The in vivo erythropoiesis, which is the generation of mature red blood cells in the bone marrow of whole organisms, has been described by a variety of mathematical models in the past decades. However, the in vitro erythropoiesis, which produces red blood cells in cultures, has received much less attention from the modelling community. In this paper, we propose the first mathematical model of in vitro erythropoiesis. We start by formulating different models and select the best one at fitting experimental data of in vitro erythropoietic differentiation obtained from chicken erythroid progenitor cells. It is based on a set of linear ODE, describing 3 hypothetical populations of cells at different stages of differentiation. We then compute confidence intervals for all of its parameters estimates, and conclude that our model is fully identifiable. Finally, we use this model to compute the effect of a chemical drug called Rapamycin, which affects all states of differentiation in the culture, and relate these effects to specific parameter variations. We provide the first model for the kinetics of in vitro cellular differentiation which is proven to be identifiable. It will serve as a basis for a model which will better account for the variability which is inherent to the experimental protocol used for the model calibration.
To better understand the mechanisms behind cells decision-making to differentiate, we assessed the influence of stochastic gene expression (SGE) modulation on the erythroid differentiation process. It has been suggested that stochastic gene expression has a role in cell fate decision-making which is revealed by single-cell analyses but studies dedicated to demonstrate the consistency of this link are still lacking. Recent observations showed that SGE significantly increased during differentiation and a few showed that an increase of the level of SGE is accompanied by an increase in the differentiation process. However, a consistent relation in both increasing and decreasing directions has never been shown in the same cellular system. Such demonstration would require to be able to experimentally manipulate simultaneously the level of SGE and cell differentiation in order to observe if cell behavior matches with the current theory. We identified three drugs that modulate SGE in primary erythroid progenitor cells. Both Artemisinin and Indomethacin decreased SGE and reduced the amount of differentiated cells. On the contrary, a third component called MB-3 simultaneously increased the level of SGE and the amount of differentiated cells. We then used a dynamical modelling approach which confirmed that differentiation rates were indeed affected by the drug treatment. Using single-cell analysis and modeling tools, we provide experimental evidence that, in a physiologically relevant cellular system, SGE is linked to differentiation.
Background Nonlinear mixed effects models provide a way to mathematically describe experimental data involving a lot of inter-individual heterogeneity. In order to assess their practical identifiability and estimate confidence intervals for their parameters, most mixed effects modelling programs use the Fisher Information Matrix. However, in complex nonlinear models, this approach can mask practical unidentifiabilities. Results Herein we rather propose a multistart approach, and use it to simplify our model by reducing the number of its parameters, in order to make it identifiable. Our model describes several cell populations involved in the in vitro differentiation of chicken erythroid progenitors grown in the same environment. Inter-individual variability observed in cell population counts is explained by variations of the differentiation and proliferation rates between replicates of the experiment. Alternatively, we test a model with varying initial condition. Conclusions We conclude by relating experimental variability to precise and identifiable variations between the replicates of the experiment of some model parameters.
The in vivo erythropoiesis, which is the generation of mature red blood cells in the bone marrow of whole organisms, has been described by a variety of mathematical models in the past decades. However, the in vitro erythropoiesis, which produces red blood cells in cultures, has received much less attention from the modelling community. In this paper, we propose the first mathematical model of in vitro erythropoiesis. We start by formulating different models and select the best one at fitting experimental data of in vitro erythropoietic differentiation. It is based on a set of linear ODE, describing 3 hypothetical populations of cells at different stages of differentiation. We then compute confidence intervals for all of its parameters estimates, and conclude that our model is fully identifiable. Finally, we use this model to compute the effect of a chemical drug called Rapamycin, which affects all states of differentiation in the culture, and relate these effects to specific parameter variations. We provide the first model for the kinetics of in vitro cellular differentiation which is proven to be identifiable. It will serve as a basis for a model which will better account for the variability which is inherent to experimental protocol used for the model calibration.vitro context, i.e. the process that takes place in cells grown in culture, is much simpler to characterize experimentally. Yet, to our knowledge, no modeling study has focused on it so far. Since the in vitro differentiation is an experimental tool of choice for the study of cellular decision-making (17,18,19) , we propose to develop a model for the dynamics of the in vitro erythropoiesis.Moreover, the current models of erythropoiesis suffer from one major drawback: the weakness of their parameterization, which can fall within three categories.A vast majority of the existing models of erythropoiesis are based on experimental parameter values from the litterature. In some cases these values are used in other contexts that those in which they were obtained (typically, in other species (12) ).In other cases, the parameter values of a model are chosen arbitrarily to reproduce a qualitative behaviour. Apart from this qualitative fit, such approaches do not provide any information regarding the validity of the values (16) .Finally, when the parameters of a model are estimated to reproduce a dataset, the precision of this estimation is seldom investigated (20) . By this, we mean that depending on the algorithmic details of the estimation, it is possible that several values of the parameters might render the same fit to the data. In this case the model is said to be unidentifiable.A model is said to be identifiable if and only if it is possible to infer a unique value for each of its parameter by comparing its output to experimental data. Otherwise it is unidentifiable. A model can be non-identifiable for several reasons (21,22) .Structural identifiability is related to the structure of the model, and the observed variables. A model is structurally unidentifi...
Background: To understand how a metazoan cell makes the decision to differentiate, we assessed the role of stochastic gene expression (SGE) during the erythroid differentiation process. Our hypothesis is that stochastic gene expression has a role in single-cell decision-making. In agreement with this hypothesis, we and others recently showed that SGE significantly increased during differentiation. However, evidence for the causative role of SGE is still lacking. Such demonstration would require being able to experimentally manipulate SGE levels and analyze the resulting impact of these variations on cell differentiation. Result: We identified three drugs that modulate SGE in primary erythroid progenitor cells. Artemisinin and Indomethacin simultaneously decreased SGE and reduced the amount of differentiated cells. Inversely, α-methylene-γ-butyrolactone-3 (MB-3) simultaneously increased the level of SGE and the amount of differentiated cells. We then used a dynamical modelling approach which confirmed that differentiation rates were indeed affected by the drug treatment. Conclusion: Using single-cell analysis and modeling tools, we provide experimental evidence that in a physiologically relevant cellular system, control of SGE can directly modify differentiation, supporting a causal link between the two.
Nonlinear mixed effects models provide a way to mathematically describe experimental data involving a lot of inter-individual heterogeneity. In order to assess their practical identifiability and estimate confidence intervals for their parameters, most mixed effects modelling programs use the Fisher Information Matrix. However, in complex nonlinear models, this approach can mask practical unidentifiabilities. Herein we rather propose a multistart approach, and use it to simplify our model by reducing the number of its parameters, in order to make it identifiable. Our model describes several cell populations involved in the in vitro differentiation of chicken erythroid progenitors grown in the same environment. Inter-individual variability observed in cell population counts is explained by variations of the differentiation and proliferation rates between replicates of the experiment. Alternatively, we test a model with varying initial condition. We conclude by relating experimental variability to precise and identifiable variations between the replicates of the experiment of some model parameters.
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