Archive for August, 2008

The blood-brain barrier

August 27, 2008

Paul Ehrlich can be considered the founder of immunology. He discovered much about the variety of different blood cells using staining techniques. In 1885 he observed that a dye injected into the bloodstream stained all body tissues with the notable exception of the central nervous system (CNS). This was later interpreted as an indication that there is a kind of partition, the blood-brain barrier (BBB), which prevents the dye from entering the brain. This was later confirmed by showing that a dye injected into the cerebrospinal fluid stains only the tissues of the CNS and nothing else.

The central nervous system consists of the brain, the spinal cord and certain particular nerves such as the optic nerve. All the other nerves in the body constitute the peripheral nervous system. What is this BBB which encloses the CNS? The walls of normal capillaries, the smallest blood vessels, consist of endothelial cells. These walls allow many kinds of chemical substances and cells to move rather freely between the bloodstream and the tissues. The smallest blood vessels adjacent to the CNS are different. There the endothelial cells are stuck together in a much stronger way through so-called tight junctions. The walls of these vessels let a much more restricted variety of cells and substances pass. This is the basis of the BBB but in fact it is a much more complicated structure. For instance, it is supported on the side facing the CNS by processes of astrocytes called ‘feet’. An idea of the complexity of the matter can be got from a review article of Hawkins and Davis (Pharmacol. Rev. 57, 173). The BBB acts to prevent pathogens such as bacteria entering the brain. It also keeps many elements of the immune system out. Breakdown of the normal function of the BBB seems to be a key element in multiple sclerosis. It results in immune cells getting access to the CNS and causing damage there. Some drugs use to treat MS have the property of sealing the BBB (see the previous post) Breakdown of the BBB has also been suggested to play a role, whether as cause or effect of the main events, in other illnesses such as Alzheimer’s disease and stroke. The BBB also has an important role to play in the treatment of diseases of the brain such as tumours. In some cases doctors would like to administer certain drugs to tissues within the brain and this is hindered by the BBB. Thus there is interest in finding controlled ways of increasing its permeability.


Cosmological perturbation theory

August 24, 2008

Cosmology is the study of the structure of the universe on the largest scales we can observe. More informally, it can be described as the study of the structure of the universe as a whole. Since it is impossible to model everything at the same time it is necessary to neglect the influence of structures on scales which are smaller than those which we are concentrating on. The simplest class of models are those which are homogeneous (all spatial points are equivalent) and isotropic (all directions at a given point are equivalent). This is formulated mathematically as the existence of a large group of symmetries, in fact a six-dimensional group (three translations plus three rotations). It has been found that this kind of highly symmetric model gives a reasonable description of observed quantities after averaging on a suitable scale. This fact and the advantage of simplicity lead to models which are homogeneous and isotropic playing a dominant role in theoretical cosmology. In the astrophysical literature there is little emphasis on rigorous mathematical analysis and so there is a lot of work to be done to bring ideas introduced in astrophysical cosmology into the domain of theorems and proofs. I find the question of the benefits which may arise from doing so very interesting but I will not comment on it further here.

It is clear that the actual universe we live in is not homogeneous or isotropic. If I look at the sky I see a star in some directions and I see none in others. How is this taken into account in the theory? The related question of the formation of structure is of great interest. We are immersed in a bath of microwaves which cannot be associated to any localized source (cosmic microwave background, CMB). This radiation has been propagating freely since a very early epoch in the development of the universe (about 300000 years after the big bang). It is highly isotropic. The variation in its temperature over directions in the sky is about one part in 100000. It can be concluded that the radiation was uniformly distributed at the time (decoupling) when it stopped interacting with other matter. Since before that there was a tight coupling between radiation and ordinary matter it follows that ordinary matter was also uniformly distributed at that time. The question is then how the inhomogeneous structures such as galaxies we observe today arose. In fact this picture is complicated by dark matter which interacts weakly with normal matter and radiation even before decoupling and may therefore consistently be highly inhomogeneous at decoupling. The task of explaining the origin of the inhomogeneity of the dark matter distribution remains.

It is evident that in order to describe the formation of localized structures it is necessary to go beyond the homogeneous and isotropic framework. If the structures develop from small deviations from homogeneity and isotropy we are dealing with quantities which are initially small. Then it is tempting to linearize the equations, discarding expressions which are quadratic or higher order in the magnitude of the deviation from homogeneity in the equations. In fact in the literature on cosmology the usual next step after considering homogeneous and isotropic models is to linearize about these. Then it is investigated whether solutions of the linear equations grow in time. In principle this a very similar procedure to what is done in the paper of Keller and Segel described in a previous post for a quite different application. In detail things are more difficult because in cosmology the equations to be solved are more complicated and the interpretation of the results presents special difficulties. In any case, this is what is meant by cosmological perturbation theory. The most useful source I have found for this subject is the book ‘Physical Foundations of Cosmology’ by V. Mukhanov. Unfortunately it is difficult to establish contact between the calculations and discussions arising in this context and rigorous mathematical statements. There is a serious problem of language at the very least. In the past I have published several papers addressing these issues in the homogeneous case. Now, in collaboration with Paul Allen, we are tackling mathematical issues raised by cosmological perturbation theory. We already have some preliminary results but I will leave a discussion of these to a later post. I will give a talk on this subject at a conference in Stockholm next week.

Alexandria, 1883

August 16, 2008

I have just finished reading the novel ‘An Imperfect Lens’ by Anne Roiphe. A central element of this book is an outbreak of cholera which took place in Alexandria in 1883. This is a work of fiction but I have the impression that it is rather close to the historical reality. I enjoyed reading this book a lot and I preferred it to the other novel centred on a cholera epidemic I have read, ‘Le Hussard sur le Toit’ by Jean Giono. One reason is the keen sense of reality, also microscopic reality, which the author manages to convey. She also introduces many ideas about the nature of science and scientists and their relations to other types of people and other fields of discourse. The book presents a vivid picture of a vibrant city where cultures mix.

The historical background is that both France and Germany sent expeditions to determine the cause of the epidemic. The action of the book centres on the French group. They were sent by Pasteur, who was himself too old to take part. His rival, Robert Koch, makes a number of brief appearances in the pages of the book. The two teams are racing against each other to find the organism causing the disease and patriotic motives are not to be neglected. At the same time it turns out that they have to race to reach their goal before the epidemic itself subsides.

In the end Koch won the race although his work needed to be completed by his later investigations in India. The French team, as it is presented in the novel, strikes me as not very professional. I thought of Scott’s expedition to the south pole – in this comparison Koch is the analogue of the more professional, and finally more successful, Amundsen.

Some years ago I read a biography of Koch and this played a big role in stimulating my present enthusiasm for medicine. Here was a practical scientist who I could wholeheartedly admire. (I pass over certain events in the later part of his life connected with tuberculin which I am not sure are so admirable.) On a couple of occasions since then I took part in open days at the Robert Koch Institute in Berlin which included, among other things, excellent lectures. In one of these the speaker described her participation in an expedition to Laos to investigate an outbreak of SARS at a time when the nature of that disease was still quite unclear. This made me feel that I had come close to the heritage of Robert Koch.

The Martiel-Goldbeter model

August 14, 2008

The Martiel-Goldbeter model (Biophys. J. 52, 807) is a system of ODE which models the production of the signalling molecule cAMP by Dictyostelium discoideum. There are a number of other competing models on the market but I will concentrate on just this one which seems to have been quite popular. A review of what this and other models can and cannot describe has been given in a recent article by Goldbeter (Bull. Math. Biol. 68, 1095) which also sketches some of the history of the problem. The applicability of this ODE model is to well-stirred cultures where there is no opportunity for pattern formation. In a situation where pattern formation is possible it is appropriate to add terms representing diffusion. This has been done in a paper by Tyson et. al. (Physica D, 34, 193) where spiral waves are found. The reason that so much effort is spent on studying this organism is that it is a model case for cell-cell communication and the development of structures from a homogeneous population of cells.

The basic biological mechanisms which are modelled by the equations concern the following processes. Extracellular cAMP can bind to a membrane receptor, thus activating it. This receptor can be phosphorylated. The activated receptor influences the enzyme adenylate cyclase which converts ATP to cAMP. There is another enzyme, phosphodiesterase, which eliminates cAMP. Martiel and Goldbeter derive a system of nine ordinary differential equations which describe the dynamics of these processes. By a quasi-steady state hypothesis, based on the smallness of certain parameters, they reduce this to a system of four equations. One of the variables in the latter system is the intracellular concentration of ATP. It is found experimentally that this concentration is almost time-independent. Numerical solutions of the four-equation system give results consistent with this. As a consequence it seems reasonable to set the concentration of ATP to a constant value in the other three equations, thus producing a system of three ODE. This three-equation system is what is usually known as the Martiel-Goldbeter system. The remaining unknowns in this system are the fraction of the receptor in the active state and the intracellular and extracellular concentrations of cAMP. For certain parameter values it is reasonable to make a further quasi-steady state assumption and reduce this system to one with only two equations. These parameter values are not appropriate for the biological system being studied and so the two-equation system cannot give useful quantitative results. It turns out, however, that solutions of the two-equation system show many of the qualitative features of solutions of the three-equation system and hence the two-equation system can be useful for obtaining a better intuitive understanding of the dynamics using phase-plane analysis.

The Martiel-Goldbeter system can be used to reproduce qualitative and quantitative features of experiments. Some of the main qualitative features are as follows. For certain parameter values the system has a stable limit cycle corresponding to spontaneous periodic production of cAMP by a cell. In this case the system is said to be oscillatory. It may be noted that oscillatory and excitable systems are often linked by bifurcations when parameters are varied, and this system is no exception. This dynamical behaviour may be important for the ‘pacemaker’ role of certain cells in the initiation of the aggregation process in Dictyostelium discoideum. For other parameter values the system is excitable and this results in cells amplifying an applied variation in cAMP concentration. This is also believed to be important in coordinating aggregation. Finally, the cells adapt in such a way that they do not react in a lasting way to a change from one time-independent value of an externally imposed cAMP concentration to another.

Excitable systems and spiral waves

August 12, 2008

I recently came across the concept of excitable systems. They appear to play a role in the signalling processes of Dictyostelium, which was my point of entry to the subject, but at the same time they seem to be of importance for modelling a wide range of systems in nature. Examples which are frequently considered include the Field-Noyes model of the Belousov-Zhabotinskii oscillatory chemical reaction and the FitzHugh-Nagumo model which is a simplified version of the Hodgkin-Huxley model of nerve conduction. Coming back to Dictyostelium, there is the Martiel-Goldbeter model for aggregation. The general set-up is as follows. There is a system of ODE with certain qualitative features which are described by the word ‘excitability’. Then diffusion terms may be added to one or more of these equations to get a system of PDE. On an intuitive level the characteristic features of excitability as a property of the ODE system are as follows. There is a stationary solution which is stable. Thus, by definition, sufficiently small perturbations of this solution stay small. On the other hand larger perturbations are such that the corresponding solutions go a long way away (whatever that means) from the stationary state before returning there. It is often said that there is a progression from an excited state to a refractory state and then back to the excited state. In the PDE context fronts occur separating an excited from a refractory region. The spatial variation within these fronts resembles the temporal variation in the ODE system. There are many discussions of excitable systems in the literature. A source which helped me to grasp some of the basic ideas are the early parts of the article ‘Pattern formation in excitable media’ by Ehud Meron (Phys. Rep. 218, 1-66). In comparison to some other accounts I have seen this paper supplies more conceptual explanation so as to allow the novice (like myself) to understand the meaning of the figures which are often presented.

One of the most interesting features of excitable systems is the occurrence of spiral waves. This concerns the case of systems of PDE in two space dimensions. The typical scenario for the formation of a spiral wave is that a plane wave (i.e. a linear front) gets interrupted in some way. Then the broken end starts to curl around, forming a tip. The front then rotates about this tip. These waves are fascinating on account of their visual form but it is even more interesting to ask how this kind of phenomenon can be described mathematically. There is an extensive literature and I have just started to scratch the surface of it. By means of a singular limit which I have not yet understood an equation can be derived which describes the motion of the front itself. This equation is related to mean curvature flow. Actually it would be more appropriate to talk about curve-shortening flow since it is this two-dimensional special case which is relevant when starting from equations in two space dimensions. In mean curvature flow a hypersurface in Euclidean space flows in the normal direction with a speed which is equal to its mean curvature. This prescription defines a parabolic equation about whose solutions much is known in the meantime. To my knowledge the relations between excitable systems and geometric evolution equations have not been explored very much up to now.

Dictyostelium aggregation revisited

August 8, 2008

In a previous post on the Keller-Segel model I raised some questions concerning the applicability of the model to the aggregation of Dictyostelium discoideum. A coherent picture of the early stages of this process is presented in a review paper of Ben-Jacob, Cohen and Levine entitled ‘Cooperative self-organization of microorganisms’ (Adv. Phys. 49, 395-554), which I found on the web page of Levine. According to the account given there there is a first phase involving spiral waves and chemotaxis only starts to play a role after that. The tips of the spiral waves define the centres of the aggregation process. See Fig. 31 of the paper. There are also interesting comments on the role of adhesion later in the process of aggregation where streams are formed. All of this is backed up by extensive references. I was interested to see that several of the topics discussed in this blog are mentioned in the paper of Ben-Jacob et. al. (Liesegang rings, chemotaxis, quorum sensing, dynamics of actin polymerization, bacterial motion, Keller-Segel model, Dictyostelium). Thus there may be more unity in my apparently rather random musings than I had realized.

The paper of Ben-Jacob at. al. appears to be a rich repository of ideas about mathematics, physics, biology and their mutual interactions. It also contains many striking pictures of the patterns which microorganisms, in particular bacteria, can produce.

Tysabri woes

August 2, 2008

Tysabri (Natalizumab) is a promising new drug for the treatment of multiple sclerosis. It is supposed to act by preventing immune cells from entering the brain via the blood-brain barrier. A loss of integrity of the blood-brain barrier is associated with relapses in MS and helping to seal this barrier is one of the mechanisms presumed to underly the beneficial effects of interferon \beta as a therapy for MS. In clinical trials Tysabri seems to compare very well with interferon \beta which is now widespread in the treatment of MS. It is not surprising that this has led to enthusiasm in the medical world and hope among patients. Unfortunately there is a catch. In 2005 two patients being treated with Natalizumab and interferon developed the dangerous disease PML (progressive multifocal leukoencephalopathy), with one of them dying. There was also a fatality in a patient being treated for Crohn’s disease with Natalizumab in combination with other drugs. The suspicion of a causal connection between the therapy and the deaths arose. The drug was removed voluntarily from the market by the manufacturers but later reapproved by the authorities with certain restrictions after an inquiry. Now two new cases of PML in patients receiving Tysabri have been reported. The share prices of the manufacturers plummeted.

It remains to be seen what these unfortunate events will mean for the future of this drug. There is another aspect of the story which I find interesting. If this substance has such powerful effects, positive or negative, can it tell us more about the normal mechanisms of multiple sclerosis or other autoimmune diseases? In particular, what can it tell us about the role of the blood-brain barrier? PML is caused by the JC virus, which is extremely widespread in the human population but usually harmless, causing no symptoms. PML occurs most frequently in patients with AIDS and transplant recipients being treated with drugs which suppress the activity of the immune system. This suggests that under normal circumstances the immune system successfully keeps the virus in check but that when the immune system is suitably impaired the virus can multiply in an uncontrolled way, causing a deadly disease. From this point on it is not hard to imagine a possible connection between drugs which seal the blood-brain barrier and PML. In this scenario the immune system would have been denied sufficient access to the central nervous system to control the virus population.

Natalizumab is a monoclonal antibody against the adhesion molecule VLA-4 (very late antigen 4), also known as CD49, which belongs to the family of integrins and occurs on the surface of leucocytes.

I have not encountered any mathematical models for the function of the blood-brain barrier, despite its importance for many medical issues.