Master regulators as order parameters of gene expression states

Cell type-specific gene expression patterns are represented as memory states of a Hopfield neural network model. It is shown that order parameters of this model can be interpreted as concentrations of master transcription regulators that form concurrent positive feedback loops with a large number of downstream regulated genes. The order parameter free energy then defines an epigenetic landscape in which local minima correspond to stable cell states. The model is applied to gene expression data in the context of hematopoiesis.

Figure 1

(a) Positive feedback loop involving master regulator (MR) and downstream regulated genes. MR concentrations are represented by the order parameter ${\varphi }_k$, and gene expression states by the spin variables $s_i$. The forward and backward couplings of the feedback loop are ${\xi }_i^k$ and ${\beta }_k{\xi }_i^k$, where ${\beta }_k$ measures the strength of the feedback. (b) The “epigenetic landscape” $V({\varphi }_1,{\varphi }_2)$ for the symmetric two-dimensional case (${\beta }_1={\beta }_2=2$). The biologically meaningful region is restricted to the positive sector ${\varphi }_k\ge 0$ with arrows indicating the stable cell states.

Figure 2

(a) Cell type correlation matrix based on expression states of $N=424$ selected genes as described in Sec. 3a. (b) Normalized expression for master regulators HOXA9, GATA3, PAX5, SPI1, and KLF1. (c) Corresponding computed order parameters for hematopoietic stem cell ($S$), T-lymphoid ($T$), B-lymphoid ($B$), myeloid ($M$), and erythroid ($E$) categories.

Figure 3

(a) Comparison of order parameters and corresponding master regulators for mixed cell types GRAN1 (neutrophil progenitor), ERY2 (erythrocyte progenitor), CMP (common myeloid progenitor), and GMP (granulocyte/monocyte progenitor). (b) Cell types projected into the $({\varphi }_M,{\varphi }_S)$ plane together with the potential $V$ in that plane. Projected minima of $V$ are shown as ($+$). (c) Graph illustrating the relationships of mixed cell types with the five cell categories $S$, $T$, $B$, $M$, and $E$. Arrows indicate the direction of development during hematopoiesis.