Albert Goldbeter and glycolytic oscillations

This Christmas, at my own suggestion, I was given the book ‘La Vie Oscillatoire’ by Albert Goldbeter as a present. This book is concerned with oscillatory phenomena in biological systems and how they can be explained and modelled mathematically. After the introduction the second chapter is concerned with glycolytic oscillations. I had a vague acquaintance with this subject but the book has given me a much better picture. The chapter treats both the theoretical and experimental aspects of this subject.

If yeast cells are fed with glucose they convert it into alcohol. Those of us who appreciate alcoholic beverages can be grateful to them for that. In the presence of a supply of glucose with a small constant rate alcohol is produced at a constant rate. When the supply rate is increased something more interesting happens. The output starts to undergo periodic oscillations although the input is constant. It is not that the yeast cells are using some kind of complicated machine to produce these. If the cells are broken down to make yeast extract the effect persists. In fact for yeast extract the oscillations go away again for very high concentrations of glucose, an effect not seen for intact cells. This difference is not important for the basic mechanism of production of oscillations. The breakdown of sugar in living organisms takes place via a process called glycolysis consisting of a sequence of chemical reactions. By replacing the input of glucose by an input of each of the intermediate products it was possible to track down the place where the oscillations are generated. The enzyme responsible is phosphofructokinase (PFK), which converts fructose-6-phosphate into fructose-1,6-bisphosphate while converting ATP to ADP to obtain energy. Now ADP itself increases the activity of PFK, thus giving a positive feedback loop. This is what leads to the oscillations. The process can be modelled by a two-dimensional dynamical system called the Higgins-Selkov oscillator. Let and denote the concentrations of substrate and product respectively. The substrate concentration satisfies an equation of the form . The substrate is supplied at a constant rate and used up at a rate which increases with the concentration of the product. (Here we are thinking of ADP as the product and ignoring other possible effects.) The product concentration correspondingly satisfies .

The Higgins-Selkov oscillator gives rise to a limit cycle by means of a Hopf bifurcation. The ODE system is similar to the Brusselator. There are two clear differences. The substance which is being supplied from ouside occurs linearly in the nonlinear term in the Higgins-Selkov system and quadratically in the Brusselator. In the Higgins-Selkov system the nonlinear term occurs with a negative sign in the evolution equation for the substance being supplied from outside while in the Brusselator it occurs with a positive sign. In the book of Goldbeter the Higgins-Selkov oscillator seems to play the role of a basic example to illustrate the nature of biological oscillations.

The NFAT signalling pathway

The role of T cells in the immune system is to recognize foreign substances and then take appropriate action. In order for this to happen information must be propagated from the surface of the cell, where the T cell receptor is, to the nucleus in order to initiate DNA transcription. The last step in this process is the binding of a suitable combination of transcription factors to the DNA. NFAT (nuclear factor of activated T cells) is one of these transcription factors. The fact that the associated signalling pathway plays an important role in the activation of T cells explains the name. In fact this substance (or class of substances – there are actually five different ones) are important for signalling in many cells of the immune system. I already mentioned the NFAT signalling pathway, its connection to calcium and a paper on the subject by Salazar and Höfer in a previous post. Now I have written a paper where I look into mathematical aspects of the activation of NFAT by means of dephosphorylation and the role of calcium in this process. Salazar and Höfer introduced a high-dimensional dynamical system and computed stationary solutions in a slightly simplified version of that system. I now proved, using chemical reaction network theory, that for each choice of the many parameters in the system there exists exactly one stationary solution of the full system for each value of the total amount of NFAT in the cell. Every solution with that total amount of NFAT converges to the stationary solution at late times. Furthermore, this solution is well approximated by the explicit solution of the simplified system under a biologically motivated assumption that certain parameters are small enough. The main tool in the proof is the Deficiency Zero Theorem.

The result just mentioned concerns the model for the dephosphorylation process with the stimulation of the cell expressed through fixed choices of the parameters. In reality the stimulation is communicated through the calcium concentration in the cytosol. This means that the parameters in the model for desphosphorylation should be replaced by time-dependent functions which themselves are the result of a dynamical process. The situation is described by Salazar and Höfer with the help of a two-dimensional dynamical system closely related to one introduced by Somogyi and Stucki to describe calcium oscillations in liver cells. In the paper I did some analysis of the model, giving criteria for the stability of the unique stationary solution for given parameter values and the existence of periodic solutions. Hopf bifurcations play a role. The model is closely related to the Brusselator and techniques of proof can be imported from that case. In particular it is important to identify explicit invariant regions for the flow. When a solution of the model for the calcium concentration is such that it tends to a constant at late times then it can be shown that the resulting configuration of the phosphorylation states of NFAT also converges to the situation with constant coefficients previously analysed. When a solution converges to a periodic solution at late times it is not clear what can be said.

IPEX and CD25

In a recent post I wrote about some ideas of Kendall Smith and his role in discovering the cytokine IL-2. On 8th December I heard him give a talk in which he presented various ideas about IL-2, its receptor and Tregs. I discussed some aspects of IL-2 in the last post. When mutant mice are engineered which cannot produce IL-2 they show a strange combination of symptoms which combine immunodeficiency (a reduced capability of the immune system to fight pathogens) and autoimmune disease (an inappropriate reaction of the immune system to host tissues). This probably has to do with the fact that IL-2 is important for the production of both effector T cells and Tregs, which act in opposite directions. Similar phenomena are seen in the disease of humans called IPEX (immune dysregulation, polyendocrinopathy, enteropathy, X-linked syndrome). It is often attributed to a lack of the transcription factor Foxp3 which is of central importance for the function of Tregs. The gene for Foxp3 is on the X chromosome and this explains the way IPEX is inherited and the term X-linked in its name. However, as pointed out by Smith in his talk, one third of patients diagnosed with IPEX have no mutation in the Foxp3 gene. In this context he referred to a paper of Caudy et al. (J. Allergy Clin. Immunol. 119, 482). What is shown in this paper is that there is a different possible cause of IPEX-like symptoms, namely mutations in the gene for CD25, a surface molecule associated to Tregs.

The paper concerns a patient (an eight year old boy) who had suffered a horrific combination of diseases. It was found that he had mutations in both copies of the CD25 gene. The mutation in one copy came from the mother and was a frame shift due to an insertion. In other words, there is a extra base in the DNA which makes the part of the gene after it look like nonsense when it is being transcribed. The mutation in the other copy came from the father and consisted of one base being exchanged. This happed to cause a stop codon so that reading stopped at that point. The combination of these circumstances meant that the boy could not produce CD25 and this was the presumed cause of his disease. His Foxp3 gene was normal. On the other hand other IPEX patients can produce CD25. Thus there appear to be two diseases with related symptoms. The gene coding for CD25 is on chromosome 10, not the X chromosome. This is why two mutations are necessary to produce CD25 deficiency.

What is the connection to IL-2? The IL-2 receptor, which was also discovered by Kendall Smith and his collaborators, consists of three chains called , and . The second and third are always present on the surface of T cells but the first is only present in variable amounts. In fact the chain of the IL-2 receptor is nothing other than CD25. The and chains together allow for some IL-2 signalling but strong signalling in response to normal concentrations of IL-2 is only possible with the help of the chain. In this case it is not only the case that the receptor signals when IL-2 is bound to it. Binding also causes the receptor to be taken into the interior of the cell and destroyed. This process is an important part of the dynamics associated to IL-2. The chain of the IL-2 receptor also forms part of the receptor for many other cytokines, for instance IL-4. The gene for this receptor is on the X chromosome. When it cannot be produced due to a mutation this leads to a disease called X-linked severe combined immunodeficiency (SCID). In this case the immune system does not function since so much of its signalling system has been disrupted. This is also known as the ‘bubble boy disease’ since children affected by it have to live in a sterile environment.

Shifting attention towards Tregs

Yesterday I heard a talk by Abul Abbas where two of the main themes were regulatory T cells (Tregs) and interleukin 2. Correctly functioning immunity is the result of a balance between effector cells and Tregs and he emphasized that in trying to develop therapies it might be more valuable to concentrate on influencing the regulatory side. He described a mouse model which he has developed for studying autoimmune disease. One criterion in developing this model was that it should concern the target tissue where an antigen is expressed and not the lymphoid tissue. Another is that the target tissue should be easily accessible for doing experiments in vivo. For this reason he chose the skin. In this transgenic model antigen expression can be turned on and off by feeding the mice with doxycyclin. When the antigen is turned on an autoimmune disease results. If it is turned off the mice recover. If it is turned on again the mice get sick again but much less than the first time. This is reminiscent of ordinary immunity which is due to memory effector cells. In this case it seems that there are memory Tregs. This suggests the idea that a possible cause of autoimmune disease in humans could be a lack of memory Tregs.

When IL-2 was first discovered it was known for causing T cells to proliferate and thus strengthening the immune response. More recently it has been found that eliminating IL-2 does not necessarily act in an immunosuppressive way. Apparently it can be replaced by something else in driving the proliferation of effector T cells. On the other hand it also drives the proliferation of Tregs and Abbas argued that this is its most essential function. In that case it cannot be replaced.

The lecturer made a number of interesting comments about themes such as immunology, therapies for immune disorders and cancer, clinical trials etc. I did not note them down and I cannot reproduce them here. Nevertheless I have the impression that a learned of lot of things which I might profit from in the future.

Calcium oscillations

There is evidence to suggest that oscillations in levels of calcium inside and outside cells are used as a signalling mechanism. A variety of mathematical models have been introduced to study this phenomenon. Here I will discuss some aspects of the subject. A more general review can be found in this Scholarpedia article. In the plasma membrane and the endoplasmic reticulum there are pumps which transport calcium ions out of the cytosol. The result is a huge concentration difference between the cytosol on the one hand and the extracellular space and the lumen of the endoplasmic reticulum on the other hand. This can be several orders of magnitude. There are also ion channels in these membranes which, when open, allow the calcium to flow down its gradient. This provides a way to change the calcium concentration in the cytosol very fast and this can cause rapid changes in the behaviour of a cell. In this context it is important that the endoplasmic reticulum has such a high surface area and is so widely distributed in the cell. One type of calcium channels in the ER reacts to the binding of the substance IP (inositol 1,4,5-trisphosphate) to the channel by opening. This effect is also modulated by the calcium concentration in the cytosol. There are calcium channels in the plasma membrane and there is also a certain amount of leakage through both membranes. Transport of calcium in and out of mitochondria can be an important effect. Some combination of these features can lead to oscillations in the calcium concentration in the cytosol. This presents a challenge for mathematical modelling. Ideally a dynamical system consisting of ODEs for the concentrations of various substances would exhibit periodic solutions. Of course a system of this kind must have dimension at least two and several two-dimensional models have been proposed. It could be that several of these models are useful since calcium signalling in different cell types may use different mechanisms. The difficult thing is not to find a model exhibiting oscillations but to find the right model for a particular type of cell. In what follows I consider one type of model. I have chosen this type for two reasons. The first is its simplicity. The second is that it may be relevant to explaining the role of calcium in the activation of T cells.

I consider first a model due to Somogyi and Stucki (J. Biol. Chem. 266, 11068). It is a two-dimensional dynamical system. The two variables are the calcium concentrations in the lumen of the ER and the cytosol, call them and . The concentration of IP is taken to be constant. The rates of change of and are given by and . The quantities are positive constants while is a positive function which describes the behaviour of the IP receptor and must be further specified to get a definite model. The inventors of the model remark that setting and causes this system to reduce to the famous Brusselator, which I have commented on elsewhere. Thus the model can be thought of as a kind of generalized Brusselator and indeed it exhibits similar qualitative behaviour. The choices which are suggested to be appropriate for the cells being studied (in this case hepatocytes) is that and is given by a Hill function, . Nice features of this system is that it has a unique stationary solution which can be written down explicitly and that it is also possible to get an explicit formula for the characteristic equation of the linearization at that stationary solution. In this way the stability of the stationary solution can be determined, with instability corresponding to the existence of a limit cycle. It is stated that a Hopf bifurcation occurs but there is no discussion of proving this. The general picture seems to be that oscillatory behaviour occurs at intermediate levels of IP stimulation and disappears at levels which are too low or too high. In this paper an alternative version of the model is introduced where in some places is replaced by . This happens when modelling effects driven by the difference of concentrations in the two compartments. Given that is normally much larger than it is plausible to replace the difference of concentrations by the concentration in the ER.

The dephosphorylation of the transcription factor NFAT during the activation of T cells has been studied in a paper of Salazar and Höfer (J. Mol. Biol. 327, 31). An important step in the activation process is an influx of calcium caused by release of IP. The calcium binds to calmodulin. It also binds to the phosphatase calcineurin which can then be activated by calmodulin. Finally calcineurin removes phosphate groups from NFAT. In this paper a model for calcium dynamics is used which is closely related to the (alternative model) of Somogyi and Stucki. There are three equations but two of them form a closed system which is more or less the Somogyi-Stucki model with a specific choice of receptor activity as a function of the concentration of IP. The last equation essentially means that the calcium level is integrated in time to give the concentration of active calcineurin.

The global attractor conjecture in chemical reaction network theory

In a previous post I wrote about chemical reaction network theory and, in particular, about a result belonging to this theory called the deficiency zero theorem. Now I have realized that in that post I claimed more than was justified. I will correct this point here. The assumptions of the deficiency zero theorem are that we have a chemical reaction network which is weakly reversible and of deficiency zero. (For the terminology I refer to the previous post on CRNT.) One conclusion for the associated dynamical system is that there is a unique stationary solution (in each stoichiometric compatibility class) where all concentrations are positive. A second conclusion is that is asymptotically stable (a local statement). The Lyapunov function used to prove the second statement also allows some further conclusions to be drawn. For solutions with positive concentrations is strictly decreasing along solutions away from . This means that a positive solution can have no positive -limit points other than . In addition tends to infinity at infinity, thus showing that each solution stays in a compact set. It can be concluded that the -limit set is compact and that unless it is it must consist of points where at least one concentration is zero. In the post just quoted I claimed that every solution converges to . Reading the original three basic papers on this subject by Horn, Jackson and Feinberg might easily give the impression that this is the case. Looking at the proofs in detail, as I have done in the meantime, reveals that this statement is not proved in those papers. There are also many later papers on CRNT where this issue is not raised. This does not mean that the problem escaped attention completely, even in the early days. In a paper by Horn from 1974 he explicitly states that the proof of the result on global stability in his paper with Jackson was not correct. He expresses the opinion that the statement is nevertheless probably true. In a paper from 2001 on the kinetic proofreading model Eduardo Sontag proved a result of this kind under some extra conditions.

Recently this issue has received renewed attention and has been given the name ‘global attractor conjecture’ by Craciun et. al. (J. Symbolic. Comput. 44, 1551). In a 2011 paper of Anderson (SIAM J. Appl. Math. 71, 1487) he writes that it ‘is considered to be one of the most important open problems in the field of chemical reaction network theory’. In that paper he proves the conjecture in the case of systems with a single linkage class and so perhaps the question is close to being resolved.

The Perron-Frobenius theorem

The Perron-Frobenius theorem is a result in linear algebra which I have known about for a long time. On the other hand I never took the time to study a proof carefully and think about why the result holds. I was now motivated to change this by my interest in chemical reaction network theory and the realization that the Perron-Frobenius theorem plays a central role in CRNT. In particular, it lies at the heart of the original proof of the existence part of the deficiency zero theorem. Here I will review some facts related to the Perron-Frobenius theorem and its proof.

Let be a square matrix all of whose entries are positive. Note how this condition makes no sense for an endomorphism of a vector space in the absence of a preferred basis. Then has a positive eigenvalue and it is bigger than the magnitude of any other eigenvalue. The dimension of the generalized eigenspace corresponding to this eigenvalue is one. There is a vector in the eigenspace all of whose components are positive. Let be the sum of the entries in the th column of . Then lies between the minimum and the maximum of the .

If the assumption on is weakened to its having non-negative entries then most of the properties listed above are lost. However analogues can be obtained if the matrix is irreducible. This means by definition that the matrix has no invariant coordinate subspace. In that case has a positive eigenvalue which is at least as big as the magnitude of any other eigenvalue. As in the positive case it has multiplicity one. There is a vector in the eigenspace all of whose elements are positive. In general there are other eigenvalues of the same magnitude as the maximal positive eigenvalue and they are related to it by multiplication with powers of a root of unity. The estimate for the maximal real eigenvalue in terms of column sums remains true. The last statement follows from the continuous dependence of the eigenvalues on the matrix.

Suppose now that a matrix has the properties that its off-diagonal elements are non-negative and that the sum of the elements in each of its columns is zero. Then the sum of the elements in each column of a matrix of the form is . On the other hand for sufficiently large the entries of the matrix are non-negative. If is irreducible then it can be concluded that the Perron eigenvalue of is , that the kernel of is one-dimensional and that it is spanned by a vector all of whose components are positive. In the proof of the deficiency zero theorem this is applied to certain restrictions of the kinetic matrix. The irreducibility property of follows from the fact that the network is weakly reversible.

The Perron-Frobenius theorem is proved in Gantmacher’s book on matrices. He proves the non-negative case first and uses that as a basis for the positive case. I would have preferred to see a proof for the positive case in isolation. I was not able to extract a simple conceptual picture which I found useful. I have seen some mention of the possibility of applying the Brouwer fixed point theorem but I did not find a complete treatment of this kind of approach written anywhere. There is an infinite-dimensional version of the theorem (the Krein-Rutman theorem). It applies to compact operators on a Banach space which satisfy a suitable positivity condition. In fact this throws some light on the point raised above concerning a preferred basis. Some extra structure is necessary but it does not need to be as much as a basis. What is needed is a positive cone. Let be the set of vectors in -dimensional Euclidean space, all of whose components are non-negative. A matrix is non-negative if and only if it leaves invariant and this is something which can reasonably be generalized to infinite dimensions. Thus the set is the only extra structure which is required.

The variability of living organisms

Today I heard a talk by John McKinney where a central theme was the variability of genetically identical bacteria. This means on the one hand the differences between individuals and on the other hand the differences in the state of one individual at different times. The organism most prominent in the talk was Mycobacterium tuberculosis (cf. my previous post concerning a talk by McKinney). This bacterium can be treated using antibiotics (95% of patients who complete treatment are cured) but requires a combination of several drugs over a long period. If a cure is possible why is it so difficult? In his studies on this question McKinney has found that variability in the population of bacteria of the kinds already mentioned must be carefully taken into account.

McKinney is now based in Lausanne, having been at Rockefeller University until a few years ago. It is natural to ask why a biomedical researcher would leave such a prestigious institution. The speaker explained a feature of his present institution which was very attractive to him. This is the expertise available there in engineering and this has allowed him to develop new experimental techniques based on microfluidics. Bacteria are grown within a microfluidic channel which is observed under a microscope over long time scales. The microfluidic system allows the conditions to be controlled very precisely. Nutrients are provided and waste removed on a continuous basis. The fate of individual bacteria can be followed very closely. The processes taking inside the cells can be followed using fluorescent labelling. It can be seen exactly when the cell divides, when and where its DNA is replicated etc. Often the population is observed in a phase of exponential growth but this kind of system could also be used as a chemostat (cf. this post) to observe a steady state population.

One of the phenomena to be understood is that the population of M. tuberculosis treated with an antibiotic is biphasic. The population decreases at an exponential rate for a while and then suddenly at a different, much smaller, exponential rate. This is a phenomenon at a population level and the new techniques can help to understand what is happening on an individual level. For example, the antibiotic isoniazid (INH) is believed to work by preventing the bacterium producing mycolic acid, a substance it needs to build its cell wall. One popular theory, the “unbalanced growth model” suggests that the volume of the bacterium grows while there is no more material available for its cell wall. As a consequence the wall thins until the bacterium bursts. The new observations on the properties of individual bacteria are inconsistent with this model. Another observation which arose while trying to understand the interaction of bacteria with antibiotics is that there is a protein which is expressed in a way whose time dependence seems to be stochastic. On the films transcription of this protein is marked by a red colour and the bacteria are seen to flash on an off in a random-looking manner,

Another film showed cells caught in a microfluidic cage. There are small connections of the fluid in the cage with the outside of diameter about a micrometer which allow nutrients and waste products to be exchanged. These connections are too small for  eukaryotic cells but on the film the cells were seen trying to squeeze their way through with apparently great energy. The aim is to cultivate bacteria in eukaryotic cells in a situation where they can be observed effectively through the microscope. The necessity of tracking the cells is avoided by locking them into the cage. This would mean that the bacteria could be observed in surroundings closer to their natural habitat.

All these observations seem to have raised more questions than they have answered but what better motor can there be for scientific progress?

Kinetic proofreading

In biological systems information is propagated from one form to another by chemical reactions. An example is the translation of mRNA into protein by the ribosome. Under certain circumstances there are limits to the accuracy of this kind of process. In a one-step process with two possible outcomes the accuracy is bounded above in terms of the difference of the free energies of the two alternative reactions. In other words, it is bounded in terms of the ratio of the reaction constants. Putting in the numbers for some important biological processes shows that this bound is exceeded by a large factor. This led to a proposal by Hopfield (PNAS 71, 4135) of a way in which this accuracy can be achieved by using more complicated reactions with several steps. He called it kinetic proofreading. (There was other related work by Ninio (Biochimie 57, 587) at about the same time.) Later McKeithan (PNAS 92, 5042) applied this idea to the question of how the T cell receptor can discriminate so accurately between different antigens. This model was studied mathematically by Eduardo Sontag (IEEE Transactions on Automatic Control, 46, 1028), who related it to chemical reaction network theory (CRNT). Here I will take Sontag’s work as starting point for my description.

Let be the concentration of T cell receptors not bound to a ligand and the concentration of peptide-MHC complexes not bound to a receptor. When a peptide-MHC complex binds to a receptor this gives the basic form of the occupied receptor and the concentration of these is denoted by . The rate constant for this process is denoted by This basic form can be modified by phosphorylation at up to sites, giving rise to quantities . There are successive phosphorylation reactions leading from to and the corresponding rate constants are denoted by . There are dissociation reactions where the peptide-MHC complex detaches from the receptor and the receptor is simultaneously completely dephosphorylated. The rate constants are denoted by . The total concentrations of T cell receptors and peptide-MHC complexes (both bound and free) are denoted by and respectively. They are conserved quantities and can be used to eliminate the variables and from the system if desired. Doing so gives the system for the variables , at the beginning of Sontag’s paper. In the terminology of CRNT this corresponds to restricting to a stoichiometric compatibility class. It is elementary to calculate the stationary solutions of the original system and there is exactly one in each stoichiometric compatibility class. In terms of CRNT the system is weakly reversible and of deficiency zero. General theory then implies that there is exactly one stationary solution in each stoichiometric compatibility class and that it is asymptotically stable. Sontag strengthens this result, proving that all solutions converge to the corresponding stationary solutions at late times.

Now I come back to the original motivation. For simplicity let us suppose that and are independent of . Let . Then it turns out, as computed by McKeithan, that the ratio of the fully phosphorylated complex to the total complex is . This means that if is not too small this ratio depends very sensitively on the value of the dissociation constant . If it is which gives rise to further signalling within the cell this gives a way of magnifying differences between the binding properties of ligands.

The life of François Jacob

I have just read the autobiography ‘La statue intérieure’ by François Jacob. I find that this is a book of high literary quality. An indication of this going beyond my personal judgement is that after the book came out Jacob was invited to talk about it by Bernard Pivot on his TV programme ‘Apostrophes’. I understand that at the time when Pivot was active an invitation from him was a kind of certificate of quality for any new book which appeared in French. Jacob is best known as a biologist but this book convinces me that he is also a very gifted writer. The book is not a monumental work, but rather a collection of anecdotes which illuminate many aspects of Jacob’s life and, more generally, many aspects of the human condition. I find it difficult to say what makes up the charm of his writing – I can only suggest reading the book in order to experience it at first hand.

The first three quarters of the book describe the part of his life before he began his career as a scientific researcher in his late twenties. He had started out studying medicine with the aim of becoming a surgeon. After less than two years this was interrupted by the outbreak of the second world war. Jacob, who is Jewish, fled France and joined the French army in exile led by de Gaulle. He spent a large part of the war in Africa. Shortly after his return to France he was very seriously wounded. This made it impossible for him to become a surgeon and left him somewhat at a loss what to do. An interesting point, which he does not emphasize in the book, is that in a sense his being wounded in this case was a result of a decision of his own. He was tending to an officer who had just been wounded when the group was bombed again. The officer could not be moved and begged Jacob not to leave him alone. Jacob could not do anything to protect the man but he nevertheless stayed with him instead of taking cover. As a result of this he was almost killed himself. His decision was very honourable but maybe not very reasonable. In any case, he ended up spending many difficult months in hospital.

Jacob’s way into research was quite indirect and dependent on a lot of chance factors. For some time he worked in an institute which was supposed to produce penicillin in France but never came close to doing so. He became involved in developing and marketing an antibiotic called tyrothricin. Somewhat later he was able to enter the research group of André Lwoff at the Institut Pasteur. This paved the way for the work for which he was awarded the Nobel prize for medicine with Lwoff and Jacques Monod. He describes how each year to commemorate the anniversary of Pasteur’s death everyone working at the institute (not just the scientists) would make a formal visit to the tomb of Pasteur in the basement of one of the buildings. This was made more vivid for me by the fact that I had visited this tomb myself a couple of years ago. I was at the Pasteur museum quite late in the afternoon when not many visitors were there. I was shown the tomb by a very friendly employee of the museum. It is an impressive structure with lots of marble. I remarked to her that Robert Koch did not have such an impressive mausoleum. She replied that the Germans do not honour their great scientists in the way the French do. I am not sure how true this is but it is at least food for thought.

A large part of the last quarter of the book is a description of the work with Monod. There are also a lot of general reflections on the way in which science is done and how the process by which scientific ideas are developed contrasts with the final product as found in research papers and textbooks. It also gives a good picture of who did what in this collaboration. A key mechanism was the interaction between what Jacob was doing on prophages and what Monod was doing on the lac operon. From a certain point on they were always looking for analogies between the two. This part of the book gives a vivid portrait of the early days of the discipline of molecular biology. It includes a description of Jacob working feverishly with Sydney Brenner at Caltech to establish the existence of messenger RNA, in an atmosphere of general scepticism. The narrative ends after the completion of the project with Monod. What happened afterwards in Jacob’s (scientific) life? According to the book ‘In the beginning was the worm’ by Andrew Brown, Jacob tried to work on Caenorhabditis elegans but without success and he failed to get funding to set up an ‘Institut de la Souris’. Jacob later wrote a book called ‘La souris, la mouche et l’homme’ and perhaps I will read that sometime. But since my summer holiday is at an end it will not be very soon.