Oscillations in a white blood cell production model with multiple differentiation stages

Abstract

In this work we prove occurrence of a super-critical Hopf bifurcation in a model of white blood cell formation structured by three maturation stages. We provide an explicit analytical expression for the bifurcation point depending on model parameters. The Hopf bifurcation is a unique feature of the multi-compartment structure as it does not exist in the corresponding two-compartment model. It appears for a parameter set different from the parameters identified for healthy hematopoiesis and requires changes in at least two cell properties. Model analysis allows identifying a range of biologically plausible parameter sets that can explain persistent oscillations of white blood cell counts observed in some hematopoietic diseases. Relating the identified parameter sets to recent experimental and clinical findings provides insights into the pathological mechanisms leading to oscillating blood cell counts.

Appendices

Supplementary calculations to the proof of Theorem 2

In the following we provide the calculations leading to Eq. (3).

$$\begin{aligned}&\, b_1\,b_2 - b_3\\&\quad = \left[ \left( 1-\frac{a_2}{a_1}\right) \,p_2 + \left( 1-\frac{a_2}{a_1}\,\left( 1-\frac{1}{2\,a_1}\right) \, \frac{1}{2-\frac{a_2}{a_1}}\right) \,d_3\right] \\&\qquad \cdot \left[ \left( 1-\frac{a_2}{a_1}\right) \, \left( 1-\frac{a_2}{a_1}\,\left( 1-\frac{1}{2\,a_1}\right) \, \frac{1}{2-\frac{a_2}{a_1}}\right) - \left( 1-\frac{1}{2\,a_1}\right) \,\left( 1-2\,\frac{a_2}{a_1}\right) \right] \,d_3\,p_2\\&\qquad - \left( 1-\frac{1}{2\,a_1}\right) \,\left( 1-\frac{a_2}{a_1}\right) \,d_3\,p_2\\&\quad = \left[ \left( 1-\frac{a_2}{a_1}\right) \,p_2 + \left( 1-\frac{a_2}{a_1}\,\left( 1-\frac{1}{2\,a_1}\right) \, \frac{1}{2-\frac{a_2}{a_1}}\right) \,d_3\right] \\&\qquad \cdot \left( 1-\frac{1}{2\,a_1}\right) \, \left( 1-\frac{a_2}{a_1}\right) \, \left[ \frac{1}{1-\frac{1}{2\,a_1}} - \frac{a_2}{a_1}\,\frac{1}{2-\frac{a_2}{a_1}} - \frac{1-2\,\frac{a_2}{a_1}}{1-\frac{a_2}{a_1}} \right] \,d_3\,p_2\\&\qquad - \left( 1-\frac{1}{2\,a_1}\right) \,\left( 1-\frac{a_2}{a_1}\right) \,d_3\,p_2\\&\quad = \left( 1-\frac{1}{2\,a_1}\right) \,\left( 1-\frac{a_2}{a_1}\right) \, \left( \left[ \left( 1-\frac{a_2}{a_1}\right) \,p_2 + \left( 1-\frac{a_2}{a_1}\, \left( 1-\frac{1}{2\,a_1}\right) \,\frac{1}{2-\frac{a_2}{a_1}}\right) \,d_3\right] \right. \\&\qquad \left. \cdot \left[ \frac{1}{2\,a_1}\frac{1}{1-\frac{1}{2\,a_1}} + \frac{a_2}{a_1}\,\frac{1}{\left( 2-\frac{a_2}{a_1}\right) \,\left( 1-\frac{a_2}{a_1}\right) } \right] -1\right) \,d_3\,p_2\\&\quad = \left( 1-\frac{1}{2\,a_1}\right) \,\left( 1-\frac{a_2}{a_1}\right) \, \left( \left[ \left( 1-\frac{a_2}{a_1}\right) \,p_2 + \beta \left( a_1,a_2\right) \,d_3\right] \cdot \gamma \left( a_1,a_2\right) -1\right) \,d_3\,p_2. \end{aligned}$$

Parameter configurations leading to Hopf bifurcation

Constellation 1:	Constellation 2:
$p_1$ increased	$p_1$ increased
$a_2$	$a_1$
decreased	increased
$d_3$ decreased ($<0.5/{\hbox {day}}$)	$d_3$ decreased (close to $0.1/{\hbox {day}}$)
Example:	Example:
$p_1=0.7171/{\hbox {day}}$	$p_1=0.9697/{\hbox {day}}$
$a_2=0.32$	$a_1=0.99$
$d_3=0.132/{\hbox {day}}$	$d_3=0.132/{\hbox {day}}$
Constellation 3:	Constellation 4:
$p_1$ increased	$p_1$ increased
$a_2$	$p_2$
increased (close to 1)	decreased
$d_2$ increased	$d_2$ increased
Example:	Example:
$p_1=0.7778/{\hbox {day}}$	$p_1=0.8687/{\hbox {day}}$
$a_2=0.99$	$p_2=0.0201/{\hbox {day}}$
$d_2=2.6644/{\hbox {day}}$	$d_2=0.2541/{\hbox {day}}$
Constellation 5:	Constellation 6:
$p_1$ increased	$p_2$ decreased (close to
$d_2$	0.01)
increased	$a_2$ increased (close to 1)
$d_3$ decreased	$d_2$ increased
Example:	Example:
$p_1=0.707/{\hbox {day}}$	$p_2=0.01/{\hbox {day}}$
$d_2=0.2541/{\hbox {day}}$	$a_2=0.99$
$d_3=0.132/{\hbox {day}}$	$d_2=0.5287/{\hbox {day}}$
Constellation 7:	Constellation 8:
$p_2$ decreased	$a_1$ increased
$d_2$	$d_1$
increased	slightly increased ($<0.1/{\hbox {day}}$)
$d_3$ decreased	$d_2$ increased
Example:	Example:
$p_2=0.01/{\hbox {day}}$	$a_1=0.95$
$d_2=0.0405/{\hbox {day}}$	$d_1=0.0405/{\hbox {day}}$
$d_3=0.132/{\hbox {day}}$	$d_2=2.7559/{\hbox {day}}$
Constellation 9:
$a_2$ increased
$d_1$
slightly increased ($<0.05/{\hbox {day}}$)
$d_2$ increased
Example:
$a_2=0.95$
$d_1=0.0405/{\hbox {day}}$
$d_2=2.5423/{\hbox {day}}$

Biologically plausible parameter regions where the Hopf-bifurcation exists are visualized in Fig. 5. The reported values for $a_1$ and $a_2$ correspond to the self-renewal fraction in presence of maximal stimulation. The self-renewal fraction at time t is given by $a_1s(t)$ and $a_2s(t)$ with $s(t)<1$.