What are the possible concentration pathways for a 3-species system?

Inspired by [this question](https://chemistry.stackexchange.com/questions/15511/can-a-multi-species-system-oscillate-around-equilibrium), which asks if a multi-species system could oscillate around its equilibrium, I thought about the possible concentration pathways the following more narrow specified system could take:


$$ \ce{A ->[\text{$k\_1$}] B} $$
$$ \ce{B ->[\text{$k\_2$}] C} $$
$$ \ce{C ->[\text{$k\_3$}] A} $$


What I was interested in are the ratio of those concentrations $A:B:C$. The initial conditions are $A(0)=1$, $B(0)=0$, $C(0)=0$. (For simplicity, $[\ce{A}] \equiv A$, and analogous for $B$ and $C$)



> 
> What are the possible concentration pathways for different values of $k\_1$, $k\_2$ and $k\_3$?
> 
> 
> 




Of course, we can't display every pathway there is. We have to select a wide variety of rate constants to get a "good map" of the parameter-space. The approach I took was this:


1. Construct the differential equations


$$\frac{\mathrm dA}{\mathrm dt} = -k\_1 A + k\_3 C$$
$$\frac{\mathrm dB}{\mathrm dt} = -k\_2 B + k\_1 A$$
$$\frac{\mathrm dC}{\mathrm dt} = -k\_3 C + k\_2 B$$


2. Solve them numerically for every possible combination of 0.1, 1 and 10 for each rate constant and the above stated initial conditions
3. This leads to 27 pathways, which are plotted in a ternary diagram


The title of the ternary diagram is $k\_1:k\_2:k\_3$, which is 0.1 : 0.1 : 0.1 in the first one.


![enter image description here](https://i.stack.imgur.com/wz1VG.png)
[![The solution.](https://i.stack.imgur.com/kvFnq.png)](https://i.stack.imgur.com/kvFnq.png)
*(click for larger image)*


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We have a **cyclic** cascade of unimolecular chemical reactions


$$\ce{A} \xrightarrow{\kappa\_1} \ce{B} \xrightarrow{\kappa\_2} \ce{C} \xrightarrow{\kappa\_3} \ce{A}$$


where $\kappa\_1, \kappa\_2, \kappa\_3 > 0$ are the rate constants. Assuming that each chemical reaction in this cascade has [mass action](https://en.wikipedia.org/wiki/Law_of_mass_action) kinetics, then we have the following system of linear ODEs


$$\begin{bmatrix} \dot a\\ \dot b\\ \dot c\end{bmatrix} = \underbrace{\begin{bmatrix} - \kappa\_1 & 0 & \kappa\_3\\ \kappa\_1 & - \kappa\_2 & 0\\ 0 & \kappa\_2 & - \kappa\_3\end{bmatrix}}\_{=: \mathrm M} \begin{bmatrix} a\\ b\\ c\end{bmatrix}$$


where $a := [\ce{A}]$, $b := [\ce{B}]$ and $c := [\ce{C}]$ are the concentrations of the species. Since $\dot a + \dot b + \dot c = 0$, integrating we obtain the following conservation law


$$a (t) + b (t) + c (t) = a\_0 + b\_0 + c\_0$$


where $a\_0, b\_0, c\_0 \geq 0$ are the initial concentrations.




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### Stability


The characteristic polynomial of matrix $\mathrm M$ is


$$\det (s \mathrm I\_3 - \mathrm M) = s \underbrace{\left( s^2 + (\kappa\_1 + \kappa\_2 + \kappa\_3) s + (\kappa\_1 \kappa\_2 + \kappa\_1 \kappa\_3 + \kappa\_2 \kappa\_3) \right)}\_{=: q (s)} = s \, q(s)$$


Since $\kappa\_1, \kappa\_2, \kappa\_3 > 0$, all three coefficients of the (monic) quadratic polynomial $q$ are positive. Hence, both roots of $q$ have negative real parts and, thus, the cyclic cascade is (internally) marginally **stable**. Given nonzero initial concentrations, the concentrations will not grow unbounded. However, due to the zero eigenvalue, the concentrations will not converge to zero either.




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### Steady state


If the concentrations of the species do not converge to zero, then to what values do they converge? At steady state, $\dot a = \dot b = \dot c = 0$. Hence, we have the following homogeneous linear system


$$\begin{bmatrix} - \kappa\_1 & 0 & \kappa\_3\\ \kappa\_1 & - \kappa\_2 & 0\\ 0 & \kappa\_2 & - \kappa\_3\end{bmatrix} \begin{bmatrix} \bar a\\ \bar b\\ \bar c\end{bmatrix} = \begin{bmatrix} 0\\ 0\\ 0\end{bmatrix}$$


where $\bar a, \bar b, \bar c$ are the steady state concentrations. Using Gaussian elimination, we eventually obtain a **line** passing through the origin that is parameterized as follows


$$\begin{bmatrix} \bar a\\ \bar b\\ \bar c\end{bmatrix} \in \left\{ \gamma \begin{bmatrix} \frac{1}{\kappa\_1}\\ \frac{1}{\kappa\_2}\\ \frac{1}{\kappa\_3}\end{bmatrix} : \gamma \in \mathbb R \right\}$$


where


$$\tau\_1 := \frac{1}{\kappa\_1} \qquad\qquad\qquad \tau\_2 := \frac{1}{\kappa\_2} \qquad\qquad\qquad \tau\_3 := \frac{1}{\kappa\_3}$$


are **time constants**. Intersecting this line with the **plane** $\bar a + \bar b + \bar c = a\_0 + b\_0 + c\_0$, we obtain


$$\gamma = \left( \dfrac{1}{\tau\_{1} + \tau\_{2} + \tau\_{3}}\right) \left( a\_0 + b\_0 +c\_0 \right)$$


and, thus, the **steady state** concentrations are


$$\boxed{\begin{array}{rl} & \\ \qquad\bar a &= \left( \dfrac{\tau\_{1}}{\tau\_{1} + \tau\_{2} + \tau\_{3}}\right) \left( a\_0 + b\_0 +c\_0 \right) \qquad \\\\ \qquad\bar b &= \left( \dfrac{\tau\_{2}}{\tau\_{1} + \tau\_{2} + \tau\_{3}}\right) \left( a\_0 + b\_0 +c\_0 \right) \qquad\\\\ \qquad\bar c &= \left( \dfrac{\tau\_{3}}{\tau\_{1} + \tau\_{2} + \tau\_{3}}\right) \left( a\_0 + b\_0 +c\_0 \right)\\ & \end{array}}$$


If, say, $\kappa\_1 \gg \kappa\_2, \kappa\_3$, i.e., if $\tau\_1 \ll \tau\_2, \tau\_3$, then $\bar a \ll \bar b, \bar c$. If $\kappa\_1 = \kappa\_2 = \kappa\_3$, then $\bar a = \bar b = \bar c$.


Assuming that at least one of $a\_0, b\_0, c\_0$ is nonzero, the **ratios** of the steady state concentrations are


$$\frac{\bar a}{\bar b} = \frac{\kappa\_2}{\kappa\_1} = \frac{\tau\_1}{\tau\_2} \qquad\qquad\qquad \frac{\bar a}{\bar c} = \frac{\kappa\_3}{\kappa\_1} = \frac{\tau\_1}{\tau\_3} \qquad\qquad\qquad \frac{\bar b}{\bar c} = \frac{\kappa\_3}{\kappa\_2} = \frac{\tau\_2}{\tau\_3}$$


Hence, the ratio of the steady state concentrations of two species is the ratio of their time constants.




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### Oscillations


Let


$$\Delta (\kappa\_1, \kappa\_2, \kappa\_3) := (\kappa\_1 + \kappa\_2 + \kappa\_3)^2 - 4 (\kappa\_1 \kappa\_2 + \kappa\_1 \kappa\_3 + \kappa\_2 \kappa\_3) \\ = (\kappa\_1 - \kappa\_2)^2 + (\kappa\_1 - \kappa\_3)^2 + (\kappa\_2 - \kappa\_3)^2 - (\kappa\_1^2 + \kappa\_2^2 + \kappa\_3^2)$$


be the discriminant of quadratic polynomial $q$. When $\Delta < 0$, the roots of $q$ have nonzero imaginary parts and, thus, the cyclic cascade will exhibit **damped oscillations**. Hence, when


$$\kappa\_1^2 + \kappa\_2^2 + \kappa\_3^2 > (\kappa\_1 - \kappa\_2)^2 + (\kappa\_1 - \kappa\_3)^2 + (\kappa\_2 - \kappa\_3)^2$$


the cyclic cascade will exhibit oscillatory behavior. Provided that the inequality above is satisfied, the concentrations will oscillate as follows


$$\begin{array}{rl} a (t) &= \bar a + \eta\_A \, e^{- \kappa\_0 t} \cos( \omega\_0 t - \phi\_A)\\ b (t) &= \bar b + \eta\_B \, e^{- \kappa\_0 t} \cos( \omega\_0 t - \phi\_B)\\ c (t) &= \bar c + \eta\_C \, e^{- \kappa\_0 t} \cos( \omega\_0 t - \phi\_C)\end{array}$$


where


$$\begin{array}{rl} \kappa\_0 &:= \frac{1}{2} \left( \kappa\_1 + \kappa\_2 + \kappa\_3 \right)\\ \omega\_0 &:= \frac 12 \sqrt{\kappa\_1^2 + \kappa\_2^2 + \kappa\_3^2 - (\kappa\_1 - \kappa\_2)^2 - (\kappa\_1 - \kappa\_3)^2 - (\kappa\_2 - \kappa\_3)^2}\end{array}$$


Each species will have its own magnitude and phase lag that are dependent on the initial conditions.


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