Centre for Mathematical Medicine & Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Wolfson Centre for Stem Cells, Tissue Engineering & Modelling, Centre for Biomolecular Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Division of Advanced Drug Delivery & Tissue Engineering, School of Pharmacy, University of Nottingham, University Park, Nottingham NG7 2RD, UK

Abstract

Background

To investigate how patterns of cell differentiation are related to underlying intra- and inter-cellular signalling pathways, we use a stochastic individual-based model to simulate pattern formation when stem cells and their progeny are cultured as a monolayer. We assume that the fate of an individual cell is regulated by the signals it receives from neighbouring cells via either diffusive or juxtacrine signalling. We analyse simulated patterns using two different spatial statistical measures that are suited to planar multicellular systems: pair correlation functions (PCFs) and quadrat histograms (QHs).

Results

With a diffusive signalling mechanism, pattern size (revealed by PCFs) is determined by both morphogen decay rate and a sensitivity parameter that determines the degree to which morphogen biases differentiation; high sensitivity and slow decay give rise to large-scale patterns. In contrast, with juxtacrine signalling, high sensitivity produces well-defined patterns over shorter lengthscales. QHs are simpler to compute than PCFs and allow us to distinguish between random differentiation at low sensitivities and patterned states generated at higher sensitivities.

Conclusions

PCFs and QHs together provide an effective means of characterising emergent patterns of differentiation in planar multicellular aggregates.

Background

Embryonic stem cells (ESCs) hold great promise as a source of cells for regenerative medicine, as they are, in principle, capable of being expanded indefinitely

Patterns of cultured embryonic stem cells

**Patterns of cultured embryonic stem cells**. Photograph showing murine embryonic stem cell aggregates (dark circular objects) adherent to tissue culture plastic and cultured under control conditions. Single cells migrated away from the aggregates, eventually forming a sheet of cells. Alcian blue staining suggests the presence of chondrogenesis.

In the context of stem-cell differentiation, theoretical models have successfully described for instance the OCT4-SOX2-NANOG system

The development of mechanistic models to describe pattern formation is a cornerstone of mathematical biology. Substantial attention has focused on systems which exhibit Turing instabilities, involving competition between short-range inhibitors and long-range activators

In this paper, we focus on two candidate mechanisms that may be responsible for pattern formation in populations of stem cells and their progeny, considering patterns which are formed by the transmission of information between cells through either diffusible morphogens or juxtacrine signalling, biasing differentiation pathways. Candidate diffusible morphogens might include TGF-

While our model is generic in the sense that we do not identify explicit morphogens or signalling pathways in our model, we can nevertheless use it to investigate the physical mechanisms that underlie experimentally observed patterns. Previous studies illustrate the complexity of this task. While juxtacrine signalling is typically concerned with pattern formation on the lengthscale of a cell

The differences between patterns arising from diffusive and juxtacrine signals therefore merit careful investigation. Given the complexity of modelling specific multi-step differentiation pathways, and their interactions with other signalling networks, we propose here a deliberately simple pattern-generating model that captures generic features in qualitative terms using minimal parameter sets. Motivated by the idea of the epigenetic landscape, we consider a model in which the state of an individual cell evolves as a flow on a two-dimensional surface

In order to analyse the patterns that emerge from our simulations, we employ statistical measures for marked or multitype spatial point processes. One common class of spatial statistics are 'second-order' characteristics, which include Ripley's K-function

In this paper, we examine two statistical measures that are particularly well suited to multicellular systems, and which could equally be applied to experimental observations. These provide a quantitative estimate of pattern length scales in populations of two cell types, distinguish 'noisy patterns' from completely random differentiation and condense image data into a small number of measures which are useful for parameter surveys. We show how PCFs can be used to assign a length-scale to patterns of differentiating cells. We also show how

Results

The model is initialised by seeding undifferentiated cells at random on a planar surface, and allowing them to push each other apart at short distances (and attract nearby cells at longer distances) to form aggregates with only minimal overlap between cells. Thereafter (for _{
n
}, _{
n
})) that bifurcates into two sub-valleys via a pitchfork bifurcation (Figure _{
n
}for cell _{
n
}that approaches the base of the sub-valley in _{
n
}> 0 (_{
n
}< 0 (

Temporal development of differentiation patterns

**Temporal development of differentiation patterns**. In (a), the left-hand diagrams show the cell states (_{n}, _{n}), with cells coloured according to the key in (b); the right-hand diagrams are the corresponding QHs for the distribution of the cell type variable _{n}. "norm. freq." is normalized frequency. Parameter values: as in Table 1. (c,d) show the potential surface for (2b), _{n }= 0 and (d) _{n }= 0.1. The cells start in a multipotent state (upper valley), but as they progress down the surface they diverge into two distinct phenotypes (lower valleys). Diffusive morphogens bias differentiation towards one of the two states ("tilting" the surface). Solid lines correspond to stable steady-states for the type equation (2b) with _{n }viewed as a constant parameter.

Signals from nearby cells tilt the landscape (Figure _{
n
}> 0) or _{
n
}< 0).

Partitioning of the cells into distinct fates is illustrated by histograms of _{
n
}(Figure

Characterising patterns

At the end of each simulation, cells are characterised by the positions of their centre and their type (

Characterising patterns with spatial statistics

**Characterising patterns with spatial statistics**. (a,d) show two representative model simulations, computed with different parameter values ((a) ^{diff }= 40, λ = 10; (b) ^{diff }= 1, λ = 40); colours indicate _{S}_{S}_{P }(i.e., nearby pairs of cells are more likely to be of the same type than two cells selected at random), the intersection point _{p }giving a quantitative estimate of pattern scale. The QH (_{S }

PCFs are represented by two functions, _{S}
_{S}
_{S}
_{S}
_{
p
}≈ 38 in this case) provides a quantitative estimate of the scale of the pattern.

QHs indicate the proportion _{
R
}of cells of type _{
q
}× _{
q
}square quadrats. If the cells differentiate at random (Figure _{
R
}has an approximately binomial form, _{
q
}
_{R }
_{
q
}, 1/2) with _{
q
}cells in each quadrat, and the type of each cell is determined randomly and independently of the others with probability _{
R
}is approximately normally distributed with _{S}
_{S}

In summary, QHs provide simple information about whether or not a pattern is present whereas PCFs provide additional information about the pattern's length-scale.

Diffusive signalling

The spatial patterns that are observed under diffusive signalling are particularly sensitive to two dimensionless model parameters: ^{diff}, which measures the response of the bias to morphogen concentrations; and the morphogen decay rate, λ. Results from individual realisations of the model for 16 pairs of parameter values are shown in Figure ^{diff }and large λ, the cells appear to differentiate randomly, as the strong decay rate inhibits communication between cells. For large ^{diff }and small λ, the patterns often contain many more of one cell type than another, and in some cases all cells adopt the same (differentiated) fate, with stochastic effects dictating whether they are all red (of type ^{diff}, we observe a transition from random differentiation to distinct patches of cells, with "noisy patches" evident for intermediate values of ^{diff}; patterning is more coherent when cells have greater sensitivity to morphogens. For fixed ^{diff }and increasing λ, the spatial scale of the patches appears to decrease, with the differentiation becoming random for sufficiently large λ.

Survey of patterns under diffusive signalling

**Survey of patterns under diffusive signalling**. Patterns generated by the diffusive signalling mechanism, for a variety of different values of the sensitivity parameter, ^{diff }= 1,3,10,40 and the morphogen decay rate, λ = 1,5,10,40, as indicated. We plot the distribution of cells at the end of a single realisation of the model for each pair of values ^{diff }and λ. For fixed ^{diff}, increasing λ decreases the scale of the patterns. For fixed λ, as ^{diff }increases there is a transition from random differentiation, through a "noisy pattern" stage, to patches of cells which are almost all of one type. The point at which this transition occurs depends upon the values of both parameters.

To identify behaviour that is consistent across multiple realisations, simulations were conducted _{sim }= 100 times for each parameter set in Figure ^{diff }and large λ (_{S}
^{diff }and small λ (_{S}
_{
p
}). The quantitative estimates of the scale of the pattern, _{
p
}, increase slightly as λ decreases (the diffusive signals act over distances proportional to ^{diff}. We report values of _{
p
}for the mean PCFs in Figure _{S}
_{S}

Pair correlation functions: diffusive signalling

**Pair correlation functions: diffusive signalling**. (a) PCFs for simulations with diffusive signalling. For each set of parameter values PCFs are calculated using the results of _{sim }= 100 realisations (corresponding to the individual realisations shown in Figure 4). The dashed line is the cross PCF _{S}_{p}, the point of intersection of _{S }^{diff }= 10 and λ = 1,5,10,40 (error bars show mean plus or minus one standard deviation). Only realisations in which the maximum value of _{S }- g

The corresponding QHs (averaged over _{sim }realisations, see Figure ^{diff }and large λ, in which the histogram has a binomial form with a peak at ^{diff }and small λ in which the majority of the quadrats contain cells which are entirely of one type (_{
R
}≈ 0,1). It is helpful to introduce a (very conservative) threshold that defines the existence of patterns: for example, if more than 10% of the quadrats have _{
R
}< 0.02 or _{
R
}> 0.98 (so lie in either of the extreme bins of the QH), then we say that well defined patterns exist. We demarcate patterned and non-patterned distributions defined by this criterion in Figure ^{24 }≈ 10^{-7}. The degree of noise in the patterns is characterised by the shape of the histograms for intermediate values of _{
R
}; the roughly uniform distribution on 0 < _{
R
}< 1 falls in magnitude as ^{diff }increases (Figure ^{diff}, appears to control the degree of noise in the patterns, whilst the morphogen decay rate, λ, controls their length-scale.

Quadrat histograms: diffusive signalling

**Quadrat histograms: diffusive signalling**. QHs for simulations with diffusive signalling. For each set of parameter values histograms are calculated using the results of _{sim }= 100 realisations (corresponding to individual realisations shown in Figure 4). Numbers show normalised frequencies (and corresponding percentages) for the bins if these are greater than 5. In those QH to the upper-left side of the red line, more than 10% of the quadrats are in either of the extreme bins (_{R }< 0.02, _{R }> 0.98), which we use as a conservative criterion for the presence of patterns.

Dimensionless parameter estimates

**Symbol**

**Description**

**Dimensionless value**

**
L
**

Computational domain size

120

_{init}

Initial number of cells

3500

_{
c
}

Typical cell radius

1

_{end}

Duration of the simulation

4

Bifurcation control parameter

5

Noise amplitude

10^{-4}

^{juxt}

Sensitivity parameter (juxtacrine)

10^{-3}

^{juxt}

Radius for juxtacrine signalling

3

^{diff}

Sensitivity parameter (diffusive)

10

_{a}, _{b}

Morphogen degradation rates

10

_{a}, _{b}

Morphogen diffusion coefficients

10^{3}

_{
s
}

Number of grid squares

120

_{
q
}

Number of quadrats in each direction

12

_{sim}

Number of simulation realisations

100

_{
g
}

Number of distance intervals for RDFs

60

d

Time step

4 × 10^{-4}

Dimensionless parameter estimates; unless otherwise stated (in the figure caption), these are the parameter values used for simulations.

Juxtacrine signalling

For the juxtacrine signalling mechanism, we consider only the effects of varying the sensitivity parameter, ^{juxt}. Simulation results (Figure ^{juxt }to small, distinct patches of cells for larger ^{juxt}. In contrast to the diffusive signalling mechanism, patch size under juxtacrine signalling is limited to approximately 20 cell radii in scale. The transition from random differentiation is evident in PCFs (_{S}
^{juxt}; _{S}
_{
p
}for larger ^{juxt}), which indicate a patch size of approximately _{p }
^{juxt}. The QHs also reflect this transition, although as the scale of the patterns is comparable to that of the quadrats, there are substantially fewer quadrats containing cells entirely of one type (_{
R
}≈ 0,1) than in the diffusive case (with large ^{diff }and small λ).

Juxtacrine signalling patterns

**Juxtacrine signalling patterns**. Patterns generated by the juxtacrine signalling mechanism, for a range of values of the sensitivity parameter, S^{juxt}. Each of the quadrat histograms and PCFs was calculated from the results of _{sim }= 100 realisations with the same parameter values.

Discussion

Heterogeneity in differentiating populations of stem cells hinders the efficient generation of specific types of differentiated cells. Whilst it seems likely that cells will always need to be sorted before being implanted

The statistical measures described here provide a robust, quantitative measure of noisy spatial patterns. We have shown, using a simple model of diffusive or juxtacrine signalling in a cellular monolayer, how QHs provide a simple measure for distinguishing binary patterns of cellular differentiation from spatially uncorrelated outcomes, and how PCFs may be used to estimate the typical lengthscale of binary patterns. As discussed below, these could be readily applied to experimental data, allowing the objective comparison of patterns associated with different culture conditions. In the future, such measures may prove useful in future for comparing the outputs of mechanistic, theoretical models with experimental outcomes. Spatial multicellular simulations often contain large numbers of parameters and generate verbose output; PCFs and QHs may prove to be useful tools for the automatic exploration of parameter space and for condensing the information into a smaller number of physically meaningful quantities.

Model extensions

The present model is deliberately simple, but sufficient to capture the fundamental dynamics (a pitchfork bifurcation with symmetry broken by signalling) that we expect to govern cell fate specification. There are many ways in which we could extend the model. For example, we could include more detailed models of the regulatory networks that govern differentiation

At present, all cells lose their "stemness" at the same, pre-determined rate. It seems plausible that individual cells could undergo a rapid, asynchronous transition from an undifferentiated stem-like state to a committed or differentiated one; our model could be extended to permit this by changing the form of the potential surface. This would also permit small numbers of partially-differentiated cells to be present in the terminal population

In addition, embryonic stem cell populations have been found to be heterogeneous, containing subpopulations which are biased towards particular lineages

More accurate models for diffusive signalling could be developed that account for realistic cell shapes in three dimensions and the details of receptor-ligand binding _{2 }tension

Cell motion can be readily included in the model, e.g. equation (1), which is here used to determine initial cell positions, could be employed and noise added to account for random cell motility. It would also be interesting to extend the model to account for cell division. However, we have concentrated on the case of static populations of non proliferating cells in order to investigate the two patterning mechanisms in a simple context.

Applications to experimental data

The positions of the cell nuclei (possibly obtained through DAPI staining and confocal imaging, followed by image segmentation and identification of the centroids of the nuclei) give a set of points in space, and if a cell type can be assigned to each point (through co-staining), the data will be of the same form as that analysed in this paper. The PCFs (and also the QHs) may be calculated in a straightforward manner using the

Conclusions

We have shown how two statistical techniques, QHs and PCFs, can be used to analyse the spatial patterns that emerge in populations of differentiating cells, when there is randomness in the spatial distribution of cells and in the superimposed patterns of differentiation. We have illustrated these techniques using data from a simple stochastic model, in which cell patterning is regulated by either diffusive or juxtacrine signals. We have shown how the size and onset of patterns can be quantified, and illustrated how patterns depend on the mechanisms controlling differentiation and the system parameters.

Our results suggest that when diffusive signalling regulates differentiation, pattern size, as characterised by the QHs and PCFs, is strongly influenced by morphogen decay rate and the degree to which the morphogen biases cell differentiation, with large-scale patterns observed when the decay rate is low and the cells' sensitivity to the morphogen is high. For juxtacrine signalling, the size of the patterns that emerge is an increasing, saturating function of the cells' sensitivity to signalling; large-scale juxtacrine patterns were not seen in our simulations. Our results also reveal how standard statistical techniques such as PCFs and the QH may be used to analyse and characterise the patterns that emerge from differentiating populations of cells in planar multicellular aggregates.

Methods

We simulate individual cells on a planar substrate. The model operates in two steps, described in detail below: undifferentiated cells are seeded at random (at

Pathways of diffusive and juxtacrine signalling

**Pathways of diffusive and juxtacrine signalling**. (a) In diffusive signalling, cells of type

Patterns of aggregation and differentiation are analysed with PCFs and QHs, as explained below.

Modelling initial spatial distribution

_{init}, generating a distribution that minimises overlapping but allows aggregate formation. Cells move due to forces between neighbouring cells that are repulsive over short distances to prevent overcrowding but attractive over longer distances to mimic adhesion.

The location of the centre of the **x**
_{
n
}, evolves according to the differential equation

Short-range repulsion and long-range attraction are simulated by the velocity

(We note that other functions having a similar quantitative form would be similarly effective.) We take the cut-off radius to be _{
v
}= 3_{
c
}, where _{
c
}is the cell radius. _{init }= 0.002, taking

Modelling cell differentiation

We parametrise the state of the _{
n
}, _{
n
}), which serves as a low-dimensional approximation to the levels of numerous transcription factors and the methylation status of many genes. The variable _{
n
}, lying in the range 0 ≤ _{
n
}≤ 1, denotes the "stemness" or degree of plasticity of the cell; each value of _{
n
}may represent a set of regulatory network activation patterns from the molecular viewpoint, and may depend on the relative abundance and subcellular localisations of proteins and RNAs as well as other types of signalling molecules.

At the start of the simulations, all cells have stemness parameter _{
n
}= 1. Over time and as the cells differentiate, _{
n
}decreases (in the present model in a deterministic manner). The variable _{
n
}(a measure of the relative expression level of specific genes) may take any real value and represents the differentiation fate of the cells. We classify the cells into two types, _{
n
}> 0 and _{
n
}< 0, respectively. (In images of simulations, cells of types _{
n
}= 0 (no preferred lineage) for all cells.

The state of the

where _{
n
}is chosen such that (with _{
n
}viewed as a parameter, and _{
n
}= _{
n
}= 1/2, with a single stable steady state for _{
n
}> 1/2, but two stable (and one unstable) steady states for _{
n
}< 1/2, associated with the two distinct cell fates (Figure _{
n
}breaks the symmetry of the pitchfork bifurcation (Figure _{
n
}= 0. Cells are assumed to remain stationary while they differentiate. We do not claim that the present model for differentiation is definitive; however, it exemplifies in a simple phenomenological way the phenotypic evolution of individual cells.

Diffusive signalling

To simulate diffusive signalling, we assume that the cells produce morphogens with concentrations (at a point **x **in space) denoted by **x**, **x**, _{
n
}> 0) produce _{n }
_{
a
}(_{
n
}, _{
n
}) and _{b}
_{
n
}, _{
n
}), respectively (Figure _{
a
}and _{
b
}, and are degraded at rates λ_{
a
}and λ_{
b
}. The concentrations

where the **x**
_{
n
}(

where _{
n
}decreases).

The influence of morphogens on cell fate in (2b) is modelled by assuming that

^{diff }being a parameter representing the sensitivity of cells to diffusive signalling. Differentiation is biased towards type

Juxtacrine signalling

To simulate signalling between cells which are in direct physical contact (represented by cells whose centres are less than a distance _{juxt }apart, where we take _{juxt }= 3_{
c
}), we define the influence function

summing over all **x**
_{
m
}- **x**
_{
n
}|< _{juxt}. The signals produced by differentiating cells (Figure

^{juxt }parametrises the sensitivity of cells to juxtacrine signalling and the constant

Parameter estimation and nondimensionalization

The governing equations can be simplified by making the model dimensionless. The parameters _{
c
}, ^{-1}, distances on _{
c
}, the cell fate variable _{
n
}on ^{1/2}
^{-1/2}, diffusive morphogen concentrations and production rates on _{
n
}on ^{3/2}
^{-1/2}. In dimensionless variables, we recover equations (2) with _{
a
}, _{
b
}replaced by _{
a
}, λ_{
b
}replaced by ^{diff }replaced by _{
c
}= 1 and ^{juxt }replaced by _{juxt }by

Estimates for the dimensionless parameters are listed in Table _{
a
}and _{
b
}are based on the diffusion coefficient for the morphogen BMP-2, which was estimated to be 10^{-8 }cm^{2}s^{-1 }in _{
a
}
_{
b
}. The typical cell radius is taken to be 10 ^{-1}. However the parameters ^{diff}
^{juxt }and λ_{
a
}, λ_{
b
}have a significant effect on the generated patterns, and therefore a wide region of parameter space is surveyed. (We note that the range of λ considered (1 ≤ λ ≤ 40) encompasses the degradation rate 2.5 × 10^{-4 }s ^{-1 }for the morphogen Dpp in Drosophila measured by _{
a
}= λ_{
b
}= λ, say.

In order to select parameter values such that the diffusive and juxtacrine mechanisms exert similar effects on differentiating cells, we estimate the maximum sizes of _{
c
}in dimensional units), so for the juxtacrine mechanism the contribution to ^{juxt}. As cells typically have 6 or fewer neighbours (close packing for discs), we estimate

where _{0 }a modified Bessel function. As _{
a
}≪ _{
a
}, we estimate

where _{
a
}= 1000, λ_{
a
}= 10, this expression is approximately 0.03^{diff}
^{juxt }is roughly 1000 times smaller than ^{diff}.

Numerical methods

Solutions to the stochastic differential equations (2) are approximated numerically using the Euler-Maruyama method

where the

The morphogen equations (3) are approximated numerically using a cell-centred finite-volume approach to discretise spatial derivatives. We denote by _{
j,k
}(_{
j,k
}(_{
s
}) the average concentration of _{
j,k
}= [(_{
s
}. Equation (3a) becomes

for 1 ≤ _{
s
}, and similarly for (3b).

Solutions to the continuous equations (3) have logarithmic singularities at the cell centres, as the cells are modelled as point sources. These singularities are regularised via the spatial discretization, which averages all quantities over a grid square, making the strength of autocrine signalling (and that between cells separated by distances which are of the order of **x**
_{
n
},t) and **x**
_{
n
},t) experienced by the **x**
_{
n
}, lies. As the system contains stochastic elements, we perform _{sim }simulation realisations for each set of parameter values.

The simulations were written in ISO C99, using the random number generator of the GSL library

**Simulation source code**. Source code for simulations of pattern generation in populations of stem cells.

Click here for file

Spatial statistics

Pair correlation functions

PCFs are 'second-order' characteristics (involving relationships between pairs of points). We first define them for sets of points which are all of one type, before extending their definitions to the multitype case.

Let Π(**
ξ
**,

In the multitype case, for each choice of ^{(2) }(**
ξ
**,

We estimate PCFs using the approach illustrated in Figure _{XY}
_{
g
}intervals of equal length _{
g
}. Setting _{
j
}
_{
g
}, we approximate _{
k
}< _{
k
}
_{+1 }by

Calculating PCFs

**Calculating PCFs**. Schematic diagram to illustrate the method used to calculate PCFs. For each distance interval (_{k}, _{k}_{+1}] and each cell with centre **x**_{m}, we count the number of (other) cells in _{k }<_{k}_{+1 }where **x**_{m}. The PCF, _{k }<_{k+}_{1 }is the mean number of cells in these annular regions normalised by _{XY}**x**_{m }to be of type _{S}_{RR}_{GG}(r)

where _{
nm
}≡ | **x**
_{
n
}- **x**
_{
m
}|, _{(s,t]}(

For each cell _{
k
}
_{k}
_{
+
}
_{1 }centred at **x**
_{
m
}, and normalise this by the expected number of cells in an area of this size were the cells to be uniformly distributed. We then average this over all _{k}
_{
+
}
_{1 }+ _{
k
})/2) (this is linearly interpolated to give a continuous line).

The cross PCFs _{XY }
_{
X
}and _{
Y
}are the numbers of cells of type _{RR}
_{GG}
_{S}

We choose to weight the two cross PCFs in proportion to the number of pairs of cells of that type, as _{S}
_{sim }realisations with the same parameter values in order to better estimate them.

Quadrat histograms

To calculate this statistic, we partition the domain [0, _{
q
}× _{
q
}squares (or quadrats) with side length _{
q
}. We calculate the proportion _{
R
}of cells of type _{
n
}> 0) in each quadrat, ignoring empty quadrats; we combine the results of _{sim }simulations with the same parameter values to generate a histogram of the distribution of _{
R
}over all quadrats and for all simulations.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

JAF developed the mathematical model in collaboration with HMB, OEJ and JRK. JAF also performed the numerical simulations and the statistical analyses of the resulting data. GRK generated the experimental results presented in Figure

Acknowledgements

This work was supported by the BBSRC/EPSRC Grant BBD0085221. OEJ acknowledges support from the Leverhulme trust. JRK also gratefully acknowledges the funding of the Royal Society and Wolfson Foundation.