Mathematical modeling of GATA-switching for regulating the differentiation of hematopoietic stem cell

Background Hematopoiesis is a highly orchestrated developmental process that comprises various developmental stages of the hematopoietic stem cells (HSCs). During development, the decision to leave the self-renewing state and selection of a differentiation pathway is regulated by a number of transcription factors. Among them, genes GATA-1 and PU.1 form a core negative feedback module to regulate the genetic switching between the cell fate choices of HSCs. Although extensive experimental studies have revealed the mechanisms to regulate the expression of these two genes, it is still unclear how this simple module regulates the genetic switching. Methods In this work we proposed a mathematical model to study the mechanisms of the GATA-PU.1 gene network in the determination of HSC differentiation pathways. We incorporated the mechanisms of GATA switch into the module, and developed a mathematical model that comprises three genes GATA-1, GATA-2 and PU.1. In addition, a novel multiple-objective optimization method was designed to infer unknown parameters in the proposed model by realizing different experimental observations. A stochastic model was also designed to describe the critical function of noise, due to the small copy numbers of molecular species, in determining the differentiation pathways. Results The proposed deterministic model has successfully realized three stable steady states representing the priming and different progenitor cells as well as genetic switching between the genetic states under various experimental conditions. Using different values of GATA-1 synthesis rate for the GATA-1 protein availability in the chromatin sites during the time period of GATA switch, stochastic simulations for the first time have realized different proportions of cells leading to different developmental pathways under various experimental conditions. Conclusions Mathematical models provide testable predictions regarding the mechanisms and conditions for realizing different differentiation pathways of hematopoietic stem cells. This work represents the first attempt at using a discrete stochastic model to realize the decision of HSC differentiation pathways showing a multimodal distribution.


Section 1. Assumptions of the mathematical model
It was proposed that each of the three genes GATA-1, GATA-2 and PU.1 forms a positive auto-regulation loop and promote its expression, where x, y and z represent GATA-1, GATA-2 and PU.1 proteins, respectively, D x , D y and D z are the corresponding DNA of each gene. Here x+ D x represents the binding reaction of protein x to its DNA promoter site D x , while x-D x denotes the molecular complex of protein x and its DNA promoter site D x . In addition, GATA-2 can bind to the GATA-1 DNA promoter site to weakly promote the expression of GATA-1 in order to maintain a moderate level of GATA-1 protein in the primed state, However, there is mutual negative regulation between the GATA genes and PU.1 gene. It has been showed that GATA-1 inhibits gene PU.1 expression via not only protein-protein interaction to prevent the binding of PU.1 to its cofactor c-Jun [1] but also physical binding of GATA-1 to the PU.1 gene at the promoter region [2]. It was assumed that GATA-2 has similar regulatory mechanisms as GATA-1 to inhibit the expression of gene PU.1. In addition, PU.1 represses GATA-1 by binding to it on DNA [3] and also forming the GATA-1-PU.1 complex to inhibit GATA-1 binding to its DNA [4]. Following the assumptions in [5], it was assumed that PU.
and protein-DNA binding reactions as well as binding reactions of hetero-dimers to the DNA promoter regions In this work the detailed transcription process and mRNA molecules were excluded from the model. We used a simplified process to represent protein synthesis, In addition, protein degradation was represented by the first order reactions Biochemical reactions in this system are classified into fast reactions that are assumed to be in an equilibrium state and slow reactions that represent transcription and degradation. Fast reactions include heterodimer formation and DNA binding reactions (Eq. 1~25). For example, reactions (Eq. 11) and (Eq. 22) can be simplified Since the total concentration of the protein remains constant, namely we develop a mathematical model (Eq. 23) to describe the expression dynamics of genes GATA-1, GATA-2 and PU.1, given by

Section 2. Stability conditions of the steady states
The steady states of the proposed model (Eq. 23) are given by the solutions of the nonlinear equation This system may have up to four steady states. One can verify that the system has the following three steady states We can verify the following conditions for the existence of stable steady state.

Theorem 1. (1)
The trivial steady state (Eq. 25) is unstable if any one of the following conditions is satisfied (2) The steady state with high expression level of gene GATA-1 (Eq. 26) is stable if the following conditions are satisfied (3) The steady state with a high expression level of gene PU.1 (Eq. 27) is stable if the following conditions are satisfied Proof.
(1) When , the Jacobian matrix of the nonlinear system is When any one of the conditions in (Eq. 28) is satisfied, at least one of the eigenvalues of the Jacobian matrix is positive.
(2) Under the conditions y=0 and z=0, the second and third equations in (Eq.

24) are equal to zero and the first equation becomes
By assuming ! ≠ 0, then state (Eq. 26) is a steady state. The Jacobian matrix of the nonlinear system (Eq. 24) for this steady state is The three eigenvalues of the Jacobian matrix are It is clear that ! ! < 0. When conditions (Eq. 29) are satisfied, we have ! ! < 0 and ! ! < 0. Therefore this steady state is stable.