  The metaphor of a potential epigenetic differentiation landscape broadly
suggests that during differentiation a stem cell follows the steepest
descending gradient toward a stable equilibrium state which represents the
final cell type. It has been conjectured that there is an analogy to the
concept of entropy in statistical mechanics. In this context, in the
undifferentiated state the entropy would be large since fewer constraints exist
on the gene expression programs of the cell. As differentiation progresses,
gene expression programs become more and more constrained and thus the entropy
would be expected to decrease. Such an entropy decrease would, in analogy to
statistical mechanics, require some form of free energy to decrease
accordingly. In order to assess these predictions, we compute the Shannon
entropy for time-resolved single-cell gene expression data in two different
experimental setups of haematopoietic differentiation. We find that the
behaviour of this entropy measure is in contrast to these predictions. In
particular, we find that the Shannon entropy is not a decreasing function of
developmental pseudo-time but instead it increases toward the point of
commitment before decreasing again. This behaviour is consistent with an
increase in gene expression disorder observed in populations sampled at the
point of commitment. Single cells in these populations exhibit different
combinations of regulator activity that suggest the presence of multiple
configurations of a potential differentiation network as a result of multiple
entry points into the committed state.
