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# Fractional calculus model of GATA-switching for regulating the differentiation of a hematopoietic stem cell

- Ahmed Alsaedi
^{1}Email author, - Alexey Zaikin
^{1, 2}, - Bashir Ahmad
^{1}, - Fuad Alsaadi
^{3}and - Moustafa El-Shahed
^{1, 4}

**2014**:201

https://doi.org/10.1186/1687-1847-2014-201

© Alsaedi et al.; licensee Springer. 2014

**Received:**16 March 2014**Accepted:**14 May 2014**Published:**24 July 2014

## Abstract

This paper deals with the fractional order model for GATA-switching for regulating the differentiation of a hematopoietic stem cell. We give a detailed analysis for the asymptotic stability of the model. The Adams-Bashforth-Moulton algorithm has been used to solve and simulate the system of differential equations.

## Keywords

- GATA-switching
- fractional order dynamic system
- stability
- numerical methods

## 1 Introduction

Hematopoiesis is a highly orchestrated developmental process that comprises various developmental stages of 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. The transcription factors PU.1 and GATA-1 are known to be important in the development of blood progenitor cells. Specifically they are thought to regulate the differentiation of progenitor cells into the granulocyte/macrophage lineage and the erythrocyte/megakaryocite lineage. 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 [1, 2].

The notion of fractional calculus was anticipated by Leibniz, one of the founders of standard calculus, in a letter written in 1695. Recently great considerations have been made to the models of FDEs in different areas of research. The most essential property of these models is their non-local property which does not exist in the integer order differential operators. We mean by this property that the next state of a model depends not only upon its current state but also upon all of its historical states [3–9].

In this paper, we consider the fractional model for GATA-switching for regulating the differentiation of a hematopoietic stem cell. We give a detailed analysis for the asymptotic stability of the model. The Adams-Bashforth-Moulton algorithm has been used to solve and simulate the system of differential equations.

## 2 Description of the model

*x*,

*y*and

*z*are the concentrations of TFs GATA-1, GATA-2 and PU.1, respectively, ${a}_{1}$, ${b}_{1}$ and ${c}_{1}$ represent the expression rates of genes GATA-1, GATA-2 and PU.1 auto-regulated by itself, respectively, ${a}_{2}$ is the expression rate of gene GATA-1 regulated by TF GATA-2, ${k}_{1}$, ${k}_{2}$ and ${k}_{3}$ are the degradation rates of TFs GATA-1, GATA-2 and PU.1, respectively. There are 23 rate constants in the proposed mathematical model (2.1). Now we introduce fractional order into the ODE model by (2.1). The new system is described by the following set of fractional order differential equations:

*α*is a parameter describing the order of the fractional time derivative in the Caputo sense defined as

## 3 Equilibrium points and stability

where $\eta ={({\eta}_{1},{\eta}_{2},{\eta}_{3})}^{T}$ and *J* is the Jacobian matrix evaluated at the equilibrium points. Using Matignon’s results [10], it follows that the linear autonomous system (3.3) is asymptotically stable if $|arg(\lambda )|>\frac{\alpha \pi}{2}$ is satisfied for all eigenvalues of matrix *J* at the equilibrium point $E=({x}_{1}^{\ast},{x}_{2}^{\ast},{x}_{3}^{\ast})$.

Following [4, 10–13], we have the proposition.

**Proposition**

*One assumes that*${E}_{1}$

*exists in*${R}_{+}^{3}$.

- (i)
*If the discriminant of*$p(x)$, $D(p)$*is positive and the Routh*-*Hurwitz conditions are satisfied*,*that is*, $D(p)>0$, ${a}_{1}>0$, ${a}_{3}>0$, ${a}_{1}{a}_{2}>{a}_{3}$,*then*${E}_{1}$*is locally asymptotically stable*. - (ii)
*If*$D(p)<0$, ${a}_{1}>0$, ${a}_{2}>0$, ${a}_{1}{a}_{2}={a}_{3}$, $\alpha \in [0,1)$,*then*${E}_{1}$*is locally asymptotically stable*. - (iii)
*If*$D(p)<0$, ${a}_{1}<0$, ${a}_{2}<0$, $\alpha >\frac{2}{3}$,*then*${E}_{1}$*is unstable*. - (iv)
*The necessary condition for the equilibrium point*${E}_{1}$*to be locally asymptotically stable is*${a}_{3}>0$.

**Theorem 3.1** *The trivial steady state* ${E}_{0}$ *is locally asymptotically stable if the following conditions are satisfied*: $\frac{{a}_{1}}{{a}_{3}}<{k}_{1}$, $\frac{{b}_{1}}{{b}_{2}}<{k}_{2}$, $\frac{{c}_{1}}{{c}_{2}}<{k}_{3}$.

*Proof*The trivial steady state ${E}_{0}$ is locally asymptotically stable if all the eigenvalues ${\lambda}_{i}$, $i=1,2,3$, of the Jacobian matrix $J({E}_{0})$ satisfy the following condition [9, 14–16]:

The eigenvalues of the Jacobian matrix $J({E}_{0})$ are ${\lambda}_{1}=\frac{{a}_{1}}{{a}_{3}}-{k}_{1}$, ${\lambda}_{2}=\frac{{b}_{1}}{{b}_{2}}-{k}_{2}$, ${\lambda}_{3}=\frac{{c}_{1}}{{c}_{2}}-{k}_{3}$.

Hence ${E}_{0}$ is locally asymptotically stable if the following conditions are satisfied: $\frac{{a}_{1}}{{a}_{3}}<{k}_{1}$, $\frac{{b}_{1}}{{b}_{2}}<{k}_{2}$, $\frac{{c}_{1}}{{c}_{2}}<{k}_{3}$. □

**Theorem 3.2**

*The steady state*${E}_{1}$

*with high expression level of gene GATA*-1

*is stable if the following conditions are satisfied*:

*Proof*The Jacobian matrix of nonlinear system (2.2) for this steady state ${E}_{1}$ is

Theorem 1 of [2] has the same results for the integer order model (and has some misprints in the first condition). They claimed that ${\lambda}_{1}$ is negative, but the sign of ${\lambda}_{1}$ depends on the quantity $(\frac{{a}_{3}{k}_{1}-{a}_{1}}{{a}_{1}})$. □

**Theorem 3.3**

*The steady state*${E}_{3}$

*with high expression level of gene PU*.1

*is stable if the following conditions are satisfied*:

*Proof*The Jacobian matrix of nonlinear system (2.1) for this steady state ${E}_{3}$ is

Under the conditions of Theorem 3.3, we can conclude that steady state ${E}_{3}$ with high expression level of gene PU.1 is stable. □

**The equilibrium points and the eigenvalues of the system**

Equilibrium point | Eigenvalues |
---|---|

${E}_{4}=(659.2192685,0,0)$ | { − 0.909177,0.435225,−0.234831} |

${E}_{5}=(322.17148,1.36478,0)$ | { − 0.909177,0.435225,−0.234831} |

${E}_{6}=(3.812275,13.9889,0)$ | { − 1.37375,0.948121,−0.560537} |

${E}_{7}=(15.99,2.19\times {10}^{-12},1.153)$ | {26.522544,−0.929339,0.0585563} |

${E}_{8}=(3.6329,13.065,1.17301)$ | { − 1.38529,−0.591193,−0.203561} |

${E}_{9}=(0.00054,0.0042,241.024)$ | { − 0.622622,−0.345295,0.0472589} |

It is clear from the table, that the equilibrium point ${E}_{8}$ is a stable point. The other points are unstable.

## 4 Numerical methods and simulations

*et al*. used the predictor-correctors scheme [11, 12] based on the Adams-Bashforth-Moulton algorithm to integrate Eq. (4.1). By applying this scheme to the fractional-order model GATA-switching for regulating the differentiation of a hematopoietic stem cell, and setting $h=\frac{T}{N}$, ${t}_{n}=nh$, $n=0,1,2,\dots $ , $N\in {Z}^{+}$, Eq. (4.1) can be discretized as follows [17]:

*t*is shown for different values of $\alpha =1,0.6$ by fixing other parameters. Figure 4 depicts GATA-2 vs. time

*t*. Figure 4 shows similar variations of GATA-1 with various values of

*α*. In Figure 5, the variation of PU.1 vs. time

*t*is shown for different values of

*α*that increase,

*α*decreases with the concentration of PU.1 gene.

## 5 Conclusions

In this paper, we consider the fractional model for GATA-switching for regulating the differentiation of a hematopoietic stem cell. We have obtained a stability condition for equilibrium points. We have also given a numerical example and verified our results. One should note that although the equilibrium points are the same for both integer order and fractional order models, the solution of the fractional order model tends to the fixed point over a longer period of time. One also needs to mention that when dealing with real life problems, the order of the system can be determined by using the collected data.

## Declarations

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.