Archive for July, 2008

The Keller-Segel model

July 30, 2008

I mentioned the Keller-Segel model in a previous post on chemotaxis. In the past I have read, and heard and thought a lot about this model but I had never actually carefully read the 1970 paper where it was introduced. (Keller, E. F. and Segel, L. A., Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399). I have done so now. I also read a paper by Evelyn Fox Keller called ‘Science as a medium for friendship: how the Keller-Segel models came about’ (Bull. Math. Biol. 68, 1033) where she describes the history of the origin of the model. As indicated in the title of the paper the main object which the model was intended to describe was the formation of concentrations of the population density of the cellular slime mould Dictyostelium discoideum under certain circumstances. The title also indicates that the relevance of this model is to the very beginning of this process. Mathematical studies of the model which follow it up to the formation of singularities, and there are many of these in the literature, are probably of little relevance to Dictyostelium. They may, however be of relevance to the population dynamics of other chemotactic organisms, for instance E. coli. References can be found for instance in a paper of Brenner, Levitov and Budrene (Physical mechanisms for chemotactic pattern formation in bacteria, Biophys. J., 74, 1677). In Dictyostelium ‘stickiness’ as Keller and Segel call it (a more dignified sounding name would be ‘adhesion’) comes into play before the cell density gets very high. The mathematical modelling of adhesion in this type of context does not seem to be well developed although some possibilities have been proposed.

Keller and Segel define a parabolic system of four coupled evolution equations. They then simplify this by assuming that some of the variables have already evolved to equilibrium. This results in a system of two equations which is the usual starting point of mathematical papers on the subject. The equations are linearized about a stationary and homogeneous state and a mode analysis is carried out for the resulting linear system. Growing modes indicate instability. The system contains a number of parameters and for certain choices of these parameters instability is found. This is interpreted as the genesis of the concentrations in the population density which are to be modelled. This procedure is very similar to what is done in many analyses of systems in physics. The authors refer to the analysis of the Benard instability. Another source they quote is the work of Turing on pattern formation. The latter work has had a huge influence in mathematical biology. I am reminded of a mechanism in astrophysics, the Jeans instability, which is invoked to explain the formation of galaxies in the early universe.

A feature of the analysis which the authors see as unsatisfactory is that there is no prediction of the spatial scale on which the concentrations occur. They do make some suggestions for overcoming this. In experiments on Dictyostelium aggregation is seen to be accompanied by pulsations. It had been suggested that these are actively controlled by some pacemaker activity of the cells. Keller and Segel raise the idea that pulsations might arise from a system of the type they discuss without the need for additional input. In this context they mention the concept of overstability which as far as I can see just means the occurrence of an eigenvalue of the linearized problem with positive real part and non-zero imaginary part. In the absence of sufficient knowledge of the more recent literature on the subject I do not know whether, or to what extent, the issues raised in this paragraph have been resolved in the meantime.


Actin motors and comet tails

July 27, 2008

Just over a year ago I attended the International Congress for Industrial and Applied Mathematics in Zurich. Among the huge number of talks given there were quite a few on biological topics. In one of these (unfortunately I do not remember the speaker or the exact topic) a video was shown of bacteria shooting around at high speed in an infected eukaryotic cell. As they move they leave trails which have been called ‘comet tails’. The effect which drives this motion is the polymerization of actin filaments. This is related to, but different from, the mechanism driving the motion of dendritic cells I mentioned in a previous post on chemotaxis. It has a thermodynamic component. Gaps between the end of the fibres and the cell which arise randomly by something like Brownian motion are used by actin monomers to squeeze in and increase the length of the chain.

There are videos of excellent lectures on this subject by Julie Theriot available on the internet. The web page which hosts these videos is a wonderful source of information on various biological and medical topics, presented in lectures by leading researchers. The details of the propulsion mechanism are not so clear and Theriot discusses various alternatives in her lectures. There has been both analytical (mathematical) modelling and numerical work. Unfortunately (from my biased point of view) it seems that it is only the numerical work which has been able to reproduce the experimental data. The biological example she discusses is that of Listeria monocytogenes but similar motion can be achieved with polystyrene beads. While a lot is said about how the bacteria move nothing is said about why they do so and I do not know the answer to the latter question.

In her last lecture Theriot introduces some very interesting ideas on the question of what is the basic difference between eukaryotes and the rest of the living world (bacteria and archaea). First the idea is introduced and discarded that the key point is the existence of a cytoskeleton in eukaryotes. In fact the main elements of the cytoskeleton (actin filaments, microtubules and intermediate filaments) have recently been found to have molecular analogues in bacteria. Next she suggests that all structures in bacteria are based on helices and that this determines the limited range of forms they produce. After emphasizing that every rule in biology has an exception she presents examples of bacteria which are (flat) squares or six-pointed stars. Explaining these forms sounds like a great challenge for geometers.

William Harvey and the circulation of the blood

July 19, 2008

Until the beginning of the seventeenth century it was generally believed by physicians that blood was produced in the liver and somehow consumed in the tissues after passing through the blood vessels. This idea had been around for 1500 years since the time of Galen. The key figure in replacing this belief by the modern view that blood circulates was William Harvey. I find it interesting that the argumentation which led him to this conclusion and which allowed him to defend it against strong opposition can be seen as an application of mathematics to medicine. Admittedly the mathematics involved is no more than elementary arithmetic but I think it is useful to consider it from this point of view. My main source of information for this story is the book ‘A brief history of medicine‘ by Paul Strathern. I can thoroughly recommend this book as an informative and entertaining account of the way in which advances in medicine have changed the way we live over the centuries.

What was Harvey’s argument? He set out to determine how much blood leaves the heart in an hour. To do this he measured the amount of blood which can be taken up by the heart of a dog and the number of heartbeats of the dog per hour. Multiplying the results and doing some extrapolation from the dog to the case of a human gives the result that the amount of blood which flows out of the heart in one hour has more than three times the weight of the whole human body. That blood could be produced and absorbed again at this rate is so far beyond the limits of plausibility that it served as the strongest argument for Harvey’s theory although it was by no means the only one he presented. Even the fact that capillaries had not yet been discovered and that as a consequence nobody knew how the blood passed from the arteries to the veins could not prevent the triumph of the new theory. This was an important step for the application of scientific arguments in medicine and indeed for the application of the scientific method in general.

Formation of black holes in vacuum

July 13, 2008

A black hole can be described by an explicit solution of Einstein’s equations, the Schwarzschild solution, which was discovered shortly after Einstein formulated general relativity. Karl Schwarzschild wrote his paper on the subject while serving on the Russian front in 1915. He died in May of the following year, possibly as a result of the autoimmune disease pemphigus vulgaris. The physical interpretation of the solution remained obscure for decades and the name ‘black hole’ arose in the 1960’s and is generally attributed to John Wheeler.

Physically the Schwarzschild solution represents an eternal black hole. Not only will it always be there, it has always been there. The maximal extension of the solution contains a white hole region which is of doubtful physical relevance. The situation of real physical interest is that where a black hole is formed by the collapse of matter. In fact, since the energy carried by the gravitational field itself gravitates in general relativity, it is possible to imagine that a black hole can be formed from an initial state with small gravitational fields without the intervention of matter. That this scenario can take place within general relativity has recently been shown in a milestone work of Demetrios Christodoulou which took the research community by surprise. The long introduction to that paper provides a beautiful introduction to the history of the mathematical study of black holes.

The Schwarzschild solution is spherically symmetric – it is a perfectly spherical configuration. As a consequence it would be very convenient if the process of formation of a black hole could also be studied in the context of spherical symmetry. Unfortunately this cannot be done in the vacuum case. The obstruction is Birkhoff’s theorem, which shows that a spherically symmetric solution of Einstein’s vacuum equations is not dynamical. Hence if it is desired to profit from the simplification of the mathematics provided by spherical symmetry matter must be included. This has been done in the past and results on the formation of black holes have been obtained by several authors including Christodoulou, Dafermos, AndrĂ©asson, Kunze and Rein.

The new work of Christodoulou quoted above concerns vacuum solutions of the Einstein equations and correspondingly has to manage without symmetry. In fact the mathematical result whose proof occupies almost all of this long text is that a trapped surface can be formed from weak initial data by the evolution defined by the Einstein equations in vacuum. The concept of a trapped surface was introduced by Roger Penrose in 1965 as a criterion for the occurrence of singularities in solutions of the Einstein equations (Penrose singularity theorem). Intuitively the presence of a singularity serves to diagnose the formation of a black hole.

So what exactly happens in the solutions whose existence is demonstrated by Christodoulou? The initial configuration is an ingoing pulse of gravitational waves of high density but short duration. Initially the condition for the presence of a trapped surface is arbitrarily far from being satisfied. As the pulse moves inwards in the course of the evolution the energy is focussed until a trapped surface is formed. A key point of the proof is to show that no spacetime singularity is formed before the trapped surface has time to develop. The techniques used are major developments of those used by Christodoulou and Klainerman in their proof of the nonlinear stability of flat space under the evolution defined by the Einstein vacuum equations.

Ubiquitin and the inflammasome

July 1, 2008

An aspect of modern molecular biology which I find attractive is the way it generates new vocabulary. Of the two examples of this included in the title of this post, the first is already quite old while the second seems to be a buzzword in certain sectors of the research community at the moment. They were brought together in a talk by Vishva Dixit which I heard yesterday. Dixit is Vice President for Research at Genentech which is one of the biggest biotech companies in the world. He has had an interesting career spanning academia and industry. This is described in an article in Nature, 428, 586.

What is ubiquitin? A classical subject in molecular biology is that of the mechanisms by which proteins are synthesized in cells. The path from the information encoded in DNA to its transcription into RNA and its translation into the sequence of amino acids which make up the protein has been studied intensively. The fact that there is a steady turnover of amino acids in the cell means that it is necessary to complement the synthesis of proteins by a process of degradation in order to maintain a steady state. The process of destruction of proteins must be carefully controlled so as to avoid damaging the cell. It was found that the controlled degradation of protein actually consumes energy. At the centre of this process is the small protein ubiquitin. Several molecules of this substance covalently attached to a target protein mark it for degradation. Ciechanover, Hershko and Rose were awarded the Nobel prize for chemistry in 2004 for the contributions of their experiments to the understanding of this process. More recently it has been found that ubiquitin plays an important role in several other processes such as the regulation of the cell cycle, inflammation and immunity.

The inflammasome is a complex of proteins which plays a role in inflammation. It has relations to the mechanisms involved in apoptosis (programmed cell death) and and it makes use of ubiquitin. Understanding how this complex of proteins works and, perhaps more importantly, what cues activate it, may be important for obtaining better insights into autoimmune diseases such as rheumatoid arthritis and lupus.