Archive for October, 2009

Manipulating cells using light

October 27, 2009

In what follows I describe another subject which was a theme in the talk of Orion Weiner mentioned in the previous post. In the meantime I am familiar with the fact that there are techniques which allow us to see details of what is going on in cells. Here the most prominent protagonist is the green fluorescent protein (GFP) which was honoured by Nobel prizes in 2008. It allows information to be exported from the cell. This is a passive process in the sense that once the system has been prepared we just watch what happens. A more active process which is sometimes shown on video is that where a neutrophil follows the moving tip of a micropipette which is releasing a substance to which the cell is chemotactic. The subject of the present post is how it is possible to actively manipulate cells by sending in light of certain wavelengths. This may mean bathing the cell in light, illuminating certain precisely defined areas with a laser or a combination of the two.

The first type of experiment involves proteins which can be located either at the cell membrane or in the cytosol and which are fluorescently labelled so that their position can be monitored. It is possible to cause these molecules to move rapidly from the one localization to the other. This can be done on a time scale of a couple of seconds and it looks likes switching on and off a light. This can be done many times in a row. Here the effect on the cell is global. The second type of experiment has to do with localizing this type of effect. It allows patterns chosen by the experimenter to be projected onto the cell. Here coloured patches are visible. Their interpretation is that concentrations of a certain substance have been fixed according to the pattern. The third type of experiment is the most striking. Here a spot of light is moved over the cell and away from it in a certain direction. There results a long projection of the cell in that direction. On the video it looks as if the the cell is being pulled by a sticky object. All these things are done by switching on certain proteins which have been made light-sensitive.The sensitivity to light is achieved by incorporating elements which are responsible for allowing certain plants to react to light. One of the plants which acts as a source here is the favourite model organism among plants, Arabidopsis thaliana. The reference to the paper describing these results is ‘Spatiotemporal control of cell signalling using a light-switchable protein interaction’, Nature 461, 997-1001 (15 October, 2009).


Spiral waves in neutrophils

October 25, 2009

A few weeks ago I heard an interesting talk by Orion Weiner from the University of California at San Francisco. This contained a lot of information and it has taken me some time to get around to processing it. One of the things he talked about establishes a surprising link between two topics I have discussed before, chemotaxis and spiral waves. The idea is that the motion of the leading edge of cells such as neutrophils are driven by spiral waves in the concentrations of certain proteins. These waves have been filmed using sophisticated techniques of microscopy. The proteins involved belong to something called the WAVE complex. The name has nothing to do with waves. The WA in the name comes from ‘Wiskott-Aldrich syndrome protein (WASp)’. More specifically the protein whose concentration shows the wavelike phenomena is hematopoietic protein 1 (Hem-1). This protein interacts with the actin which is involved in the mechanics of the motion. However the waves are not visible in the concentration of the actin itself. More information about this and an interview with Weiner can be found here.

The cosmic no hair theorem

October 15, 2009

The last post was sparked off by a talk I heard at a conference in Oberwolfach. Here I will write about a topic where another talk at that conference looks like a big step forward. This was by Jared Speck. He was describing work of his with Igor Rodnianski which is not yet fully written up.

These days there is a wide consensus among astrophysicists that there is strong observational evidence to indicate that the expansion of the universe is accelerated. In other words it is not only the case that all distant galaxies are moving away from us (and from each other) but the velocity of recession is actually increasing. In the standard view this is only consistent with general relativity if there is a positive cosmological constant or some exotic matter called dark energy. For convenience I will not distinguish between these two in what follows. Dark energy leads to accelerated expansion and accelerated expansion causes spatial irregularities to be damped. The
geometry of spacetime and the matter distribution are smoothed. This kind of idea can be turned into a precise mathematical statement (maybe not uniquely) called the cosmic no hair theorem. From a mathematical point of view this is rather a conjecture than a theorem – at least it has been that way for most of the time it has existed. The name originates from a phrase of John Wheeler, ‘a black hole has no hair’. The idea of this was that a particular solution of the Einstein equations describing a black hole, the Kerr solution, should be attractor for the evolution of more general solutions containing a black hole. In other words a general class of solutions should evolve so as to look more and more like the Kerr solution. The Kerr solution depends only on two parameters. Thus in this scenario all the details get lost dynamically, leaving a very simple object with no complicated features, no hair. In models for an expanding universe with positive cosmological constant the smoothing process mentioned above also seems to drive all solutions towards an attractor, the de Sitter solution. It is this analogy which gave rise to the name ‘cosmic no hair theorem’.

The mathematical formalization of the cosmic no hair theorem says that a solution of the Einstein-matter equations with positive cosmological constant converges to the de Sitter solution at late times in a suitable sense. A weaker statement is that this should be true for solutions which start close to the de Sitter solution. The latter version can also be thought of as a kind of stability statement for de Sitter space. In the case of the vacuum Einstein equations the stability of de Sitter space was proved by Helmut Friedrich in 1986. Since our universe is certainly not empty the relevance of this result to cosmology is not immediately obvious. It turns out, however, that there are reasons to believe that the cosmological constant can often have a dominant effect on the late-time cosmological expansion which tends to make the effect of the matter into a higher-order correction. It is important to confirm these ideas by a theorem which includes the effect of matter. The most commonly used matter model in cosmology is a perfect fluid with linear equation of state. It contains a parameter which is often restricted by an inequality corresponding to perfect fluids which are less stiff than radiation. The result of Rodnianski and Speck is a form of the cosmic no hair theorem for precisely this class of matter models. The proofs build on previous work of Hans Ringström. Friedrich’s proof uses a technique (the conformal method) which is very powerful but rather rigid. It is difficult to see how to modify the proof to include matter such as a fluid, or indeed to replace the cosmological constant by some other kind of dark energy, such as a nonlinear scalar field. Ringström introduced more flexible methods which allowed him to obtain a version of the cosmic no hair theorem for dark energy modelled by certain types of nonlinear scalar field. His methods open up the perspective of including matter and this is what Rodnianski and Speck have now done. These methods use energy estimates, the workhorse of the theory of nonlinear hyperbolic equations, in a clever way. (I might say more clever than I am, since I once tried very hard to do this, without success.)

The result of Rodnianski and Speck is restricted to the case of irrotational fluids. I see no fundamental reason why this should be necessary. Nevertheless there is a clear technical reason – in the irrotational case the Euler equation of the fluid is equivalent to a nonlinear wave equation. On the level of formal power series the case with rotation works out, as shown in a paper of mine (Ann. H. Poincare 5, 1041). Another question is what happens for large data. In that case there are various restrictions.For sufficiently large data it is to be expected that black holes would be formed (even in the vacuum case). Moreover, the fluid can be expected to form shocks which means that the solution cannot be continued, at least in the realm of smooth solutions. I find it remarkable that the expansion caused by a positive cosmological constant is strong enough to suppress formation of shocks in a small data regime. There is just one result available on this subject for large initial data and inhomogeneous solutions. In this work, due to Blaise Tchapnda and myself (arising from Blaise’s PhD thesis, Class. Quantum Grav. 20, 3037) we treated plane-symmetric solutions of the Einstein-Vlasov system with positive cosmological constant. In this case the symmetry prevents formation of black holes and the choice of matter model allows any analogue of shocks to be avoided.

The Newtonian limit of general relativity

October 14, 2009

This week I am at a workshop on mathematical relativity at the Mathematical Research Institute in Oberwolfach which I am organizing together with Piotr Chrusciel and Jim Isenberg. I was a co-organizer of similar conferences here in 2000, 2003 and 2006. The institute organizes workshops fifty weeks in the year on all areas of mathematics and participation is generally by invitation only. The isolated setting of the institute in the Black Forest tends to create an intense research environment. Work is also stimulated by the fact that the institute has the best mathematics library in Germany which is no doubt also one of the best in the world.

There have been a lot of excellent talks here. One of these which was of particular interest to me personally was by Todd Oliynyk. His subject was connected with the Newtonian limit of general relativity. I mentioned this topic in a previous post as having been something whose importance was emphasized by Jürgen Ehlers. Unfortunately Jürgen did not live long enough to see some of his questions answered by the work which Oliynyk has been doing recently.

So what is the Newtonian limit? General relativity is, among other things, a theory of gravity which is fully relativistic. In standard textbooks on the subject we can read that Newtonian physics arises as a limit of general relativity when typical velocities in the system are small compared with the speed of light. Unfortunately it is quite unclear what this means mathematically. For instance, in general relativity gravity is described by the metric, a tensor with ten components, while in Newtonian gravity it is described by a scalar function. How can the former converge to the latter? The conceptual basis of the Newtonian limit was elucidated in work by many people over many years and these ideas were synthesized by Ehlers. On this basis I was able to prove a theorem about convergence to the Newtonian limit in 1994. This concerned asymptotically flat spacetimes (in physical terms isolated systems) and the matter was described by kinetic theory (Vlasov equation). I chose this type of matter since a more commonly used description, the perfect fluid, suffers from technical difficulties. This is because the equations degenerate when the fluid density becomes small and in an isolated system the density has to become small somewhere. The Vlasov equation is immune to these difficulties.

The key mathematical problems involved in the analysis of the Newtonian limit should be independent of the details of the matter model chosen. We just need some matter model which does not place obstacles in our way. In a way Oliynyk took this more literally than I did myself. I had proved an existence theorem for certain types of fluid bodies in general relativity by extending ideas introduced by Tetu Makino in the Newtonian case. These were far away from the generality which would be desirable from a physical point of view but they are good enough to play the role of matter sources when studying the Newtonian limit. This has been exploited by Oliynyk who used ‘Makino fluids’ as matter source in his results. The formulation of the Newtonian limit involves a family of solutions of he Einstein-matter equations depending on a parameter \lambda, roughly corresponding to c^{-2} where c is the speed of light. The Newtonian limit is then the limit \lambda\to 0. What I proved in 1994 was the existence of families which are continuous in \lambda at \lambda=0. It is also interesting to know how smooth the family is at \lambda=0. The derivatives, when they exist, define higher order approximations to general relativity called the post-Newtonian approximations (nPN for n derivatives). I only got 0PN. Oliynyk has in the meantime reached 2PN. Results have been obtained for the asymptotically flat case which is the one most frequently considered in physics. It is well known that after 2PN the simple expansion breaks down. How can this be understood? My explanation (a little vague) is as follows. We are trying to approximate something which is bounded in its dependence on the spatial variables. Unfortunately above the 2PN level the approximation is not uniform and the coefficients in the expansion want to be unbounded. If you try to force them to be bounded by assumption the expansion breaks down. These coefficients are supposed to solve Poisson equations but the right hand sides have poor decay. The physicists typically try to represent the solutions of the Poisson equations in terms of the fundamental solution and get divergent integrals.

In his talk here Oliynyk reported that he has been able to treat post-Newtonian expansions of arbitrary order in the cosmological (spatially compact) case. It was a pleasant surprise for me that this works at all. When solving the Poisson equations in the cosmological case the right hand sides must have integral zero. It is remarkable that this works out at all. Once it is known that the procedure works at all, even for low orders, the intuition presented above makes it plausible that the obstructions familiar from the asymptotically flat case will not come up.

Returning to the asymptotically flat case, methods based on the post-Newtonian approximations are used to do the theoretical modelling for gravitational wave detectors whose cost is of the order of a billion dollars. It is an interesting comment on the role of mathematics in applications that nobody seems to worry too much about the almost entire lack of a rigorous mathematical foundation for these methods. In any case, the work I have been reporting on here represents the first steps on the road to changing this.