range inhibitor of this factor (e.g., representative of Wnt inhibitors such as Dkk and SFRPs). We find that the progression of the tumors and their response to treatment is controlled by the spatiotemporal dynamics of the signaling processes. The model predicts the development of spatiotemporal heterogeneous distributions of the feedback factors (Wnt, Dkk and TGFβ) and tumor cell populations with clusters of stem cells appearing at the tumor boundary, consistent with recent experiments. The nonlinear coupling between the heterogeneous expressions of growth factors and the heterogeneous distributions of cell populations at different lineage stages tends to create asymmetry in tumor shape that may sufficiently alter otherwise homeostatic feedback so as to favor escape from growth control. This occurs in a setting of invasive fingering, and enhanced aggressiveness after standard therapeutic interventions. We find, however, that combination therapy involving differentiation promoters and radiotherapy is very effective in eradicating such a tumor.

Biological and mathematical modeling of melanocyte development

We aim to evaluate environmental and genetic effects on the expansion/proliferation of committed single cells during embryonic development, using melanoblasts as a paradigm to model this phenomenon. Melanoblasts are a specific type of cell that display extensive cellular proliferation during development. However, the events controlling melanoblast expansion are still poorly understood due to insufficient knowledge concerning their number and distribution in the various skin compartments. We show that melanoblast expansion is tightly controlled both spatially and temporally, with little variation between embryos. We established a mathematical model reflecting the main cellular mechanisms involved in melanoblast expansion, including proliferation and migration from the dermis to epidermis. In association with biological information, the model allows the calculation of doubling times for melanoblasts, revealing that dermal and epidermal melanoblasts have short but different doubling times. Moreover, the number of trunk founder melanoblasts at E8.5 was estimated to be 16, a population impossible to count by classical biological approaches. We also assessed the importance of the genetic background by studying gain- and loss-of-function β-catenin mutants in the melanocyte lineage. We found that any alteration of β-catenin activity, whether positive or negative, reduced both dermal and epidermal melanoblast proliferation. Finally, we determined that the pool of dermal melanoblasts remains constant in wild-type and mutant embryos during development, implying that specific control mechanisms associated with cell division ensure half of the cells at each cell division to migrate from the dermis to the epidermis. Modeling melanoblast expansion revealed novel links between cell division, cell localization within the embryo and appropriate feedback control through β-catenin.

Operator Splitting Implicit Integration Factor Methods for Stiff Reaction-Diffusion-Advection Systems

For reaction-diffusion-advection equations, the stiffness from the reaction and diffusion terms often requires very restricted time step size, while the nonlinear advection term may lead to a sharp gradient in localized spatial regions. It is challenging to design numerical methods that can efficiently handle both difficulties. For reaction-diffusion systems with both stiff reaction and diffusion terms, implicit integration factor (IIF) method and its higher dimensional analog compact IIF (cIIF) serve as an efficient class of time-stepping methods, and their second order version is linearly unconditionally stable. For nonlinear hyperbolic equations, weighted essentially non-oscillatory (WENO) methods are a class of schemes with a uniformly high-order of accuracy in smooth regions of the solution, which can also resolve the sharp gradient in an accurate and essentially non-oscillatory fashion. In this paper, we couple IIF/cIIF with WENO methods using the operator splitting approach to solve reaction-diffusion-advection equations. In particular, we apply the IIF/cIIF method to the stiff reaction and diffusion terms and the WENO method to the advection term in two different splitting sequences. Calculation of local truncation error and direct numerical simulations for both splitting approaches show the second order accuracy of the splitting method, and linear stability analysis and direct comparison with other approaches reveals excellent efficiency and stability properties. Applications of the splitting approach to two biological systems demonstrate that the overall method is accurate and efficient, and the splitting sequence consisting of two reaction-diffusion steps is more desirable than the one consisting of two advection steps, because CWC exhibits better accuracy and stability.