The recent advances in the mathematical modelling of human pluripotent stem cells
L E Wadkin, S Orozco-Fuentes, I Neganova, M Lako, A Shukurov, N G Parker
NNature Applied Sciences manuscript No. (will be inserted by the editor)
The recent advances in the mathematical modelling of humanpluripotent stem cells
L E Wadkin ∗ · S Orozco-Fuentes · I Neganova · M Lako · A Shukurov · N G Parker Received: date / Accepted: date
Abstract
Human pluripotent stem cells hold greatpromise for developments in regenerative medicine anddrug design. The mathematical modelling of stem cellsand their properties is necessary to understand andquantify key behaviours and develop non-invasive prog-nostic modelling tools to assist in the optimisation oflaboratory experiments. Here, the recent advances inthe mathematical modelling of hPSCs are discussed,including cell kinematics, cell proliferation and colonyformation, and pluripotency and differentiation.
Keywords human pluripotent stem cells · mathemati-cal modelling Human pluripotent stem cells (hPSCs) have the abil-ity to self-renew indefinitely through repeated divisions( mitosis ) and can differentiate into any bodily cell type(the pluripotency property). The latter property under-pins their promising clinical applications in drug dis-covery, cell-based therapies and personalised medicine[1,2]. Amongst others, cardiomyocytes [3], pancreaticcells [4] and corneal cells [5] have all been successfullycreated from hPSCs. In the lab, hPSCs are grown inmono-layer colonies of up to thousands of cells (Figure 1)from which they can be directed for specific experimentsor therapies, or expanded to produce further hPSCcolonies. They occur either as human embryonic stemcells (hESCs) derived from the early embryo, or humaninduced pluripotent stem cells (hiPSCs) which are de-rived by the genetic reprogramming of differentiated *E-mail: [email protected] [1] School of MathematicsStatistics and Physics, Newcastle University, UK, [2] Insti-tute of Cytology, RAS St Petersburg, Russia, [3] Institute ofGenetic Medicine, Newcastle Univeristy, UK.
Fig. 1: Microscopy images of hESCs showing growingcolonies from (a) a few cells up to colonies of (b) hun-dreds and (c) thousands.cells [6]. The latter approach, which received the 2012Nobel Prize in Medicine or Physiology for its discovery,offer patient-specific hPSCs without the ethical issuesassociated with hESCs.Emerging biomedical technologies require the effi-cient, large-scale production of hPSCs [7]. Furthermore,applications of hPSCs in the clinic require great con-trol over the pluripotency, clonality (the proportion ofidentical cells that share a common ancestry) and dif-ferentiation trajectories in-vitro . However, the existingprocedures for large scale experiments remain inefficientand expensive due to low cloning efficiencies of 1% to27% (the percentage of single cells seeded that form aclone) [8,9]. Understanding factors which promote theefficient generation and satisfactory control of hPSCcolonies (and their derivatives) is a key challenge.Mathematical and computational modelling allowsthe identification of generic behaviours, providing aframework for rigorous characterisation, prediction ofobservations, and a deeper understanding of the under-lying natural processes. The application of mathematicsto biology [10] has led to many significant achievementsin medicine and epidemiology (for example, predictingthe spread of ‘mad cow’ disease [11,12] and influenza[13]), evolutionary biology [14] and cellular biology (de- a r X i v : . [ q - b i o . CB ] S e p L E Wadkin ∗ et al. scriptions of chemotaxis [15] and predicting cancer tu-mour growth [16]). Similarly, mathematical models area powerful tool to further our understanding of hPSCbehaviours and optimise crucial experiments.The first mathematical model of stem cells, a stochas-tic model of cell fate decisions [17], has since been ex-tended to include many other aspects of cell behaviour[18,19,20,21,22]. In particular, when such mathemat-ical models are rigorously underpinned and validatedon experimental observations, the reciprocal benefit forexperimentation can be profound: an example is the de-velopment of an experimentally-rained model of hiPSCprogramming, which led in turn to strategies for markedimprovements in reprogramming efficiency [23].Coherent mathematical models of hPSC propertiesmay provide non-invasive prognostic modelling toolsto assist in the optimisation of laboratory experimentsfor the efficient generation of hPSC colonies. Statisticalanalysis of experimental data allows the quantificationof stem cell behaviour which can then inform the devel-opment of these models. Here we shall discuss recentadvances in the mathematical modelling of hPSCs andtheir impact.This review focuses mostly on hESCs, with somelimited discussion of hiPSCs. We first outline some ofthe key properties of hPSCs before focussing on recentdevelopments in mathematical models of the key prop-erties: – Section 2: Key biological properties of hPSCs – Section 3: Cell kinematics . The movement of cellsalone, in relation to one another and within hPSCcolonies. – Section 4: Colony growth . Models capturing cell pro-liferation, with and without a spatial component. – Section 5: Cell pluripotency . Pluripotency regulationmodels, both intra-cellular and at the colony scale.Finally, in Section 6 we provide a summary of the modelsdiscussed, their impact on biological experiments andthe next steps for model development.
The satisfactory understanding and control of hPSC evo-lution remains elusive due to their complex behaviourover multiple scales: the intra-cellular scale (processeshappening within cells), the cellular or micro-environmentscale (the environmental effects on individual cells) andthe colony scale (collective cell behaviours throughoutcolonies), as illustrated in Figure 2. Advances in imag-ing and molecular profiling (classification based on geneexpression) have identified the core processes within theevolving colony [8,24,25,26]. Here we outline some of Fig. 2: Scales of hPSC behaviour: (a) Intra-cellular scalee.g., cell cycle, division, inheritance of pluripotency fac-tors, PTF. (b) Cell micro-environment e.g., interactionwith other cells, the medium and substrate. (c) Colony-scale phenomena e.g., patterning of differentiated cells.these key biological properties across these scales andtheir relevance for mathematical modelling.2.1 Intra-cellular scaleThe key intra-cellular behaviours integral to hPSC mod-elling are the cell cycle and pluripotency regulation. Thecell cycle is the timed series of events controlling DNAreplication and resulting in a cell division. The phases ofthe cell cycle are: G1 (growth phase), S (synthesis phasein which DNA is replicated), G2 (further growth) andM (mitosis, the cell division). The G1 phase is shortenedfor hPSCs, leading to more rapid proliferation than forsomatic cells [27].The maintenance of pluripotency depends on thestable inter-regulation of pluripotency transcription fac-tors (PTFs) [28], mainly by the genes OCT4, SOX2and NANOG [29]. Fluctuations of the PTF abundancesare believed to cause the variation in pluripotency indifferent sub-populations [28]. Destabilisation and theinteraction of these PTFs with chemical signalling path-ways triggers differentiation , the departure from thepluripotent state [28,30] towards specific cell fates [31].The cell cycle also affects pluripotency and cell fate[32] and vice versa [29,33,34]. Moreover, recent work athematical modelling of hESCs 3 suggests that the PTFs are inherited asymmetrically asa cell divides [35], biasing the fate of the daughter cellsand contributing to colony heterogeneity.2.2 Micro-environmentAs in the embryo, the local environment of the cell iskey to its in-vitro evolution. One of the leading envi-ronmental factors affecting hPSCs is the substrate onwhich they are grown. Substrates may either consist ofa layer of mouse or human ‘feeder’ cells or a protein sub-strate, with the latter growing in popularity for clinicalapplication since they avoid the risk of genetic contami-nation. The substrate influences pluripotency [36] andmobility [37] through its growth factors and adhesionforces. Low cell motility improves clonality by suppress-ing cross-contamination of colonies [38], although itsrole in colony heterogeneity is yet to be established.As well as the substrate, cell-cell interactions arealso important. hPSCs benefit from being in colonieswhere they exhibit higher viability and pluripotency [39].hPSCs apparently sense each other up to a distanceof around 150 µ m (of order 5 cell diameters) [40,41].Meanwhile, as the colony grows and becomes denser,the mutual mechanical pressure of the hPSCs can affectthe cell cycle [42].2.3 Colony scalePerhaps most intriguing, yet least understood, are be-haviours that emerge on a colony scale. The promotionof pluripotency in larger colonies [43,44] shows that sin-gle cells are influenced by the whole colony. Indeed, ithas been suggested that pluripotency is a collective sta-tistical property of cells [45], rather than a well-definedproperty of individual cells.Further colony-scale effects are evident in the spatialpatterning of the cell fates after differentiation. Mechan-ical forces and chemical signals operating over distanceslarger than the cell separation influences single-cell ge-netic expression to form bands of differentiated cells[46] (illustrated in Figure 2); these structures are en-hanced under imposed boundaries, emphasizing the roleof mechanical forces [47,48]. With further understanding,mechanical effects and boundaries could be harnessedto engineer specific desired differentiated cells [49].Incorporating these complex behaviours over multi-ple scales into mathematical models is challenging. Akey goal is to develop coherent models which capturethe individual cell behaviours, e.g., cell kinematics andthe inter-cellular maintenance of pluripotency, and lead to the observed collective effects on the colony scale,e.g., collective migration and the spatial patterning ofpluripotency and differentiation. Motility is an intrinsic property of hPSCs; they can in-crease their migratory activity under certain conditions[50]. Their migration is achieved through adaptationsin cell morphology via the reorganisation of the actincytoskeleton to form a leading edge pseudopodia [51].Unregulated cell migration in-vitro can cause clonalityloss as the cell population grows, undesirable when a ge-netically identical clonal population is required [52,53].Furthermore, anomalous cell migration has been linkedto deviations in the undifferentiated state of hiPSCs [54].A thorough understanding of the migration of hESCs isneeded to optimise in-vitro clonality and facilitate thedevelopment of therapies for migration related disorders.Here we discuss the kinematics of isolated cells and theirpairs as well as cell migration within colonies.3.1 Kinematics of isolated cells and pairshPSCs are often seeded at low density to preserve theclonal purity of the emerging colonies. Migration ofindividual cells between the incipient colonies can resultin clonality loss. It is important therefore to quantifythe migration of individual cells upon a growth plate.The unconstrained motion of cells on a 2D plane canoften be described as a 2D random walk, the simplestbeing Brownian motion [55,56]. Random walks can bebiased by an external source giving preference to move-ment in a particular direction (a biased random walkor BRW). A correlated random walk (CRW) involves acorrelation in the direction of the next step in relationto the previous step, i.e., persistence, where the nextstep is more likely to be in the direction of the previousstep, or anti-persistence, where the next step is morelikely to be in the opposite direction. CRWs often occurin cell kinematics in the absence of external biases [57,58,59].The diffusive nature of a random walk can be quanti-fied by considering the mean square displacement (MSD)of cell trajectories. The MSD is a measure of the trajec-tory of a particle from its starting position over time, (cid:104) r (cid:105) = (cid:104) ( x − x ) (cid:105) where x ( t ) is the position of theparticle, x is the initial position at t = 0 and angularbrackets denote the average taken over all trajectories.For a typical diffusive particle, the MSD increases lin-early with time, (cid:104) r (cid:105) ∝ Dt , where D is the diffusioncoefficient. The root mean square displacement is given L E Wadkin ∗ et al. by (cid:104) r (cid:105) / = √ Dt , from which D can be calculated. If (cid:104) r (cid:105) ∝ Dt α , with α < α > µ m away from any neighbouring cells. As the sep-aration distance decreases the cell movements becomemore directed towards each other, with motility-inducedre-aggregation occurring in 70% of instances when thedistance been two hESCs is less than 6 . µ m [40]. Aminority of isolated single cells exhibit super-diffusivebehaviour, contributing heavily to the motility relatedclonality loss [8,40,60]. Example experimental trajecto-ries for cells exhibiting typical diffusive behaviour andsuper-diffusive migration are shown in Figure 3. (a) -10 -5 0 5 10 x ( m)-10-50510 y ( m ) (b) x ( m)0100200 y ( m ) (c) Fig. 3: Example single cell trajectories for (a) isotropicmotion around a central point and (b) a directed walk.The initial and final cell centroid positions are shownas a circle and a square respectively (note that thesepoints are not representative of cell or nucleus size). (c)A single hESC migrating backwards and forwards alonga local axis. The blue dot shows the cell nucleus andthe black arrow the direction of instantaneous velocity.The scale bars are 30 µ m in length [41].Our study containing further experimental analysisof hESCs [41] has shown evidence of correlated randomwalks of individual isolated stem cells. Single hESCs (more than 150 µ m away from any neighbouring cells,as in [40]) tend to perform a locally anisotopic walk,moving backwards and forwards along a preferred lo-cal direction correlated over a time scale of around50 minutes, becoming more persistent over time. Themotion is also aligned with the axis of cell elongation(Figure 3) which could suggest an attempt to locateother neighbouring cells. Further experiments shouldquantify how the presence of multiple neighbours affectsthis anisotropic movement.Our study also found that pairs of hESCs in closeproximity tend to move in the same direction, with theaverage separation of 70 µ m or less and a correlationlength (the length scale of communication) of around25 µ m. Often the pairs of cells remained connected bytheir pseudopodia, even at larger distances ( > µ m)when they exhibited independent movements. For thecorrelated pairs, it is not known whether the movementcorrelation is facilitated by the physical connection orthe coordination is due to cell-cell chemical contactalone.There is evidence that cell migration in 3D doesnot follow a persistent random walk and new modelswill need to be developed to accurately describe thismotion [61]. These experimental results further informthe development of individual based models for cellmigration as a random walk and can be integrated intomore complex models of cell movement within colonies.3.2 Colony kinematicsStem cells also exhibit motion as part of larger groupsand colonies. The coordinated migration of large num-bers of hPSCs in-vivo is essential in tissue generation[62] and wound healing [63]. The modelling of such largergroups and colonies of hPSCs is more complex, as bothcollective and individual behavioural effects are involved[64].Popular agent-based models have been developedto incorporate these results into colony models, butthe challenges still remain to fully capture the experi-mental behaviours, especially collective aspects and cellmigration in 3D. These agent-based migration modelsare often combined with models of colony growth andproliferation [65,66].hPSCs show coordinated intra-colony movementswhich cease upon differentiation [67]. Cell movementspeed varies within colonies, with higher average speedat the periphery and lower in the central region [54].Recently, a two-dimensional individual-based stochasticmodel was developed of cell migration, cell-cell con-nections and cell-substrate connections and captures athematical modelling of hESCs 5 Fig. 4: The migration of cells can be modelled either ona lattice or in continuous space.well these experimental observations [65]. The modelintroduces the energies of cell-cell and cell-substrate con-nections. Any energy released by breaking and formingthese connections allows cell migration to one of theeight directions on a square lattice. The direction ofmovement is determined at random based on a probabil-ity related to the cell’s energy and a spatial weightingwhich favours a side rather than a diagonal direction (asdescribed in [68]). Cell proliferation and quiescence (thereversible state of a cell in which it does not divide) arealso included. The model suggests that cell division isa leading factor in the increased mobility at the colonyedges, and will be useful for studying behaviours ofhiPSCs and improving experiments.Modelling cell movement on a discrete lattice iswidely used, e.g., for mesenchymal stem cell tissue differ-entiation [69] and cancer stem cell driven tumour growth[70]. Some models allow many lattice nodes per cell asin the Potts model [71]. There is also a range of agent-based continuous models where cell movement is notrestricted to a grid but a cell can move continuously inany direction as illustrated in Figure 4 [72,73]. Here themovement is described using forces or potentials withpositions obtained from differential equations of motionfor each cell. In centre based models (CBM), each cellis represented by a simple geometrical object, such asa circle, whereas in vertex models a cell is defined by anumber of connected nodes [74]. These models will bediscussed in more detail in Section 4.There are also models which focus on the cells’ chang-ing morphology. For example, a model has been devel-oped for mesenchymal stem cells which includes therandom formation, elongation and retraction of pseu-dopodia, resulting in dragging forces which lead to cellmovement [66]. However, the model of Ref. [66] showsmore ballistic and accelerated dynamics than experi-mental results [75].
Colonies of hPSCs are formed by repeated mitosis inwhich two genetically identical daughter cells are pro-duced from the division of the mother cell. The cellcycle is the sequence of events that occur in a cell inpreparation for the division as described in Section 2.1.The simplest mathematical models incorporate cell pro-liferation probabilistically, with the division time foreach cell drawn at random from a suitable probabilitydistribution [65]. Others go a step further by movingcells through each cell cycle phase according to tim-ings based on experimental data [76] or as cell volumeincreases [66]. Sometimes divisions do not occur; thisprobabilistic nature of self-renewal can be incorporatedwhen the end of the cell cycle is reached [77]. There arealso more complex models which describe the relation-ship between inter-cellular processes based on growthfactors (proteins that regulate cell growth) [73] and moresophisticated mathematical models describing the cellcycle in terms of limit cycles [78].The doubling time of stem cells number varies andcan be affected by various environmental and chemicalfactors, including cell density and the colony maturity[8,79,80,81]. Models of colony growth can be dynamical-system type models that address the time evolution ofthe colony size, or spatial models which track individualcells and the growing colony in space and time.4.1 Population dynamics modelsPopulation models have been used to understand theprocess by which blood cells are formed [22], cancer tu-mours grow [82] and the impact of hPSC colony growthon clonality [83]. Early population dynamics models forstem cells were based on stochastic birth-death processes[17] involving systems of ordinary differential equations[84]. One of the most popular models for hPSCs includestwo populations of dividing and non-dividing cells, witha term for accounting for cell loss through death ordifferentiation (often referred to as the Deasy model,which is a development of the Sherley model to includecell loss) [85,86]. The evolving number of cells over time N ( t ) is obtained as N ( t ) = N (cid:34)
12 + 1 − (2 α ) t/D t +1 − α ) (cid:35) − M, (1)where N is the initial number of cells, α is the mitoticfraction, D t is the cell division time, and M is thenumber of lost cells.More recently, hyperbolastic growth models (a newclass of parameter model for self-limited growth be- L E Wadkin ∗ et al. haviours [87]) have been introduced for both adult andembryonic stem cells [88]. These growth models providemore flexibility in the growth rate as the populationreaches its carrying capacity and have been demon-strated to capture experimental data well [87,88]. Thepopulation in this case is governed by a non-linear dif-ferential equation dN ( t ) dt = ( L − N ( t )) (cid:20) δγt γ − + θ √ θ t (cid:21) , (2)with the initial condition N (0) = N , and the parame-ters L (representing the limiting value, or carrying ca-pacity of the population), δ (the intrinsic growth rate), γ (a dimensionless allometric constant) and θ (additionalterm allowing for the variation in the growth rate). Thismodel can be used to describe both proliferation andcell death rates more accurately than Equation (1) [88].Our most recent work develops a population modelof the growth for hESC colonies based on experimen-tal data [83]. We analysed the evolution of the colonypopulations and found that the distribution of colonysizes was multi-modal, corresponding to colonies formedfrom a single cell and colonies formed from pairs of cellsas shown in Figure 5. The colony populations can bedescribed using a stochastic exponential growth model,with the growth rates of colonies emerging from singlecell and cell pairs being drawn from normal distributions: (cid:40) N A = e γ A t , γ A ∼ N( µ A , σ A2 ) , probability α,N B = 2 e γ B t , γ B ∼ N( µ B , σ B2 ) , probability β , (3)with µ A = 0 .
039 and σ = 0 . , µ B = 0 . σ =0 . , α = 0 .
77 and β = 0 .
23 inferred from the fittingto the experimental data shown in Figure 5. The growthrate for colonies emerging from pairs of cells is greaterthan for colonies founded by single cells. This means thatcolonies that have grown from cell pairs are larger notonly due to the initial condition but also because theirproliferation rate is larger. This is consistent with ob-servations that hPSCs proliferate more effectively whenin close proximity to other cells [39,89]. This differenceis important when the clonality of a colony needs to beassessed non-invasively, e.g., from its size.The model can be used to predict hPSC colonygrowth and to calculate the time scales over which colonysize no longer predicts the number of founding cells basedon their seeding density. This model can also be usedto simulate colony growth in space which is discussedin the next section. (a) N (72) pd f (b) pd f Fig. 5: (a) The colony populations at 72 h after seedingwith a lognormal mixture model fitting for the singlefounding cell population (blue) and the pair foundingcell population (orange). (b) The growth rate probabilitydistributions for both populations. Adapted from [83].4.2 Spatial modellingColony growth can also be modelled spatially and, aswith cell migration, the models can either be set on a reg-ular or irregular lattice or in continuous space. Each cellcan be modelled individually in an agent-based model,or for large numbers of cells where agent-based modelsbecome computationally challenging, using continuummodels. A thorough summary of these different modeltypes, along with their advantages and disadvantageswith a view to tissue mechanics is provided in [74]. Herethe recent attempts to model hPSC colonies using avariety of these techniques will be discussed.Our multi-population model, Equation (3), can beimplemented to explore the impact of colony growthon clonality [83]. Generating homogeneous populationsof clonal cells is of great importance [52,53] as clon-ally derived stem cell lines maintain pluripotency andproliferative potential for prolonged periods [90]. Toachieve this, cross-contamination and merger of colonies(illustrated in Figure 6(a)) should be avoided.Assuming that, intially, the cells are randomly scat-tered in a growth area with a particular seeding density(the average number of cells per unit area), each cell (orgroup of cells) proliferates according to Equation (3).Each colony is then approximated by a circle, with acertain position in space (the geometric centre of thefounding cells) and a radius based on the populationsize and an assumed cell are of 250 µ m [91]. The timeat which a colony begins to merge with its neighbour, τ , is the time at which the perfect clonality is lost asillustrated in Figure 6. The simulation leads us to anequation, consistent with experimental data, from whichwe can estimate the time taken for the first colony merge athematical modelling of hESCs 7(a) (b) Fig. 6: (a) An example of two colonies merging from ex-perimental images. The two colonies, shown in blue andorange are beginning to merge at 5 days after seeding.The scale bar represents 100 µ m. (b) Diagram illustrat-ing initially seeded cells and the colonies at time τ , thefirst time at which the two growing colonies touch eachother from a simulation of the cell seeding model. Theorange cells are classed as a pair and grow accordinglyfaster. From [83].to occur τ ≈ − n
140 cm − , (4)where n is the initial seeding density of cells beforetheir attachment to the substrate in cells/cm . Theseresults can be used to achieve the best outcome forhomogeneous colony growth in-vitro by choosing theoptimal cell seeding density.Other spatial models consider each individual cell’sposition in space. Common vertex based models foradult stem cell proliferation use Voronoi tessellation todescribe cell position and areas. The colony area is di-vided so that the area occupied by a cell is obtained bytracing straight lines between the position of a cell andall its neighbours and drawing a perpendicular line inthe middle as shown in Figure 7(a). These lines form aconvex polyhedron called the Voronoi cell. The Voronoicells are not uniform in shape and their number of sidesvaries. The tessellation can be constructed from experi-mental images using the cell centroid or cell nuclei posi-tions, as shown in Figure 7(b) [91]. Voronoi tessellationhas been used to model adult stem cells in intestinalcrypts in 2D [92,93] and is now being transferred to hESCs. The model uses an agent-based approximationin which each cell is represented as a Voronoi tessellationof the space [94,93]. The domain grows according to thepressure flow due to mitotic divisions in the colony. Thedynamics between the cells are described by an elasticpotential acting on each cell i as V ( r i , t ) = k v α i ( t ) − α ( t )] + k c r i ( t ) − r i ( t )] (5)with k v and k c elastic constants, α i the area of eachcell, α the equilibrium area and r i the initial positionsof the cells, which do not necessarily correspond to thecentroids denoted with r i . The first term in the righthand side of Eq. (5) tends to enforce uniform cell sizeand the second one gives the shape of the cells. Since theforces are conservative, applying the gradient operatorto Eq. (5) and adding a drag force, the total force actingon each cell is obtained.The boundary of the colony is modelled using ‘ghostcells’ whose only function is to bound the domain. Fig-ure 7(c) shows a simulated colony undergoing a celldivision. Cells in the middle of the colony experience ahigher pressure and show mitotic arrest, i.e. they do notdivide.Spatially modelling each individual cell in a colony inthis way raises an important question about the physicalprocess involved in cell division: how does the colony re-arrange to make space for new cells? In Voronoi tessella-tion models [93,94] the cells re-accommodate themselvesaccording to the potential from the neighbouring cellsor the crypt walls. In most square or hexagonal lattice-based models, one daughter cell is placed in the sameposition as the mother cell while the other is put in aneighbouring position, chosen at random [95], isotropicmitosis. If there is no free position available next to thedividing cell, the neighbouring cells are re-arranged intoother available free spaces stochastically until there isa free space next to the dividing cell [65] or, if this isimpossible, mitosis is suppressed (quiescence) [69,96].Further experimental time-lapse image data is neededto clarify exactly how the new cells are placed in realcolonies.Proliferation also depends on spatial and environ-mental factors. There is evidence that high cell densityreduces cell proliferation [42], which has been capturedin a model showing preferential cell division at the colonyedge [65]. Self-organisation of cells has also been ob-served, where the newly divided (smallest) cells clustertogether in patches, separated from larger cells at thefinal stages of the cell cycle [91]. This segregation bycell size allows the interchange of neighbours as thecolony grows and could directly influence cell-to-cellinteractions and community effects. L E Wadkin ∗ et al.(a) (b)(c) Fig. 7: (a) Voronoi diagram illustrating how colony areais split into tessellated cells. (b) The Voronoi tessella-tion obtained from the centroid positions of cells in anexperimental microscopic image [91]. (c) Voronoi tes-sellation to simulate a proliferating hESC colony. Thecells divide and give rise to two daughter cells undersuitable conditions, see highlighted cells in yellow. Left:the colour bar shows the elastic field in Eq. 5 with theyellow cell highly stressed due compression from theirneighbours. Right: the colour bar shows the same colonywith cells coloured according to the stage of their cellcycle, from early in the cell cycle (red) to late (blue).Spatial models of hPSCs become increasingly com-plex with colony size, and it is difficult to successfullyincorporate many properties of colony growth along withany collective migratory effects. The question of howcolonies re-arrange upon cell divisions requires moreexperimental investigation to elucidate the best models.The development of these models has already had animpact in understanding the growth of cancer tumours[97] and wound healing [98].
Pluripotency is the defining characteristic of stem cells,often referred to as a cell’s ‘stemness’. It is hPSCs pluripotency that gives them the capability of differen-tiating into any type of specialised cell in the humanbody. However, hPSCs can undergo spontaneous differ-entiation which is undesirable for further experimentalapplications. Mathematical models of pluripotency aredeepening our understanding of how pluripotency isregulated, leading to the optimisation and control ofpluripotency in the laboratory.The decision of a stem cell to remain pluripotent orto differentiate into a particular specialised cell is knownas its fate decision. It is not known when a cell makes thisdecision. Even clonal cells under the same conditionsmake different fate decisions and it remains unclearhow much fate choice is lead by inherited factors versusenvironmental factors and intracellular signalling. [99].There are several thorough reviews of the computationalmodels of cell fate decisions [100,101,102]. Here we focuson the regulation of pluripotency and spatial patterningwithin colonies.Biomedical and clinical applications of hPSC coloniesdemand tight control of colony pluripotency and homo-geneity [43], yet this remains challenging. At a single-celllevel, pluripotency is inherently stochastic; indeed, ithas been proposed that pluripotency is only definedstatistically within a population [45]. Cells are regu-lated by their local environment [54,103], notably theirbeneficial interactions with neighbours [44,46]. Coloniesexhibit heterogeneous subpopulations of cells with differ-ing levels of PTF expression [28,30] suggesting a play-offbetween disruptive single-cell and regulatory communityeffects. Such heterogeneity is undesirable, biasing evolu-tion the trajectories and leading to spatially disordereddifferentiation [47]. Here we will consider intra-cellularmodels of pluripotency based on PTFs, and the spatialorganisation of pluripotency at the colony level.5.1 Fluctuating PTFsThe positive-feedback regulation between PTFs (thetransciption factors which regulate pluripotency, see Sec-tion 2.1) was first described as a first order differentialequation model using the Hill equations [104]. However,the parameters of such a model are difficult to estimateaccurately [105]. More recently, PTFs have been mod-elled through branching processes [106]. A thoroughreview of the models of pluripotency is available [18],along with a review of computational modelling of thefate control of mouse embryonic stem cells, with manymodels transferable to hPSCs [102].Recent experimental work has investigated how thePTFs vary over time, and how maternal PTFs are trans-mitted and distributed between the daughter cells [35].The OCT4 abundance in the cells was tracked over time athematical modelling of hESCs 9 before and after the addition of an agent which inducesdifferentiation (BMP4). The cell fates were also recorded.The OCT4 values over time for all cells, organised bycell fate (pluripotent, unknown or differentiated), areshown in Figure 8(a). (a) -40 -30 -20 -10 0 10 20
Time (h) O C T (b) Fig. 8: (a) OCT4 values over time, coloured by cellfate - pluripotent cells (red), unknown (yellow) anddifferentiated (green). Time zero is the time the BMP4is added to the cells. Figure reproduced from [35]. (b)The OCT4 splitting ratio between daughter cells beforeand after BMP4 addition. Figure from [35].We are currently working on modelling the trendsand fluctuations in pluripotency over time based on theexperimental OCT4 data in [35]. First we quantified thenature of the persistence of the OCT4 time series. TheHurst exponent, H is a measure of the the long-termmemory of a time series, with H = 0 . < H < . . < H < µ m.els are well established, however their application tomodelling pluripotency is novel.As the general OCT4 levels is inherited after cell divi-sion, pluripotency levels are most similar among closelyrelated cells even when a reasonable level of randomnessis allowed for [35]. The analysis in [35] also shows thatOCT4 is not always equally allocated between daughtercells upon cell division with the split being sometimesasymmetric, as shown in Figure 8(b). Models of pluripo-tency inheritance should take into account this variationin the splitting ratio upon cell division. This study alsosuggests that a cell’s decision to differentiate is largelydetermined before the differentiation stimulus is addedand can be predicted by a cell’s pre-existing OCT4 sig-nalling patterns. These results imply that the choicebetween developmental cell fates can be largely prede-termined at the time of cell birth through inheritanceof a pluripotency factor [35].These results highlight the important properties formodels of hPSC pluripotency to capture at the individ-ual cell level: the stochastic inheritance of PTFs, theanti-persistence or self-regulation of pluripotency andthe pre-determined cell fate decision. Suitable modelscan then be developed to not only represent the be-haviour on a individual cell scale, but also the colonyscale.5.2 Spatial organisationPluripotency also shows spatial variation on the colonyscale. Preliminary experiments monitoring the OCT4levels in colonies grown from single cells at 72 h postseeding show that pluripotency is clustered, with highlypluripotent cells grouped together, as shown in Figure 9. ∗ et al.(a) (b) Fig. 10: (a) Phase (top) and immunostaining images(bottom) of hESC colonies before and after BMP4 addi-tion. (b) Analysis of expression of a pluripotency markerSOX2 and differentiation marker (AP2 α ) 3 days afterBMP4 treatment. Fluorescent intensity is plotted as afunction of distance from the colony edge and normal-ized to the maximum intensity of each colony [n= 20colonies, p < . α (green) and SOX2 (red) levels between distance 35 µ mand 175 µ m from the edge using a two-tailed pairedt-test]. Error bars represent standard deviations fromthe mean. Adapted from [46].The differentiation of hPSCs also shows distinc-tive spatial patterning [46,47]. Experiments monitoringthe pluripotency marker SOX2 and the differentiationmarker AP2 α have shown that differentiation occurspreferentially at the colony periphery in a band of con-stant width, independent of colony size, as illustratedschematically in Figure 2(c) and shown in Figure 10[46]. These differentiated cells originate from the outerthird of the colony, and remain at the edge. This pro-vides important information for modelling the spatialpatterning of the pluripotent state.This within-colony spatial patterning behaviour ofthe differentiation has been captured by a mechanicalbidomain model [107], a continuum model first devel-oped to describe the elastic behaviour of the cardiactissue [108]. The model predicts that differentiation andtraction forces occur within a few length constants ofthe colonies edge, consistent with the experimental re-sults for differentiation in hPSCs [46,47]. The modelassumes that differences in displacement are responsi-ble for any mechanotransduction (chemical processesthrough which cells sense and respond to mechanicalstimuli) and describes both the intra and extra-cellularspaces in colonies with relationships between stress,strain and pressure forces. The basic equation for thedifference between the intra and extra-cellular displace- ments for changing distance from the colony centre r , u r and w r respectively as u r − w r = − T σ ν exp (cid:26) r − R σ (cid:27) , (6)where T is a uniform stress caused by the growth andcrowding of cells, ν is the shear modulus, σ is a lengthconstant and R is the colony radius. This model showsthat if the difference between the intra-cellular andextra-cellular displacements drives the differentiation,then differentiation is confined to the edge of the colony.This model could be further developed to include morecomplicated geometries as currently the colony is as-sumed to be circular to allow analytical solutions to themodel equations. Furthermore, it is worth investigatingwhether the cell growth represented by the tension T isa function of u r − w r alone, as observations for hESCssuggest distinct actin organization and greater myosinactivity near the colony edge, implying that T could benon-uniform [46].Further experiments are needed to collect data onthe pluripotency of cells across colonies. Analysis of thedata using techniques common in spatial statistics willallow the continued development of pluripotency modelson the colony scale. Mathematical and computational models of hPSC growthare essential in formulating non-invasive predictive tools.Although we have focussed on hPSCs here, it is worthnoting that similar models are used to describe the repro-gramming of somatic cells into iPSCs, which is still a low-yield process with the underlying processes of cell fatedecision uncharacterised [109]. As the reprogramming isa stochastic process, most mathematical models in thisarea probabilistic [23]. A model describing cell types asa set of hierarchically related dynamical attractors rep-resenting cell cycles has lead to the identifications of twomechanisms for reprogramming in a two-level hierarchy:cycle-specific perturbations and a noise-induced switch-ing [21]. These reprogramming protocols make specificpredictions concerning reprogramming dynamics whichare broadly in line with experimental findings. Anotherreprogramming model using a two-type continuous-timeMarkov process with a constant reprogramming rate hasrevealed two different modes of cellular reprogrammingdynamics: TF expression alone leads to heterogeneousreprogramming while TFs plus certain other factorshomogenise the dynamics [110].Here we have discussed some key properties of hPSCs:cell kinematics, cell proliferation and cell pluripotency. athematical modelling of hESCs 11
However, there are other important factors which couldbe included in modelling, e.g., environmental factors,cell-cell signalling, intra-cellular properties and collectivemigration. Models isolating a few of these key propertieshave often captured experimental results well. For exam-ple, focussed migration models have lead to a greater un-derstanding of the behaviour of isolated cells [40,41,60]and the movement of cells within colonies [65,66]. Thereare many population models for colony proliferation,taking into account cell divisions and deaths, providinga distinct computational advantage over more complexspatio-temporal models. Models of colony growth havebeen used to investigate the impact of colony expan-sion on clonality [83], cell regeneration within intestinalcrypts [92,93] and tumour growth [97].Many current efforts focus on modelling cell pluripo-tency and cell fate, as applications of hPSCs requiregreater control over pluripotency and differentiation tra-jectories. The stochastic nature of pluripotency at thesingle cell level [45], along with regulatory communityeffects leads to heterogeneous sub-populations acrosscolonies [28,30]. Recent studies of the fluctuations ofPTFs throughout colonies [35] and spatial patterningof differentiation [46,47] are being used to inform thedevelopment of models of pluripotency and cell fate.Developing comprehensive models of hPSCs remainschallenging, due to their many complex properties acrossmultiple scales, and not yet characterised collective be-haviour effects. It is also difficult to match parame-ters with experimental observations. Model refinementshould be based on a two-way interaction with experi-ments; model parameters should be informed by experi-mental results, and models should influence experimen-tal design. Such models have already helped provide aninsight into tissue formation, wound healing, tumourgrowth and the reprogramming of iPSCs and will nodoubt continue to do so as these models progress.
Acknowledgements
We acknowledge financial support fromNewcastle University, and European Community (IMI-STEMBANCC,IMI-EBISC, ERC
Conflict of interest
The authors declare that they have no conflict of interest.
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