So here we are a month since I last posted. I would explain but that would feel like giving excuses and I'm trying to get out of the habit of feeling guilty for things which were out of my hands. I do however have something for you to read and perhaps enjoy.
My most recent deadline has been for a draft masters thesis for my project. This is available for you to read via this link.
What I have written covers about a third of the area of work I have been doing. It is very minimalist and basic with very little of the style or continuity that I duly put in my work. My introduction and background are perhaps not as full as I would like but maybe you can provide some thoughts on that since it is aimed at bringing people of different ranges of knowledge on the subject up to speed before I share my analysis. If there is any point of logic missing please do let me know.
The version I am sharing is the version I would have like to share with the examiners but my supervisor had other ideas so I actually submitted a stripped down version that is even worse so I will not be sharing that.
One point which you should all love is all the images generated that I have included in the report. In my time I have analysed the Julia set changes, the stability changes, the manifolds and most recently the full bifurcations in one half with the other half due to be completed this week. I have finally reached the crux of my examination of the map which is a strange one. The circle hole in one half has an important bearing on the location of the attractor which "disappears" and yet still points remain in the set - this is only possible with an attractor still present yet at some point there doesn't even exist a higher period attractor, just saddles and repellors. Thus this week my question to answer is whether an attractor still appears to exist for the points under the mapping despite the properties attributed to it or if the boundary of the hole itself becomes the attractor or neither of these things (which would mean either a waste of half a year's work or something totally cool and new).
I'll let you know next week hopefully and also provide you will the full bifurcation diagram.
Showing posts with label bifurcation. Show all posts
Showing posts with label bifurcation. Show all posts
Saturday, 5 March 2011
Sunday, 12 December 2010
A Biological Twist
Since I am an engineer you might wonder what I mean by the title. As I said previously, chaos theory is applicable in every science. Biological systems especially exhibit chaotic behaviour. The system under examination today involves DNA within a cell. Some background on DNA and the DNA - transcription - mRNA - translation - protein cycle is available in an interactive form here.
My document for this week is available here.
The Goodwin Oscillator describes a gene's behaviour when it is self-repressing i.e. as it is transcribed and translated it recognises if it is already present at a certain concentration and prevents any more being made. Exactly how this happens biologically I have very little idea. As an chaos mathematician and an engineer all I need is the equations used to mathematically describe the system. From there, as you will see in the paper, I can build a view of what the system does under different parameter values.
In the paper I have used two software packages: MATLAB and XPP. XPP is the best for newcomers to the chaos scene if you want to play with systems. A tutorial and the free download are available here. MATLAB is a (definitely not free), matrix based, mathematical tool provided by Mathworks. A lot of people don't like to use it but I have worked with is so long I can now avoid most of the annoyances and can get it to do what I want now. It is good at crunching lots of numbers for chaotic systems and displaying their behaviour diagrammatically.
In the paper I have focused on the bifurcations of the system. A bifurcation is a point where the system qualitatively changes behaviour dramatically and instantaneously. A steady system can suddenly become oscillatory under the right parameters. In the paper the system of gene self-repression in the cell is bistable - it has a period two oscillation; it oscillates between two values - for all parameter values that I could calculate. The unusual behaviour comes just before that - the behaviour before it settles down to bistable for small x/alpha values is rather strange even for chaotic system. If I could have furthered the paper I would have tested that area more thoroughly and worked out what happens there. However, I was limited by time and pages.
The best thing that the system I examined in the paper shows is that even very simple systems can exhibit highly unusual behaviour as well as perfectly understandable behaviour. It also demonstrates how understanding the mathematics leads to better understanding of the biology. Even without knowing the biology or chemistry to flesh out the details I can tell you that a self-repressing gene will allow itself to be 'switched on' when the concentrations of the mRNA and protein reach certain low levels and will then 'switch off' itself when the concentrations reach certain high levels with the levels being determined by the particular physically characteristics of the system (parameter values). I can tell you that interesting research (in both the biological and the mathematical sense) would be in the area for very small parameter values. I can tell you that I feel that I now understand DNA transcription and translation behaviour better because of this research.
You might think that what I described happening is perfectly reasonable and of course is what happens! But the paper can specify which cells (if exact enough measurements can be taken) will display which behaviour for that gene. And it can specify which cells will not. And it can go further and build from the simple, constrained system to a more complex one taking the mathematical and biological understanding further. Biology is one field that is now accepting chaos theory with open arms because it opens new doors to research and furthers understanding of biological phenomenon by being able to generalise as well as specify. While the field may have arisen from Physics, it is Biology that is currently receiving most of the benefits. In my department, four out of fourteen lecturers have research specialisms in applying chaos to biological systems. That's 28.6% of the department - almost a third.
My final thought for you is for you to consider, if even genes behave chaotically, what other biological processes could do so also?
Further reading for bifurcations: Strogatz (if you have it) covers them really well in ch 3 (basics) and ch 8 (much more interesting!) also a good starting point is the NLDC web page under 'Nonlinear dynamics in action'.
My document for this week is available here.
The Goodwin Oscillator describes a gene's behaviour when it is self-repressing i.e. as it is transcribed and translated it recognises if it is already present at a certain concentration and prevents any more being made. Exactly how this happens biologically I have very little idea. As an chaos mathematician and an engineer all I need is the equations used to mathematically describe the system. From there, as you will see in the paper, I can build a view of what the system does under different parameter values.
In the paper I have used two software packages: MATLAB and XPP. XPP is the best for newcomers to the chaos scene if you want to play with systems. A tutorial and the free download are available here. MATLAB is a (definitely not free), matrix based, mathematical tool provided by Mathworks. A lot of people don't like to use it but I have worked with is so long I can now avoid most of the annoyances and can get it to do what I want now. It is good at crunching lots of numbers for chaotic systems and displaying their behaviour diagrammatically.
In the paper I have focused on the bifurcations of the system. A bifurcation is a point where the system qualitatively changes behaviour dramatically and instantaneously. A steady system can suddenly become oscillatory under the right parameters. In the paper the system of gene self-repression in the cell is bistable - it has a period two oscillation; it oscillates between two values - for all parameter values that I could calculate. The unusual behaviour comes just before that - the behaviour before it settles down to bistable for small x/alpha values is rather strange even for chaotic system. If I could have furthered the paper I would have tested that area more thoroughly and worked out what happens there. However, I was limited by time and pages.
The best thing that the system I examined in the paper shows is that even very simple systems can exhibit highly unusual behaviour as well as perfectly understandable behaviour. It also demonstrates how understanding the mathematics leads to better understanding of the biology. Even without knowing the biology or chemistry to flesh out the details I can tell you that a self-repressing gene will allow itself to be 'switched on' when the concentrations of the mRNA and protein reach certain low levels and will then 'switch off' itself when the concentrations reach certain high levels with the levels being determined by the particular physically characteristics of the system (parameter values). I can tell you that interesting research (in both the biological and the mathematical sense) would be in the area for very small parameter values. I can tell you that I feel that I now understand DNA transcription and translation behaviour better because of this research.
You might think that what I described happening is perfectly reasonable and of course is what happens! But the paper can specify which cells (if exact enough measurements can be taken) will display which behaviour for that gene. And it can specify which cells will not. And it can go further and build from the simple, constrained system to a more complex one taking the mathematical and biological understanding further. Biology is one field that is now accepting chaos theory with open arms because it opens new doors to research and furthers understanding of biological phenomenon by being able to generalise as well as specify. While the field may have arisen from Physics, it is Biology that is currently receiving most of the benefits. In my department, four out of fourteen lecturers have research specialisms in applying chaos to biological systems. That's 28.6% of the department - almost a third.
My final thought for you is for you to consider, if even genes behave chaotically, what other biological processes could do so also?
Further reading for bifurcations: Strogatz (if you have it) covers them really well in ch 3 (basics) and ch 8 (much more interesting!) also a good starting point is the NLDC web page under 'Nonlinear dynamics in action'.
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