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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.

在上一集中我們已經看過，如何用7維的聯立微分方程組模擬造血系統的動態，以及調整造血幹細胞的死亡速率會讓系統從無週期→有週期→無週期。但很顯然的，這樣的系統既複雜又不好分析，所以我們要把它簡化成二維的系統來探討。

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

我們回頭看看上次的7維系統：

%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:

我們知道，MyB, ProM, MyC, MetaM只是把HSC的動態往下傳遞到NO而已，所以其實這些方程式都可以丟掉或等效吸收到其他參數裡面去。甚至連G-CSF的方程式其實也可以丟掉，因為他只是反映NO的數量，負回饋到HSC而已。所以我們可以把上面的方程式化簡成像下面這樣：

%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.

這個簡化的二維系統的動態行為和上面的七維系統「基本上」會類似(實際上這樣的化簡會有一個最重大的影響是──它剝奪了quasiperiodic motion和chaotic motion的可能性，這兩種運動型態在連續微分方程組中，系統必須大於等於三維才可以發生。)

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

但這樣簡化的CP值還是非常好的。我們可以非常容易解出這個系統的fixed point(就是對時間微分等於零的點)是：

%[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:

要注意的是，這些參數並不是完全自由的，要讓這個系統足以代表造血系統的動態行為，他們的值域會受到一些限制。例如K>0, b>0, c>0, 0<f<1, d>0。從這裡我們就可以知道，如果我們希望存在一個NO & HSC都>0的fixed point，唯一的可能性是：

(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:

但知道fixed point的存在還不夠，我們必須知道他的穩定性才行。要知道fixed point的線性穩定性(linear stability)，我們需要用到微分方程組的Jacobian matrix。(**這裡必須強調「線性」穩定分析才行，對更高維度的系統以下方法未必能告訴你正確的結果。)對一微分方程組如下：

The Jacobian matrix could be defined as

我們定義他的Jacobian matrix為：

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

所以套用到我們的微分方程組，我們就可以得到他的Jacobian為：

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.

我們回頭看看我們的限制，我們就會發現，J12<0, J21>0, J22<0，唯一可能變號的只有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：

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.

因此我們就可以看出來，影響系統行為最重要的參數就是a, f, b, c，而最簡單調整系統行為的就是b&c。適當的調整這些參數就可以讓fixed point從穩定的變成不穩定的，而不穩定的fixed point就可能伴隨著週期性運動的存在。(*註：這裡用簡單的方式帶過，實際上這個不穩定的fixed point必須是一個node或focus才可以，不可以是一個saddle。)

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.

最後我們回過頭來看週期性嗜中性球缺少症，最常見的突變是elastase發生了toxic gain of function mutation，造成造血幹細胞(其實應該是granulocyte progenitor cell)的死亡，也就是調整了上面的c值，影響造血系統的動態。希望以上的討論可以讓大家對這個疾病的動態行為有更多的了解囉！

在上一集中我們已經看過，如何用7維的聯立微分方程組模擬造血系統的動態，以及調整造血幹細胞的死亡速率會讓系統從無週期→有週期→無週期。但很顯然的，這樣的系統既複雜又不好分析，所以我們要把它簡化成二維的系統來探討。

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

我們回頭看看上次的7維系統：

%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:

我們知道，MyB, ProM, MyC, MetaM只是把HSC的動態往下傳遞到NO而已，所以其實這些方程式都可以丟掉或等效吸收到其他參數裡面去。甚至連G-CSF的方程式其實也可以丟掉，因為他只是反映NO的數量，負回饋到HSC而已。所以我們可以把上面的方程式化簡成像下面這樣：

%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.

這個簡化的二維系統的動態行為和上面的七維系統「基本上」會類似(實際上這樣的化簡會有一個最重大的影響是──它剝奪了quasiperiodic motion和chaotic motion的可能性，這兩種運動型態在連續微分方程組中，系統必須大於等於三維才可以發生。)

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

但這樣簡化的CP值還是非常好的。我們可以非常容易解出這個系統的fixed point(就是對時間微分等於零的點)是：

%[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:

要注意的是，這些參數並不是完全自由的，要讓這個系統足以代表造血系統的動態行為，他們的值域會受到一些限制。例如K>0, b>0, c>0, 0<f<1, d>0。從這裡我們就可以知道，如果我們希望存在一個NO & HSC都>0的fixed point，唯一的可能性是：

(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:

但知道fixed point的存在還不夠，我們必須知道他的穩定性才行。要知道fixed point的線性穩定性(linear stability)，我們需要用到微分方程組的Jacobian matrix。(**這裡必須強調「線性」穩定分析才行，對更高維度的系統以下方法未必能告訴你正確的結果。)對一微分方程組如下：

The Jacobian matrix could be defined as

我們定義他的Jacobian matrix為：

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

所以套用到我們的微分方程組，我們就可以得到他的Jacobian為：

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.

我們回頭看看我們的限制，我們就會發現，J12<0, J21>0, J22<0，唯一可能變號的只有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.

因此我們就可以看出來，影響系統行為最重要的參數就是a, f, b, c，而最簡單調整系統行為的就是b&c。適當的調整這些參數就可以讓fixed point從穩定的變成不穩定的，而不穩定的fixed point就可能伴隨著週期性運動的存在。(*註：這裡用簡單的方式帶過，實際上這個不穩定的fixed point必須是一個node或focus才可以，不可以是一個saddle。)

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.

最後我們回過頭來看週期性嗜中性球缺少症，最常見的突變是elastase發生了toxic gain of function mutation，造成造血幹細胞(其實應該是granulocyte progenitor cell)的死亡，也就是調整了上面的c值，影響造血系統的動態。希望以上的討論可以讓大家對這個疾病的動態行為有更多的了解囉！

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