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Stem Cell Fate Decision Making: Modeling Approaches Alexander A. Spector, and Warren L. Grayson ACS Biomater. Sci. Eng., Just Accepted Manuscript • DOI: 10.1021/acsbiomaterials.6b00606 • Publication Date (Web): 11 Jan 2017 Downloaded from http://pubs.acs.org on January 12, 2017

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ACS Biomaterials Science & Engineering

Stem Cell Fate Decision Making: Modeling Approaches

Alexander A. Spector1-3* & Warren L. Grayson1-4

1

Department of Biomedical Engineering, Johns Hopkins University School of Medicine, Baltimore MD, USA 2

Translational Tissue Engineering Center, Johns Hopkins University School of Medicine, Baltimore MD, USA 3

Institute for Nanobiotechnology (INBT), Johns Hopkins University Whiting School of Engineering, Baltimore MD, USA

4

Department of Material Sciences & Engineering, Johns Hopkins University, Whiting School of Engineering, Baltimore MD, USA

Key words Cell differentiation, dynamical system, mechanobiology, stochastic methods,

* Correspondence: Alexander A. Spector Johns Hopkins University  Department of Biomedical Engineering Traylor 411 Baltimore, MD 21205, U.S.A. [email protected]

Warren L. Grayson Johns Hopkins University Department of Biomedical Engineering Smith 5023 Baltimore, MD 21287, U.S.A. [email protected] 1

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Abstract Mathematical (computational) modeling approaches can be effective tools in providing insight into cell-fate decisions. In this article, several major approaches to the modeling of embryonic, hematopoietic, adipose-derived, cancer, and neural stem cell differentiation are discussed. First, the population dynamics approach is considered. Then, the models described as bifurcating dynamical systems that result in bistability or periodic oscillations are discussed. Also, spatiotemporal models of cell differentiation, including continuum and discrete (agent- and rule-based) approaches, are reviewed. Further, the effects of the mechanical factors are discussed, including the convergence of the differentiation and mechanotransducton pathways and computational analysis of the extracellular matrix (surrounding tissue). Finally, the stochastic models that take into account the molecular noise of internal and external origins are reviewed. The effectiveness of the modeling in the creation of the improved differentiation platforms, elucidation of various pathological conditions, and analysis of treatment regiments has been demonstrated.

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INTRODUCTION Stem cells have the potential for use in a variety of medical applications. However, understanding the mechanisms regulating their differentiation pathways is critical to the effectiveness of cell-based therapies. Mathematical (computational) modeling approaches can be effective tools in providing insight into cell-fate decisions. Readily available methods of applied mathematics, thermodynamics, and mechanics can be applied to facilitate quantitative analyses of multiscale differentiation-related processes that occur in vitro and in vivo. These techniques may be effectively employed to test hypotheses underlying cell lineage-fate choices by simulating various outcomes in silico according to governing rules of cell behavior thereby significantly reducing the numbers of costly in vitro and in vivo studies1, 2 . Modeling approaches are capable of effectively describing the multi-scale nature of stem cell differentiation processes, e.g., the analysis of

cell differentiation deals with different time scales of transcription

factor/DNA binding, protein expression, and cell morphological changes, and the analysis of tissue regeneration involves different, molecular, cellular, and organ, spatial scales . Models have the unique capacity to synthesize disparate experimental observations into contextual ‘wholes’ in order to adequately describe systems. The ‘systems’ incorporate the cells and their immediate microenvironments, inclusive of neighboring cells, extracellular matrix, growth factors, and biophysical stimuli. These microenvironmental factors influencing cell fate are all modeled as parameters. The modeling approach can describe a comprehensive analysis of the system’s parameters, including parameter sensitivity and optimization. The proposed model can be a basis of the design of additional physical experiments to better elucidate the process of stem cell kinetics and optimize their differentiation along a particular lineage.



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In this article, we review several major approaches to the modeling of stem cell fate choice. First, we consider deterministic approaches, starting with the use of the dynamical systems. The process of stem cell differentiation can be considered as a progression through several stages from the original stem cells through terminally differentiated cells. These stages are characterized by expression of particular proteins (factors) and by cell morphological changes. This progression can be described by systems of ODEs (dynamical system). The dynamical systems have featured properties that can be interpreted as describing the cell fate choice. Among them, there are the system bifurcations resulting in the appearance of new solutions upon the system parameters reach critical values. Here, we consider two types of such solutions, steady states and limit cycles resulting in bistability and periodic oscillations, respectively, and review applications to the fate decision-making of embryonic stem cells (ESCs), hematopoietic stem cells (HSCs),

adipose-derived stem cells (ASCs), bone marrow-derived stem cells

(BMSCs), and neural stem cells (NSCs). We then consider spatio-temporal problems and review continuous (PDE-based) and discrete (agent- and rule-based) methods mainly applied to stem cell differentiation in development and cancer. The importance of the mechanical factors (that are part of the in vivo microenvironment of many cells) to stem cell fate has been commonly recognized, and we discuss an effect of ECM stiffness on differentiation of a number of stem cells as well as important particular cases from orthopedics and myogenesis. Finally, we discuss some stochastic methods revealing the noise-related phenomena of stochastic switches and analyze the effects of different type of noise. Some important topics such as computational methods dealing with high throughput data or cellular reprogramming are not covered in this article, and we refer to several available reviews 1-3. DYNAMICAL SYSTEMS 4 ACS Paragon Plus Environment

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Cell population dynamics. The process of stem cell fate decision-making can be presented in terms of time evolution of a system that includes various factors specifying different ‘states’ of the cell. Such states can be defined as a progression of increasingly committed intermediate stages on the cell’s path towards a terminally differentiated state. To quantitatively implement this concept, the population profiles (or fractions of cells in each stage relative to the total cell number) are used as the main variables, and the model is formulated in terms of their time rates. The model equations express these time rates as the sums of cell fluxes from the previous stage, into the current sage, and on to the next stage.

The time evolution of cell states is determined

by cell proliferation and death as well as signaling among cells in different states or from the cellular microenvironment. The molecular signaling provides non-linear components of the model, and can significantly affect the properties of the kinetic (ODE) system under consideration. The stages of cell differentiation can be described in terms of the expression of paricular proteins (factors) or a group of them. This approach is binary (i.e. a particular protein is either expressed-or not expressed), thus the information on the level of the protein expression requires an additional analysis.

One important example of this approach is the analysis of ESC pluripotency that can be described via two states: the naïve, undifferentiated state (supported by cell self-renewal) and the terminally differentiated state. Prudhomme and Lauffenburger4 have considered the differentiation of murine ESCs and defined the undifferentiated vs. differentiated states in terms of the presence or loss of the transcription factor Oct-4. Assuming the cells in the original state can either grow or transform to the differentiated state with certain rates and the cells in the differentiated state divide with a given rate, the model reduces to two linear ODEs. Selekman et 5 ACS Paragon Plus Environment


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al.

5

have considered the process of differentiation of human ESCs (hESCs) into endothelial

cells and introduced four stages (states) of this process. These stages are defined in terms of the expression or loss of three factors, Nanog, K14 (keratin 14), and K18 (keratin 18). In each stage, cell either self-renew, die, or transform to another state, and these transitions are described in terms of certain rate constants (Fig. 1) , and their effects are considered via a sensitivity analysis of the model. The system of ODEs includes a non-linear factor due to a threshold of the total number of cells.

Cell population dynamics approaches have also been applied to the modeling of HSC populations. Marciniak-Czochra et al. 6,7 and Stiehl and Marciniak-Czochra8 have introduced six stages from undifferentiated HSCs to mature blood cells (Fig. 2). The process can simulate the kinetics of blood cell generation after HSC transplantation following a chemotherapy treatment. It was assumed that in each stage cells asymmetrically divided resulting in partial self-renewal and partial differentiation into the next stage. The mature (final) stage is not associated with further differentiation. The evolution of the cell number in each stage is described in terms of cell proliferation, self-renewal, and death rates. An important component of the model resulting in its non-linearity is a feedback signal which is assumed to be a decreasing function of the total number of cells in the mature stage. Several feedback scenarios are considered, including the signal affecting self-renewal rates (Fig. 2), proliferation rates, or both. It has been shown that the first scenario (Fig. 2) corresponded more accurately with the experimental data of the cell numbers in various stages over time. More complicated conditions are considered in ref. 8 : the stem cells can differentiate toward two distinct mature states. Each of these mature states can generate particular signals affecting the renewal rates in earlier stages, and the stages common to 6 ACS Paragon Plus Environment

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both paths receive two signals. Further analyses of a similar ODE system under different definitions of the feedback functions were developed in ref. 9.

A similar approach has been used to mathematically describe the myogenic differentiation. Cell myogenesis is a critical part of muscle development, function, and regeneration. Different types of stem cells are used in cell therapies of skeletal muscle dysfunctions, including muscular dystrophy, severe trauma, and aging. Stem cell myogenesis follows the path of satellite cells providing muscle regeneration under normal conditions which consists of a transition through several stages specified morphologically and by the expression of structural and transcription factors. Among other stem cells, ASCs are a promising approach to muscle therapy because of these cells’ accessibility and abundance in the body of a patient. The modeling of ASC myogenesis has been analyzed in

10-12

. Six typical stages of ASC myogenesis were defined in

terms of the expression of five markers (Pax7, Desmin, myoD, myogenin, and MHC) (Fig. 3). Each subsequent stage is associated with the expression of a new factor which remains expressed through the following stages (with the exception of the factor Pax7, a requirement for the activation of quiescent stem cells that becomes an obstacle to the later differentiation).The fluxes of cells (time rates of cell numbers) in each stage are associated with cell proliferation, differentiation, and death. Two mechanisms, consistent with the experiment10,11, are assumed to be associated with the early and late stages of ASC myogenesis. In the former case, the stage progression occurs via asymmetric division with particular proliferation, self-renewal (differentiation) and death rates for each stage. In the late stages, the transition occurs via the direct differentiation: the corresponding differentiation coefficients are derived assuming the roles of transcription factors, myoD and myogenin, as the actuators in the expression of 7 ACS Paragon Plus Environment


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myogenin and MHC, respectively. It results in the differentiation coefficients entering equations for stage 4 and 5 to be Michaelis-Menten functions of the cell number in stage 3 and 4, respectively. A sensitivity analysis of the kinetics to the variation of the differentiation parameters is developed in ref. 12. Also, additional conditions were used to control the kinetics of ACS myogenesis. They include a lower limit on the number of cells expressing PAX7 to be met in order for myoD to be expressed and an upper threshold on the number of cells expressing myoD to be met in order for the expression of myogenin to start. Finally, a non-linear feedback as a function of the total cell number is imposed on the rates of cell numbers in all stages. More discussion of the model of stem cell myogenesis and its application to the analysis of the effect of a unidirectional stretch is presented below in the section Mechanobiology and Stem Cell Differentiaton. Bifurcations and bistability. The dynamical system of the kinetic equations describing the stem cell evolution has particular solutions providing important information on the stem cell fate choice. One important approach is the analysis of steady state solutions of the system and the dependence of their number and stability on the system parameters. The system bifurcations occur when a system parameter reaches a critical value associated with a change in the number of steady state solutions. In particular, the phenomenon of bistability is related to parameter intervals of two stable steady state solutions. Depending on the initial conditions, the systems trajectories are attracted to one or the other steady state points. Chickarmane et al.13 have considered the fate decision (self-renewal vs. differentiation) of ESCs. The gene-network model includes the main sub-system of three interacting factors, OCT4, SOX2, NANOG, connected to downstream target genes. This network is governed by two signals: if the first signal is on, it triggers self-renewal and pluripotency of stem cells and when the second signal is off, it results 8 ACS Paragon Plus Environment

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in stem cell differentiation. The resulting distributions of the NANOG factor and OCT4-SOX2 complex vs. strength of the first signal exhibit two bifurcation points with a bistable regime for the signal level being between those points. The NANOG and OCT4-SOX2 dependence on the second signal also exhibits two bifurcation points and bistability (Fig. 4). Also, the cases with a single bifurcation point are considered. This results in NANOG and OCT4-SOX2 being irreversibly on (off) even if the first (second) signal is removed. Wang et al.

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have considered

memory effects in bone marrow stem cells (BMSCs) and clone a colony of such cells exhibiting myogenic factors. The MAPK activity is necessary for myogenic differentiation, and an agent was applied to inhibit it. This treatment resulted in the expression of osteogenic factors, and the cells were returned to myogenesis by removing the MAPK-inhibitory agent. This evolution of stem cell fate was described by a single nonlinear ODE in terms of a “lineage factor.” The nonlinear component of the model was associated with a feedback term expressed by a Hill equation. The solution exhibits two bifurcation points and bistability as a function of the externally applied agent inhibiting MAPK. Huang et al.

15

have studied the dynamic of two-fate

(erythroid vs. myelomonocyte) decision of stem cells go by two equations, governed by GATA1 and PU.1 proteins. The solution of the corresponding dynamical system show a bistability regime provided by the autoregulation of each protein. Further analysis of the GATA1-PU.1 dynamical system can be found in ref. 16 . Limit cycles and oscillations. Another common type of particular solutions (attractors) of dynamical systems is limit cycles that correspond to oscillatory (periodic) solutions as functions of time. Such features were observed in hematopoeitic processes, including pathological conditions of “periodic” hematological diseases. It has been shown17 that a model of two ODEs in terms the numbers of proliferating and non-proliferating cells with delay terms can explain 9 ACS Paragon Plus Environment


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oscillatory regimes. Further development in this direction can be found in refs.

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18,19

where the

model has four distinct compartments representing the pluripotent stem cells and the circulating leukocytes, platelets and erythrocytes. Adimy and Crauste

20

have presented a rigorous

mathematical model of hematopoietic stem cell dynamics, in which proliferation and apoptosis are controlled by growth factor concentrations. Cell proliferation is positively regulated, while apoptosis is negatively regulated. The resulting model is reduced to a system of three ODEs, with three independent delays, and the existence of bifurcation resulting in a periodic solution is investigated. This approach was further developed in ref. 21where different assumptions on the system delay, a discrete delay, a continuous distributed delay, and a state-dependent delay, are considered. It was shown that in each case the system can have oscillating solution and their periods were estimated. Other stem cells (e.g. neural progenitor cells) and Notch effector Hes1 and factor Ngn2 exhibit oscillatory behavior

22,23

. Li et al.

23

have considered the dynamics of

NSC fate decision as the interplay between two factors, Hes1 and miR-9. It has been shown that the system exhibits oscillatory regimes (has limit cycles) that, for certain ranges of the parameters, degenerate to steady state solutions. Shimojo and Kageyama24 have analyzed the oscillatory regimes for two interacting NSCs by using a system of two ODEs with negative feedback loops and delays.

SPATIO-TEMPORAL ANALYSIS Continuum approaches. In many cases, stem cell spatial distribution and its evolution in time are governed by the gradients of external factors, or geometry of the tissue, or interaction with other cells. One of the approaches to the analysis of such conditions is the use of diffusion-like


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terms in the rate equations for the cell number in different states which can be mathematically implemented in systems of PDEs. An advantage of the continuum approach is the availability of analytical and computatiotional tools of PDEs. As an example, Bailon-Plaza et al.25 present a two-dimensional mathematical model of the bone healing process. The inflammatory and tissue regeneration stages of healing are modeled by mesenchymal cell migration; mesenchymal cell, chondrocyte and osteoblast proliferation and differentiation; and extracellular matrix synthesis and degradation affected by a growth factor. Moore et al.26 have developed a mathematical model to predict the tissue genesis by periosteum (a membrane covering the outer surface of a bone)-derived stem cells within a bone defect surrounded by periosteum or a periosteum substitute. Under the action of a growth factor, the original stems differentiate into osteoblasts and chondrocytes which in turn contribute to the bone and cartilage, respectively. As a result, the densities of two types of cells as functions of time and the radial distance from the bone center are computed and compared with the experiment. Hillen et al.27 have considered the tumor dynamics in terms of the interaction and kinetics of two types of cells, cancer stem cells and tumor non-stem cells and applied a model with additional integral terms to the traditional rate terms. These integral terms relate the rate of stem and non-stem cell densities at the tumor point under consideration with cell densities along all the tumor area, and the model take the form of a system of integro-differential equations. As a result, a tumor growth paradox (when tumor grows, while the cells die) has been explained. Piso et al.28 have proposed a novel approach to the modeling of stem cell evolution and related the differentiation into specialized cells and cell division to changes in the cell mass. This concept was applied to the conditions with two distinct specialized cells each of them having its own differentiation pathway governed by a particular growth factor (e.g., bone healing process with differentiation into osteoblast and chondrocyte).



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The mathematical model was presented as a PDE system in terms time- and mass-dependent mass distribution function. Doumic et al.

29

have developed another approach to stem cell

differentiation which is a continuum generalization of a multi-stage approach of population dynamics, and it was assumed that cells to fall into three categories, stem cells that asymmetrically divide, (precursor) cells of different maturity, and the terminally differentiated cells that neither differentiate nor divide. The novelty of this approach is in the assumption that the precursor cells can go through continuum states of differentiation determined by the maturity level. In addition, they proliferate and die at rates also depending on this factor. Moreover, the proliferation, differentiation, and death rates depend on a feedback parameter previously discussed as part of HSC modeling in the section Dynamical Systems. The continuum of states of cell differentiation results in a diffusion-like PDE where the role of spatial coordinates previously discussed in this section is played by the maturity factor. The proposed model is consistent with the approach of ref. 12 where the process of stem cell myogenesis can be divided into early and late parts with different mechanisms of differentiation. The authors

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further

investigate the relationship between the continuum model and finite-state models and show that the former one can be approximated by a set of finite-state models corresponding to small increments in the maturity factor. Discrete Agent- and Rule-Based Approaches. To understand the evolution of multi-stage multicellular systems, discrete approaches, where each “agent” (cell or subcellular component) along with its states and interaction with neighboring agents is explicitly considered, can be more effective than continuum methods. Such discrete methods have been successfully used to describe stem cell performance, including their migration, proliferation, differentiation, and death, and applied to such areas as tumor growth and metastases in cancer and tissue 12 ACS Paragon Plus Environment

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development. The spatial aspect of such methods is typically based on regular or irregular lattices, although lattice-free versions are also considered. A key component of such models is a set of rules that governs the transitions to other states, movement to one of the neighboring spots of the grid, cell disappearance (death), various forms of a feedback, etc. Enderling et al.30 have proposed an individual agent-based model of early tumor growth on the basis of the dynamics of cancer stem cells and their progeny tumor cells with limited proliferation capacity. Tumor cells come to occupy adjacent lattice points by either migration or proliferation. A cell can randomly migrate to one of the eight neighboring lattice points, if one is available, vacating the original lattice point, or it can proliferate, with a daughter cell randomly occupying an adjacent lattice point, if one is available. If all the eight lattice points are occupied, a cell attempting to migrate will do nothing, and a cell attempting to divide will become quiescent instead. Proliferation of a cell is ultimately limited by its proliferation capacity. Cancer stem cells (CSCs) have unlimited replicative potential and self-renewing ability. With probability, ps, a new CSC is produced, and with probability, 1-ps, a non-stem progeny cancer cell (CC) is produced that, with each division, loses proliferation capacity until proliferation is exhausted and the cell dies. Further, Enderling et al.31 have applied their cell-based model to the analysis of the effectiveness of the radiotherapy of tumors. The main results are the computed dynamics of cancer cell eradication as a function of the parameters of cancer and non-cancer cells in the tumor. Thus, the effectiveness of radiotherapy for a tumor of a given size can be predicted. Poleszczuk et al.

32

have further

analyzed the evolution of tumor and its cellular content by using a version of agent-based model from ref.

30

. Fig. 5 shows hierarchy of cell in the tumor determined by their proliferative

potential. All types of cells but cancer stem cells proliferate symmetrically where each division



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reduces their proliferative capacity until they stop proliferating. The cancer stem cells proliferate asymmetrically, and the probabilities of self-renewal and differentiation are specified.

White et al.33 have considered a problem of the development of multicellular aggregates, embryoid bodies, via losing pluripotency of the original stem cell, the process associated with the loss of the expression of the protein Oct4. The process of the evolution (growth) of embroyid body is described by a set of transition rules, several scenarios are considered depending on the imposed rules of transition (Fig. 6). In the first scenario, an Oct4+cell loses it pluripotency with a prescribed probability, in the second scenario, the Oct4+ to Oct4- transition is effected by positive feedback from the group of cells lost the pluripotency, and in the last scenario, the change in the expression of Oct4 is affected by a competition between the activation by Oct4- cells and the inhibition by Oct4+ cells. As a result, spatio-temporal analyses have been developed for the composition of the embroyid body and its time course, depending on the initial conditions and assumed rules of the pluripotency loss (Fig. 7). Setty et al.

34

have considered 4-D (spatio-

temporal) dynamics of pancreatic organogenesis. For each cell, the model included several “agents” (membrane and nucleus) into consideration and the conditions of cell proliferation or differentiation is defined in terms of the state of the agents. Sun et al.

35

have applied an agent-

based model to the analysis of stem cell differentiation into mature skin cells (there are two more distinct stages between the initial and terminal ones). The model proposes a set of rules describing cell division, differentiation, and migration. The components of the model include all interacting cells as well as cell environment characterized by the calcium level. Another agentbased approach to skin remodeling by differentiation of keratinocytes was proposed in ref.

36

.

Several scenarios of keratinocyte asymmetric division were considered and long-term 14 ACS Paragon Plus Environment

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differentiation pattern has been simulated. The cells were presented as frictionless rigid spheres and a specialized mechanical solver is used to simulate force balance conditions for moving cells. Galvao et al.

37

have developed an agent-based method to simulate the chronic chagasic

cardiomyopathy regeneration after bone marrow stem cell transplantation. The model has different types of agents to describe the chagasic cardiac tissue. In this model, each type of agent represents a different type of cell. An agent is located to the environment where different agents can interact.

MECHANOBIOLOGY IN STEM CELL DIFFERENTIATION Mechanical factors are part of stem cell niche, and they have a significant effect on stem cell differentiation. It has been shown that such external microenvironmental stiffness

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, area of cell spreading

39

factors as ECM

, applied cyclic strains and forces10 , and substrate

topography40 can both enhance stem cell differentiation and re-direct it to a different lineage commitment. The modeling of such problems is challenging since it requires the understanding of the interaction of differentiation and mechanotransduction pathways as well as the knowledge of the internal mechanical factors in 2-D or 3-D matrixes that are locally sensed by the cells and ultimately determine their differentiation.

.One key question associated with a recent observation in stem cell mechanobiology and differentiation is what is the mechanism of stem cell sensing the ECM stiffness38. Zemel et al.41 have proposed a mechanism behind this observation in terms cytomyosine stress fiber orientation dependence on the ECM stiffness. The cell is represented by an “active” ellipsoid (3-D case) 15 ACS Paragon Plus Environment


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and ellipse (2-D case) where cell’s active properties are described in terms of a dipole force distribution representing actin-myosine interaction in stress fibers. The model predicts the fiber distribution for a given stiffness of ECM, and this result is supported by the experiment with immunostained myosine in hMSCs. Another modeling study

42

investigates the typical distance

at which a stem cell the stiffness of ECM. This distance is defined by the magnitude of the strain along the cell/ECM contact area, and FEM (cell active properties are modeled via a prestess) simulaton is used to estimate the surface boundary strain for different stiffness and thickness of the ECM. This study has shown that the distance of cell sensing the ECM is about a fraction of the cell size, and also the stem cell sensitivity is greater than that of the differentiated cells.

A criterion of stem cell differentiation in terms of the local mechanical factors is an important aspect of orthopedic models. Based on biphasic nature of tissues, Prendergast et al.43 have proposed a criterion of stem cell differentiation and used it in orthopedic applications. This criterion includes two factors, the shear strain associated with the solid component and the fluid velocity associated with the liquid component of the tissue. Andreykiv et al.

44

have studied the

effect of surface geometry on peri-implant tissue differentiation. The model consists of biological and mechanical parts. In the biological part, the cell migration, proliferation and tissue production is considered. Four types of cells are, mesenchymal stem cells, osteoblasts, chondrocytes, and fibroblasts are considered via PDEs in terms of cell densities. In addition, the time evolution of bone, cartilage, and fibrous tissues volume fractions are considered via a system of ODEs. The mechanical part of the model is aimed at the calculation of the stimulus P in the tissue. All the tissues are modeled as poroelastic with a total Lagrangian formulation and neo-Hookean hyperelastic properties for the solid phase. Fig. 8 shows the simulation of tissue 16 ACS Paragon Plus Environment

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differentiation within a time interval of 28 days. The lightness vs. darkness of the points of the tissue correspond to low vs. high level of the differentiation stimulus P. The other panel sketches the differentiation coefficients (associated with chondrocytes, fibroblasts, osteoblasts) in the equation of the matrix production. Byrne et al. 45 have analyzed the fracture healing in the tibia including bone cell differentiation. The computational model includes differentiation of mesenchymal stem cells depending on the level of the local stimulus P determined by the shear strains and fluid velocity: high level of the stimulus corresponds to stem cell differentiation into fibroblasts, intermediate level corresponds to differentiation into chondrocytes, and a low level corresponds to differentiation into osteoclasts. Several different criteria of local mechanoregulation of cell differentiation have been proposed, and Khayyeri et al.

46

have discussed the

advantages and disadvantages of each of them. Moreo et al.47 have proposed a model of tissue ingrowth on the surface of a dental implant. The model analyzes spatio-temporal development of the tissue, including several cell and matrix components. Among cells, several osteogenic cells, are considered. Among tissue components, fibrin network volume fraction, woven bone volume fraction, and lamellar bone volume fraction are considered. In addition, two growth factors produced in response to the surgical procedure and causing proliferation and migration of the cell are also involved in the model. The factors enter the flux equations for cells via MichealisMenten functions. The cell migration is described by continuous diffusion operators. The model was implemented as a subroutine of the commercial finite element package Abaqus 6.6. The second part of this paper48 has proposed a simplification of the original version by keeping just one growth factor. This allowed a mathematical analysis of the solution and prediction of its qualitative behavior. In addition, the simplified model could be solved just by using the Matlab package. The multistage analysis of stem cell myogenesis first presented in the sub-section Cell



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Population Dynamics has been further developed (Deshpande et al.

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12

) to better understand the

mechanism of the applied strains. In the proposed model12, the strains enter both equations of the first, proliferation stages and those of the later, direct differentiation stages (Fig. 3). The model interprets the effect of external strain on stem cell myogenesis in terms of a hypothetical signaling molecule, S (Fig. 9) that affects the states of the transcription factors myoD and myogenin (for the simplicity, the consideration was limited to a single signaling molecule). It is further assumed that a fraction of the free myoD affected by the signaling molecule binds the promoter of myogenin. Similarly, a fraction of myogenin affected by the molecule S binds the promoter of MHC. Mathematically, it results in Michaelis-Menten dependences on S that enter the differentiation functions.

STOCHASTIC MODELS Bistable Switches: Effects of Noise. In the deterministic model of stem cell differentiation described above, the cell could be in one of two stable states, depending on the initial condition: attracted to one of the states the cell keeps its commitment. However, when the state of the system is determined by a limited number of components (protein, transcription factors, etc.), the stochasticity of the interactions among them (internal noise) has significant effects on system’s behavior. Such effects were revealed by considering stochastic versions of the description of bistability. Nitzan et al. 49 have considered a representative system of two genes regulating each other’s expressions which exhibits bistability if formulated deterministically in terms of the rates of the components. The stochastic version of this system is described in terms of the probability rates of interactions between the components, and its solution switches between two stable states


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with the time of occupation determined by the system parameters. This feature is qualitatively new compared to the deterministic analysis. Lai et al.50 have considered bistable switch in the sonic hedgehog (Shh) signaling system which controls stem cell fate choices in many systems. Noise can be also introduced as an external factor to control the cellular behavior. Jaruszewicz et al.

51

have considered a model with an autoregulatory gene with a positive feedback that exhibit

bistability in the deterministic formulation. The authors add three different types of internal noise, transcriptional and translational (due to the small number of gene product molecules and the gene switching noise (due to gene activation and inactivation transitions). They have demonstrated that the type of noise in addition to the noise magnitude dictates the allocation of probability mass between the two stable steady states. In particular, the authors found that when the gene switching noise dominates over the transcriptional and translational noise (which is characteristic of eukaryotes), the gene preferentially activates, while in the opposite case, when the transcriptional noise dominates (which is characteristic of prokaryotes) the gene preferentially remains inactive. Hasty et al.

52

have considered a single network derived from

bacteriophage λ and added stochastic additive and multiplicative perturbations into kinetic equations of the system. They have shown that noise can qualitatively change the cell behavior, including acquired switching between the stable states. Balazsi et al. 53have reviewed the effects of internal noise in a variety of biological systems (from viruses to mammalian cells) that can be described in terms of a genetic network with a feedback and demonstrated new features compared to deterministic models. Hematopoietic cells. In hematopoiesis, the generation of blood cells, a series of gene switches has been found to determine the differentiation path of hematopoietic stem cells and to direct the ratio of the mature blood cells. The most prominent example in this context is the mutual 19 ACS Paragon Plus Environment


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inhibition of Gata-1 and Pu.1, two transcription factors responsible for the development of erythroid and myeloid blood cells from common myeloid progenitors. Strasser et al.

54

have

studied the stability and dynamics of a two-stage gene expression switch within a probabilistic framework inspired by the properties of the Pu/Gata toggle switch in myeloid progenitor cells. Tian and Smith-Miles

55

proposed a stochastic model to study the mechanisms of the GATA-

PU.1 gene network in the determination of HSC differentiation pathways. Also, Dingli et al.56 have proposed a stationary compartmental probabilistic model of hematopoiesis where cells proceed from stage to stage with certain probabilities. Peixoto et al.

57

have applied a similar

model of hematopoiesis to both normal and pathological (chronic myeloid leukemia, cyclic neutropenia, and paroxysmal nocturnal hemoglobinuria) conditions.

SUMMARY Mathematical (computational) modeling of ESCs, HSC, and ASCs has contributed to a better understanding of the cell lineage-fate determination processes underlying development, red blood cell differentiation, and myogenesis. Modeling has revealed the effects of cell system’s noise on cell fate decision. Models may be used to provide a comprehensive understanding of cellular-fate decisions based on information gleaned from diverse experimental data sets and protocols. They may also be used to assess the influence of key microenvironmental parameters on cell fate and the sensitivity of culture systems to parameter variation. Hence, modeling can be used in the development of effective in vitro protocols for stem cell differentiation. Of primary importance is the use of the stem cell modeling in elucidating the mechanisms of various diseases and helping in their treatments. A probabilistic model has been used to describe


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“periodic” hematopoietic pathological conditions, and a multi-stage kinetic model has described hematopoiesis in development. A cell-based model, involving CSCs has been applied to the analysis of the effectiveness of the tumor radiotherapy resulting in the prediction of the treatment effect on the tumor of a given size. Computational spatio-temporal models based on a local criterion of cell differentiation combined with poroelastic analyses of the mechanics of the surrounding tissues have been developed and used in the effective simulation of orthopedic and dental procedures. The modeling methods reviewed in this paper are summarized in the table below.

ACKNOWLEDGMENTS The authors thank Daniel Yuan and Rajiv Deshpande for their help in the preparation of the manuscript.



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Figure Legends Fig. 1 A four-stage model of hESC differentiation into epithelial cells. Each stage is characterized by a particular combination of protein-markers, Nanog, K14, and K18. A) The sketch, which is a basis of the dynamical (ODE) system, shows the paths of self-renewal and further differentiation in each stage; the k-parameters are the corresponding self-renewal and differentiation rates. B) Four stages of hESC transition to the terminally differentiated epithelial cells. (From Selekman et al.5 , with permission). Fig. 2 A multi-stage model of differentiation of HSCs. A) Sketch of six major stages with selfrenewal and differentiation. The feedback signaling from the terminal state affects self-renewal in each of earlier stages. B) Different scenarios of cell divisions, symmetric division with two daughter cells identical to the parent cell, asymmteric division with one daughter cell belonging to the next stage, and symmetric differentiation with both daughter cells belonging to to the next stage (From Marciniak-Czochra et al. 7 ). Fig. 3

A six-stage model of ASC myogenic differentiation. Each stage is characterized by

expression of a combination of particular markers, PAX7, desmin, MyoD, myogening, and MHC. The dashed line separates the early stages of stage transition via asymmetric division and the late stages of stage transition via direct differentiation. The process of differentiation is affected by signaling of different types. First, myogenic media is applied. Second, the ESM interact of the cells via externally-applied strains. Third, cell-cell interaction results in signaling if cell density (total cell number) reaches a threshold. (From Deshpande and Spector12 ). Fig. 4 Bifurcations and bistability in the dynamical system of ESC control. A) A network model13 of three interacting factors, Nanog, Oct 4, and Sox2 regulated by a signal (B ). B) 26 ACS Paragon Plus Environment

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Correponding bifurcation diagram of the Nanog expression as a function of the level of the control signal. Solid lines correspond to stable states. The dashed line correspons to unstable states separating two stable states. System bifurcations occur at signal intensities c1 and c2.where the system changes its fate. The bistable region of two possible fates (between two bifurcation points) is shown in grey (From Herberg and Roeder 2 ).

Fig. 5 An agent-based model of CSC (cancer stem cells) transition into cancer cells (CCs) and 2D migration. A) CSCs divide asymmetrically with a given probability of self-renewal. CCs can divide and their further capacity for division decreases until they dye. B) Both types of cells can migrate to the neighboring spaces in a 2-D lattice. C) sketch of CSC division ( From Poleszczuk et al.32). Fig. 6 A rule-based model of ESC loss of pluripotency with three scenarios of stem cell transition to the differentiated state (differentiation is determined by the loss of the expression of the marker, Oct14). A) Differentiation occurs with a prescribed probability. B) Positive feedback. Probability of differentiation is positively influenced by the number of neighboring differentiated cells. C) Competing feedback. Probability of differentiation is positively influenced by the number of neighboring differentiated cells and is negatively influenced by the number of neighboring undifferentiated cells. (From White et al. 33) Fig. 7 Spatio-temporal analysis of ESC loss of pluripotency obtained by using a based-rule model. A) and B) Trajectories of both types of cells, differentiated and undifferentiated, and 3-D images of the computed cellular spherical body, respectively, in the case of the prescribed probability of state transition. C) and D) Trajectories of both types of cells, differentiated and



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undifferentiated, and 3-D images of the computed cellular spherical body, respectively, in the case of the positive feedback (Fig. 5). E) and F) Trajectories of both types of cells, differentiated and undifferentiated, and 3-D images of the computed cellular spherical body, respectively, in the case of the competing feedback (Fig. 5). (From White et al. 33) Fig. 8 Simulation of peri-implant tissue regeneration. Tissue “damage’ was made by the attached rigid particle. A) Color maps of the distribution of the stimulus, P, value that is determined by the local shear stress of the tissue solid fraction and shear velocity of the local tissue liquid fraction. The simulation covers 28 days of regeneration. The higher values of P correspond to the darker shade. B) The P-dependence of the differentiation coefficients, Fob, Fcc and Fbl , associated with osteoblasts, chondrocytes, fibroblasts, respectively, that enter the model equation of the matrix production (From Andreykiv et al.39,with permission). Fig. 9 Sketch of the direct differentiation part of ASC myogenesis, involving myoD, myogenin, and MHC. MyoD affects the myogenin production, and the latter factor affects the MHC production. The transcriptional activity of myoD and myogenin is affected by a signaling molecule S generated by the applied strain. (From Deshpande and Spector12 ). Table Summary of the methods and applications reviewed. Copy of ref. 12


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Fig. 1



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B

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Fig. 9



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Modeling Approach

Brief Description

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Applications

Dynamical Cell systems. population dynamics.

Compartmentalization of cell population. Time transition through several stages. Described in terms of ODEs. Spatial effects are not covered.

• • •

Development/loss of pluripotency 4,5 Hematopoiesis 6-9 Myogenesis 10-12

Bistability.

A specific feature of ODEs when the system’s parameters reach critical values resulting in the appearance of two stable steady state solutions. Such solutions can describe two alternative fates of a stem cell.

•

Development13-16

Limit cycles.

A specific feature of ODEs when the system’s parameters reach critical values resulting in the appearance of periodic oscillatory solutions. Such solutions can describe alternative lineage of a stem cell.

• •

Hematopoietis17-21 Neurogenesis22-24

Continuous methods.

Stem cell differentiation is described by a system of PDEs with the spatial component presented by diffusion terms. Can describe effects of factor spatial gradients and cell migration.

• •

Orthopedics 25,26 Cancer27

Rule- and agentbased methods.

Discrete computational approaches • where the agents under consideration • can be the whole cell or subcellular components, and the cell transition in time and space is described by a set of rules. A versatile approach effective in the description of spatio-temporal evolution of cellular systems.

Spatiotemporal analysis.

Mechanical It is biologically based on the factors convergence of the differentiation and mechanotransducton pathways. Includes computational analysis of the 38 ACS Paragon Plus Environment

• • •

Cancer 30-32 Development33

Orthopedics 43-46 Dentistry 47,48 Myogenesis10-12

Page 39 of 51

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surrounding tissue. Explains the effects of ECM stiffness and viscoelasticity, surface topography, cyclic strains.

Stochastic approaches

Used when the system has a limited number of factors and the noise (stochasticity) of their interaction has a significant effect. This effect can result in qualitatively different behavior of the system; e.g., in the deterministic bistability, the system attracts to one of the steady state solution (depending on the initial condition). In contrast, noise results in a switch when the solution alternates between two steady state points.


•

Broad applications in biology from viruses to mammalian cells.4954


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Fig. 1



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Fig. 2


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Fig. 3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Fig. 4


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Fig. 5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Fig. 6


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Fig. 7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 48 of 51

Fig. 8

B

Fbb

Fcc

Ffb

Fbma

Fcma

Ffma

x

x

x

Fbmi n

Pmin

P

Pmin Pmax


P

Pmin Pmax

P

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Fig. 9



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Modeling Approach

Brief Description

Page 50 of 51

Applications

Dynamical Cell systems. population dynamics.

Compartmentalization of cell population. Time transition through several stages. Described in terms of ODEs. Spatial effects are not covered.

  

Development/loss of pluripotency 4,5 Hematopoiesis 6-9 Myogenesis 10-12

Bistability.

A specific feature of ODEs when the system’s parameters reach critical values resulting in the appearance of two stable steady state solutions. Such solutions can describe two alternative fates of a stem cell.



Development13-16

Limit cycles.

A specific feature of ODEs when the system’s parameters reach critical values resulting in the appearance of periodic oscillatory solutions. Such solutions can describe alternative lineage of a stem cell.

 

Hematopoietis17-21 Neurogenesis22-24

Continuous methods.

Stem cell differentiation is described by a system of PDEs with the spatial component presented by diffusion terms. Can describe effects of factor spatial gradients and cell migration.

 

Orthopedics 25,26 Cancer27

Rule- and agentbased methods.

Discrete computational approaches  where the agents under consideration  can be the whole cell or subcellular components, and the cell transition in time and space is described by a set of rules. A versatile approach effective in the description of spatio-temporal evolution of cellular systems.

Spatiotemporal analysis.

Mechanical It is biologically based on the factors convergence of the differentiation and mechanotransducton pathways. Includes computational analysis of the 10 ACS Paragon Plus Environment

  

Cancer 30-32 Development33

Orthopedics 43-46 Dentistry 47,48 Myogenesis10-12

Page 51 of 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


surrounding tissue. Explains the effects of ECM stiffness and viscoelasticity, surface topography, cyclic strains.

Stochastic approaches

Used when the system has a limited number of factors and the noise (stochasticity) of their interaction has a significant effect. This effect can result in qualitatively different behavior of the system; e.g., in the deterministic bistability, the system attracts to one of the steady state solution (depending on the initial condition). In contrast, noise results in a switch when the solution alternates between two steady state points.




Broad applications in biology from viruses to mammalian cells.4954

Stem Cell Fate Decision Making - American Chemical Society

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