### #65 The oscillation in peripheral blood-III 血球的數量消長_下

In the previous episode we have used a 7-dimensional system of ODE to simulate the dynamics of hematopoiesis, and we adjusted the death rate of hematopoietic stem cells to produce the cyclic behavior that observed in patients with cyclic neutropenia. However, a 7-dimensional system is too complex to be analyzed easily, so today we are going to simplify it into a 2-dimensional system.

Let's see again the 7-dimensional system we have dealt with last time:

%HSC' = +a*(1-f)*GCSF*HSC + b1*HSC - d_HSC*HSC
%MyB' = +a*f*GCSF*HSC + b2*MyB - c1*MyB - d_MyB*MyB
%ProM' = +c1*MyB +b3*ProM -c2*ProM - d_ProM*ProM
%MyC' = +c2*ProM + b4*MyC - c3*MyC - d_MyC*MyC
%MetaM' = +c3*MyC - c4*MetaM - d_MetaM*MetaM
%NO' = +c4*MetaM - d_NO*NO
%GCSF' = +GCSF_prod*(1 - NO/(K + NO)) - d_GCSF*GCSF

It's obvious that all what MyB, ProM, MyC & MetaM did was just transferring the dynamical behavior of HSC down to NO. So they could be discarded or absorbed into some parameters equivalently. Even the ODE accounting for the dynamical behavior of G-CSF could be discarded, since it merely reflected the negative feedback effect of NO onto HSC. We therefore simplify our system of ODE into something like this:

%HSC' = +a*(1-f)*HSC*(1 - NO/(K_NO)) + b*HSC - c*HSC
%NO' = +a*f*HSC*(1 - NO/(K+NO)) - d*NO

This simplified system has "BASICALLY" the same dynamical behavior with the above 7-dimensional system. However, this simplification would eliminate the possibility of quasiperiodic motion & chaotic motion. Both of them are only possible in dynamical systems whose dimension are at least 3.

Even with these shortcoming, such simplification is still profitable. We could easily solve the fixed point of the system:

%[NO]ss = -K[(b-c) + a(1-f)]/(b-c)
%[HSC]ss = (d/af)*[NO]ss*(1+[NO]ss/K)

It should be noticed that these parameters are not freely adjustable. To make the system suitable to represent the dynamics of hematopoietic system, their ranges are somewhat limited. For example, K>0, b>0, c>0, 0<f<1, d>0. If we hope that there is a fixed point with NO > 0 & HSC > 0, it's necessary that:

(b-c) < 0   &   a(1-f) > (c-b)

However, merely knowing the existence of the fixed point is not enough. We need to know its stability to predict the dynamical behavior of the system. We need the Jacobian matrix of he ODE system to deduce its (linear) stability. For an ODE system:

The Jacobian matrix could be defined as

We adopt this formula into our ODE system and we will get its Jacobian matrix:

Let's the consider the limits we set previously. We will then discovered that J12<0, J21>0, J22<0. The only element that could change from negative to positive or the opposite is J11.

The trace and the determinant of the Jacobian matrix would determine the (linear) stability of the fixed point, as shown below:
Jacobian matrix的trace (= J11 + J22) 和determinant (= J11J22 - J12J21)會決定fixed point的穩定性，像下面這張圖所示：
The properties of the fixed points vary with the trace and the determinant of the Jacobian matrix of our ODE system. Figure originally from：https://commons.wikimedia.org/wiki/File:Phase_plane_nodes.svg

Therefore we could recognize "b & c" as the most important parameters of the system. Properly adjust b & c could most easily change the dynamical behavior of the system.

Finally, let's look back to the cyclic neutropenia. The most common mutation shared by patients with cyclic neutropenia is the toxic gain of function of the gene of elastase, which cause the apoptosis of granulocyte progenitor cells. This is equivalent to adjusting the parameter "c" above and therefore change the dynamical behavior of the ODE system most drastically. We hope the above discussion could make all of you understand more about the dynamical behavior of this disease.