In our model cell differentiation is defined as the loss of stem cell properties. Cell differentiation is quantified by a continuous state variable α that can adopt values between zero (full stem cell competency) and one (completely differentiated cell). Each value of α may represent a set of regulatory network activation patterns. From the molecular point of view, α may depend on the abundance and sub-cellular localization of proteins and RNAs, as well as other types of signalling and metabolic molecules 17 . Cell differentiation is assumed to occur independently of cell proliferation 18 .

The model assumes that each cell's α-value fluctuates randomly with a state dependent noise amplitude σ(α). From its current α value a cell adopts a new value α' with a transition rate R. α' is drawn from a Gaussian distribution p(α'| α), centred around α with standard deviation σ(α). According to this assumption, cells tend to accumulate in low noise states. The state dependence of σ(α) is further assumed to be determined by the environment. Hence, a differentiation-inducing environment reduces noise in high α states causing an accumulation of cells in differentiated states (see Figure 1a).

MSC differentiation involves lineage-priming 19 . This process implies particular cellular decisions, which can be modelled considering a second state variable 14 . We here assume that differentiation and de-differentiation dynamics do not depend on these decisions. However, a switch from one into another specific lineage may require a defined degree of stemness as suggested for the chondrogenic lineage 14 . In this case, differentiation stabilises lineages and the described capability of de-differentiation is synonymous with MSC plasticity in general. We here focus on that kind of MSC plasticity.

An important environmental factor during MSC expansion is oxygen 14 20 . In our model, we assume an oxygen dependent control of the state fluctuations. Increasing oxygen tension reduces the state fluctuations in differentiated states, thereby inducing unspecific differentiated, non-proliferative cells. This was implemented assuming the following dependence of the noise-amplitude σ(α) on the oxygen tension pO₂ :

where σ₀ denotes the fluctuation strength in stem cell states and f is a Hill function approaching 0 and 1 at low and high pO₂, respectively.

Cell proliferation is assumed to depend on the differentiation state α of a cell. In our model, it is restricted to intermediate differentiation states α_p with: 0 < α_s < α_p < α_d < 1 (Figure 1b). These states are termed 'progenitor states' in the following. For these proliferative states we assume an identical doubling rate r = 1/τ and average growth time τ. 'Stem cells' (α < α_s) and 'differentiated cells' (α > α_d) do not proliferate. The state fluctuations cause the cells to switch frequently between proliferative and non-proliferative state, which results in an effective average growth time larger than τ.

In order to simulate the spatio-temporal dynamics of MSC populations we use an IBM where the cells are modelled as elastic adhesive spheres 21 . We assume that the cell volume in suspension cannot be smaller than a minimum value V₀. A cell can move actively by migration and passively by being pushed, it can deform, adhere to other cells or a substrate, and it can grow and divide. A proliferating cell divides if its volume has grown to twice the volume V₀.

Assuming that cells can approximately be described by an isotropic homogenous elastic solid, cell-cell and cell-substrate interaction are modelled by a modified Hertz-Potential, consisting of the classic Hertz-Potential and an adhesion term 22 :

In the first term on the right hand side ν_i denotes the Poisson's ratio of the interaction partner i (i = 1,2), E_i its Young modulus, R_i its radius (substrate radius R = ∞) and δ the surface deformation. The second term models adhesion proportional to the Hertz contact area, where ε₁₂ is the anchorage given as adhesion energy per unit area.

Cell proliferation is modelled assuming a two phase cell cycle: During the interphase, a cell doubles its volume by stochastic increments. During the mitotic phase, a cell divides into two daughter cells of equal volume. This growth process results in an approximately Γ-distributed growth time τ of the cells 23 . A cell undergoes a growth arrest if the sum of the deformation forces on it exceeds a critical value F_c.

We simulate cell motion by using a Langevin equation for each cell 21 . The small Reynolds numbers in the regime of single cells allows us to neglect inertia, leading to a linear system of stochastic equations for the cell displacements. Thereby, the displacement dx _i of cell i is given by:

where the sums run over all neighbouring cells j in direct contact to cell i. F _ij ^Hertz denotes the Hertz force between cell i and cell j and F_i ^stoch the stochastic Langevin force on cell i. The friction coefficients γ_is and γ_ij describe friction between cell i and the substrate and between cell i and cell j, respectively. These coefficients are assumed to be proportional to the respective contact areas. Details can be found in 14 .

In addition to the IBM we pursue a theoretical population dynamics model as previously described 15 . Here, we use this model for studying the population average of dynamic properties of individual cells; therefore proliferation is not included. The model is then equivalent to a master equation for a Markov process 24 describing the dynamics of the average number of cells N(α) in state α:

with transition probability and constant randomization rate R. Transition times ϑ(α) from an initial α into the regimes of stem cells (α < α_s) or differentiated cells (α > α_d) were computed using an absorbing boundary approach 24 .

Our model of MSC differentiation dynamics depends on parameters describing intracellular regulation; the randomization rate R, the stem cell state fluctuation strength σ₀, the parameters of the Hill function (n and k) and those specifying the proliferation rate (r and α_s with α_d = 1-α_s). The IBM of spatio-temporal organisation of growing MSC populations depends on parameters specifying cell-cell and cell-substrate interaction, as the Poisson's ratio, the Young modulus, and the friction constants. Combining these models in a particular application one has to adjust a large parameter set.

Recently, we have applied the combined multi-scale model to ovine MSC expansion at low (5%) and high (20%) oxygen tension 14 . These former investigations enable us to use an experimentally validated set of model parameters in the present study. These parameters are summarized in Table 1. We used this parameter set in all simulations if not further specified.

Using the IBM the fates of individual cells in growing populations can be monitored. We simulated individual α-trajectories and compared the cell differentiation dynamics at low (5%) and high (20%) oxygen concentrations. The genealogies of two selected clones in α space are shown in Figure 1c and 1d, for low and high oxygen, respectively.

In order to quantify the degree of plasticity that is inherent in MSCs we calculated the average time required to adopt specific cellular phenotypes. The average transition times of a cell to reach stem cell states (0.0 < α < α_s = 0.15) and differentiated states (α_d = 0.85 < α < 1.0) were calculated as follows: 100.000 cells with α-values equally distributed in the interval [0,1] were subjected to state fluctuations. Throughout the simulations cells that reached the specified subpopulation for the first time were counted and histograms about their initial state were derived. From these histograms we calculated the i) average transition times (Figure 2a, b) and ii) the fractions of cells that successfully transferred within a defined time.

Our results demonstrate that at low oxygen a frequent exchange between the subpopulations occurs on a time scale of about 2 days. At high oxygen the average transition time for stem cells into the pool of differentiated cells increases to about 4 days. Transition times for differentiated cells into the stem cell pool at high oxygen are much larger (>100 days), indicating quasi-deterministic cell differentiation behaviour. We confirmed our results using the master equation approach. In Figure 2c the fraction of cells having entered the stem cell pool at 20% pO₂ is shown as a function of the initial α value and the simulation time. Only in this particular case, the fraction of absorbed cells grows too slowly to calculate the average transition times. In the three other cases, they were computed with high precision (less than 10^-12 of all cells remain to be absorbed).

Since stem cell states are more easily accessible at low oxygen compared to high oxygen we predict MSC plasticity to be more pronounced under these conditions.

In vitro validation of the above results would require single cell tracking of MSCs and techniques to identify the differentiation state of the tracked cells. Currently, considerable effort is taken in order to establish tracking techniques for stem cell systems 25 26 . Unfortunately, MSCs are particularly hard to track, because they tend to aggregate; a phenomenon known as mesenchymal condensation 27 28 . Thus, in the following we present results on MSC plasticity as seen on the population level which can be validated in simpler experimental setups.

Chang et al. 7 studied how fast the distribution of differentiation marker expression within a cell population regenerates from subpopulations with defined expression level. They performed the following experiment: a population of precursor cells was generated under standard conditions and characterised by the expression level of a particular differentiation marker. Subpopulations of cells with defined expression levels of the differentiation marker were separated. These subpopulations were cultivated under standard conditions and regeneration of the distribution of expression levels in the population was monitored over time by FACS.

We simulated this population regeneration experiment as follows: Starting from a population that was grown at low density, i.e. which shows no signs of contact inhibition of growth, we selected 200 stem cells and 200 differentiated cells and followed their development over 5 days in secondary cultures. In order to characterise the environmental dependence of the regeneration process, we compared the MSC behaviour at low and high oxygen tension. Figure 3 shows the results for a selected realisation.

At low oxygen the population structure is roughly regenerated by stem cells and by differentiated cells within about 1 day. At high oxygen the population is regenerated in about 2 days by stem cells but it takes about 8 days when starting with differentiated cells. This is still a surprisingly short time taking into account the large transition times for differentiated cells into the stem cell pool. This phenomenon can be understood by analysing the clone sizes of the 200 selected clones. The distributions of clone sizes after 5 days for all considered cases are shown in Figure 4. Except for regeneration from a differentiated subpopulation at high oxygen the distribution peak is located at about 50-100 cells per clone, demonstrating that most of the clones started growing. If regeneration started from differentiated cells at high oxygen, most of the cells remained quiescent throughout the observation time (137 out of 200 in Figure 4c) and only a few cells started to proliferate and formed large clones. This means the regeneration is driven by the progeny of these few cells only.

At the centre of expanding MSC clones proliferation becomes contact inhibited. The quiescent region grows with colony size until all cells will stop proliferation, when an expanding in vitro culture becomes confluent. Such changes in proliferation activity affect the population structure of MSC colonies. Figure 5 compares the α-distributions of different MSC populations at high oxygen (20% pO₂). Shown are the α-distributions in a low-density population without any sign of contact inhibition, in growing clones with weak and strong contact inhibition induced by variation of the cell-substrate friction constants and in a confluent and thus quiescent population. The fraction of differentiated, non-proliferative cells (α > α_d) increases from about 25% in the low density population to about 90% in the confluent population. A comparable induction of spontaneous differentiation in MSC can be observed in vitro (per. communication, A. Stolzing).

These simulation results implicate that if regeneration refers to the growth of a few large clones, as in the case of differentiated cells at high oxygen, the effect of contact inhibition becomes more relevant for population regeneration. The α-distribution in large clones significantly differs from that of a low-density culture. Moreover, due to the increased number of differentiated cells, these populations show a lower CFU capacity (compare 14 ).

In general, 'self-renewal' of the stem cell population appears in our model as steady occupancy of stem cell states due to a particular population dynamics. Thus, additional information on MSC organisation in vitro can be obtained by performing the regeneration experiments described above in parallel for all subpopulations. Splitting the mother population into a number of subpopulations according to the expression of a differentiation marker, applying the 'regeneration protocol' suggested above to each of these subpopulations and quantifying the number of stem cells in each subpopulation after a fixed regeneration time would allow to quantify the fraction of stem cells in a MSC population descending from a particular subpopulation.

In additional simulations, we followed this concept. However, instead of splitting the mother population into subpopulations, we separated each individual cell of the mother population and followed expansion of the clones generated by the individual cells. For different time points we quantified the clonal composition of the common stem cell pool (0 < α < α_s = 0.15) of all clones in terms of the initial α values of the cells that induced the clones. Figure 6 shows this clonal composition of the stem cell pool after 5 days of clonal expansion. At low oxygen (5% pO₂) the fraction of stem cells that originate from stem cells is about 11%. At high oxygen (20% pO₂) this fraction decreases to only 5%. In both cases, most of the cells in the stem cell pool originate from progenitor states. At low oxygen tension, all progenitor states equally contribute to this pool, while at high oxygen tension most cells originate from progenitor states with a high α value between 0.7 and 0.8.