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MBW:Oscillatory behavior in enzymatic control processes

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This wiki page presents a model that depicts how the frequency of a system of two coupled biological oscillators can be varied across a wide range. This model specifically investigates the coupled oscillations of the concentrations of mRNA strands and the proteins expressed from them. A differential equation can describe a single oscillation. This model uses a system of two differential equations to describe the oscillations of the concentrations of a single species of mRNA and the corresponding protein, and then combines two of these systems to form another system of four differential equations describing the coupled oscillations of two linked oscillating mRNA-protein systems. Each oscillator is then controlled by negative feedback from it and the other oscillator. Parameters in the model include the coupling constants, which can be varied to control the frequency of the coupled biological oscillator system. The variation of these parameters allowed the model to be analyzed graphically, while their variation by an organism allows it to tune its natural rhythms to their optimal frequency.


Understanding the mechanisms by which life can be sustained is a problem that scientists have been trying to gain insight on for a long time. A topic of interest pertaining to this subject during the first half of the 19th century was whether or not the molecular control processes in living cells involved oscillatory behavior. Traditionally it was thought that molecules in the cell would move towards steady state concentrations. These concentrations determined by the environment inside and around the cell. Experimental data however was beginning to show evidence of biological “clocks” inside of living cells that would determine when processes would happen[2]. This evidence was an indicator that there must be oscillatory motion within a cellular organism. The trouble was that there are many cyclic processes in a cell that have different cycle periods (sleep, photosynthesis, puberty, etc. In the 1965, Brian C. Goodwin published a paper “Oscillatory behavior in enzymatic control processes”[1] that described how two coupled biological oscillators could create a gamut of frequencies by varying the coupling constant. This provided a model for how two oscillators could control the timing for several biological processes all on different timescales. This discovery helped pave the way for our current understanding of both cellular organisms and of coupled oscillators. This paper will present a simplified model of a single biological oscillator and then couple two biological oscillators to show that different timescales can be created with such a system. The paper will be modeled after work done in Goodwin’s 1965 work.

Figure 1- mRNA creating a protein in the cell

Single biological oscillator model

Figure 2- Compartment model of a single negative feed back oscillator.

We will first consider a system showing oscillatory behavior between concentrations of messenger ribonucleic acid (mRNA), and proteins in a cellular organism. A video showing how mRNA molecules create proteins can be found here [1]. We denote concentrations of mRNA at time step i by X_{{1}}, and the concentration of proteins by Y_{{1}}. The job of the mRNA is to produce proteins and to make sure the cell has enough proteins. Therefore it makes sense that a cell with a lot of proteins will have negative feedback to suppress production of mRNA molecules. The system of equations that govern the concentrations of these two species within the cell are given by

{\begin{array}{c}{\frac  {dX_{{1}}}{dt}}={\frac  {a_{{1}}}{A_{{1}}+k_{{1}}Y_{{1}}}}-b_{{1}}\\\\{\frac  {dY_{{1}}}{dt}}=m_{{1}}X_{{1}}-B_{{1}}\end{array}}

Notice that the number of mRNA molecules produced is inversely proportional to the number of proteins present. This is the negative feedback that will allow our system to oscillate. This behavior shows one mechanism by which a cell can keep track of time. This system was modeled by Goodwin with


These values were chosen such that the concentrations of variables in the calculation stayed in a region that the computer being used could handle. The computer was one of the first analogue computers built at the Massachusetts Institute of Technology (MIT). The results of this calculation can be seen below. The larger of the two oscillations represents the protein concentrations in a cell and the smaller amplitude oscillation represents mRNA concentrations. Though these oscillations are non-linear, they still show that low protein concentrations lead to an increase in mRNA which boosts protein concentrations as expected.

Figure 3- A single oscillator behaving in a non-linear fashion.

We have thusfar neglected the fact that there are large numbers of proteins and mRNA molecules inside of any one individual cell. In order to keep track of such large numbers of molecules, we study the system using statistical mechanics [3] and can define an effective temperature or talandic temperature of the system \theta . This allows us to create a probability distribution of how many mRNA molecules there are with respect to how many protein molecules there are in any given cell. We now find the steady state equations of a single oscillator to be

{\begin{array}{c}\\X_{{1}}^{{*}}={\frac  {B_{{1}}}{a_{{1}}}}\\\\Y_{{1}}^{{*}}={\frac  {1}{k_{{1}}(\left(a_{{1}}/b_{{1}}\right)-A_{{1}})}}.\end{array}}

For the parameter values chosen, we expect the steady state to occur when both mRNA and proteins have a concentration of 0. This is an unstable equilibrium. It was interesting to read that when the initial conditions X_{{1}}(t=0)=Y_{{1}}(t=0)=0 noise from the 1960's computer caused oscillations to come out of the steady state equations, a story that highlights the lack of stability. It is shown below that this behavior and in this figure it can be seen that the amplitude of protein oscillation is larger than that of mRNA oscillation. These small values however are closer to the equilibrium values of the system and we therefore see nice sine waves meaning the system is behaving linearly. We will now try to predict this using the talandic temperature and techniques of statistical mechanics. Utilizing Boltzmann statistics and the parameter values listed above, we find that the average positive value of our concentrations are

{\begin{array}{c}\\X_{{{\mathrm  {avg}}}}=4.8\theta \\\\Y_{{{\mathrm  {avg}}}}=18\theta .\end{array}}

This suggests, and as the simulations show, that the amplitude of the oscillations of protein concentrations is larger than the mRNA oscillations by a significant amount.

Figure 3- A single oscillator with a linear response.

Coupled oscillators

Figure 5- Compartment model of two coupled oscillators.

There are many types of mRNAs all making different types of proteins simultaneously in a living cell. With so many proteins being created by and inhibiting the production of mRNA, it is to be expected that these oscillatory cycles should interact with each other. For example a high population of a protein can inhibit the production of an mRNA that is designed to make a different protein. This behavior can be modeled as two oscillators each creating negative feedback for itself and for the other oscillator. Such a system is known as a coupled oscillator. Though there are many proteins mRNA pairs in any given cell, we begin by examining the simplest possible system of two coupled oscillators. The behavior of these two oscillators is given by

{\begin{array}{c}{\frac  {dX_{{1}}}{dt}}={\frac  {a_{{1}}}{A_{{1}}+k_{{11}}Y_{{1}}+k_{{12}}Y_{{2}}}}-b_{{1}},{\frac  {dY_{{1}}}{dt}}=m_{{1}}X_{{1}}-B_{{1}}\\\\{\frac  {dX_{{2}}}{dt}}={\frac  {a_{{2}}}{A_{{2}}+k_{{22}}Y_{{2}}+k_{{21}}Y_{{1}}}}-b_{{2}},{\frac  {dY_{{2}}}{dt}}=m_{{2}}X_{{2}}-B_{{2}}\\\end{array}}

where k_{{21}}, and k_{{12}} describe the coupling of the two oscillators. If these two variables are zero, then the oscillators are no longer coupled. Goodwin begins to examine the system in this limit using the following parameter values:


Figure 6- Two uncoupled oscillators oscillate independently of each other.

As shown above these two uncoupled oscillators are free to oscillate independently of each other with relative resonant frequencies theoretically given by

{\begin{array}{l}\\{\frac  {\omega _{{2}}}{\omega _{{1}}}}={\frac  {11}{10}}.\\\end{array}}

We can now begin to couple these oscillators by setting k_{{12}}=0.3 so that the faster oscillator is now driving the slower one and as shown below, our oscillators have loved their frequencies in with each other and oscillate coherently.

Figure 7- Two oscillators with their frequencies matched via coupling.

If we now reverse the coupling so that the slower oscillator is diving the faster one (k_{{21}}=0.3), then we find that the frequencies do not sync up, and we find that oscillator 1 has a beat frequency in its amplitude envelope. The frequency of this amplitude is given by the normalized frequency of the system (\omega _{{{\mathrm  {beat}}}}=\omega _{{2}}/11=\omega _{{1}}/10).

Figure 8- Two oscillators can be coupled to create beat frequencies.

We can continue the investigation by returning to an uncoupled system k_{{21}}=k_{{12}}=0, and using the above parameters, we will only change values a_{{1}}=0.6,a_{{2}}=2.0 so that the ratio

{\frac  {\omega _{{2}}}{\omega _{{1}}}}={\frac  {13}{7}} shown below.

Now we observe mutual coupling between the oscillators (k_{{12}}=1.2,k_{{21}}=0.4). The result shown below is known as a subharmonic resonance. This curve shown that both oscillators now run at the same frequency which is much smaller than either of their original frequencies. Y_{{2}} has its original frequency superimposed on the subharmonic oscillation and is shown below in the lower curve. Some interesting features of this subharmonic property is that it causes the concentrations of both Y_{{1}} and Y_{{2}} to nearly double and puts the oscillations of the two oscillators \pi radians out of phase.

Figure 9- Two coupled oscillators creating a subharmonic frequency.

We have seen that the possibilities for cell behaviors when oscillators begin to couple to each other is far greater than the frequencies defined by the free running independent oscillators. Things such as subharmonics and beat frequencies shown above could account for many long biological rhythms such as the circadian (24 hour) cycle. A cell can also create primary and subharmonics and pick out which ones it wants to monitor by changing substrate concentrations. This will change the coupling and therefore change which oscillator (1, or 2) has a larger amplitude.

Theoretical calculations have shown that the subharmonic behavior is only stable in specific regions of parameter space which is divided into stable and unstable regions. The bifurcation values of this system occur at the roots of


Single oscillator with damping

{\begin{array}{c}{\frac  {dX_{{1}}}{dt}}={\frac  {a_{{1}}}{A_{{1}}+k_{{1}}Y_{{1}}}}-b_{{1}}\\\\{\frac  {dY_{{1}}}{dt}}=m_{{1}}X_{{1}}-B_{{1}}\\\\{\frac  {dZ_{{1}}}{dt}}=\gamma _{{1}}Y_{{1}}-\delta _{{1}}Z_{{1}}\\\end{array}}

There are many interpretations of these equations which are described in detail in Goodwin's paper. They have do do with delays occurring at different points in the oscillation cycle. The only differences in the interpretations given in the paper are the parameter values that should be chosen for simulation. He does not state which model he uses, but analyzes the system using parameter values

a_{{1}}=360,A=43,k_{{1}}=1,b_{{1}}=2,m_{{1}}=1,B_{{1}}=0,\gamma _{{1}}=1,\delta _{{1}}=0.8,

the initial concentrations are not listed. The oscillations for these parameters are shown below whereX_{{1}}, and Y_{{1}} are the smaller valued oscillations with X_{{1}} leading Y_{{1}} in phase. Z_{{1}} has a large amplitude and has an even further delay behind Y_{{1}}. The phase portrait at the bottom of this page shows that only in a limited region of parameter space does this system behave as an oscillator. The region where it is an oscillator however is stable. This is good for cells maintaining oscillatory behavior in a range of environments.

Figure 10- Concentrations of protein, mRNA, and Z in time for a single damped oscillator.
Figure 11- The phase plane for a single damped oscillator.


It has been shown by Goodwin that negative feedback loops in biological mRNA protein cycles can create biological clocks that can create a range of frequencies from which the living organism can keep track of time. This is one possible theory that is able to explain the data taken which suggests that oscillations in a cell are taking place at many frequencies simultaneously. In order to understand this theory, the paper presented a thermodynamic representation of a biological oscillator, observed beat frequencies, subharmonic resonances, and damping processes in the oscillations. This theory has become very popular in biological physics since in publication, and this paper has been cited over 300 times.

Discussion of Citation

Goodwin's paper on which this project is based was recently cited in the paper "On the analysis of complex biological supply chains: From process systems engineering to quantitative systems pharmacology" [4]. This paper discusses the development of more sophisticated mathematical models of biological systems using principles of chemical process engineering. Adding mechanistic information to a model can potentially give deep insight into the behavior of the system being modeled, but often makes the model specific to a certain scale. Principles of chemical process engineering are used to create models that incorporate the system's mechanism but remain applicable to multiple scales. Part of this paper dealt with analyzing the circadian oscillation of cortisol, a hormone involved in inflammation. Since it takes into account the state of the system, on which this behavior depends, they used Goodwin's negative feedback oscillator instead of a simple oscillatory function to model it. This enabled them to understand how varying some of the system parameters affects cortisol oscillation. Changes in cortisol oscillation are a real phenomenon that occurs particularly in chronic inflammatory diseases, and studying its causes may lead to novel treatment methods.


1. B. C. Goodwin. Oscillatory behavior in enzymatic control processes. Adv. Enzyme Regula tion, 3:425�-4384 438, 1965.

2. J. L. CLOUDSLEY, ON, Rhythmic Activity in Animal Physiology and Behaviour, Academic Press, New York and London (1961).

3. B. C. Goodwin. A statistical mechanics of temporal organization in cells, Syrup. Soc. Expor. Biol. 18, 301-326 (1964).

4. R. T. Rao, M. L. Scherholz, C. Hartmanshenn, S. B. Bae, I. P. Androulakis. On the analysis of complex biological supply chains: From process systems engineering to quantitative systems pharmacology, Computers & Chemical Engineering, 107, 100-110 (2017).